Cooper L.N., Feldman D. (Eds.) BCS: 50 Years
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Superconductor-Insulator Transitions 271
of strong disorder, the other limit of weak disorder and high critical fields
contains different physics in which a quantum metallic state is the central
feature. The high disorder films exhibit a substantial magnetoresistance
peak. It is suggested that the regime of high resistance is a Bose insulating
regime, which is more extensive in magnetic field, the more disordered the
film. The high field behavior is always Fermi like in character.
The exponent product for the more disordered films of 2.33, which is
close to 7/3, suggests that the transition is a quantum rather than classical
percolation transition and that the results are consistent with the dirty Boson
picture of the SI transition, at least for the case of the strongly disordered
branch of the data.
9. Discussion
Despite a long history it is clear that there are many open questions relating
to SI transitions. These include the following: What different physical mod-
els govern the various SI transitions, which have different exponent products
and different critical resistances? Why are the insulating states of various
transitions so very different? What are the ingredients necessary for a large
resistance peak? Is this regime one in which there are localized Cooper pairs
in an essentially homogenous system, or in particular is disorder on a length
scale greater than atomic scales necessary for such a peak? Is an interme-
diate metallic regime, or a two-phase regime always an intrinsic feature of
some transitions or is the former an artifact of the failure of electrons to
cool? Assuming that there are examples in which there is no intermediate
metallic and/or two-phase regime phase, can one explore parameter space
of the SI transition and determine critical exponents in the asymptotic limit
so as to accurately ascertain the universality classes of these transitions?
The work of Steiner, Brezney and Kapitulnik purports to do this for In
x
O
y
,
but its results, in particular the claim that the critical resistances of highly
disordered films are universal are hard to understand given the requirement
of charge-flux duality.
34
Acknowledgments
The author would like to thank numerous individuals for enlightening dis-
cussions over the years. These include, M. P. A. Fisher, Steven Girvin,
Aharon Kapitulnik, Zvi Ovadyahu, Aviad Frydman, James Valles, Wenhao
Wu, Boris Shklovskii, Leonid Glazman, Robert Dynes, Herve Aubin, He
August 24, 2010 17:25 World Scientific Review Volume - 9.75in x 6.5in ch12
272 A. M. Goldman
would also like to thank students who have worked in his laboratory on this
problem. These include Bradford Orr, Heinrich Jaeger, Ying Liu, Anthony
Mack, Nina Markovic, Cathryn Christiansen, Gloria Martinez-Arizala, Luis
Hernandez, Kevin Parendo, Sarwa Tan, and Yen-Hsiang Lin.
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September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
NOVEL PHASES OF VORTICES IN
SUPERCONDUCTORS
Pierre Le Doussal
CNRS-Laboratoire de Physique Th´eorique,
Ecole Normale Sup´erieure, Paris 75238 Cedex 05, France
ledou@lpt.ens.fr
An overview is given of the new theories and experiments on the phase
diagram of type II superconductors, which in recent years have progressed
from the Abrikosov mean field theory to the “vortex matter” picture. We
then detail some theoretical tools which allow to describe the melting of
the vortex lattice, the collective pinning and creep theory, and the Bragg
glass theory. It is followed by a short presentation of other glass phases of
vortices, as well as phases of moving vortices.
1. Introduction
With the discovery of high-T
c
superconductors, the physics of vortices in
superconductors has experienced a revolution. Because of high temperature
and the anisotropy of the new compounds, fluctuations are much enhanced
and the mean field picture of a well ordered vortex lattice made of parallel
straight vortices became vastly insufficient. A host of new phenomena have
been discovered, most of them arising from the interplay of the enhanced
fluctuations and pinning. Describing them has been a challenge to the the-
ory which has considerably benefited from the advances in related fields such
as the physics of glasses and disordered systems. As a result, new concepts
in vortex physics have emerged. The development of powerful new experi-
mental tools has allowed to image and probe the behavior of the vortices to
unprecedented high precision from the microscopic to mesoscopic scale, and
also to create and control new pinning environments for the vortices. This
has allowed for tests of the new theories and for a fruitful interplay with
experiments.
The field has grown much too vast for any exhaustive review in such
short space, so we have chosen instead to focus on the sub-topic of the phase
277
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
278 P. Le Doussal
diagram of vortices. We first give an overview (Sec. 2) with a historical
bent on how the phase diagram has metamorphosed from the mean field
(MF) picture to the more recent (and in many respect still evolving) “vortex
matter” picture. This part is more qualitative, easy to read and describes
the interplay between experiments and theory. Next, we develop two special-
ized theoretical topics. The first is the melting of the vortex lattice (Sec. 3),
with two dual approaches, one based on the elasticity theory and the sec-
ond on the lowest Landau level within the Ginzburg–Landau theory. Phase
coherence and decoupling phenomena are briefly evoked. The second topic
is the description of the vortex lattice in the presence of pinning disorder
(Sec. 4), and what is the fate of the Abrikosov lattice. It contains a short
presentation of modern methods in the theory of disordered systems, which
have been crucial tools to develop new concepts such as the glass phases of
vortices. We then give a description of the Bragg glass phase, and, in Sec. 5,
a brief description of the vortex glass, the Bose glass, as well as moving
phases of vortices.
For the conventional picture we refer here to the classical articles, books
and reviews
1–3
for all details and references. For the numerous developments
in the last decades, we apologize in advance to many of our colleagues for
omissions, and refer to the many excellent reviews on vortex physics
4–10
for
a more exhaustive presentation and for different perspectives. Some of the
present material already appeared in Ref. 11 but has been notably updated
and expanded.
Notations: Here x is 3D coordinate (more generally, d dimensional), x =
(r, z) where r lives in 2D. x
1
designates one particular direction. r denotes
either the vector or its norm, when no ambiguity arises. Fourier transforms
are not noted differently, q = (q
⊥
, q
z
). c is the speed of light, and c an elastic
constant. T denotes kT .
2. Overview
2.1. Conventional picture, basics
The conventional, also called mean-field, equilibrium phase diagram of type
II superconductors in an external field H is obtained by the minimization
of Ginzburg–Landau (GL) energy functional. It predicts a Meissner phase
for H < H
c1
(T ), where the complex superconducting order parameter ψ is
uniform and B = 0, and a mixed phase H
c1
(T ) < H < H
c2
(T ) where the
magnetic induction B penetrates the bulk in the form of vortex lines. Each
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Novel Phases of Vortices in Superconductors 279
May 3, 2010 17:58 World Scientific Review Volume - 9.75in x 6.5in chapternewv9
New phases of vorticesin superconductors 3
spacing a
0
= (4/3)
1/4
a for a triangular lattice. The GL theory predicts that both
the coherence length ξ and the London length λ are temperature dependent with
ξ ∼ (1−T/T
c
)
−1/2
near T
c
and κ = ξ/λ being T -independant
a
. The vortex spacing
decreases from a
0
∼ λ at B ∼ H
c1
∼ Φ
0
/λ
2
to a
0
∼ ξ at H = H
c
2
= Φ
0
/2πξ
2
,
leading to the MF phase diagram pictured in ??.
Fig. 1.1.
In practice superconductors are almost never in equilibrium, and one must worry
about vortex dynamics, pinning and surfaces. In presence of a current density j,
each vortex segment aligned along unit vector n is submitted to a force per unit
length f. In the simplest and most frequent situation (small Hall angle) it is equal
to the Lorentz force f =
Φ
0
c
j ×n where j is the local current density. In the absence
of other forces a vortex thus moves in presence of a uniform current with velocity v
given by:
ηv = f (1.2)
where η is the friction coefficient from scattering of normal core electrons with the
substrate. The resulting time-dependent induction generates an (in-plane) electric
field E = B × v/c, hence, from (1.2), a resistive response (to in- plane currents)
E ∼ j. In the vortex lattice, each vortex also feels the (repulsive) Lorentz forc es from
the screening currents of its neighbors. These however cancel for the equilibrium,
or for the uniformly moving vortex lattice, leading to E = j
ext
/ρ
ff
in presence of
an external transport current j
ext
, with ρ
ff
= Φ
0
B/ηc
2
. Matching at B ≈ H
c2
a
while GL is in principle valid near T
c
, as any good theory its range of validity is considerably
larger with choices of parameters from phenomenology or microscopic theo ry (when available)
May 3, 2010 17:58 World Scientific Review Volume - 9.75in x 6.5in chapternewv9
New phases of vorticesin superconductors 5
Fig. 1.4.
Fig. 1.5.
be pinned. Pinning is provided by impurities in the material. Since they weaken
superconductivity locally, they are energetically favorable positions for the vortex
core to s it. The conventional description of pinning replaces (1.2) by:
ηv =(f − f
c
)θ(f − f
c
) (1.3)
Fig. 1. Left: The vortex solution. Right: Mean-field phase diagram.
vortex line carries a quantum of magnetic flux Φ
0
= hc/2e and spans the
entire system, being aligned along H. An isolated vortex consists of a normal
cylindrical region of radius ξ, the core, where ψ(r) is reduced (it vanishes at
the center), surrounded by another cylindrical region of radius λ where the
induction is screened by super-currents (B(r) ∼ e
−r/λ
and |ψ|
2
= cst away
from the vortex). Its energy cost per unit length, e
l
= H
c1
Φ
0
/4π =
0
ln(λ/ξ)
is overcome by its magnetic energy gain −HΦ
0
/4π in the mixed phase. The
important energy scale (per unit length):
0
=
Φ
0
4πλ
2
(1)
determines both the self-energy and mutual interactions between vortex
lines. From minimization of the GL functional, Abrikosov predicted that
these vortex lines form a regular lattice (later found to be triangular in stan-
dard systems), observed later in decoration experiments. This is natural if
one thinks of vortex lines as identical mutually repelling objects. From flux
quantization, the area of the unit cell is a
2
= Φ
0
/B in simple relation to the
averaged induction B, with a spacing a
0
= (4/3)
1/4
a for a triangular lattice.
The GL theory predicts that both the coherence length ξ and the London
length λ are temperature dependent with ξ ∼ (1 − T/T
c
)
−1/2
near T
c
and
κ = ξ/λ being T -independent.
a
The vortex spacing decreases from a
0
∼ λ
at B ∼ H
c1
∼ Φ
0
/λ
2
to a
0
∼ ξ at H = H
c
2
= Φ
0
/2πξ
2
, leading to the mean
field phase diagram pictured in Fig. 1.
In practice, superconductors are almost never in equilibrium, and one
must worry about vortex dynamics, pinning and surfaces. In presence of a
a
While
GL is in principle valid near T
c
, as any good theory its range of validity is
considerably larger with choices of parameters from phenomenology or microscopic
theory (when available).
September 2, 2010 15:31 World Scientific Review Volume - 9.75in x 6.5in ch13
280 P. Le Doussal
current density j, each vortex segment aligned along unit vector n experi-
ences a force f per unit length. In the simplest and most frequent situation
(small Hall angle), it is equal to the Lorentz force f =
Φ
0
c
j × n where j is
the local current density. In the absence of other forces a vortex thus moves
as a result of a uniform current with velocity v given by:
ηv = f (2)
where η is the friction coefficient from scattering of normal core electrons
with the substrate. The resulting time-dependent induction generates an
(in-plane) electric field E = B × v/c, hence, from (2), a resistive response
(to in-plane currents) E ∼ j. In the vortex lattice, each vortex also feels
the (repulsive) Lorentz forces from the screening currents of its neighbors.
These, however, cancel for the equilibrium, or for the uniformly moving vor-
tex lattice (VL), leading to E = ρ
ff
j
ext
in presence of an external transport
current j
ext
, with ρ
ff
= Φ
0
B/ηc
2
. Similarly at B ≈ H
c2
to the resistivity
of the normal metal yields the Bardeen–Stephen estimate for the friction
η ≈ Φ
0
H
c2
/ρ
n
c
2
. Note that near a surface, the surface screening current
(which pushes the vortex inside) and the image vortex (which attracts it)
conspire to create, for H
c1
< H < H
c
a surface barrier to vortex entry.
Thus, to achieve a true dissipationless transport current one needs that
vortices be pinned. Pinning is provided by impurities in the material. Since
they weaken superconductivity locally, they are energetically favorable posi-
tions for the vortex core to sit. Schematically, the conventional description
of pinning replaces (2) by
b
:
ηv = (f − f
c
)θ(f −f
c
) (3)
represented in Fig. 2, i.e. the vortex line remains pinned for f < f
c
and
recovers a linear v–f characteristics for f > f
c
. Here f
c
is the critical force,
qualitatively the maximum pinning force per unit length
c
that the impurities
can exert. It corresponds to a critical current density j
c
= cf
c
/Φ
0
, with
j
c
< j
0
∼ c
0
/ξΦ
0
the depairing current. Equation (3) immediately yields
to irreversible, far from equilibrium behavior, and to the Bean critical state.
Consider, for example, a superconductor occupying the x
1
> 0 half-space
and H along z. From Maxwell’s equation, j = −
c
4π
dB
dx
1
ˆx
2
. Since B (averaged
over a mesoscopic region) is proportional to the vortex density, a given value
b
Valid for a particle in a potential V (u) = f
c
|u|. More realistic potentials with finite
barriers lead to qualitatively similar v − f curves, though asymptotic to ηv = f at
large f, as shown in Fig. 2.
c
Below we denote the critical force per unit volume F
c
= f
c
/a
2
not to be confused
with an energy.