Interplay of Anisotropy and Disorder in the Doping-Dependent
Melting and Glass Transitions of Vortices in Bi2Sr2CaCu2O8+δ
H. Beidenkopf,1, ∗ T. Verdene,1 Y. Myasoedov,1 H. Shtrikman,1 E. Zeldov,1 B. Rosenstein,2 D. Li,3 and T. Tamegai4
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China
Department of Physics, Peking University, Beijing 100871, China
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
(Dated: November 11, 2018)
We study the oxygen doping dependence of the equilibrium first-order melting and second-order
glass transitions of vortices in Bi2Sr2CaCu2O8+δ. Doping affects both anisotropy and disorder.
Anisotropy scaling is shown to collapse the melting lines only where thermal fluctuations are dom-
inant. Yet, in the region where disorder breaks that scaling, the glass lines are still collapsed. A
quantitative fit to melting and replica symmetry breaking lines of a 2D Ginzburg-Landau model
further reveals that disorder amplitude weakens with doping, but to a lesser degree than thermal
fluctuations, enhancing the relative role of disorder.
PACS numbers: 74.25.Qt, 74.25.Dw, 74.72.Hs, 64.70.Pf
Elasticity, thermal energy, disorder, and inter-layer
coupling are some of the closely competing energy scales
in the intricate H − T phase diagram of the vortex
matter in the layered high temperature superconductor
Bi2Sr2CaCu2O8+δ (BSCCO) [1, 2, 3, 4, 5, 6]. The low-
temperature part of the equilibrium phase diagram of
BSCCO was recently made accessible to experiment by
vortex shaking [7, 8, 9]. It unveiled a first-order (FO) in-
verse melting line, which continues the thermal melting
line from high temperatures [8] separating low-field or-
dered phases from high-field amorphous ones. A second-
order (SO) transition line, at which low-temperature
glassy phases get thermally depinned, was subsequently
reported [9]. In this letter we study the oxygen doping
dependence of these transition lines. We show that the
SO line scales with material anisotropy even where the
FO line does not, and that effective disorder weakens
with doping, but gains relative dominance over thermal
fluctuations.
We present measurements of optimally doped (OPD),
slightly over doped (SOD), over doped (OVD) and highly
over doped (HOD) BSCCO crystals [10, 11] with critical
temperatures Tc =92, 90, 88.5 and 86 K, respectively,
corresponding to hole concentrations of 0.171, 0.180,
0.184 and 0.190 [12]. Various crystal geometries were
studied with typical sizes of ∼300×300×40 µm3. They
were mounted on 10 × 10 µm2 Hall sensor arrays, fabri-
cated in a GaAs/AlGaAs heterostructure. At low tem-
peratures we utilized a 350 Oe in-plane ac shaking field
of 10 Hz to relax the pancake vortices towards their equi-
librium configuration [7, 8, 9]. Conjugating local probes
with shaking yielded the equilibrium reversible magneti-
zation of the samples.
Figure 1 shows the local induction, B(H,T ), measured
in the SOD sample by sweeping the temperature, T , at a
constant out-of-plane field, H , in presence of an in-plane
shaking field. A linear term α
T + β
was subtracted
FIG. 1: (color online) The local induction shows FO (dashed-
dotted) and SO (dashed) transition lines in the H − T phase
diagram. The induction in the high-temperature depinned
liquid and solid phases was artificially flattened by a linear
subtraction α
T + β
(hence the constant color). The low-
temperature BrG and glassy phases have thus a finite positive
slope. (a) Values of α
and β
used. (b) FO-SO-FO tran-
sition sequence at a 380 Oe temperature sweep on the back-
ground of the colorbar. (c) Discontinuous steps in dB/dT ,
manifesting the SO nature of the transition.
from each temperature sweep (Fig. 1a) to flatten the orig-
inally increasing (α
> 0) local induction. As a result,
the FO and SO singular behavior can be readily traced
throughout the H − T phase diagram.
The FOmelting transition is manifested by a discontin-
uous step in the magnetization along the dashed-dotted
line in Fig. 1. It separates the high-field amorphous glass
and liquid phases from the low-field quasi-long-range-
ordered Bragg glass (BrG) [5, 13]. The melting line be-
comes nonmonotonic at a certain temperature [8, 9]. Its
http://arxiv.org/abs/0704.1902v1
0.4 0.5 0.6 0.7 0.8 0.9 1
t=T/T
0.4 0.45 0.5
t=T/T
bisection
FO−SO
maximum
FIG. 2: (color online) FO and SO transition lines (solid
and open symbols, respectively) measured with OPD (◦),
SOD(�), OVD (♦) and HOD (△) samples. The inset shows
the doping dependence of the maximum of the FO line and
its bisection with the SO line. Dotted and dashed lines are
second-degree polynomial fits.
inverse-melting part is believed to be an order-disorder
transition, induced mainly by quenched disorder and not
by thermal fluctuations [1, 2, 3, 4, 5, 14, 15, 16].
The SO phase transition is manifested by a break in
the slope of the magnetization at Tg, marked in Fig. 1 by
the dashed line. It separates the fully flattened high-
temperature magnetization (dB/dT |T>Tg − α ≈ 0) of
constant color from a low-temperature region of nonzero
slope (dB/dT |T<Tg − α > 0), whose color varies with
temperature. The nonanaliticity at Tg is demonstrated
in Fig. 1c, which shows ∼40 mG/K steps in dB/dT for
temperature sweeps lying below as well as above the
FO line. The location of the SO line above melting
resembles earlier dynamic irreversibility measurements
[17, 18, 19, 20, 21] and theoretical models of a glass
transition [1, 2, 3, 4, 5, 22, 23, 24]. Few dynamic mea-
surements found a similar line of thermal depinning be-
low melting [9, 25, 26, 27, 28, 29, 30], which separates
the BrG phase below it from a thermally depinned solid,
whose detailed characteristics are not yet certain.
We mapped three more samples of various doping lev-
els that yielded the B−T phase diagram plotted in Fig. 2.
The melting line shifts as a whole to higher fields with
doping due to the decrease in anisotropy, which implies
stronger inter-layer coupling and higher stiffness of the
pancake vortex stacks [10, 30, 31, 32]. As the FO line
shifts to higher fields, its maximum shifts towards in-
creasingly higher temperatures (Fig. 2 inset). Identify-
ing these maxima with the crossover from disorder to
thermally dominated behavior [1, 2, 3, 4, 5] may sug-
gest that doping enhances disorder. The SO transition
line also shifts upwards, further signifying that disorder
is enhanced with doping. However, we will show be-
low that this conclusion is inaccurate. Moreover, arguing
naively that the balance between characteristic energies
constrains the SO line to bisect the FO one at its maxi-
mum [1] is also too simplistic. The inset of Fig. 2 clearly
shows that in over-doped samples the bisection point re-
sides below the FO maximum temperature (the overlap
of the two in the OPD sample is apparently accidental).
Therefore, a naive description of thermal depinning can-
not account for both the SO line and the FO maximum
behavior (for instance, the latter is directly affected also
by elasticity) [2, 3, 4].
We now present a quantitative scaling analysis of
the doping dependence, parameterized by the sample
anisotropy ratio ε2 = mab/mc (mi is the electronic effec-
tive mass in the ith direction). For high-κ superconduc-
tors, such as BSCCO, the Ginzburg-Lanadau (GL) free
energy functional can be recast into an isotropic (ε = 1)
form by rescaling its parameters [33]. We focus on one
such transformation [34] that rescales space, magnetic
induction, the penetration depth λ0 and the coherence
length ξ0. Models for high-temperature melting are usu-
ally independent of ξ0, and find λ0 to enter with some
model-dependent power as a proportionality factor. For
definiteness we resort to a specific model [6, 35] for a FO
evaporation line with no disorder from vortex solid to
pancake gas BE(t) ∝ (ε2/λ20d)(1− t2)/t, where t = T/Tc
and d is the inter-plane separation. The scaling transfor-
mation [34] indeed renders BE(t) isotropic.
Figure 3 shows the rescaled FO and SO lines. The
high-temperature parts of the FO lines are perfectly col-
lapsed by dividing their induction axes by a constant
[30]. BE(t) (dashed-dotted line) fits the collapsed melt-
ing lines precisely, asserting that the multiplicands in this
procedure are (ε◦/ε)
2, normalized by the OPD ε−1
≈ 500
[36]. Anisotropy scaling collapses the data down to
tth ≈ 0.58. Below tth, which appears to be independent
of anisotropy, the rescaled FO lines disperse again, and
the fit to BE(t) breaks. The flattening of the FO line to-
wards an inverse-melting behavior results from quenched
disorder, which gains dominance with decreasing tem-
peratures [1]. Accordingly, above tth the FO transition is
purely thermally-induced, and completely unaffected by
disorder [14]. Just below tth disorder becomes a relevant,
though not yet a dominant, energy scale.
This counterparts the extremely low-temperature be-
havior, where thermal energy becomes negligible relative
to pinning, resulting in a flat temperature-independent
behavior of the FO lines [1]. Indeed the FO lines in
Fig. 3 tend to flatten towards their ends, below which
350 Oe - 10 Hz shaking is insufficient for detecting a
reversible melting step. We thus conjectured a similar
doping-independent threshold temperature td ≈ 0.25, be-
low which thermal energy becomes irrelevant. We fitted
the low-temperature order-disorder lines by a leading or-
der expansion BOD(t & td) ∼ BOD(td)+Λ(t−td)2, which
agrees very well with measurement. We can, therefore,
estimate BOD(td), at which elasticity is balanced solely
by the disordering potential. It increases monotonically
with doping (inset of Fig. 3), stating that with reducing
anisotropy elasticity gains dominance also over disorder.
Yet, the most remarkable outcome of the anisotropy
scaling shown in Fig. 3 is the simultaneous collapse of the
SO transition lines (with zero freedom). The SO lines re-
side in a temperature region where anisotropy scaling of
the melting lines fails due to effects of disorder. Still, the
same scaling transformation somehow succeeds in render-
ing the glass-transition isotropic, even though this tran-
sition is believed to be intimately related with the com-
petition between disorder and thermal fluctuations.
To gain deeper understanding of the low-temperature
behavior we fit the measured transition lines to those
predicted by a recent calculation [37], which gives access
to the doping dependence of the model’s free parameters.
It incorporates thermal, disordering and elastic energies
to yield bisecting FO and SO lines. The pancake vortex
system is modeled by the 2D GL theory. We interpret
this single layer model as the outcome of an integration
of all other layers out of a complete 3D theory, resulting
in an effective 2D model whose renormalized coefficients
may still depend on anisotropy ε of a 3D mass tensor.
The free energy functional is averaged over gaussian
disorder in the coefficients of the quadratic and quar-
tic terms using the replica method. Taking the lowest-
Landau-level (LLL) approximation yields for the repli-
cated partition function
Ψ1...Ψn
G0(Ψa) +
R̃|Ψa|2|Ψb|2
G0(Ψ) =
aT |Ψ|2 +
κ2(b − h)2
2Gibt
where R̃ = a2T bR/32π
2, R is the disorder amplitude in the
coefficient of the quadratic LLL term, b = B/H̃c2, aT =
(b+t−1)/(2π2b2t2Gi)1/4 is the LLL parameter, Gi is the
effective Ginzburg number that generally scales thermal
fluctuations, and κ is the GL parameter. Randomness in
the coefficient of the quartic LLL term, although crucial
for obtaining the non-analyticity of SO, has negligible
effect on the locus of the transition lines, and is therefore
neglected in the present context.
The FO line is calculated by equating the energies of
the homogeneous and crystalline states under the influ-
ence of disorder in gaussian approximation. This ex-
tends beyond the earlier calculation, which treated dis-
order perturbatively [38]. The glass line is found from
the stability analysis of the replica symmetric solution.
The replica symmetry breaking (RSB) is continuous. The
corresponding Parisi function describing the hierarchial
structure of the glassy state and its detailed derivation
can be found in Ref. [37].
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=T/T
1 1.3 1.6 1.9
ε / ε
FIG. 3: (color online) Data collapse of the over-doped high-
temperature (t > tth) FO lines (solid symbols) onto the OPD
line, which also collapses the SO lines (open symbols). The
fit to BE(t) (dashed-dotted) sets the anisotropy ratios ε/ε◦
to 1, 1.20, 1.54 and 1.85 for the OPD (◦), SOD (�), OVD
(♦) and HOD (△) samples, respectively. The inset shows
the linear dependence of the characteristic BOD(td) on ε/ε◦
(dotted line).
The model’s free parameters are R, Gi, T̃c, and H̃c2 =
. However, due to a hidden symmetry, given
by Gi/H̃2c2 and Gi/R
2, the theory effectively has only
three independent fitting parameters. We therefore fixed
H̃c2 =100T, for simplicity, consistent with our limited
range of hole concentrations [12]. Figure 4 presents two
fitting strategies motivated by different physical behav-
iors. The first (thick lines) is optimized to fit the col-
lapsed parts of the phase diagram - SO lines and FO ones
for t > tth. Interestingly, this set simultaneously provides
a collapsed FO line also for t < tth. This results from the
above symmetry of the model in which Gi and ε take
the same role, defining a relation R ∝
Gi along which
the calculated phase diagram remains unchanged under
anisotropy rescaling. The thick lines in Fig.4 insets show
the resulting parameters Gi = 3.91(ε0/ε − 0.37)2 and
R = 1.54(ε0/ε − 0.37). The departure of the calculated
FO line (thick dashed) from the measured ones below
tth may indicate that theory lacks a symmetry-breaking
mechanism once anisotropy scaling breaks in experiment.
An alternative fit (thin lines), faithful to the com-
plete FO lines, dictates the slightly different set of values
Gi = 4.77(ε◦/ε − 0.38)2, R = 0.75(ε◦/ε − 0.2) (insets,
thin lines). The excellent fit of the FO lines (dashed)
suggests that the effective 2D model still captures the es-
sential physics involved. Its validity should break down
at very low temperatures, where indeed it misses the flat-
tening of the FO lines as td is approached, and close to Tc
due to critical fluctuations, which may explain the 10%
overestimate in the fitted bare T̃c values [39]. Substitut-
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  
t=T/T
0.5 0.75 1
FIG. 4: (color online) Fits of FO (dashed) and SO (solid) lines
of the LLL 2D GL model with H̃c2 =100 T, T̃c =103, 101, 98
and 95 K for the OPD (◦), SOD (�), OVD (♦) and HOD
(△) samples, respectively. The values of Gi (a) and R (b)
were fixed either by fitting the SO lines and FO ones above
tth (thick lines) or by fitting the full FO lines (thin lines).
ing this set of values in the calculation of the RSB lines
(solid) with zero freedom also produces a good fit that
improves with doping, but misses the anisotropy rescal-
ing of the measured SO lines. Nevertheless, the discrep-
ancy between the two sets of values of the different fitting
strategies is small, and vanishes in the high doping limit.
In Fig. 4a the reduced dominance of thermal fluctua-
tions with doping is clearly captured by the decrease in
Gi. Its dependence on anisotropy is closer to the 3D 1/ε2
than to the 2D ε-independent behavior. Remarkably, the
disorder amplitude in Fig. 4b also decreases with doping.
This contrasts the result of Ref. [40], which found that
oxygen addition increases the defects concentration. The
apparent contradiction is resolved by noting that the dis-
order amplitude is affected also by the anisotropy depen-
dence of the effective pinning potential, which decreases
with doping as the vortex stacks get stiffer. The over-
all decrease of R suggests that the latter mechanism is
dominant, not the rising defect concentration.
Last, while both Gi and R decrease with doping, Gi
has a stronger dependence on anisotropy. This clarifies
the true mechanism that shifts the maximum of the FO
line and its bisection with the SO line towards higher
temperatures with doping (Fig. 2). Although the magni-
tude of the disorder amplitude decreases with increased
doping, disorder still gains relative dominance over ther-
mal fluctuations, which decrease faster with ε.
In summary, ε2 scaling accounts for the reduced
anisotropy of BSCCO samples with doping down to tth.
Below tth, where disorder becomes relevant, ε
2 scaling
still collapses the SO lines. From the quantitative agree-
ment with the LLL 2D GL model we deduce that disor-
der and thermal-fluctuations weaken relative to elasticity
with increased doping, which shifts the FO line towards
higher fields. Disorder still gains relative dominance over
thermal fluctuations, which concurrently shifts the max-
imum of the FO line and its bisection with the SO line
towards higher temperatures.
We thank V.M. Vinokur, and G.P. Mikitik for stim-
ulating discussions. This work was supported by the
Israel Science Foundation Center of Excellence, by the
German-Israeli Foundation G.I.F., by Grant-in-aid for
Scientific Research from the Ministry of Education, Cul-
ture, Sports, Science, and Technology, Japan, by the Na-
tional science foundation of China, and by the MOE ATU
Program of R.O.C. NSC952112M009048. BR acknowl-
edges the support of the Albert Einstein Minerva Center
for Theoretical Physics and EZ the US-Israel Binational
Science Foundation (BSF).
∗ Electronic address: haim.beidenkopf@weizmann.ac.il
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  We study the oxygen doping dependence of the equilibrium first-order melting
and second-order glass transitions of vortices in
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. Doping affects both anisotropy and
disorder. Anisotropy scaling is shown to collapse the melting lines only where
thermal fluctuations are dominant. Yet, in the region where disorder breaks
that scaling, the glass lines are still collapsed. A quantitative fit to
melting and replica symmetry breaking lines of a 2D Ginzburg-Landau model
further reveals that disorder amplitude weakens with doping, but to a lesser
degree than thermal fluctuations, enhancing the relative role of disorder.

Interplay of Anisotropy and Disorder in the Doping-Dependent
Melting and Glass Transitions of Vortices in Bi2Sr2CaCu2O8+δ
H. Beidenkopf,1, ∗ T. Verdene,1 Y. Myasoedov,1 H. Shtrikman,1 E. Zeldov,1 B. Rosenstein,2 D. Li,3 and T. Tamegai4
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
Electrophysics Department, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China
Department of Physics, Peking University, Beijing 100871, China
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
(Dated: November 11, 2018)
We study the oxygen doping dependence of the equilibrium first-order melting and second-order
glass transitions of vortices in Bi2Sr2CaCu2O8+δ. Doping affects both anisotropy and disorder.
Anisotropy scaling is shown to collapse the melting lines only where thermal fluctuations are dom-
inant. Yet, in the region where disorder breaks that scaling, the glass lines are still collapsed. A
quantitative fit to melting and replica symmetry breaking lines of a 2D Ginzburg-Landau model
further reveals that disorder amplitude weakens with doping, but to a lesser degree than thermal
fluctuations, enhancing the relative role of disorder.
PACS numbers: 74.25.Qt, 74.25.Dw, 74.72.Hs, 64.70.Pf
Elasticity, thermal energy, disorder, and inter-layer
coupling are some of the closely competing energy scales
in the intricate H − T phase diagram of the vortex
matter in the layered high temperature superconductor
Bi2Sr2CaCu2O8+δ (BSCCO) [1, 2, 3, 4, 5, 6]. The low-
temperature part of the equilibrium phase diagram of
BSCCO was recently made accessible to experiment by
vortex shaking [7, 8, 9]. It unveiled a first-order (FO) in-
verse melting line, which continues the thermal melting
line from high temperatures [8] separating low-field or-
dered phases from high-field amorphous ones. A second-
order (SO) transition line, at which low-temperature
glassy phases get thermally depinned, was subsequently
reported [9]. In this letter we study the oxygen doping
dependence of these transition lines. We show that the
SO line scales with material anisotropy even where the
FO line does not, and that effective disorder weakens
with doping, but gains relative dominance over thermal
fluctuations.
We present measurements of optimally doped (OPD),
slightly over doped (SOD), over doped (OVD) and highly
over doped (HOD) BSCCO crystals [10, 11] with critical
temperatures Tc =92, 90, 88.5 and 86 K, respectively,
corresponding to hole concentrations of 0.171, 0.180,
0.184 and 0.190 [12]. Various crystal geometries were
studied with typical sizes of ∼300×300×40 µm3. They
were mounted on 10 × 10 µm2 Hall sensor arrays, fabri-
cated in a GaAs/AlGaAs heterostructure. At low tem-
peratures we utilized a 350 Oe in-plane ac shaking field
of 10 Hz to relax the pancake vortices towards their equi-
librium configuration [7, 8, 9]. Conjugating local probes
with shaking yielded the equilibrium reversible magneti-
zation of the samples.
Figure 1 shows the local induction, B(H,T ), measured
in the SOD sample by sweeping the temperature, T , at a
constant out-of-plane field, H , in presence of an in-plane
shaking field. A linear term α
T + β
was subtracted
FIG. 1: (color online) The local induction shows FO (dashed-
dotted) and SO (dashed) transition lines in the H − T phase
diagram. The induction in the high-temperature depinned
liquid and solid phases was artificially flattened by a linear
subtraction α
T + β
(hence the constant color). The low-
temperature BrG and glassy phases have thus a finite positive
slope. (a) Values of α
and β
used. (b) FO-SO-FO tran-
sition sequence at a 380 Oe temperature sweep on the back-
ground of the colorbar. (c) Discontinuous steps in dB/dT ,
manifesting the SO nature of the transition.
from each temperature sweep (Fig. 1a) to flatten the orig-
inally increasing (α
> 0) local induction. As a result,
the FO and SO singular behavior can be readily traced
throughout the H − T phase diagram.
The FOmelting transition is manifested by a discontin-
uous step in the magnetization along the dashed-dotted
line in Fig. 1. It separates the high-field amorphous glass
and liquid phases from the low-field quasi-long-range-
ordered Bragg glass (BrG) [5, 13]. The melting line be-
comes nonmonotonic at a certain temperature [8, 9]. Its
http://arxiv.org/abs/0704.1902v1
0.4 0.5 0.6 0.7 0.8 0.9 1
t=T/T
0.4 0.45 0.5
t=T/T
bisection
FO−SO
maximum
FIG. 2: (color online) FO and SO transition lines (solid
and open symbols, respectively) measured with OPD (◦),
SOD(�), OVD (♦) and HOD (△) samples. The inset shows
the doping dependence of the maximum of the FO line and
its bisection with the SO line. Dotted and dashed lines are
second-degree polynomial fits.
inverse-melting part is believed to be an order-disorder
transition, induced mainly by quenched disorder and not
by thermal fluctuations [1, 2, 3, 4, 5, 14, 15, 16].
The SO phase transition is manifested by a break in
the slope of the magnetization at Tg, marked in Fig. 1 by
the dashed line. It separates the fully flattened high-
temperature magnetization (dB/dT |T>Tg − α ≈ 0) of
constant color from a low-temperature region of nonzero
slope (dB/dT |T<Tg − α > 0), whose color varies with
temperature. The nonanaliticity at Tg is demonstrated
in Fig. 1c, which shows ∼40 mG/K steps in dB/dT for
temperature sweeps lying below as well as above the
FO line. The location of the SO line above melting
resembles earlier dynamic irreversibility measurements
[17, 18, 19, 20, 21] and theoretical models of a glass
transition [1, 2, 3, 4, 5, 22, 23, 24]. Few dynamic mea-
surements found a similar line of thermal depinning be-
low melting [9, 25, 26, 27, 28, 29, 30], which separates
the BrG phase below it from a thermally depinned solid,
whose detailed characteristics are not yet certain.
We mapped three more samples of various doping lev-
els that yielded the B−T phase diagram plotted in Fig. 2.
The melting line shifts as a whole to higher fields with
doping due to the decrease in anisotropy, which implies
stronger inter-layer coupling and higher stiffness of the
pancake vortex stacks [10, 30, 31, 32]. As the FO line
shifts to higher fields, its maximum shifts towards in-
creasingly higher temperatures (Fig. 2 inset). Identify-
ing these maxima with the crossover from disorder to
thermally dominated behavior [1, 2, 3, 4, 5] may sug-
gest that doping enhances disorder. The SO transition
line also shifts upwards, further signifying that disorder
is enhanced with doping. However, we will show be-
low that this conclusion is inaccurate. Moreover, arguing
naively that the balance between characteristic energies
constrains the SO line to bisect the FO one at its maxi-
mum [1] is also too simplistic. The inset of Fig. 2 clearly
shows that in over-doped samples the bisection point re-
sides below the FO maximum temperature (the overlap
of the two in the OPD sample is apparently accidental).
Therefore, a naive description of thermal depinning can-
not account for both the SO line and the FO maximum
behavior (for instance, the latter is directly affected also
by elasticity) [2, 3, 4].
We now present a quantitative scaling analysis of
the doping dependence, parameterized by the sample
anisotropy ratio ε2 = mab/mc (mi is the electronic effec-
tive mass in the ith direction). For high-κ superconduc-
tors, such as BSCCO, the Ginzburg-Lanadau (GL) free
energy functional can be recast into an isotropic (ε = 1)
form by rescaling its parameters [33]. We focus on one
such transformation [34] that rescales space, magnetic
induction, the penetration depth λ0 and the coherence
length ξ0. Models for high-temperature melting are usu-
ally independent of ξ0, and find λ0 to enter with some
model-dependent power as a proportionality factor. For
definiteness we resort to a specific model [6, 35] for a FO
evaporation line with no disorder from vortex solid to
pancake gas BE(t) ∝ (ε2/λ20d)(1− t2)/t, where t = T/Tc
and d is the inter-plane separation. The scaling transfor-
mation [34] indeed renders BE(t) isotropic.
Figure 3 shows the rescaled FO and SO lines. The
high-temperature parts of the FO lines are perfectly col-
lapsed by dividing their induction axes by a constant
[30]. BE(t) (dashed-dotted line) fits the collapsed melt-
ing lines precisely, asserting that the multiplicands in this
procedure are (ε◦/ε)
2, normalized by the OPD ε−1
≈ 500
[36]. Anisotropy scaling collapses the data down to
tth ≈ 0.58. Below tth, which appears to be independent
of anisotropy, the rescaled FO lines disperse again, and
the fit to BE(t) breaks. The flattening of the FO line to-
wards an inverse-melting behavior results from quenched
disorder, which gains dominance with decreasing tem-
peratures [1]. Accordingly, above tth the FO transition is
purely thermally-induced, and completely unaffected by
disorder [14]. Just below tth disorder becomes a relevant,
though not yet a dominant, energy scale.
This counterparts the extremely low-temperature be-
havior, where thermal energy becomes negligible relative
to pinning, resulting in a flat temperature-independent
behavior of the FO lines [1]. Indeed the FO lines in
Fig. 3 tend to flatten towards their ends, below which
350 Oe - 10 Hz shaking is insufficient for detecting a
reversible melting step. We thus conjectured a similar
doping-independent threshold temperature td ≈ 0.25, be-
low which thermal energy becomes irrelevant. We fitted
the low-temperature order-disorder lines by a leading or-
der expansion BOD(t & td) ∼ BOD(td)+Λ(t−td)2, which
agrees very well with measurement. We can, therefore,
estimate BOD(td), at which elasticity is balanced solely
by the disordering potential. It increases monotonically
with doping (inset of Fig. 3), stating that with reducing
anisotropy elasticity gains dominance also over disorder.
Yet, the most remarkable outcome of the anisotropy
scaling shown in Fig. 3 is the simultaneous collapse of the
SO transition lines (with zero freedom). The SO lines re-
side in a temperature region where anisotropy scaling of
the melting lines fails due to effects of disorder. Still, the
same scaling transformation somehow succeeds in render-
ing the glass-transition isotropic, even though this tran-
sition is believed to be intimately related with the com-
petition between disorder and thermal fluctuations.
To gain deeper understanding of the low-temperature
behavior we fit the measured transition lines to those
predicted by a recent calculation [37], which gives access
to the doping dependence of the model’s free parameters.
It incorporates thermal, disordering and elastic energies
to yield bisecting FO and SO lines. The pancake vortex
system is modeled by the 2D GL theory. We interpret
this single layer model as the outcome of an integration
of all other layers out of a complete 3D theory, resulting
in an effective 2D model whose renormalized coefficients
may still depend on anisotropy ε of a 3D mass tensor.
The free energy functional is averaged over gaussian
disorder in the coefficients of the quadratic and quar-
tic terms using the replica method. Taking the lowest-
Landau-level (LLL) approximation yields for the repli-
cated partition function
Ψ1...Ψn
G0(Ψa) +
R̃|Ψa|2|Ψb|2
G0(Ψ) =
aT |Ψ|2 +
κ2(b − h)2
2Gibt
where R̃ = a2T bR/32π
2, R is the disorder amplitude in the
coefficient of the quadratic LLL term, b = B/H̃c2, aT =
(b+t−1)/(2π2b2t2Gi)1/4 is the LLL parameter, Gi is the
effective Ginzburg number that generally scales thermal
fluctuations, and κ is the GL parameter. Randomness in
the coefficient of the quartic LLL term, although crucial
for obtaining the non-analyticity of SO, has negligible
effect on the locus of the transition lines, and is therefore
neglected in the present context.
The FO line is calculated by equating the energies of
the homogeneous and crystalline states under the influ-
ence of disorder in gaussian approximation. This ex-
tends beyond the earlier calculation, which treated dis-
order perturbatively [38]. The glass line is found from
the stability analysis of the replica symmetric solution.
The replica symmetry breaking (RSB) is continuous. The
corresponding Parisi function describing the hierarchial
structure of the glassy state and its detailed derivation
can be found in Ref. [37].
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t=T/T
1 1.3 1.6 1.9
ε / ε
FIG. 3: (color online) Data collapse of the over-doped high-
temperature (t > tth) FO lines (solid symbols) onto the OPD
line, which also collapses the SO lines (open symbols). The
fit to BE(t) (dashed-dotted) sets the anisotropy ratios ε/ε◦
to 1, 1.20, 1.54 and 1.85 for the OPD (◦), SOD (�), OVD
(♦) and HOD (△) samples, respectively. The inset shows
the linear dependence of the characteristic BOD(td) on ε/ε◦
(dotted line).
The model’s free parameters are R, Gi, T̃c, and H̃c2 =
. However, due to a hidden symmetry, given
by Gi/H̃2c2 and Gi/R
2, the theory effectively has only
three independent fitting parameters. We therefore fixed
H̃c2 =100T, for simplicity, consistent with our limited
range of hole concentrations [12]. Figure 4 presents two
fitting strategies motivated by different physical behav-
iors. The first (thick lines) is optimized to fit the col-
lapsed parts of the phase diagram - SO lines and FO ones
for t > tth. Interestingly, this set simultaneously provides
a collapsed FO line also for t < tth. This results from the
above symmetry of the model in which Gi and ε take
the same role, defining a relation R ∝
Gi along which
the calculated phase diagram remains unchanged under
anisotropy rescaling. The thick lines in Fig.4 insets show
the resulting parameters Gi = 3.91(ε0/ε − 0.37)2 and
R = 1.54(ε0/ε − 0.37). The departure of the calculated
FO line (thick dashed) from the measured ones below
tth may indicate that theory lacks a symmetry-breaking
mechanism once anisotropy scaling breaks in experiment.
An alternative fit (thin lines), faithful to the com-
plete FO lines, dictates the slightly different set of values
Gi = 4.77(ε◦/ε − 0.38)2, R = 0.75(ε◦/ε − 0.2) (insets,
thin lines). The excellent fit of the FO lines (dashed)
suggests that the effective 2D model still captures the es-
sential physics involved. Its validity should break down
at very low temperatures, where indeed it misses the flat-
tening of the FO lines as td is approached, and close to Tc
due to critical fluctuations, which may explain the 10%
overestimate in the fitted bare T̃c values [39]. Substitut-
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1  
t=T/T
0.5 0.75 1
FIG. 4: (color online) Fits of FO (dashed) and SO (solid) lines
of the LLL 2D GL model with H̃c2 =100 T, T̃c =103, 101, 98
and 95 K for the OPD (◦), SOD (�), OVD (♦) and HOD
(△) samples, respectively. The values of Gi (a) and R (b)
were fixed either by fitting the SO lines and FO ones above
tth (thick lines) or by fitting the full FO lines (thin lines).
ing this set of values in the calculation of the RSB lines
(solid) with zero freedom also produces a good fit that
improves with doping, but misses the anisotropy rescal-
ing of the measured SO lines. Nevertheless, the discrep-
ancy between the two sets of values of the different fitting
strategies is small, and vanishes in the high doping limit.
In Fig. 4a the reduced dominance of thermal fluctua-
tions with doping is clearly captured by the decrease in
Gi. Its dependence on anisotropy is closer to the 3D 1/ε2
than to the 2D ε-independent behavior. Remarkably, the
disorder amplitude in Fig. 4b also decreases with doping.
This contrasts the result of Ref. [40], which found that
oxygen addition increases the defects concentration. The
apparent contradiction is resolved by noting that the dis-
order amplitude is affected also by the anisotropy depen-
dence of the effective pinning potential, which decreases
with doping as the vortex stacks get stiffer. The over-
all decrease of R suggests that the latter mechanism is
dominant, not the rising defect concentration.
Last, while both Gi and R decrease with doping, Gi
has a stronger dependence on anisotropy. This clarifies
the true mechanism that shifts the maximum of the FO
line and its bisection with the SO line towards higher
temperatures with doping (Fig. 2). Although the magni-
tude of the disorder amplitude decreases with increased
doping, disorder still gains relative dominance over ther-
mal fluctuations, which decrease faster with ε.
In summary, ε2 scaling accounts for the reduced
anisotropy of BSCCO samples with doping down to tth.
Below tth, where disorder becomes relevant, ε
2 scaling
still collapses the SO lines. From the quantitative agree-
ment with the LLL 2D GL model we deduce that disor-
der and thermal-fluctuations weaken relative to elasticity
with increased doping, which shifts the FO line towards
higher fields. Disorder still gains relative dominance over
thermal fluctuations, which concurrently shifts the max-
imum of the FO line and its bisection with the SO line
towards higher temperatures.
We thank V.M. Vinokur, and G.P. Mikitik for stim-
ulating discussions. This work was supported by the
Israel Science Foundation Center of Excellence, by the
German-Israeli Foundation G.I.F., by Grant-in-aid for
Scientific Research from the Ministry of Education, Cul-
ture, Sports, Science, and Technology, Japan, by the Na-
tional science foundation of China, and by the MOE ATU
Program of R.O.C. NSC952112M009048. BR acknowl-
edges the support of the Albert Einstein Minerva Center
for Theoretical Physics and EZ the US-Israel Binational
Science Foundation (BSF).
∗ Electronic address: haim.beidenkopf@weizmann.ac.il
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