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Resistance in Superconductors 221
In standard BCS superconductors, the minimum energy to create a quasi-
particle is given by the energy gap ∆. The value of ˜σ
n
is proportional to
the number of excited quasiparticles, which goes to zero as e
−∆/T
at low
temperatures. However, one can also have superconductivity without an
energy gap, as in the presence of magnetic impurities. In such cases, the
normal fluid conductance will not vanish exponentially at low temperatures,
and a more complicated analysis is necessary. See e.g. Refs. 87 and 88. Of
course for frequencies larger than 2∆/ there would be power absorption
even at T =0.
5. Systems of Ultra-cold Atoms
Recent experiments with ultra-cold alkali atoms have opened a new chapter
in the study of nonequilibrium dynamics of superfluids. Two features of such
systems make them particularly suitable for studying dynamical phenom-
ena: complete isolation from the environment and characteristic frequencies
of the order of kHz, which are readily accessible to experimental analysis.
Dynamics of atoms in optical lattices
89
has particularly close connection to
the issues that we discussed earlier in the context of superconductors. Here
we restrict our discussion to bosonic atoms although interesting experiments
have also been done with fermions (see e.g. Ref. 90).
Optical lattices are created using standing waves of laser beams that pro-
vide an artificial periodic potential for the atoms. The strength of the optical
potential can be controlled and atomic systems can be tuned between the
regimes of a weak lattice, where kinetic energy dominates, and strong lattice,
where repulsive interactions between atoms play the dominant role.
89,91
In
the former case, one finds BEC of weakly interacting atoms and a macro-
scopic occupation of the state with quasimomentum k = 0. In the latter
case, the system is in the Mott insulating state which has no long range
phase coherence and strongly reduced number fluctuations. The transition
between the two phases is an example of the quantum fluctuations driven
phasetransitionwhichwediscussedinSecs.3.3and3.5. Suchatransition
was first observed by Greiner et al.
92
by measuring momentum occupation
numbers in the so-called time-of-flight experiments.
Experimentally, one can also prepare cold atoms systems moving with
respect to the optical lattice and study the decay of the current. Such ex-
periments are very similar in spirit to the critical current measurements in
the case of superconductors. In the weakly interacting regime, one expects
the critical current to be determined by the inflection point of the single
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222 B. I. Halperin, G. Refael and E. Demler
particle dispersion.
93
–
95
Beyond the inflection point, the effective mass
becomes negative, which is equivalent to a change of interaction from repul-
sive to attractive, so small density fluctuations become amplified making the
system unstable to fragmentation. Close to the superfluid-to-Mott transition
we expect the critical current to go to zero continuously. So the question
is how to connect the two regimes. This was analyzed both theoretically
96
and experimentally,
97
–
99
and direct signatures of both thermal and quantum
phase slips have been observed.
6. Conclusion
We have seen that the mechanisms for production of resistance in supercon-
ducting materials are understood in broad outline. However, there remain
many open questions, particularly at very low temperatures, when quan-
tum fluctuations are important. In such situations, the dynamics of phase
slips or vortex motion will be sensitive to couplings to sources of dissipation,
internal or external, possibly at multiple frequencies, and we have only a
limited understanding of how this occurs in actual experiments. Open ques-
tions exist even for presumably classical problems, such as the resistance of
a superconducting wire close to near T
c
, where the simple time-dependent
Ginzburg–Landu model seems to work much better than it should.
There are many open questions regarding the role of disorder in the clas-
sical as well as quantum regimes. We understand only partially the collective
pinning that results from the interplay of disorder and vortex-vortex repul-
sion for a type-II superconductor in a strong magnetic field. Issues of how to
increase pinning and decrease flux creep are of great importance for practical
applications of superconductors in the areas of power transmission and high
field magnets.
Review articles and books, as well as articles in the original literature,
point to open issues in the field. Among the helpful examples are Refs. 49,
54, 60, 100–103, 104.
Acknowledgments
The authors have benefited from discussions with many people on topics of
this review, including, in recent years, M. Tinkham, Y. Oreg, A. Bezryadin,
I. Aleiner, D. S. Fisher, M. P. A. Fisher, V. Galitskii, and C. L. Lobb.
They acknowledge support from the Packard Foundation, a Cottrell Fellow-
ship from the Research Corporation, and NSF grants DMR-0906475 and
DMR-0705472.
September 14, 2010 9:18 World Scientific Review Volume - 9.75in x 6.5in ch10
Resistance in Superconductors 223
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October 4, 2010 10:37 World Scientific Review Volume - 9.75in x 6.5in ch11
COOPER PAIR BREAKING
P. Fulde
Max Planck Institute for the Physics of Complex Systems,
N¨othnitzer Str. 38, 01187 Dresden, Germany
and
Asia Pacific Center for Theoretical Physics,
POSTECH, Pohang, Korea
fulde@pks.mpg.de
An overview is given of a number of pair-breaking interactions in super-
conductors. They have in common that they violate a symmetry of the
pair state. In most cases pairs are formed from time reversed single-
particle states, a noticeable exception being antiferromagnetic supercon-
ductors. When time reversibility is broken by an interaction acting on the
electrons, the time evolution of the time-reversal operator plays an im-
portant role. Depending on whether it is nonergodic or ergodic, we deal
with pair weakening or pair breaking. Numerous different interactions are
analyzed and discussed. Unifying features of different pair-breaking cases
are pointed out. Special attention is paid to the Zeeman effect and to scat-
tering centers with low-energy excitations. The Kondo effect and crystalline
field split rare-earth ions belong in that category. Modifications caused by
strongly anisotropic pair states are pointed out. There is strong evidence
that in some cases intra-atomic excitations lead to pair formation rather
than pair breaking for which an explanation is provided.
1. Introduction
During the second part of the ’50s several groups in G¨ottingen and at
Bell Laboratories realized that small amounts of paramagnetic impurities,
i.e. transition-metal or rare-earth ions have profound effects on the transition
temperature T
c
of superconductors.
49,50,58,64
This temperature was found
to decrease rather rapidly with increasing impurity concentration n
I
. For
example, 0.6% of Gd are sufficient to suppress superconductivity in LaAl
2
(T
c
= 3.2 K) completely. This stimulated F. Reif and the student M. Woolf
to start work on tunneling on quenched In and Pb films in order to study
227
October 4, 2010 10:37 World Scientific Review Volume - 9.75in x 6.5in ch11
228 P. Fulde
the behavior of the density of states as function of impurity concentration.
59
The results were truly surprising. Not only was a rapid decrease of the su-
perconductivity energy gap observed as T
c
decreased with increasing n
I
of
Gd and Fe, but it was also found that for a range of impurity concentrations
the gap disappeared completely despite the fact that the probes were still
superconducting. The experiments revealed clearly that a gap in the exci-
tation spectrum is not a prerequisite for the existence of a superconducting
state. Rather, it is the order parameter which distinguishes it from the nor-
mal state. This is by now common knowledge. But the phenomenon of gap-
less superconductivity was in 1962 rather unexpected. Not long thereafter it
was realized that there already existed a theory by Abrikosov and Gorkov
2
which predicted what was found experimentally. Since at that time commu-
nication between physicists in the Soviet Union and their western colleagues
were rather limited, that theory had remained rather unnoticed. Another
reason might have been that the theory was formulated in an elegant Green’s
function formalism in which only a very small number of condensed matter
theorists in western countries were familiar with at that time. This changed
very fast after the potential of that approach was realized (see, e.g. Refs. 4,
39, 40, 68). Nevertheless, it was of help to many working in the field that
an alternative approach to gapless superconductivity based on the time evo-
lution of the time-reversal operator was presented by de Gennes.
17
He drew
attention to the fact that the exchange interaction between conduction elec-
trons and magnetic ions changes sign when a time-reversal transformation
is applied to the conduction electron system. The theory is very physical
and simple but not as easily applicable to a wide range of problems as is the
Green’s function approach.
It was very soon realized that pair breaking and gapless superconduc-
tivity are rather general phenomena. Any external perturbation acting on
Cooper pairs will lead to pair breaking when it is not time-reversal invari-
ant. In the case of paramagnetic impurities it is the exchange interaction be-
tween the conduction electrons and the impurity which violates time-reversal
symmetry. The reason is apparent. Since Cooper pairs are usually formed
from time-reversed one-particle states,
5,20
any perturbation acting differently
on the partners of a pair will destabilize the paired state and break it. (Note
that pairing does not always take place in time reversed states. In an anti-
ferromagnetic superconductor electrons are paired in states which are time
reversed followed by a lattice translation connecting the two sublattices.
6
)
Another example of time symmetry breaking is an external magnetic field ap-
plied either perpendicular to the surface of a type II superconductor
15,18,41
October 4, 2010 10:37 World Scientific Review Volume - 9.75in x 6.5in ch11
Cooper Pair Breaking 229
so that Abrikosov vortices form or applied parallel to a thin film.
39,41,75
Details of pair breaking vary considerably here, depending on mean free
path and coherence length of the system.
17
Since a magnetic field acts on
the orbit of an electron as well as on its spin (Zeeman effect), the relative
size of the two effects plays an important role (Maki parameter).
43,76
Also
the interface between a superconducting and a ferromagnetic film acts as a
pair breaker and so does a supercurrent under certain conditions.
Numerous experiments have been done in which several pair-breaking
perturbations act at the same time on a superconductor. An example are
paramagnetic impurities and a magnetic field.
14,30
The magnetic field can
spin-polarize the impurities and spin-polarized impurities act differently on
Cooper pairs than unpolarized ones. Nevertheless, there are many situations
where we may simply add the characteristic pair-breaking parameters of the
different perturbations in order to describe their combined effect.
25
Let us return to the starting point, i.e. to the pair-breaking properties of
magnetic impurities. When the magnetic ions are subject to the Kondo effect
they form singlet states with the conduction electrons. In the presence of
Cooper pairing the singlet formation is affected. This may result in reentrant
superconductivity, an effect pointed out by M¨uller-Hartmann and Zittartz
57
and measured by Maple et al.
33,47
Thereby a superconducting state appears
at a transition temperature T
c1
but disappears again at a lower temperature
T
c2
< T
c1
. Eventually it can reappear again at an even lower temperature
T
c3
< T
c2
.
When we deal with paramagnetic rare-earth ions, e.g. Pr
3+
ions their
incomplete 4f shell is in a Hund’s rule ground-state multiplet, in the present
case with total angular momentum J = 4. Under the influence of the crys-
talline electric field (CEF) of the neighboring ions this manifold is split into
different CEF levels, with level spacings of order few to tens of meV. When
those ions are in a superconductor, T
c
and the CEF splittings are of simi-
lar magnitude and influence each other. Thereby the form of the dominant
interaction between the conduction electrons and the CEF split energy lev-
els plays a decisive role. When the exchange interaction is most important
we expect Cooper pair breaking since the interaction is not time-reversal
invariant with respect to the conduction electrons. However, when aspherical
Coulomb charge scattering is more important we expect support of Cooper
pair formation, because the interaction is time-reversal invariant. In that
case the inelastic intra-atomic scattering processes have a similar effect on
pair formation as Einstein oscillators, i.e. localized phonons.
23
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230 P. Fulde
A number of interesting effects result from the Zeeman term in a mag-
netic field.
10,13,46,61
If it is small it causes a Zeeman splitting of the BCS
density of states, an effect pioneered by Meservey and Tedrow.
52,72
With
increasing field strength eventually Cooper pairs are broken. But before the
system goes into the normal state it may go over into a gapless inhomo-
geneous state.
22,37
We face the problem of how to pair electrons when the
constituents have a different chemical potential (FFLO state). In order for
this state to be formed, requirements on the mean-free path and on the Maki
parameter have to be met. Since that topic is covered in another contribu-
tion to this volume, we omit it here. Related to the Zeeman-interaction term
is also the Jaccarino-Peter effect.
34
It is based on a partial compensation
of the pair-breaking exchange interaction of polarized magnetic impurities
and the Zeeman term due to an applied field. Therefore it may happen that
superconductivity is no longer completely suppressed by a given concentra-
tion of magnetic ions and will reappear when the field is sufficiently high.
At even higher fields, superconductivity is eventually destroyed due to the
pair-breaking effect of the field. This reentrance of superconductivity was
indeed observed by Meul et al.
53
and Hiraki et al.
31
In particular in the Ginzburg-Landau regime but not only there, nu-
merous physical effects caused by pair-breaking external perturbations have
been calculated and tested by experiments. Examples are ultrasonic atten-
uation, Knight shift, electrical and thermal conductivity, and nuclear-spin
relaxations to name a few. What is astonishing is the high degree of accu-
racy with which experiments could be explained and partially predicted. It
is rarely matched by other theories in condensed matter physics, quantum
Hall physics excluded. The robustness and accuracy of the BCS theory is
the basis of these findings.
As pointed out above, the interactions between CEF-split magnetic ions
and conduction electrons contain parts, e.g. a spherical charge scattering,
which are time-reversal invariant and therefore favor pair formation instead
of pair destruction. However, they are very often dominated by the time-
reversal symmetry violating exchange interaction. But there is one clear-cut
exception: the filled skutterudite PrOs
4
Sb
12
has a T
c
of 1.85 K, while the
corresponding compound LaOs
4
Sb
12
has a T
c
of 0.74 K only. The only
difference between the two systems are the 4f
2
electrons of Pr
3+
which
are absent in La. Therefore the 4f electrons must provide the additional
glue for Cooper pairs.
11
An even more surprising case is UPd
2
Al
3
, where
there is clear evidence that non-time-reversal invariant magnetic scattering of
itinerant electrons on localized ones results in superconductivity.
51
However,