Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors
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11 Universal Properties of Cuprate Superconductors 489
superconductors that are close to a metal–insulator
boundary.A glanceat Fig.11.34 shows that the corre-
lation between
c
(
0
)
and
DC
c
T
+
c
is particularly re-
warding. In various classes of layered superconduc-
tors, including organics, transition metal dichalco-
genides and cuprates,
c
(
0
)
is systematically sup-
pressed with increase of normal state conductivity
DC
c
T
+
c
. Without thermal and quantum fluctua-
tions the DC conductivity is given by
DC
c
=
ne
2
m
c
tr
=
c
2
tr
4
2
c
(
0
)
. (11.88)
tr
is the mean scattering time of the normal elec-
trons in transport properties with number density
per unit volume n and effective mass m
c
.Provided
that the variations of
tr
are small, superconductors
where fluctuations can be neglected (mean-field su-
perconductors) are then characterized by the dotted
line in Fig. 11.34, which is
c
(
0
)
∝
DC
c
−1/2
. (11.89)
This agrees rather well with the data of mean-field
superconductors. Interestingly enough, MgB
2
and
Sr
2
RuO
4
appear to fall into this class as well. Promi-
nent and systematic deviations from this behavior
occur for the highly anisotropic organics and under-
doped cuprates. As
c
(
0
)
increases with the reduc-
tion of the normal stateconductivity
DC
c
,weobserve
that the data tends to scatter around two branches,
consistent with
c
(
0
)
= §
s
DC
c
−3/4
,whichisthe
scaling relation (11.58) with z = 1, describing the
critical behavior close to 2D–QSI critical points. The
most prominent deviations from these trends occur
for overdoped cuprates (see also Fig. 11.23). In par-
ticular the data for overdoped YBa
2
Cu
3
O
7−ı
(YBCO)
exhibit an upturn, signaling the crossover to 3D–
QSN criticality (see (11.63)). Since organics un-
dergo superconductor to insulator transitions [155],
are highly anisotropic and can be varied over a
rather extended interval, i.e.
T
c
= 180 for -(ET)
2
Cu[N(CS)
2
]Br,
T
c
= 350 for -(ET)
2
Cu(NCS)
2
[156]
and
T
c
=2×10
3
for˛- (BEDT-TTF)
2
NH
4
Hg(SCN)
4
,
it becomes clear that the organics line in Fig. 11.34
indicates the flow of this class of superconductors to
2D–QSI criticality, where the scaling relation (11.58)
applies.
Thus unlike the organics the cuprates undergo a
doping tuned crossover from 2D–QSI to 3D–QSN
criticality, where
DC
c
T
+
c
and
c
(
0
)
tend to in-
finity, while T
c
vanishes. These critical points are
attained in the underdoped and overdoped limit,
respectively, where T
c
vanishes. In this crossover
the cuprates either pass the organics or the dot-
ted mean-field line, spanned by conventional su-
perconductors (see Fig. 11.34).YBa
2
Cu
3
O
7−ı
(YBCO)
and La
2−x
Sr
x
CuO
4+ı
(LSCO) cross the dotted mean-
field line in the neighborhood of TaS
2
,Sr
2
RuO
4
and
MgB
2
. In this region of the
c
(
0
)
-
DC
c
T
+
c
plane
fluctuations do not play a dominant role. Indeed,
Sr
2
RuO
4
exhibits Fermi liquid normal state proper-
ties [157]. However, overdoped cuprates flow to 3D–
QSN criticality,where disorder destroys the phase co-
herence and the pairs. On the contrary, the organics
evolve rather smoothly from the mean-field regime,
to which Pb, Nb, Nb-junctions, ˛-Mo
1−x
Ge
x
and the
dichalcogenides belong. Thus, these flows provide a
perspective of the regimes where fluctuations and
disorder are essential.
Acknowledgements
It is a pleasure to thank H. Keller and K.A. M¨uller for
numerous helpful discussions.
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12 Vortex Matter
G. Blatter Theoretische Physik, ETH Z¨urich, Switzerland
V. B. Geshkenbein Theoretische Physik, ETH Z¨urich, Switzerland
12.1 Introduction ............................................................................496
12.2 Ginzburg–Landau and London Theories ...................................................502
12.3 Vortex Lines .............................................................................505
12.3.1Elasticity..........................................................................507
12.3.2Dynamics.........................................................................508
12.3.3Green’sFunctions..................................................................512
12.3.4FreeEnergyandAction.............................................................512
12.4 Vortex Lattice ...........................................................................512
12.4.1Elasticity..........................................................................513
12.4.2Dynamics.........................................................................517
12.4.3Green’sFunctions..................................................................517
12.4.4FreeEnergyandAction.............................................................518
12.5 Layered Materials ........................................................................518
12.5.1Lawrence–DoniachModel...........................................................519
12.5.2JosephsonVortices.................................................................520
12.5.3PancakeVortices...................................................................522
12.5.4RenormalizedSuperfluidStiffness...................................................527
12.5.5VortexLattice......................................................................527
12.6 Anisotropic Scaling Theory ...............................................................532
12.7 Statistical Mechanics .....................................................................538
12.7.1Fluctuationsin2DThinFilmandLayeredSuperconductors............................540
12.7.2LindemannAnalysisofVortexLatticeMelting.........................................542
12.7.3LindemannAnalysisofLayerDecoupling.............................................549
12.7.4VortexLatticeMeltingin2DThinFilms..............................................552
12.7.5ElectromagneticallyCoupledLayeredSuperconductors................................553
12.7.6CharacteristicsofFirst-OrderMelting................................................560
12.7.7VortexLiquid......................................................................562
12.7.8ConstitutiveRelation...............................................................565
12.8 Quenched Disorder: Pinning and Creep ....................................................566
12.8.1StrongPinning.....................................................................567
12.8.2WeakCollectivePinning............................................................569
12.8.3WeakVersusStrongPinning.........................................................570
12.8.4VortexCreep.......................................................................573
12.9 Uncorrelated Disorder: Collective Pinning and Creep ........................................575
12.9.1DirtyElasticManifolds.............................................................576
12.9.2SingleVortexPinning...............................................................579
12.9.3VortexLatticePinning..............................................................582
12.9.4CollectiveCreep ...................................................................585
496 G. Blatter and V.B. Geshkenbein
12.9.5ThermalDepinning................................................................596
12.9.6AnisotropicMaterial................................................................601
12.9.7LayeredSuperconductors...........................................................601
12.9.8PinningandCreepDiagrams........................................................604
12.10 Correlated Disorder ......................................................................605
12.10.1ColumnarDefects.................................................................605
12.10.2TwinBoundaries..................................................................609
12.11 Surface and Geometrical Barriers .........................................................610
12.11.1SurfaceBarriers...................................................................611
12.11.2GeometricalBarriers..............................................................612
12.12 Vortex Glasses ...........................................................................615
12.12.1Disorder-InducedMelting..........................................................616
12.12.2Vortex-GlassScaling...............................................................619
12.13 Concluding Remarks .....................................................................621
References..............................................................................625
12.1 Introduction
The discovery of high-temperature superconductiv-
ity by Bednorz and M¨uller in 1986 [1] has triggered
an enormous research activity with various goals:
unraveling the microscopic mechanism producing
such high transition temperatures, understanding
the phenomenological properties of these layered
and strongly type II superconductors, and develop-
ing materials ready for technological applications.
This chapter deals with the phenomenological as-
pect of strongly fluctuating type II superconductors
in general and materials with high transition tem-
peratures in particular; as such the main goal is to
develop the physics of vortices as a new state of mat-
ter,hence the title“Vortex Matter”.In fact,thanks to a
set of lucky circumstancesthe new high-temperature
superconductors set the stage for the physical re-
alization and experimental accessibility of the en-
tire statistical mechanics of vortices and thus, from
a more general point of view, of interacting elastic
strings. Furthermore, the coupling of the vortices to
external currents and fields allows to exert forces on
these strings, opening up the wide field of dynam-
ical behavior of the vortex system. Finally, vortices
are susceptible to disorder and thus the statistical
and dynamical properties of this system have to be
investigated in the presence of a random potential
environment.
The phenomenology of superconductivity is
based on the Ginzburg–Landau theory [2] which
provides the starting functional (free energy) for
the charged superfluid coupled to electromagnetism.
The two fields determining the physics of the sys-
tem are the superconducting order parameter ¦ and
the vector potential A; the complex order parame-
ter ¦ = |¦ |exp(i' ) describes the density and the
phase of the quantum fluid.The resulting mean-field
version of the H–T phase diagram has been derived
by Abrikosov back in 1957 [3] and has proved to
describe accurately the phenomenological behavior
of all the conventional low-temperature supercon-
ductors.This mean-field H–T phase diagram,shown
in Fig. 12.1, comprises a Meissner–Ochsenfeld phase
characterized by a complete flux expulsion at low
magnetic fields H < H
c
1
, separated from the mixed
(or Shubnikov) phase [4] at higher fields H > H
c
1
,
where the magnetic field penetrates the supercon-
ductor in the form of flux lines (or vortices). The
lower critical field H
c
1
is mainly determined by the
12 Vortex Matter 497
Fig. 12.1. Mean-field phase diagram for a type II super-
conductor: below the lower critical field H
c
1
the magnetic
fieldisexpelledfromthesample;thisistheMeissner–
Ochsenfeld effect. In the intermediate field regime H
c
1
<
H < H
c
2
the magnetic field penetrates the sample via flux
lines or vortices; this is the Shubnikov phase with vortices
arranged in a triangular Abrikosov lattice. Above the up-
per critical field H
c
2
superconductivity is destroyed and the
sample is in the normal metallic phase
penetration depth ,which is the length scale associ-
ated with the electromagnetic response of the super-
conductor.Sincethe superconductingstate is a quan-
tum fluid, the magnetic flux enclosed in a vortex is
quantized in units of ¥
0
= hc /2e ≈ 2.068 10
−7
Gcm
2
,
the flux quantum.With increasing field the density of
flux lines (which form a triangular lattice) increases
until the vortex cores overlap when the upper critical
field H
c
2
is reached; beyond this field we recover the
normal metallic state. The upper critical field H
c
2
is
determined by the coherence length of the super-
conductor which represents the second fundamental
length scale in the system associated with the re-
sponse of the macroscopic quantum fluid ¦ .
Applying an external current density j to the vor-
tex system, the flux lines start to move under the
action of the Lorentz force F
L
= j ∧ B/c.Within
a homogeneous system the driving Lorentz force is
counteracted by the friction force F
=−v alone,
where v is the steady state velocity of the vortex sys-
tem, v = j ∧ B/c,and is the Bardeen–Stephen
viscous drag coefficient [5]. The flux motion pro-
duces an electric field E = B ∧ v/c directed parallel
to the driving current density j;asaresultthepower
P =(j ∧ B)
2
/c
2
is dissipated in the system and the
superconducting property of dissipation free current
flow is lost. Hence, while the Meissner–Ochsenfeld
phase is a true superconductor,the Shubnikov phase
exhibits a normal response under the application of
a transverse current j ⊥ H. On the other hand, the
longitudinal(j H) response in the Shubnikov phase
is superconducting. Finally, the normal-metal phase
exhibits an ohmic response in all directions.
In order to recover the desired property of
dissipation-free current flow the flux lines have to be
pinned such that v = 0 in the presence of a forcefield
F
L
= 0. In this case the driving Lorentz force is coun-
teracted by the pinning force F
pin
[6–10].Fortunately,
any static disorder affecting the superconducting or-
derparameterwill contributeto a finitepinning force
density F
pin
and thereby the technological usefulness
of the type II superconductors is reestablished.How-
ever, dissipation-free flow now has become a matter
of optimizing the pinning force density F
pin
since in-
creasing the current density j beyond j
c
= cF
pin
/B
(we assume j ⊥ B) leads to the depinning of the vor-
tices and hence to the reappearance of dissipation.
The current density j
c
is the critical depinning cur-
rent density, which is always bounded by the depair-
ing current density j
0
∼ cH
c
/ determined by the
thermodynamic critical field H
c
. The dimensionless
critical current density ratio j
c
/j
0
is a measure of the
strength of the pinning force density F
pin
and as such
can be conveniently used as the fundamental param-
eter in the phenomenology of type II superconduc-
tors characterizing the strength of the quenched dis-
order in the system.
Going beyond the traditional mean-field picture,
thermal fluctuations have to be taken into account
and the statistical physics of vortex matter has to be
developed. Large thermal fluctuations (of the order
of the lattice constant a
0
≈ (¥
0
/B)
1/2
)ofthevortex
positions will lead to a melting transition and pro-
duce a new vortex-liquidphase [11–13],see Fig. 12.2.
498 G. Blatter and V.B. Geshkenbein
Fig. 12.2. Schematic phase diagram for a strongly fluctu-
ating type II superconductor. The vortex lattice undergoes
a first-order melting transition transforming the vortex
solid into a vortex liquid.Transverse screening produces an
exponentially weak interaction between the vortex lines at
low magnetic fields, leading to a reentrance of the melting
line.The flux lines in the vortex-liquid are entangled result-
ing in an ohmic longitudinal response, hence the vortex-
liquid and normal metallic phases are separated by a mere
crossover at H
c
2
Near the normal-metal–superconductor transition
not only the phase but also the modulus of the order
parameter is subject to large thermal fluctuations,
resulting in a sizeable critical region.
Besides these thermodynamic consequences, the
inclusion of thermal fluctuations into the phe-
nomenology of type II superconductors also affects
the dynamical behavior of the vortex system in var-
ious ways. First of all, the vortex lines can move at
small current densities j < j
c
due to thermally acti-
vated jumps over the pinning barriers [14], leading
to the famous creep phenomenon in type II super-
conductors. Creep of vortex lines produces a small
but finite directed motion of the vortex lines and
thus reestablishes dissipation in the system.One then
could ask if true “transverse” superconductivity ex-
ists at all in the thermodynamic sense (note that lon-
gitudinal superconductivity persists as current den-
sities j B produce a force-free configuration). The
crucial question is whether creep persists down to
the limit of zero driving force: Does the dissipa-
tion and hence the resistivity vanish in the limit
j → 0 or does a superconductor in the mixed phase
finally end up in a resistive state (note that probing
the superconductor with a vanishing current den-
sity j → 0 amounts to investigating the thermody-
namic state of the vortex system). The kind of re-
sponse the system develops for j → 0 then deter-
mines whether the vortex system constitutes a new
vortex-glass phase [15,16], characterized by infinite
barriers against creep [17] and hence (j → 0) → 0,
or whether the vortex system is in a liquid state with
finite creep barriers allowing vortex motion at any
nonzero temperature producing a finite resistivity
(j → 0) > 0.
Another hint to the appearance of a glassy phase is
thedestruction of thetriangular vortex lattice:Ashas
been shown by Larkin [18], in less than four dimen-
sions quenched disorder always destroys the transla-
tional long-range order of the Abrikosov vortex lat-
tice,giving way to a glassy/liquid phase.Aslong as the
elastic properties of the vortex lattice are preserved,
i.e., in the absence of dislocations, the destruction of
the lattice-order is weak: with a displacement field
growing only logarithmically with distance [19–21]
the vortex lattice remains quasi-ordered and the re-
sulting phase has been termed a “Bragg-glass” [21].
Increasing the effect of disorder the quasi-ordered
lattice will undergo a disorder-driven melting tran-
sition [22] taking the vortex system into a more dis-
ordered glassy or frozen liquid state, cf. Fig. 12.3.
The interplay between quenched disorder and
thermal fluctuations not only manifests itself in the
phenomenon of creep. Thermal fluctuations of the
individual vortex lines lead to a dynamical sampling,
and hence thermal averaging, of the disorder poten-
tial over the spatial extent of the mean thermal dis-
placement u
2
1/2
th
of the vortex lines. Thermal dis-
order hence opposes quenched disorder also via a
smoothing of the disorder potential and thereby re-
duces the critical current density j
c
in the system.
The smoothing of the quenched disorder potential
due to thermal fluctuations has become known un-
der the name of thermal depinning [23], of thermal
12 Vortex Matter 499
Fig. 12.3. Ina defected crystal the vortexlines are pinned by
the disorder potential.The vortex solid is transformed into
a vortex-glass, while the vortex-liquid remains a (viscous)
liquid. At low fields the solid is only weakly disordered;
the Bragg-glass is free of dislocations and exhibits a super-
conducting response in all directions. The more strongly
disordered vortex-glass (possibly a frozen vortex-liquid)
carries a finite density of dislocations
though care should be taken that the phenomenon
depinning is not an abrupt transition but rather a
continuous crossover from a pinned to an unpinned
situation.
It turns out that the fundamental parameter gov-
erning the strength of thermal fluctuations in all
of the above phenomena is the Ginzburg number
Gi =(T
c
/H
2
c
V
coh
)
2
/2, which measures the relative
size of the (T = 0) condensation energy H
2
c
V
coh
within a coherence volume V
coh
with respect to the
critical temperature T
c
(we set the Boltzmann con-
stant to unity, k
B
= 1). The layered structure of
thecopper-oxidesuperconductors reduces thecoher-
ence volume by the anisotropy factor " =(m/M)
1/2
,
V
coh
= "
3
,wherem and M ( m) denote the effec-
tive masses of the charge carriers parallel and per-
pendicular to the superconducting planes.
At low temperatures the role of thermal fluctua-
tions is taken over by their quantum analog. Similar
to thermal fluctuations,quantum fluctuations can af-
fect superconductivity via fluctuations in the order
parametermodulus andphase,e.g.,viaquantummo-
tion of vortices (quantum creep [24]).Again, the im-
portance of quantum fluctuations can be expressed
by a dimensionless parameter; in the limit of dissi-
pative vortex motion this parameter is determined
by the normal state resistivity
n
of the material,
Qu =(e
2
/)(
n
/").
With the three fundamental parameters quan-
tifying quenched disorder (j
c
/j
0
), thermal fluctua-
tions (Gi), and quantum fluctuations (Qu)weare
in a position to express the difference between the
conventional low-T
c
superconductors and the high-
temperature superconductors in a compact way:
In conventional superconductors pinning is usually
strong, j
c
/j
0
∼ 10
−2
–10
−1
, whereas both thermal
and quantum fluctuations are weak, Gi ∼ 10
−8
and
Qu ∼ 10
−3
. In generic high-T
c
superconductors such
as YBa
2
Cu
3
O
7−y
(YBCO)ortheBi-andTl-basedma-
terials (e.g., Bi
2
Sr
2
Ca
1
Cu
2
O
8+y
(BiSCCO)) pinning is
usually weak, j
c
/j
0
∼ 10
−3
–10
−2
,whereasthermal
and quantum fluctuations are large, Gi ∼ 10
−2
and
Qu ∼ 10
−1
. This change in the relative importance of
static and dynamic disorder is a consequence of the
extreme material parameters characterizing the ox-
ide superconductors: First of all, the transition tem-
perature T
c
≈ 100 K is very large and hence the co-
herence length ∼ v
F
/T
c
is small, ≈ 16 Å in
YBCO and ≈ 25 Å in BiSCCO (v
F
denotes the
Fermi velocity). Since the oxide superconductors are
doped insulators rather than generic metals the elec-
tronic density is small, hence v
F
is small and the re-
sistivity
n
as well as the penetration depth tend
to be large (provided we can still use Fermi-liquid
concepts for these crude estimates): the penetration
depth ≈ 1400 Å and ≈ 2000 Å in YBCO and BiS-
CCO, respectively. In addition, the layered structure
of the oxides (with layer separation d ≈ 12 Å and
d ≈ 15 Å in YBCO and BiSCCO, respectively) intro-
duces a large uniaxial anisotropy in the system which
reduces the coherence length along the anisotropy
axis (the c-axis of the material) by an additional fac-
tor " =(m/M)
1/2
< 1; " ≈ 1/8-1/5 in YBCO while