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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 291

VOLUME 76, NUMBER 14 P H Y S I C A L R E V I E W L E T T E R S 1 APRIL 1996

The local magnetization measurements were performed in

applied ﬁeld H

jjzjjc axis using arrays of 10 3 10 mm

GaAs/AlGaAs Hall sensors [24,25].

Figure 1 illustrates typical local magnetization loops as

measured by the sensors. The ﬁrst-order transition [10]

is clearly seen as a small positive sharp step in B

2 H

at elevated temperatures as demonstrated for the as-grown

crystal at 64 K [Fig. 1(a)] and for the overdoped crys-

tal at 70 K [Fig. 1(b)]. Figure 2 shows the mapping of

the ﬁrst-order transition as a function of temperature for

three crystals of different oxygen stoichiometry, i.e., dif-

ferent anisotropy. We ﬁnd that the transition line shifts

signiﬁcantly with anisotropy and moves to higher ﬁelds

for more isotropic crystals. In addition, in all the crys-

tals the ﬁrst-order transition terminates abruptly at a sam-

ple dependent critical point (indicated by arrows) in the

range of 40 to 50 K [10]. Two scenarios have been pro-

posed for the ﬁrst-order phase transition, the melting of

the vortex lattice [5] and the decoupling of the vortex-

line liquid into uncorrelated vortex pancakes [4]. In both

cases, the predicted transition line is very sensitive to

anisotropy ´ [1], and scales as ´

. Therefore, the ob-

served anisotropy effect (Fig. 2) is consistent with both

theories and does not resolve this ambiguity. The temper-

ature dependence, however, should be different. Accord-

ing to the melting theory [5], an approximate power-law

behavior B

! B

1 2 T #T

is expected, where

a#2. The best ﬁt for the optimally doped crystal results

in a high value of a $ 2.28 and unrealistic T

$ 104 K.

As-grown crystals give a $ 1.57, T

$ 94.6 K, and the

overdoped one results in a $ 1.51 and T

$ 88 K. The

melting ﬁts produce too high values of T

for all the sam-

ples and a sample dependent a. Therefore we cannot

derive a consistent estimate of ´ from these ﬁts. The

decoupling scenario, on the other hand, results in much

more consistent ﬁts as shown by the solid lines in Fig. 2.

The theory [4] predicts B

! B

2 T

#T , where

FIG. 1. Local magnetization loops B

2 H

measured in the

central part of BSCCO crystals as a function of local magnetic

induction. At elevated temperatures a ﬁrst-order transition is

manifested by equilibrium magnetization step at B

presented

for the as-grown crystal at T ! 64 K (a) and overdoped crystal

at T ! 70 K (b). At low temperatures a sharp, possibly

second-order, transition is observed at the onset of the second

magnetization peak B

as shown for the optimally doped

crystal at T ! 25 K (c).

% a

#&4pl

d. Here a

$ 0.1 is a uni-

versal parameter, and d $ 15 Å is the interlayer spacing

that varies by less than 0.1 Å with changing the annealing

temperature [17,20]. Note that this equation depends on

the combination of ´ and l!0". Since we know l!0" from

[21], we are able to estimate the change in anisotropy

due to variation in oxygen stoichiometry. The obtained

values are ´

$ 100 for the overdoped sample (lower

anisotropy), ´

$ 120 for the as-grown, and ´

$ 130

for the optimally doped (higher anisotropy). We conclude

that the optimally doped sample has indeed the largest

anisotropy. Currently, there is unfortunately no reliable

method for a direct quantitative measurement of such high

. It is interesting to note that a recent analysis of

the melting behavior in layered superconductors at low

ﬁelds [6] predicts a power-law dependence of B

!T " with

a ! 1.5 and B

that scales with ´#l

rather than ´

Also in this case, using the above estimates of l, our data

in Fig. 2 imply a similar relative variation of ´ for the

studied crystals. A simultaneous and independent deter-

mination of both ´ and l

is therefore of major im-

portance for determination of the origin of the observed

ﬁrst-order phase transition. Note also that neither melting

nor decoupling theories account for the ﬂattening of the

transition line near the critical point and the critical point

itself (see Fig. 2).

We now discuss the behavior at temperatures below

the critical point. As demonstrated in Fig. 1(c), a large

second magnetization peak is observed at ﬁelds above

!T ". The magnetization changes abruptly at B

opposite directions on increasing and decreasing ﬁelds

in contrast to the behavior at B

. Analysis of the ﬁeld

proﬁles across the sample shows a transition from smooth

dome-shaped proﬁles due to surface and geometrical

barriers at low ﬁelds to Bean proﬁles due to the onset

FIG. 2. The ﬁrst-order phase transition B

for three

BSCCO crystals of different anisotropy. The solid lines are

ﬁts to the decoupling transition B

! B

2 T

#T with

$ 84.8 K, B

$ 946 G for the overdoped sample

, T

91.0 K and B

$ 397 G for as-grown

, and T

$ 89.5 K,

$ 275 G for the optimally doped crystal

. The arrows

indicate the position of the critical point.

2556

Fig. 8. Magnetization loops in BSCCO. Left top and bottom: High T , reversible

jump at melting. Right: Low T , ﬁshtail or second magnetization peak, from Ref. 65.

this line.

Several scenarios have been suggested to explain

the second magnetization peak, including ﬁeld dependent

pinning,

dynamic effects,

and dimensional crossover

However none of these mechanisms can explain the

sharpness of the observed transition.

Figure 4!a" shows the

FOT line B

(T) together with the second peak line B

(T)

in the as-grown crystal. These two lines merge at a critical

point T

, where the FOT manifested by the equilibrium

magnetization step terminates. Figure 4!a" also shows the

position of these lines, both after a low dose of electron

irradiation and after a second repeated irradiation of the same

crystal, thereby doubling the dose. The selected low doses

produce sufﬁciently weak point disorder that the FOT step is

preserved and the position of the FOT line is unaffected at

high temperatures. It also results in some increase of bulk

pinning at very low temperatures. The main effect of this

weak point disorder is, however, a signiﬁcant shift of the

(T) line toward smaller ﬁelds as reported previously.

Further, the FOT line shows downward curvature at interme-

diate temperatures and an associated shift which matches that

of the B

line. This remarkable behavior !the irreversibility

or depinning line is generally expected to shift upwards with

disorder in contrast to a downward shift here" was conﬁrmed

on a second different batch of crystals.

We argue that these ﬁndings demonstrate that !i" phase B

is a well ordered vortex lattice, which we will refer to as the

quasilattice state, !ii" phase C is a highly disordered vortex

solid, and !iii" B

(T) is a disorder-driven solid-solid transi-

tion. These conclusions are consistent with other experimen-

tal ﬁndings; small angle neutron diffraction measurements in

phase B show clear Bragg peaks which are associated with

an ordered state.

The ordered structure of phase B is also

supported by

and decoration

experiments. Figure

4!a" conﬁrms this conclusion by demonstrating that phase B

is unstable with respect to weak point disorder, as expected

for an ordered state, and its stability region on the B-T phase

diagram shrinks with increased point disorder. The existence

of a ﬁnite long-range order and a lattice symmetry in phase B

is also strongly supported by the fact that this phase melts

through a FOT at elevated temperatures. Furthermore, we

ﬁnd that a low dose of columnar defects does not destabilize

the quasilattice. This implies that the vortices in this phase

are rather straight linelike objects. Point disorder, in contrast

to the columnar defects, induces distortions or wiggling of

the lines and leads to the destruction of the ordered lattice.

Phase C, unlike phase B, is a highly disordered state accord-

ing to the neutron diffraction and

SR measurements.

Fig-

ure 4!a" further emphasizes this conclusion by demonstrating

that phase C is in fact stabilized by point disorder and its

range on the B-T diagram expands when disorder is in-

creased. This disordered state melts through a continuous

transition where it becomes mobile at the depinning line of

Fig. 1.

15,27

It would be very surprising if the B

transition did not

share a common mechanism with the FOT, since these two

lines always form one continuous transition line across

which the quasilattice state is destroyed. We argue that this is

indeed the case; there are two sources for destruction of long

range correlations in the vortex lattice. These are thermally

induced vortex deformations and point disorder induced dis-

tortions respectively. As ﬁeld or temperature are increased

the elastic stiffness of the lattice becomes softer and the

quasilattice becomes more susceptible to destruction. At high

temperatures thermally induced deformations are large,

while the effects of weak point disorder are strongly sup-

pressed by thermal smearing. As a result at high tempera-

tures the quasilattice is destroyed through thermal melting,

and the position of the transition line is practically indepen-

dent of disorder $see Fig. 4!a"%. At low temperatures the

situation is just the opposite since thermal deformations are

very weak. Here the quasilattice is destroyed at some char-

acteristic ﬁeld through disorder-induced transition, and the

position of the transition line is practically independent of

temperature. With increased point disorder this transition is

shifted downwards. At intermediate temperatures, the two

mechanisms add up and tail smoothly into one another,

forming one continuous quasilattice phase boundary. The

critical point T

marks where the transition changes from

predominantly disorder induced to thermally induced charac-

ter, and thus shifts to higher temperatures with increased

disorder $Fig. 4!a"%. At temperatures slightly above T

dis-

order still contributes weakly by assisting the thermal de-

struction of the quasilattice, thereby resulting in the peculiar

downward curvature of the FOT line towards T

. At the

highest temperatures the point disorder is completely

smeared out by thermal ﬂuctuations and hence no effect on

the FOT line is observed.

In the above discussion we have neglected the possibility

that point disorder induced by electron irradiation can

modify the penetration depth &, anisotropy

, and T

, and

thus cause an artifactual shift of the transition lines. Numeri-

FIG. 4. !a" FOT lines B

(T) !open symbols" together with the second

peak transition lines B

(T) !ﬁlled symbols" for BSCCO crystals irradiated

with different indicated doses of 2.5 MeV electrons. Enhanced point disor-

der destabilizes the quasilattice and causes a downward shift of the

(T) transition line. !b" Comparison of B

(T) and B

(T) lines for dif-

ferent oxygen stoichiometry. Note the relative change in slope of the

(T) lines at high temperature in contrast to the behavior in !a".

RAPID COMMUNICATIONS

56 R519VORTEX-MATTER PHASE TRANSITIONS IN...

Fig. 9. Phase lines in BSCCO from Ref. 65, as in Fig. 6 upon electron irradiation

(top) or changing the doping (bottom): the melting line and second peak line move

together. The solid region is reduced upon increasing disorder as expected from

an instability line of the Bragg glass to dislocations (correlated disorder has the

opposite eﬀect — see below).

ture of a maximum in j

, or at least in the persistent (i.e. relaxed) current as

a function of B. Hence it was unclear whether it was some broad maximum

related to ﬁeld variation of elastic coeﬃcients (due to their non-locality),

as often argued to explain the so-called peak eﬀect observed in low T

, or

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

292 P. Le Doussal

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265

270

275

280

285

30 35 40

B (G)

T (K)

Vortex liquid

Vortex

lattice

Amorphous

20 40 60 80

100

200

300

Vortex liquid

Vortex lattice

(T)

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1.2

1.4

1.6

1.8

2.0

–0.8

–0.4

0.0

0.4

0.8

35 40 45

!S (k

per vortex per layer)

!B (G)

T (K)

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Fig. 10. Left: Hysteretic magnetization is observed below the transition (open

circles). Application of H

ac⊥

removes the residual hysteresis and enhances the

magnetization step. Right: Inset, the ﬁrst-order B

(T ) line (ﬁlled circles) along

with the second magnetization peak line B

(T ) (open circles). Main panel: The

ﬁrst-order transition was observed at all temperatures around T

at which fully

reversible magnetization was achieved by shaking. From Ref. 69.

some dimensional crossover, etc. Remarkably, it was then discovered that

the second peak line was perfectly correlated with the position of the melt-

ing curve, appearing to be merely its continuation at low T (see Fig. 9). A

proof of that was obtained in a remarkable series of experiments. By shak-

ing the vortex lattice using a transverse ac ﬁeld, the magnetization could be

made reversible down to low T . As a result, a FOT was revealed below the

point where the melting curve has a reentrant behavior, called inverse

melting

(see Fig. 10). This made T

unlikely to be a tricritical point, but

rather only a point of rapid variation of the eﬀective disorder due to thermal

screening. It was also shown that B

at low T corresponds to a ﬁrst order

transition line between a low j

phase and a high j

one (phases B and C in

Fig. 6). Coexistence of these two phases in the same sample near B

, and

their domain wall motions, was revealed by the Bean proﬁle.

Note that

larger deformations in the high ﬁeld less ordered glass (or pinned liquid) is

expected to result in a higher j

A universal phase diagram was thus emerging, with a low-ﬁeld weak dis-

order vortex solid phase, consistent with a Bragg glass (more evidence for

that is discussed in Sec. 4), unstable via a ﬁrst order transition along a unique

line which gradually changed from thermal melting to disorder-induced melt-

The E-j characteristics of the two phases are predicted

and observed to cross:

the persistent currents depends on the observation time, resulting in more or less

pronounced peaks.

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

Novel Phases of Vortices in Superconductors 293

ing as it crosses the irreversibility line. The BrG instability line was found

reentrant at very low ﬁelds near H

. Similar phase diagrams were ob-

served in many high T

materials with a wide range of anisotropy: thallium

compounds, La

2−x

CuO

(LSCO),

YBCO,

1−x

BiO

(BKBO),

and by a variety of probes (e.g. muons

). It was also consistent with nu-

merical simulations.

In BSCCO, layer decoupling of pancake vortices is quite likely to play

a role. This is indicated by Josephson Plasma Resonance experiments

77,78

which probe the superconducting phase coherence between adjacent layers.

The latter is found to be strongly reduced across the melting line. A decou-

pling transition was predicted in the absence of disorder.

19,20,108

It should

occur already inside the vortex solid, and be triggered by proliferation of

vacancy-interstitial defects inside each layer, also called quartets.

79,80

The

corresponding picture for more isotropic superconductors is the transition

from the usual vortex line crystal to a supersolid, i.e. a solid of entangled

vortex lines.

Since a vacancy is a bound pair of dislocations, this decoupled

solid, or supersolid, is still a topologically ordered crystal with long range

translational order. In presence of pinning disorder, an analogous decou-

pling transition is predicted

21,82,83

within the Bragg glass phase, between a

coupled BrG (also called a Josephson glass) and a decoupled one. An ex-

planation for the second peak line in terms of such a decoupling was also

proposed.

Until now, there seems to be no strong evidence for a decoupled

solid (or supersolid, or decoupled BrG) so it is quite possible that melting

and decoupling occur simultaneously. In terms of the extended Lindemann

criterion it means that L

(the translational order length along z) reaches

∼ d, the interlayer spacing, before R

reaches a (i.e. the criterion becomes

hδu(z = d)

i ∼ c

). It is then likely that the unbinding of dislocation loops

starts with formation of quartets. In the end, it is the universality of the

structure of the phase diagram for all compounds, from the most anisotropic

to isotropic one, which is the most convincing argument for a single scenario.

This motivated a reexamination of low T

superconductors and eluci-

dation of the famous “peak eﬀect”, a pronounced but broad maximum in

(T ) before it vanishes near H

It was thus found that in NbSe

, in

which thermal ﬂuctuations are very small, the liquid is replaced by a pinned

disordered phase with a high j

Transport measurements in Corbino ge-

ometry (i.e. vortices do not cross sample edges) revealed

a ﬁrst-order jump

in j

(see Fig. 11) in agreement with the Bragg glass to disordered phase

universal phase diagram, a feature previously masked by the phase coex-

istence unavoidable in the conventional stripline geometry commonly used

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

294 P. Le Doussal

4.4 4.8 5.2 5.6

[mA]

T [K]

2 kOe

dis

Strip AC

Corbino

Strip DC

OP in Corbino,

coexistence in Strip

DP in both

geometries

ord

-C

-S

-C

-V

4.2 4.5 4.8 5.1 5.4 5.7

T [K]

H [kOe]

0.8 1.2 1.6 2.0

[mA]

H [kOe]

4.2 K

Corbino Strip

Paltiel et al., Fig. 2

VOLUME 80, NUMBER 4 P H Y S I C A L R E V I E W L E T T E R S 26 JANUARY 1998

FIG. 1. Shown are the q value and reﬂectivity of the ﬁrst

diffraction peak at T ! 4.6 K (open symbols, dashed line) and

6.4 K (solid symbols, full line) versus ﬁeld. For h ! H !H

0.5, the sample is in the intermediate mixed state. Analysis of

the SANS data was performed for h . 0.65, where the sample

is in the mixed state.

diffraction out to (3,0) are seen. Such large reﬂectivity

leads to substantial multiple scattering corrections for

higher order reﬂections, so our analysis is derived entirely

from the ﬁrst order peaks. Analysis of the form factor

was performed using ﬁrst order Ginzburg-Landau (GL)

corrections to the London model [7]. Separate ﬁts using

the Clem model [8] gave k

"T ! 6.4 K# ! 0.82 and

"T ! 4.6 K# ! 1.34, consistent with the GL analysis

and in agreement with thermodynamic studies. Typical

of these models, the individual values of j ! 29.7 nm

and l ! 25.2 nm are both about 30% lower than the

thermodynamics data.

Three FLL correlation lengths can be determined [9]

from a SANS experiment. Fits to the azimuthal and radial

widths of data in the plane of the detector, such as are

shown for three different ﬁelds in Fig. 2, give information

on two correlation lengths perpendicular to the ﬂux lines.

The radial width, corrected for instrumental resolution,

gives the positional correlation length j

, parallel to

the reciprocal lattice vector G, and is related to the

compressive displacements of the FLL. Similarly, the

azimuthal width gives j

, perpendicular to G, which is

related to shear displacements. The last length, derived

from the width of the rocking curve in our geometry, is

the longitudinal correlation length j

, a measure of the

correlations along the ﬂux lines, which is related to tilt

displacements. Our experimental setup is optimized for

, where the resolution is roughly 2 orders of magnitude

better than for either j

or j

. Macroscopic ﬁeld

gradients, which can inﬂuence j

and j

were minimized

using both a ﬁeld cooled procedure and masking of the

perimeter of the sample. In addition, such effects were

checked for using analysis of higher order diffraction

peaks and scans at different neutron wavelengths, and

were negligible.

FIG. 2(color). Shown are diffraction patterns from the FLL

in three regimes at T ! 4.6 K. Top: FLL crystal (H !

1.1 kOe, h ! 0.47). Middle: hexatic FLL just below the peak

effect (H ! 2.15, h ! 0.91). Bottom: ring of scattering from

amorphous FLL (H ! 2.25, h ! 0.96). The radial width of the

peaks determines the positional correlation length ($1a

), the

azimuthal width of the peaks gives the orientational correlation

length, and rocking through the Bragg gives the longitudinal

correlation length ($100a

). Nonuniform peak intensities are

instrumental artifacts.

The results of the SANS experiments at T !

6.4 K and T ! 4.6 K are summarized in Fig. 3. The

data are plotted versus reduced ﬁeld h ! H!H

, where

"T ! 6.4 K# ! 1.55 kOe and H

"T ! 4.6 K# !

2.35 kOe. The positional correlation length, normalized

to the measured FLL spacing a

"mm# ! 0.134B

21!2

"kG#,

is shown in the middle panel. At low ﬁelds, all correlation

lengths are long, and the FLL is a well ordered crystal as

shown in Fig. 2, top. As a precursor to the peak effect,

we ﬁnd that j

$ 2 at the minimum in the criti-

cal current, where strong deviations from J

h ! const

begin at both temperatures. As the critical current rises,

we ﬁnd j

! 1 (Fig. 2, center). Commensurate with

$ 2 is the onset of a rapid reduction in j

, shown

in the bottom panel of Fig. 3. A uniform ring of scat-

tering (Fig. 2, bottom), with j

! 1 and j

! 1,

is seen only at the peak in the critical current. This

dramatic and complete positional disordering is in accord

with the widely accepted scenario for the peak effect: It

is a static, structural disordering of the FLL.

Pioneering studies of the positional correlation length

were ﬁrst performed by Lippmann et al. [10]. In those

data, the focus was on the doping dependence, and the

regime of the peak effect was not measured. More

recently [11], Lynn et al. studied the two transverse

correlation lengths j

and j

, interpreting the reduction

of these length scales in terms of FLL melting, although

834

Fig. 11. Left, from Ref. 86: The conventional peak eﬀect in a strip is found to

result from coexistence of an ordered solid (OP) and a disordered pinned (DP)

phase, the ﬁrst order jump at the OP/DP disordering transition (DT) (at T

) is

revealed in the Corbino geometry. The phase diagram H

(T ) is plotted and has

the universal form, the liquid phase being replaced by the DP phase. H

is deﬁned

as 0.1 the normal resistance. Right, from Ref. 87: Bragg peaks disappearing into a

ring in NbSe

through the peak eﬀect.

for critical current measurements. The topological order of the OP phase

over distances ∼100a

was conﬁrmed by neutron diﬀraction,

providing a

consistent picture in NbSe

and in Nb.

We close this overview by the eﬀect of correlated disorder on the phase

diagram. It arises either naturally from twin planes or dislocation lines, and

can be created and controlled as columnar disorder from heavy ion irradia-

tion.

In the latter case, using the ﬂux line to quantum boson analogy, a

new, Bose glass (BoG) phase was predicted

where ﬂux lines are localized

along columns when these are parallel to the external ﬁeld. This being a very

eﬃcient way to pin the vortices, even a small density of columns results in a

dramatic increase in the critical current. A high concentration of columnar

defects pushes the melting line to much higher ﬁelds and results in a phase

diagram with a single glass phase, the BoG, that melts into a liquid phase. In

this case, the BoG/liquid transition is predicted to be continuous, very much

as the vortex glass, but with anisotropic scaling

90,91

as observed experimen-

tally.

92,160

At suﬃciently low concentration of columnar defects, however,

the transition was observed to remain ﬁrst order.

Columnar defects are

thus of technological importance, and are now included in superconducting

tapes in applications, e.g. as nanorods.

Their eﬀect can even be increased

by introducing a small dispersion in their direction along the c-axis, leading

to the splay glass phase,

where localization and entanglement eﬀects were

predicted and observed

to conspire to a slowdown in the creep relaxation.

Correlated disorder is developed in Sec. 5.2.

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

Novel Phases of Vortices in Superconductors 295

Let us now mention some important open questions. A ﬁrst one is the na-

ture of the amorphous strongly disordered pinned phase (region C in Fig. 6).

As discussed again in Sec. 5.1, the vortex glass scenario is still not fully es-

tablished. On the theoretical side, the gauge glass phase was found unstable

to screening eﬀects (ﬁnite London length). On the experimental side, ﬁts

to vortex glass scaling in BSCCO seem to fail to lead to universal values

for the exponents. In YBCO, the question was raised whether some of the

previous vortex glass scaling may have been due to twins, since correlated

disorder also leads to similar (though anisotropic) scaling and since the FOT

is more visible in untwinned YBCO crystals. Thus other scenarios have been

proposed, such as a polymer glass or pinned liquid,

a vortex molasses,

decoupled solid,

a frozen mesh of dislocations.

A second open question

stems from recent observations in BSCCO that the disordered solid/liquid

depinning line extends as a rather sharp thermal depinning line within the

ordered solid.

It has been proposed as a true transition,

100

or described

in terms of a Josephson glass,

however, since a perfect solid is just ruled

out by present theories of pinning (in d = 3 it would require diverging elas-

tic coeﬃcients), it may also be simply a (rather sharp) thermal crossover

within the Bragg glass phase (see e.g. Ref. 194). Even so, a more accurate

description of such a sharp crossover is clearly needed.

Finally, we left out new developments in moving phases of vortices. A

short overview is given in Sec. 5.3.

3. Melting of the Vortex Lattice

Let us recall the GL free energy, F = F

kin

mag

in terms of ψ = |ψ|e

iφ

kin

|Dψ|

ψ|

(6)

xα|ψ|

|ψ|

, F

mag

8π

(7)

where D = ∇ + i

2π

A, and one also often uses the Gibbs energy G = F −

4π

xHB. One has |Dψ|

= (∇|ψ|)

+ |ψ|

(Dφ)

, with the super-current

j =

∗

|ψ|

Dφ.

We are interested in the ﬂuctuation free energy F = −T ln Z with Z =

DψDAe

−F/T

and we denote the free energy density as F

. We now brieﬂy

describe the two main approaches, either from computing the elastic matrix

of the vortex lattice from the GL theory (or more simply, the London theory),

and applying the Lindemann criterion, or directly from using the lowest

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

296 P. Le Doussal

Landau level approximation on GL and comparing the free energies of the

liquid and the solid.

3.1. Interactions between vortices and line energy (isotropic)

For large κ one can use constant |ψ| and the London equations except in the

(small) core region. The kinetic energy becomes F

kin

≈ 2π

x(λ/c)

where λ

= mc

/(4π(e

∗

)

|ψ|

). In presence of vortex lines at positions

(z) = (r

(z), z) the local ﬁeld B(r, z) is determined from:

(1 −λ

∇

)B(r, z) = Φ

T (r, z) , T (r, z) =



(z)

, 1



(r −r

(z)) (8)

with q · T (q) = 0 from ﬂux line conservation. The energy, neglecting the

core, is the sum of the magnetic and kinetic energy from the currents, using

∇ × B =

4π

F =

8π

x[B

+ λ

(∇ × B)

] =

(2π)

V (q) |T (q)|

(9)

with:

V (q) =

4π

1 + λ

(10)

of 3D fourier transform V (x) = 

−x/λ

interaction between line elements.

For some quantities, one must use V

(q) (one choice is V

(q) = e

−ξ

V (q))

which is cutoﬀ at small distance. One ﬁnds the line energy e

= F/L of a

single straight vortex line of length L as e

⊥

(2π)

⊥

, 0) ≈ 

ln κ =

/(4π), and the interaction energy W (r) = 2

(r/λ) per unit length,

between two straight ﬂux lines separated by r.

3.2. Elasticity of a vortex line and of the vortex

lattice (isotropic)

For a weakly distorted single vortex line, rotational invariance yields the total

energy F = e

L ≈ F

+ F

and the elastic energy F



dr(z)



For the vortex lattice, one denotes the vortex positions r

(z) =

+ u

(z), where R

denote the perfect 2D Bravais lattice. One de-

ﬁnes u(q) = a

(z)e

⊥

+iq

from which one retrieves u

(z) =

q,BZ

u(q)e

−iq

⊥

−iq

, with the limit

q,BZ

≡

−d

→

(2π)

(BZ des-

ignates integration over the ﬁrst Brillouin zone). From (9) the total energy

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

Novel Phases of Vortices in Superconductors 297

is:

F =



1 +

(z)

)



(z) −x

)) (11)

Expanding to second order one ﬁnds:

q,BZ

αβ

(q)u

(−q) (12)

with an elastic matrix:

αβ

(q) = a

−2

dz[V

αβ

(R, z)(1 − e

⊥

R+iq

) + δ

αβ

V (R, z)e

⊥

R+iq

]

= a

−4

αβ

(K + q) − M

αβ

(K)) , M

αβ

(q) = (q

+ δ

αβ

)V (q)

(13)

using twice Poisson formula

f(R) =

−2

rf(r)e

iKr

. For the

triangular lattice one ﬁnds:

αβ

(q) = Φ

(q)P

αβ

⊥

) + Φ

(q)P

αβ

⊥

) (14)

(q) = c

(q)q

⊥

+ c

(q)q

, Φ

(q) = c

(q)q

⊥

+ c

(q)q

(15)

with P

αβ

(q) =

, P

αβ

(q) = δ

αβ

−

and the bulk and shear moduli, for

 B < 0.4H

(q) ≈ c

(q) ≈

4π

1 + λ

, c

≈

(8πλ)



(16)

from K = 0 and K 6= 0 respectively. The tilt modulus c

(q) = c

(q)+c

(q)

with c

(q) from the contribution of K 6= 0. One has c

(q) = B(H −B)/(4π)

for q → 0 and c

(

, q

) ∼



ln(κ

/(1 + λ

+ q

)) and K

= 2

√

π/a

the radius of the circular BZ with same area. An estimate of the density of

vortex line times the elastic energy of a single line gives c



ln κ which

is valid for q

⊥

∼ π/a. For smaller q

⊥

the VL is much stiﬀer. Note that

 c

and transverse modes are dominating.

The elastic coeﬃcients were also computed for larger ﬁelds B > 0.6H

using GL beyond the above London approximation.

5,29,30

One eﬀect is that

λ → λ

= λ/

√

1 − b with b = B/H

(T ), with however c

≈



(1 − b)

At very small ﬁelds, near H

, the vortex lattice is dilute and for B <

/ ln κ one has c

∼ 3c

∼ 

1/2

−3/2

−a/λ

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

298 P. Le Doussal

3.3. Anisotropic elasticity

The above calculation, when extended to the anisotropic case using

anisotropic London theory

101

yields an elastic matrix which depends on θ,

the angle of H from the c-axis. Here we only discuss θ = 0. c

is found

unchanged and (see e.g. Ref. 4):

(q) ≈

4π

1 + λ

(q) , c

(q) ≈

4π

1 + q

⊥

+ q

(17)

while c

(q) grows from ∼



ln(κ/)/2a

for λq

 1 (the Josephson cou-

pling contribution) to ∼ 

/(2λ

) for λq

 1 (the electromagnetic con-

tribution). For a single vortex line F

|r(q

where the

line tension e

) becomes strongly dispersive and increases from a value

˜

≈ 



ln(κ/) for λq

 1 to ≈

for λq

 1. Anisotropy reduces ˜

and

thus strongly enhances the tendency to ﬂux line wandering: in the solid, the

individual ﬂux lines are kept place by their neighbors, but in the liquid this

results in entanglement of the ﬂux lines.

More generally it was argued

that for Q a quantity such as volume V ,

energy E, temperature T , or ﬁeld, one can relate the anisotropic system to

the isotropic one using:

Q(θ, H, T, ξ, λ, ∆) = s

iso

(

H, T/, ξ, λ, ∆/) (18)

where 

= 

sin

θ + cos

θ and θ is the angle of H from the c-axis. One has

= s

= , s

= s

= 1/

. Here ∆ is the disorder strength (see

below). This again shows that ﬂuctuations, both thermal and from disorder,

are enhanced by anisotropy.

3.4. Melting of the elastic vortex lattice

We now consider the thermal ﬂuctuations, described by the canonical par-

tition function Z =

Due

−F

[u]/T

. To estimate the melting line from

the Lindemann criterion one usually neglects the longitudinal mode since

(k)  c

and write:

i ≈ T

⊥

(2π)

2π

⊥

+ c

(q)q

(19)

If one neglects the dispersion of c

and uses a circular BZ:

i ≈ T

4π

√

, T

≈ 2

√

(20)

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

Novel Phases of Vortices in Superconductors 299

If one instead uses the expression:

(q) =

4π(1 + λ

+ λ

⊥

)

(21)

one ﬁnds (20) with c

→ c

eﬀ

= c

(0)/F (y

, y

)

with y

= λ

(0)

with F (y

, y

) =

dx(1 + y

)

1/2

(1 + y

)

−3/2

. The

combination y

= λ

is large and using c

= 

(1 − b)

/4a

with

b = B/H

(T ) = 2πξ

one ﬁnds y

1−b

and

i ≈ a



√

π(1 + y

√

1 + y

)

(1 − b)

−3/2

(22)

Note that thermal ﬂuctuations are large since T and λ

are large. It is the

region q

⊥

≈ K

that contributes most. A slightly more detailed calcula-

tion

29,30

gives the melting curve b

(T ) as the solution of:

√

1 − b

√

1 − t

√

2 − 1)

√

1 − b

+ 1

2πc

√

(23)

Near T

(T )  1 and one ﬁnds:

(T ) = 5.6

(0)(1 − t)

(24)

Although we did not discuss here melting in 2D geometries, such as ﬁlms,

we may indicate a few among numerous studies.

102,103

3.5. Landau

Ginzburg approach to melting

Another approach is the lowest Landau level approximation (LLL) within

GL. Here we follow the presentation in Ref. 40, to which we refer for details

(see also Ref. 36). Consider (6) with α(T ) = −α

(1 − t) and rescale r →

ξ(0)r, z → ξ(0)z where ξ(0) =

2mα

and ψ

→

2α

, b =

(0)

with

(0) =

2πξ(0)

. One then obtains:



|Dψ|

|∂

ψ|

−

1 − t

|ψ|

(b − h)



(25)

with ω = T β/(4α

ξ(0)

) = πt

√

2Gi. For large κ one can neglect the ﬂuc-

tuations of the electromagnetic ﬁeld, hence we can consider a uniform B

and use the Landau gauge, i.e. D

= ∂

− iby, D

= ∂

. The LLL condi-

tion −D

ψ = bψ implies

|Dψ|

= b

|ψ|

. Further rescaling r → r/

√

z → z



bω

4π

√



−1/3

, ψ

→



bω

4π

√



2/3

and neglecting the last term yields

September 2, 2010 15:31 World Scientiﬁc Review Volume - 9.75in x 6.5in ch13

300 P. Le Doussal

the ﬂuctuation free energy density F

(0)

2πκ



bω

4π

√



4/3

in terms of the

reduced free energy density f

= −

4π

√

DψDψ

∗

exp



−

f[ψ]

4π

√



(26)

f[ψ] =



|∂

ψ|

|ψ|



, a

=−



bω

4π

√



−2/3

1−t−b

(27)

being the volume in the rescaled units. Let us start with the mean ﬁeld

theory

which simply minimizes (27), i.e. f

= f[ψ

min

]/V

. For a

> 0

the minimum is ψ = 0 corresponding to the normal phase. For a

< 0,

i.e. B < H

(T ) = H

(0)(1 − t), a non-trivial minimum is searched for in

the LLL (exact at a

≈ 0) i.e. it obeys:

((∂

− iy)

+ ∂

+ 1)ψ = 0 (28)

hence it must be a superposition of the form ψ(r) =

(r) with

(r) = π

−1/4

x−

(y−k

)

. The triangular vortex lattice is described

choosing x along lattice direction k

= k

= 2πn/a

with C

n+2

= C

and

= iC

leading to the Abrikosov lattice (AL) wave function ψ(r) = Cφ(r)

where

φ(r) = 3

1/8

∞

`=−∞

1/4

1/2

`x)−

(y−3

1/4

1/2

(29)

is normalized in the unit cell, and f

= min

[|C|

|C|

] = −

2β

where:

= h|φ(r)|

u·c

−

)

≈ 1.16 (30)

for any lattice

with vectors d

, d

, is minimum for the triangular lattice,

hence selected. Since f

= 0 for a

< 0, it leads to the mean ﬁeld speciﬁc

heat jump ∆C/T = H

/8πκ

. The (reduced) structure factor of the

perfect AL, associated to the density |ψ(r)|

is found as

S(q

⊥

, z = 0) =





4π

−K

δ(q

⊥

− K).

From (27) the ﬂuctuation free energy of the liquid was obtained in an

improved perturbation theory

in the |ψ|

term around ψ = 0 leading to:

liq

= 4µ(1 + g(x)) ≈

→−∞

1/2

+ ·· (31)

− a

µ − 4 = 0 , x = 1/(2µ

) (32)