Bennemann K.H., Ketterson J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors; Volume 2: Novel Superconductors

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620 G. Blatter and V.B. Geshkenbein

glass describing a disordered granular superconduc-

tor [487,488],

H =−J



i,j

cos('

− '

− A

) ,

2



A · dl , (12.482)

where the variables '

denote the phases of the indi-

vidual grains and A is the vector potential; the ran-

domness and frustration is introduced through the

potentials A

, usually chosen as quenched random

variables uniformly distributed between 0 and 2.

Recent numerical simulations (via a parallel temper-

ing MonteCarlo technique) of the three-dimensional

gauge glass provide evidence for a glass transition

at T

≈ (0.47 ± 0.03)J with characteristic expo-

nents  ≈ 1.39 ± 0.20 and z ≈ 4.2 ± 0.6 [489].

However, including screening into the model (by

adding a ﬂuctuating gauge field a , A

→ A

+ a

/,

H → H +(1/2)



[∇∧a]

, ∇a =0)appearsto

destroy the finite temperature gauge glass transi-

tion [490,495]. On the other hand, the magnetic field

penetrating the (weakly disordered) superconduc-

tor in the form of vortex lines naturally introduces

an anisotropy into the system — it remains unclear

what effect this change of symmetry has on the ex-

istence and nature of the vortex-glass transition in a

3D bulk disordered superconductor.

The above scaling analysis can be extended to an

anisotropic situation as naturally provided by the

vortex system and definitely required for the Bose-

glass transition in the presence of correlated disor-

der [55]; in this case two diverging length scales

⊥

= l

loc

(T) ∝ (T

− T)

−

⊥

and l



∼ l

⊥

(12.483)

are introduced, with D

a short distance “diffusion”

constant. In the simplest scenario, D

remains finite

on approaching the phase transition, such that the

scaling exponent 



takes the value 



=2

⊥

.The

dynamic scaling exponent z



is defined via 

∝ l



⊥

An (experimental) analysis of the current-voltage

characteristicin the vicinityof the transitionintothe

glass phase can then be used to extract the scaling ex-

ponents z (z



)and (

⊥

) (see Fig. 12.36): The slope

Fig. 12.36. Current-voltage characteristic of a superconduc-

tor in the mixed state. The plot provides the vortex-glass

scalinglawsforE(j),(T),and j

(T)and theglassexponent

. At the glass transition T

the characteristic is algebraic,

E ∝ j

,with˛ =(z +1)/(d −1).AboveT

the system is

a liquid on large distances with  ∝ (T − T

)

 (z+2−d)

and

critical at small distances. Below T

we have a glassy re-

sponse at large distances E ∝ exp[−c(j

/j)



]andacritical

response at small scales. The crossover current densities

separating critical from liquid/glassy response scale as

∝|T − T

v(d−1)

of the power-law characteristic at T

in a log–log plot

provides a measure of the dynamic exponent z.De-

termining the crossover current densities j

and j

−

allows for an estimate of the correlation length ex-

ponent  . Finally, a measurement of the resistivity

(T)aboveT

provides a consistency check for the

two exponents z and  . The vortex-glass exponent 

determining the response below T

is usually diffi-

cult to measure. The first scaling analysis of this type

providing experimental evidence for a continuous

transition into a vortex-glass phase has been carried

out on epitaxial thin films of YBa

7−y

by Koch et

al. [35]. The large critical current densities j

(T =77

K, H =0)> 10

Acm

−2

indicate the presence of

12 Vortex Matter 621

rather strong pinning (large disorder) in these sam-

ples. Magnetic field values ranging from 0.5 T to 4 T

have been studied. The exponents z and  as deter-

mined from the power-law characteristic at T

and

the crossover current densities j

and j

−

above and

below the transition arez ≈ 4.8±0.2,

≈ 1.7±0.4,

and 

−

≈ 1.5–2.0. The consistency check on (T)

provides z ≈ 4.8,  ≈ 1.7, in good agreement with

the previous independent measurements of z and  .

The vortex-glass exponent  as obtained by fitting

the characteristic to a form (12.480) takes the value

 ≈ 0.4 ± 0.2. One should remark that a large dy-

namic range is needed in order to distinguish the

vortex-glass behavior (12.480) from a simple power-

law characteristic. A careful scaling analysis of the

resistive transition on a twinned YBCO single crystal

provides the Bose-glass exponents z



≈ 6 ± 1and



⊥

≈ 1 ±0.2 [476].Originally,the scaling analysis of

the irreversibility line in (both, electron and proton

irradiated) untwinned YBCO single crystals at high

fields has proven unsuccessful [473, 474]; however,

increasing the proton irradiation dose above a crit-

ical value, good resistivity scaling with an exponent

s =  (z +2−d) ≈ 5.1 ± 0.5 has been observed in

a recent experiment [475] within a field range be-

tween 1 and 8 T. In the end, it appears still unclear

what are the generic requirements for the existence

of a vortex-glass transition in 3D bulk supercon-

ductors: Dirty samples exhibit a continuous transi-

tion/crossover from an irreversible low-temperature

to a reversible high-temperature phase. Improving

the sample quality the Bragg-glass to vortex liquid

transformation proceedsvia a sharp first-order tran-

sition and there is no second-order glass transition.

While strong enough irradiation seems to bring the

glass transition back, the generic requirements for

its presence are still unclear at present.

Finally,we notethatno glass transitionis expected

in a 2D system as point-like vacancies and inter-

stitials cutting the diverging barriers are present at

any finite temperature; also, the absence of Bragg-

glass order in 2D thin films due to the presence

of a finite density of dislocations has been shown

in [21,492,493] (this density, however, is extremely

small when disorder is weak). Experiments on 2D

YBCO films seem to confirm these predictions [494].

Asa further consequence,thedecoupled phase in lay-

ered superconductors exhibits normal-metallic re-

sponse and true superconducting (glassy) response

is only established in the 3D superconductor below

the decoupling line.

12.13 Concluding Remarks

In the disorder-free situation the Vortex Matter in

strongly ﬂuctuating superconductors is character-

ized by the thermodynamic H–T phase diagram,

see Figs. 12.2, 12.11, 12.13, 12.15, and 12.17. In an

anisotropic material a first-order melting line sep-

arates the vortex lattice from an entangled vortex-

liquid phase, Figs. 12.2 and 12.11. The mean-field

line has turned into a crossover marking the

binding of charged carriers into Cooper pairs as ob-

servable through a broad maximum in the specific

heat. However, no phase stiffness has been estab-

lished yet in the vortex-liquid below the crossover

at H

and the material is still in the metallic phase

with its characteristic ohmic response. The true

phase transition breaking the translational symme-

try through creation of a vortex latticeand introduc-

ing a phase stiffness parallel to the magnetic field

appears only at the first-order melting/solidification

line. The phase transition line is given by the expres-

sion T

=0.105 ""

√

/B (or, alternatively B

0.069 H

(T)(""

/T)

, cf. (12.192)) and involves the

superﬂuid density n

(T) via the penetration depth

(T) — the knowledge of the material anisotropy

" and the superﬂuid density allows for an accurate

prediction of the location of the melting line, cf.

Sect. 12.7.2. The characteristic jumps at this first-

order transition are well understood,too,with a typi-

cal entropy jump S

≈ 0.4 k

and a positive jump in

the induction(few0.1 Gauss,see Fig.12.18) following

from the backward slope dH

/dT < 0ofthemelting

line and the Clapeyron equation B =−4s/@

hencethe vortexliquidis denserthan the vortex solid

and the vortex lattice melts like ice (cf. Sect. 12.7.6).

The melting line reenters at low fields and gives way

to a narrow sliver of vortex-liquid (order of 10 G

wide, see Figs. 12.2 and 12.11); the narrow extent of

this phase and its susceptibility to disorder render

the observation of this phase a difficult task.Thermal

622 G. Blatter and V.B. Geshkenbein

motion of the vortex lines also renormalizes the con-

stitutive B(H) relation; in strongly anisotropic and

layered superconductors large thermal ﬂuctuations

produce an appreciable attractive ﬂuctuation force

in the range  < R < /" leading to a van der Waals

type low-field phase diagram.

Inlayeredsuperconductors themelting lineisdra-

matically decreased as a consequence of the extreme

softness of theVortex Matter.In the absence of inter-

layer Josephson coupling the pancake vortex system

canbeviewedasanarrangementof2Dvortexlattices

coupled by the residual electromagnetic interaction

along the third dimension. The melting line then in-

terpolates between the 2D shear-dominated melting

at high fields/low temperatures and the evaporation

of isolated pancake-vortex stacks at T

BKT

in the low-

field limit (before reentering near T

BKT

). Away from

these limits, the melting process then can be under-

stood as the melting of a 2D pancake-vortex sys-

tem subject to a substrate potential which itself is

formed from the 2D pancake-vortex systems in the

other layers; the corresponding self-consistent anal-

ysisprovides us with a low-fieldinstabilityat B

inst

(T)

(cf. Fig. 12.14 and the comparison of the free ener-

gies of the solid and liquid phases shows that the real

melting line is close by, cf. Fig. 12.15.

Including a finite Josephson coupling into the

analysis, the pancake vortices bind into line ob-

jects and the melting line is rapidly pushed upwards,

crossing over into the continuous anisotropic result,

cf. Fig. 12.11. In turn, thermal ﬂuctuations can de-

stroy this interlayer coupling and an additional de-

coupling transition appears in the low-temperature-

high-field portion of the phase diagram and possi-

bly also close to T

, see Figs. 12.13 and 12.17.As the

interlayer coupling goes to zero (implying " → 0)

the decoupling transitionturns into a topologicalde-

fect transition (cf.Fig. 12.17)and the decoupling line

close to T

vanishes as T

→ T

, cf. (12.213) and

Fig. 12.13.

Adding weak point-like disorder destroys the vor-

tex lattice; at low fields this destruction is “soft” as

the ordered lattice is turned into a quasi-long-range

ordered Bragg-glass phase. As a consequence, the

first-order nature of the melting line is preserved,

at least at low enough fields. In anisotropic super-

conductors (YBCO) the jumps in magnetization and

entropy decrease with increasing magnetic field and

the first-order line seems to terminate [37,244,459].

At presentit is unclear how many phase lines emanate

from this (multi) critical point; possible scenaria in-

volve two ((frozen) vortex-liquid and Bragg-glass),

three (vortex-liquid, Bragg-glass, and vortex-glass),

see Fig. 12.3, or even more vortex phases appear-

ing in the vicinity of this point. The persistence of

a specific heat step above the (multi) critical point

does provide thermodynamic evidence for the con-

tinuation of the phase line at higher fields, an ob-

vious candidate being a second-order vortex-liquid

to vortex-glass transition. However, in a recent ex-

periment this specific heat step has been located

away from the irreversibility line and an interpre-

tation in terms a vortex-liquid to vortex-liquid tran-

sition has been suggested [496]; different types of

vortex-liquid phases have been postulated in the past

based on symmetry considerations and duality argu-

ments [287,288]and on numerical simulations[497].

Although some agreement has been reached regard-

ing the entangling of the vortex system upon melting,

the final answer regarding the existence of additional

vortex-liquid phases has not been given yet.

An interesting recent development is the manip-

ulation of the Vortex Matter phase diagram through

deliberate introduction of additional pinning cen-

ters. Starting with a clean single-crystal YBCO sam-

ple exhibiting a first-order melting transition, disor-

der is enhanced through irradiation with electrons,

protons, or heavy ions. With increasing proton irra-

diation dose the first-order transition retreats both

from low and high fields [498],finallygiving way to a

second-order glass transition[475].Columnartracks

produced by irradiation with heavy ions shift the up-

per criticalpointup towards higherfields [499],while

point-disorder (introduced via electron/proton irra-

diation) leads to a downward shift [498]. Columnar

tracks produce localization, while electron irradia-

tion leads to line wandering, hence the above shifts

in the critical point are suggestive of an entangled

high field “glassy” or frozen-liquid phase.

Subjecting the pancake-vortex system in layered

superconductors (e.g., BiSCCO) to disorder has a

number of interesting consequences. Again, weak

12 Vortex Matter 623

disorder preserves the first-order nature of the melt-

ing line.Adding a few strong pins through irradiation

with a low dose of heavy ions clearly establishes the

lattice nature of the low-field phase: With a match-

ing field B

= 5 G and an applied field of order

200–400 G each vortex pinned by a column is sur-

rounded by 40–80 free vortices on average.The intro-

duction of these pins has essentially no effect on the

high-temperature phase which remains reversible.

On the contrary,the low-temperature phase exhibits

a marked increase in the width of the magnetic hys-

teresisand inthe criticalcurrent density [500].These

findingsare consistentwith theappearanceof a finite

shear modulus in the low-temperature phase; with a

finite c

, pinning a few vortices is sufficient to pin

all the vortex lattice. Hence a consistent interpreta-

tion of the first-order phase transition is in terms of

a melting line [500]. Increasing the irradiation dose

to matching fields above 40 G leads to the transfor-

mation of the sharp melting transition into a contin-

uous (possibly Bose-glass) transition line shifted up

to higher temperatures and fields.

The addition of point-like disorder favors a dis-

ordered phase and the melting line ﬂattens at low

temperatures, apparently turning into a field-driven

order–disorder transition [477], see Fig. 12.37, pos-

sibly capturing both decoupling and melting of

the pancake-vortex structure. Recently, a smooth

crossover has been observed between the thermal

and disorder-induced melting lines, deferring a pu-

tative (multi) critical point to low temperatures

[470, 477]: making use of a vortex-shaking tech-

nique[308],the first-ordertransition couldbe traced

[470] far into the low-temperature regime where the

disorder-induced melting is usually identified with

the help of the “second peak” in the magnetization.

The continuous evolution of the thermally induced

melting line at high temperatures into a disorder-

induced first-order melting/decoupling line at low

temperatures then provides a natural interpretation

of these findings. Note that all these observations

are restricted to temperatures above ∼ 20–25 K. The

absence of a “second peak” at lowest temperatures

(below ∼ 20 K) casts doubts on an interpretation

of the “second peak” line in terms of a jump in the

critical current density,as such a jump should be vis-

Fig. 12.37. Schematic phase diagrams for a strongly ﬂuc-

tuating layered superconductor in the presence of disor-

der. The (first-order) melting line H

at high tempera-

tures smoothly crosses over into a (first-order) disorder-

induced melting/decoupling transition (H

m,dis

).The nature

of the low-temperature-high-field phase is presently un-

clear: Shown are two scenaria where the depinning line

dep

denotes a phase transition into a generic vortex-glass

state different from the low-field Bragg-glass and the high-

temperature vortex-liquid phases (top)andwherethede-

pinning line H

dep

terminates in a critical point leaving the

Bragg-glass phase as the only thermodynamic phase dif-

ferentfromthe vortex-liquid/normal-metalphase(bottom;

note that the depinning line may be contracted to zero)

624 G. Blatter and V.B. Geshkenbein

Fig. 12.38. Schematic phase diagram for a low-T

supercon-

ductor as inferred from experimental studies [469]. With

thermal ﬂuctuations remaining small, the melting line is

expected to close up with the H

line. On the other hand,

quenched disorder still can induce a sharp (first-order)

phase transition taking the vortex system between an or-

dered vortex solid (Bragg-glass) and a disordered (amor-

phous vortex-glass or frozen-liquid) vortex phase

ible down to the lowest temperatures. On the other

hand, this observation does lend support to a sce-

nario based on two states of VortexMatter character-

ized by different creep barriers [477–479], departing

from one another at low drives.

Disorder also affects the decoupling line: indeed,

in layered superconductors the solidification into

a still decoupled array of (disordered) 2D lattices

at high magnetic fields does not produce a glass

phase as mobile pancake vortex defects can circum-

vent the diverging creep barriers and produce ohmic

response. Glassiness is only established at the de-

coupling line: the coupling of the layers into a 3D

coherent superconductor prohibits the appearance

of point-like defects and all creep barriers diverge

at small drive. It is unclear at present whether the

decoupling in disordered layered superconductors

joins up with melting into the disorder-induced first-

order line at low temperatures (with a joint melting

and decoupling at the“second-peak”line) or whether

the decoupling appears as a separate phase transi-

tion. Experimentally, an additional depinning line is

observed at low temperatures and high fields, ex-

tending upward from the critical point and sepa-

rating (thermal-ﬂuctuation dominated) liquid and

(disorder dominated) glassy/frozen phases [501]; at

present it is unclear whether this linemarks a contin-

uous transitionor a mere crossover.Note thatin these

layered systems pinning is strong at low tempera-

tures and due to point defects in the bulk; at elevated

temperatures strong thermal ﬂuctuations efficiently

smear the pinning potential and the irreversibility is

due to surface and geometrical barriers [501].

Finally, these new theoretical and experimental

concepts developed within the program to under-

stand strongly ﬂuctuating high temperature super-

conductors find their application to low-T

materi-

als as well. In a clean sample, the melting line is ex-

pected to close up with the H

line and no further

new vortex phases are expected. The presence of dis-

order, however, again allows for a disorder-induced

sharp (first-order) phase transition taking the vor-

tex system between an ordered vortex solid (Bragg

glass) and a disordered (amorphous vortex glass or

frozen liquid) vortex phase [468,469], see Fig. 12.38.

The sharpness of this transition can be observed in

experiments carried out in a Corbino disc geome-

try avoiding vortex entry from the sample bound-

ary. Such vortex entry through an imperfect surface

is expected to “contaminate” the Bragg-glass phase

with a disordered vortex phase. The interplay be-

tween the ordered and the disordered vortex phase is

at the originof the peak effect [502–506]and numer-

ous dynamical phenomena (such as history depen-

dent response [507–509], memory effects [510,511],

low-frequency noise [512–515], slow voltage oscilla-

tions [514,516]) observed in many low T

supercon-

ductors.

An interesting question is about the status of the

line — is the H

line a phase-transition or

crossover line? Combining our knowledge gained

in the studies of anisotropic (YBCO), layered (BiS-

CCO), and low-T

(NbSe

) superconductors we can

summarize the situation as follows: experiments in-

12 Vortex Matter 625

dicate to us the existence of a depinning line, contin-

uing upward from the (multi) critical point in YBCO

and departing upward from the (disorder–induced)

melting line in BiSCCO — the corresponding line in

NbSe

should be identified with the H

line. The

crucial question that future experiments may an-

swer is, whether this depinning line is a phase tran-

sition line or a mere crossover into a“more metallic”

phase. On the theoretical side we know about the

existence of the Bragg-glass phase, a generic ther-

modynamic phase different from the normal metal

and characterized by its specific structural (quasi-

long-range order) and dynamical (glassy response)

properties. The Bragg-glass is a quite delicate phase,

easily destroyed by thermal ﬂuctuations (via a first-

order transition to the vortex liquidwhich is equiva-

lent to the normal metal) or via strong disorder. The

crucialquestion then is about the natureof the phase

reached from the Bragg glass by increasing the disor-

der strength: is the resulting phase equivalent to the

normal-metal — then the depinning line would be a

mere crossover and type II superconductors develop

only one generic phase, the Bragg-glass (we ignore

here the possibility of multiple liquid phases). Or is

the resulting phase different from the normal metal

(a vortex glass, an amorphous vortex solid, etc.), in

which case the depinning line should mark a ther-

modynamic phase transition and the Vortex Matter

in type II superconductors involves more than one

thermodynamic phase.We hope that this puzzle will

be resolved in the near future.

Acknowledgments

We thank Matthew Dodgson, Thierry Giamarchi,

J´erˆome Koopmann, Sergey Korshunov, Alexei

Koshelev, Anatoly Larkin, Henrik Nordborg, and

Manfred Sigrist for illuminating discussions.

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