Measurements and analysis of the upper critical field H
c2 of underdoped and
overdoped La2−xSrxCuO4 compounds
D. H. N. Dias and E. V. L. de Mello
Instituto de F́ısica, Universidade Federal Fluminense,
Niterói, R. J., 24210-340, Brazil
J. L. Gonzalez and H. A. Borges
Dept. de F́ısica, Pontif́ıcia Universidade Católica de Rio de Janeiro,
R. Marques de São Vicente 225, R. J., 22453-900 Brasil.
A. Gomes
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil.
A. Loose
FU-Berlin. Berlin. Germany
A. Lopez
Universidade do Estado de Rio de Janeiro, Rio de Janeiro, Brazil.
F. Vieira and E. Baggio-Saitovitch
Centro Brasileiro de Pesquisas F́ısicas, Rio de Janeiro, Brazil.
(Dated: November 21, 2018)
The upper critical field Hc2 is one of the many non conventional properties of high-Tc cuprates.
It is possible that the Hc2(T ) anomalies are due to the presence of inhomogeneities in the local
charge carrier density ρ of the CuO2 planes. In order to study this point, we have prepared good
quality samples of polycrystalline La2−xSrxCuO4 using the wet-chemical method, which has been
demonstrated to produce samples with a good cation distribution. In particular, we have studied the
temperature dependence of the upper critical field, Hc2(T ), through magnetization measurements
on two samples with opposite average carrier concentration (ρm = x) and nearly equal critical
temperatures, namely, ρm = 0.08 (underdoped) and ρm = 0.25 (overdoped). The results close to Tc
do not follow the usual Ginzburg-Landau theory and are interpreted by a theory which takes into
account the influence of the inhomogeneities.
PACS numbers: 74.72.-h, 74.80.-g, 74.20.De, 02.70.Bf
I. INTRODUCTION
High critical temperature superconductors (HTSC)
display many non-conventional properties which remain
to be explained by a concise physical picture[1, 2, 3].
This state of affairs might be due to the fact that, differ-
ently from low temperature superconductors, these ma-
terials have a large degree of intrinsic inhomogeneities.
Although its origin is unknown, there are many evidences
from different experiments that they do not have a homo-
geneous doping level[4, 5]. For instance, recent Scanning
Tunneling Microscopy (STM)[6, 7, 8, 9] measurements
have detected strong variations in the density of states as
the tip travels over a clean and sharp surface. Also, neu-
tron diffraction data have revealed a complex structure of
the charge distribution that has become known as ”stripe
structure”[10]. Nuclear Quadruple Resonance (NQR)[11]
and Angle Resolve Photo-Emission (ARPES)[12] have
also detected inhomogeneities in the local environment
and in the charge distribution.
Based on these experimental evidences we argue that
this behavior and the inhomogeneities are possibly due
to a phase separation transition (PST) connected with
the anomalies seen at the upper pseudogap temperature.
Such PST would bring the system to a disordered charge
distribution state, with the formation of islands or re-
gions of distinct doping levels[13]. Since any PST de-
pends on the chemical mobility, this approach may shed
some light on the reason why some compounds appear
to be more homogeneous or, at least, do not display
any gross inhomogeneity[14, 15], although the phase di-
agrams of cuprates seem to be universal. Therefore, we
think it is possible to formulate a unified theory for the
HTSC despite of their different degrees of disorder.
In this paper we explore this possibility by showing
that the anomalies of the upper critical field Hc2(T ) as
function of the temperature T are in agreement with
charge carriers inhomogeneities in two samples with com-
pletely different average doping levels: one underdoped
and other in the far overdoped region of the super-
conducting phase diagram. In order to achieve this
goal this work is threefold: i) we prepared samples of
http://arxiv.org/abs/0704.1425v2
La2−xSrxCuO4 (LSCO), ii) we made several sets of
Hc2(T ) measurements and iii) perform a theoretical in-
terpretation of the data. The non conventional fea-
tures of Hc2(T ), like a positive curvature and absence
of saturation at low temperatures, are well known from
many previous experiments[16, 17, 18]. We concentrat
in two samples in the underdoped and overdoped dop-
ing levels, and measured Hc2(T ) to detect qualitative
doping dependent behavior. We interpreted the results
through a theory that takes into account the different
contribution from stripes of different local charge den-
sity. These calculations are based on the Cahn-Hilliard
(CH) theory of phase separation[19, 20] for compounds
with ρm ≤ 0.20[2, 13] and on a Gaussian charge fluc-
tuation around the average for ρm ≥ 0.20. In both
cases the resulting inhomogeneous systems are studied
with a Bogoliubov-deGennes approach to a disordered
superconductor [13]. Indeed there are several different
approaches to deal with the inhomogeneities in HTSC,
like the method of Ghosal at al[21] of a local disorder in
the chemical potential, Nunner et al[22] that deals with
out of plane chemical disorder, and Cabo et al[23] that
introduced an in plane Gaussian disorder around de av-
erage doping, to mention just a few of what can be found
in the literature. So far, all these approaches succeeded
in explain some HTSC features, what shows that the in-
homogeneities are important, but only new and refined
experiments will be able to determine the correct way to
deal with them.
II. SAMPLE PREPARATION
Several polycrystalline samples belonging to the
La2−xSrxCuO4 system were prepared by the wet-
chemical method according to reference[24, 25]. Pure
(99,99%) oxide and carbonate compounds, namely
La2O3, CuO, and SrCO3 were dried at 150
0C and
weighted with adequate stoichiometrical proportions.
The powders were dissolved into 50 ml of ultra pure acetic
acid (CH3COOH) and the final solution was dried and
after heated at 9000C during 24 hours in flowing oxygen.
After that the powders were quenched at room temper-
ature, then the mixtures were reground and pressed into
pellets. Finally the pre-sintered samples were sintered at
10500C during 50 hours.
Fig.(1) shows the x-rays diffraction performed in both
samples prepared according the wet-chemical method.
The x-ray diffraction (XRD) patterns were obtained in a
(XPert PRO PANalytical) powder diffractometer using
CuKα radiation (λ = 1.5418Å). Data were collected by
step-scanning mode (200 ≤ 2θ ≤ 900) and 2 s counting
time in each step at room temperature. Orthorhombic
(Bmab) and tetragonal (F4mmm) structure space groups
were assumed in the Rietveld analysis for the 0.08 and
0.25 strontium percent samples respectively.
Once we have characterized the sample, we have per-
formed magnetic measurements by a SQUID magnetome-
20 30 40 50 60 70 80 90
20 30 40 50 60 70 80 90
25 % Sr
8 % Sr
FIG. 1: X-rays diffraction spectra at room temperature for
both samples used in the experiment. The principal peaks
were identified according to Rietveld analysis.
ter, as described in the next section.
III. EXPERIMENTAL PROCEDURE
5 10 15 20 25
-8,0x10-5
-6,0x10-5
-4,0x10-5
-2,0x10-5
18 19 20 21 22 23 24 25 26 27 28
Tc(3kOe)=21.15
Tc(1kOe)=22.6 K 1 kOe
 3 kOe
 7 kOe
 10 kOe
T (K)
25 % Sr
FIG. 2: (Color on line) The zero-field cooling magnetic mo-
ment at some applied magnetic fields as a function of tem-
perature for the sample with 25% of strontium. The insert
shows how the values of Tc were determined for two values of
the applied field.
Magnetization curves as a function of temperature
were obtained with a SQUID magnetometer in a conven-
tional DC mode. The measurements were performed in
zero field-cooling conditions with moderate applied fields
ranging between zero and two Tesla. Fig.(2) contains sets
of M(T) curves for the sample La1.75Sr0.25CuO4 (25 %
of strontium) is presented. In the inset we show details of
how Tc(H) were obtained for two selected fields, namely
3kG and 1 kG. By taking the measured values of Tc(H),
we obtain the plot shown in Fig.(3), from which we can
extrapolate the value of Tc(Hc2 = 0). For the present
case, we obtain Tc(Hc2 = 0) = 24.1K and for the under-
doped sample, we get Tc(Hc2 = 0) = 31.1K. A similar
set of curves was obtained for the 8% strontium sample.
In this manner we got the value of Tc(Hc2 = 0) for both
samples.
18 19 20 21 22 23 24 25 26 27
12000
16000
20000
T (K)
FIG. 3: (Color on line) Values of Tc(H) calculated as shown
in inset of Fig.(2) ( for La1.75Sr0.25CuO4). The value of
Tc(Hc2 = 0) is taken where the curve extrapolates to zero.
In this case we get Tc(Hc2 = 0) = 24.1K, value that will be
used further on (Fig.(7)).
On the other hand, the critical temperatures Tc(H =
0) for both samples were determined from many sets
of resistivity data, by taking the maximum of the
first derivative of the resistivity vs temperature curves,
namely Tc(8%Sr) = 22.8K and Tc(25%Sr) = 19.9K.
From these data, the widths of the superconducting
transitions were estimated at half-maximum of the first
derivative, as displayed in Fig.(4). For the sample with
25% of strontium the ∆Tc was about 4.5 K while in the
0.08% sample the transition was wider pointing out to
the presence of strong inhomogeneities in the sample.
The presence of these disorder will be important in the
discussions presented in the next sections.
10 20 30 40
0 10 20 30 40
∆ΤC=4.5 K
FIG. 4: Resistivity as function of temperature. The insert
shows the first derivative of the data and the maximum is
taken as the superconductivity transition temperature for this
sample at Tc(25%Sr) = 19.9K
IV. PST RESULTS AND DISCUSSION
As discussed in the introduction, there are many exper-
imental evidences showing that some cuprates are highly
inhomogeneous in their charge distribution while others
are not, but all of them display the same phase diagram.
To deal with this non trivial problem, we have intro-
duced the idea of a PST that, depending on the mobility
of the ions, can generates various degrees of decomposi-
tion. This phase segregation process can form patterns
on the sample, as the stripes[10] or patchwork[6], gener-
ating islands with different values of the charge density,
or can merely form small fluctuations around an average
doping level. Applying a Bogoliubov-deGennes (BdG)
theory to these systems we were able to calculate the
local superconducting pairing amplitude at a given site
”i” in a cluster ”l”. Thus, in our calculations, a given
sample with average or mean charge density ρm may be
composed of local regions with local densities ρ(l). Here
we show results on a 14× 14 matrix, that is, 14 clusters
on a stripe form (each stripe has 14 sites) (l = 1to14)
and 196 sites (i runs from 1 to 196).
In general, regions with larger charge densities or dop-
ing levels usually may become superconducting, and
those with very low doping level are insulators and never
become superconducting. This anomalous behavior is de-
tected by the superconducting pairing amplitude ∆(l, T )
which, as the temperature is decreased, starts at Tc(l),
increases and saturates at low temperatures. This allows
us to define a local superconducting temperature Tc(l).
Here we exhibit simulations on a square mesh 14× 14
that display stripe inhomogeneities similar to the exper-
imental results[10], but derived from a CH phase separa-
FIG. 5: (Color on line) Temperature evolution of the local
pairing amplitude ∆(i, T ) at each stripe on a square of 14×14,
that is, with 196 sites ”i”, for different samples. Because of
the inhomogeneities, the sites in the left have ρ(i) ≈ 0 and the
ones in the right ρ(i) ≈ 2ρm. As the systems are cooled down,
more regions become superconducting and they percolate at
Tc(ρm). These percolation threshold temperatures for each
compound are indicated in their respective panel. The panel
with ρ = 0.22 does not have the stripe structure, because it
has a random distribution of doping values, which are sorted
following a Gaussian distribution.
tion theory applied to the LSCO series[13]. According to
the CH results[19, 20], the square mesh phase separates
into a bimodal distribution of charge. For underdoped
samples the phase separation is essentially total and leads
to two halves where the 7 stripes at the left are character-
ized by local doping ρ(l) ≈ 0 and the 7 at the right side
have ρ(l) ≈ 2ρm as showed in the top panel of Fig.(5)
(for ρ = 0.05). As ρm increases, the charge fluctuations
also increases, changing the properties of a compound
into metallic, and superconducting at low temperatures.
Thus, for compounds with 0.06 < ρm < 0.20, at low tem-
peratures, as one can see in Fig.(5), there are also stripes
with non-vanishing densities at low density regions. No-
tice that the values of ∆ are constant in a given stripe,
in all its sites ”i”, and that is why we may plot ∆(i, T )
at each site, although that is meaningful only at a given
stripe ∆(l, T ). As the density of charge carriers increases,
it produces the ∆(i, T ) at the very low doping sites, i.e.,
in the left region of Fig.(5). When this occurs at low
temperatures, it is possible that the system becomes su-
perconductor at Tc(ρm) by the percolation of all the local
superconducting regions[26, 27] and it can hold a dissipa-
tionless current. The percolation among the local regions
where ∆(l, T ) in non-zero may occur either directly or by
Josephson coupling and the associated temperature, or
the superconducting temperature Tc(ρm), is also shown
in the panels of Fig.(5). Notice that the compound with
ρ = 0.22 does not have the stripe structure, it has just a
Gaussian distribution of doping values, since it is larger
than the PST threshold of ρ ≈ 0.2.
Above Tc(ρm), depending on the value of ρm, the com-
pounds may be formed by mixtures of superconducting,
insulator and normal domains and above the pairing for-
mation temperature (the onset temperature which some-
times is called the lower pseudogap), they are a disor-
dered metal with mixtures of normal (ρ(l) > 0.05) and
insulator (ρ(l) ≤ 0.05) regions. For ρm ≥ 0.20 the
charge disorder is practically zero, with a small fluctu-
ation around ρm. From these calculations, we identify
Tonset(ρm) as the highest temperature (Tc(l)) which in-
duces a ∆(l, T ) in a given compound which is easily seen
in the panels of Fig.(5). Tonset may be also identified
with the onset of Nernst signal[13]. To make clear how
the values of Tonset of a given compound are obtained,
we show the larger values of Tc(i) and Tc(l) in Fig.(6) for
the sample with ρ = 0.15.
FIG. 6: (Color on line) To explain the concept of local super-
conducting temperature and specially how certain regions de-
velops a non-vanishing pairing amplitude, we plot the ∆(i, T )
or ∆(l, T ) for each site ”i” of our two dimensional 14 × 14
array. The onset temperatures values for which these pairing
amplitudes develop for each stripe are clearly indicated in the
figure, and the highest value, namely, T = 90K, is taken as
the lower pseudogap temperature of this compound.
V. Hc2 RESULTS AND CALCULATIONS
It is well known that the Ginzburg-Landau (GL) up-
per critical field of a homogeneous superconductor is a
linear function of the temperature near Tc and falls to
zero at this temperature[28]. This behavior is not ob-
served by our measurements, showing another departure
from conventional properties. As we can see in Fig.(7),
the Hc2(T ) experimental points for both samples are lin-
ear only near and below T/Tc ≤ 0.9. As the tempera-
ture increases it performs an upturn curvature falling to
zero further beyond Tc. Consequently, we need some new
ideas or theories to interpret these results.
From these Hc2(T ) curves, we can estimate that the
upper critical field goes to zero at about 31.1 K, which is
substantially higher than the value Tc = 22.8K of the 8
%-Sr sample (from resistivity measurements as discussed
in section II). Similarly, Hc2(T ) falls to zero at 24.1K for
the 25 %-Sr sample which has a Tc = 19.9K. Thus we
see that Hc2(T ) vanishes at temperatures 8-22% larger
than Tc. This effect, the nonzero value of Hc2 above
Tc, is quite unusual for a normal superconductor but it
was also observed in LSCO and Bi-2201 cuprates by the
group of Wang et al[29]. We will show below that this
result may be explained as a consequence of the intrinsic
disorder in HTSC, namely the presence of regions with
different local dopings and distinct superconducting local
temperatures Tc(l).
In order to provide an interpretation to these results,
we applied a generalization of the GL Hc2(T ) expression
following along the lines described by Caixeiro et al[28]:
The GL upper critical field near Tc of a homogeneous
superconductor may be written as
Hc2(T ) =
Tc − T
. (T < Tc) (1)
At a temperature T , we take each superconducting re-
gion in the sample characterized by a ρ(l) as the source
that generates a ∆(l) to produce a magnetic response
and displays a local Hc2, provided that T ≤ Tc(l). As
discussed, these regions can be in stripe or others forms,
but the important point is that they are characterized by
a region of fairly constant density ρ(l) that at T ≤ Tc(l)
may shield the applied magnetic field. Thus each of such
given local region has a local superconducting tempera-
ture Tc(l) and will contribute to the upper critical field
with a local linear upper critical field H l
(T ) near Tc(l)
according to the usual GL approach. This is justified be-
cause each region has a constant density, like a type II
low temperature superconductor, and should posses its
own Hc2 that is expected to vanish linearly at Tc(l).
Consequently the total contribution of the local su-
perconducting regions to the whole sample upper critical
field is the sum of all the H l
(T )’s,
Hc2(T ) =
Tc(l)− T
Tc(l)
(T ). (T < Tc(l) ≤ Tonset(ρm)) (2)
WhereW is the total number of superconducting regions,
stripes or islands with its local Tc(l) ≥ T . The maxi-
mum value of Tc(l) is the pseudogap temperature iden-
tified above as the Tonset(ρm). For the LSCO series a
coherence length of ξab(0) ≈ 22Å is used, in accordance
with the measurements[28]. This value of ξab(0) leads to
Hc2(0)=Φ/2πξ
(0)=64T. Due to the limitations of the
GL approach, we expect the result of this equation to be
accurate only near and above the system Tc.
Fig.(7a) shows both the Hc2 results of the generalized
GL calculations together with the experimental values
for underdoped La1.92Sr0.08CuO4 compound. In the cal-
culations on this compound, we used a maximum tem-
perature of superconducting formation, Tonset ≈ 90K
FIG. 7: Experimental points (connected by a thin line)
against the reduced temperature t = T/Tc and the calcu-
lated curve (thick line) of Hc2 considering inhomogeneous
samples with a stripe distribution of local superconducting
temperature Tc(l) for ρ = 0.08 and a similar calculation with
a Gaussian distribution for ρ = 0.25.
(the maximum Tc(l) for this sample) from pseudogap
estimates[5] and from our calculations shown in Fig.(5).
This is the reason why the calculated curve falls to zero
at large values of the reduced temperature t = T/Tc. The
measured critical temperature is, by the first derivative
of the resistivity, Tc(ρm) = 22.8K. Thus, using no ad-
justable parameters, only values taken from experiments,
we are able to obtain very reasonable agreement with
the Hc2 experimental values and, more importantly, a
clear explanation why it does not vanish at Tc: at tem-
peratures just above Tc there are some superconducting
islands that do not percolate, leading to a finite resis-
tivity, but they are still large enough to produce a clear
magnetic response. The sum of such local magnetic re-
sponses is clearly seen in our experiments and in other
Hc2 measurements[29]. The magnetic contributions from
non percolated islands above Tc was also measured in the
form of an anomalous magnetization[23, 30] for under-
doped compounds. Such results were interpreted within
the framework of the critical state model on a charge
disordered superconductor made up of islands[31], very
close to the above approach.
Fig.(7b) also shows the data and the calculations on
the overdoped La1.75Sr0.25CuO4. Perhaps the PST line
or upper pseudogap vanishes at ρm = 0.20[2], what
is in agreement with many experiments that indicate
more ordered behavior to overdoped than the under-
doped compounds[13, 32]. Thus, we considered just a
small Gaussian variation in the local charge density which
yielded also a small variation in ∆Tc(ρm) ≈ 18%. This
calculation, with small fluctuations instead of large stripe
like variations, is in agreement with the Fermi liquid
behavior of overdoped samples. Accordingly, the vari-
ations of ∆Tc(l) are very similar to the compound with
ρm = 0.22, showed in the last panel of Fig.(5). As a con-
sequence, the calculated Hc2(T ) curve of the overdoped
sample falls to zero just 18% above Tc, while the under-
doped vanishes at a much larger temperature.
VI. CONCLUSION
We observed that the measuredHc2(T ) curves for both
underdoped and overdoped La2−xSrxCuO4 compounds
display several non-conventional features like the positive
curvature and non-vanishing values above Tc.
The original GL approach to Hc2 near Tc fails to repro-
duce these behaviors. However it is possible to describe
qualitatively well the observed behavior by a generaliza-
tion of the GL theory that takes into account the intrinsic
charge inhomogeneities. As an additional step towards
an unified description, we assumed that such disorder
was originated from a phase separation transition, possi-
bly near the upper pseudogap temperature. The calcula-
tions were done in connection with the BdG formalism to
obtain the distribution of local superconducting temper-
ature Tc(l), that is the onset of local pairing amplitude,
on clusters with local charge density ρ(l).
The measurements yield stronger non-conventional be-
havior for the underdoped sample which may be an indi-
cation of a larger degree of inhomogeneity, in agreement
with other experiments[8, 32]. The different degrees of
disorder were taken into account by the phase separa-
tion and this unified approach reproduced well the Hc2
results. Thus we conclude that the observed unusual fea-
tures associated with Hc2 for both samples are consis-
tent with the presence of charge inhomogeneities in the
La2−xSrxCuO4, which depending on the value of x or
ρm, appear either in the form of stripes or in the form of
small fluctuation around the average doping level.
J. L. González and A. Lopez acknowledge financial sup-
port from Faperj. E. V. L. de Mello is grateful to CNPq
for partial financial support.
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http://arxiv.org/abs/cond-mat/0107347

  The upper critical field $H_{c2}$ is one of the many non conventional
properties of high-$T_c$ cuprates. It is possible that the $H_{c2}(T)$
anomalies are due to the presence of inhomogeneities in the local charge
carrier density $\rho$ of the $CuO_2$ planes. In order to study this point, we
have prepared good quality samples of polycrystalline $La_{2-x}Sr_xCuO_{4}$
using the wet-chemical method, which has demonstrated to produce samples with a
better cation distribution. In particular, we have studied the temperature
dependence of the second critical field, $H_{c2}(T)$, through the magnetization
measurements on two samples with opposite average carrier concentration
($\rho_m=x$) and nearly the same critical temperature, namely $\rho_m = 0.08$
(underdoped) and $\rho_m = 0.25$ (overdoped). The results close to $T_c$ do not
follow the usual Ginzburg-Landau theory and are interpreted by a theory which
takes into account the influence of the inhomogeneities.

Measurements and analysis of the upper critical field H
c2 of underdoped and
overdoped La2−xSrxCuO4 compounds
D. H. N. Dias and E. V. L. de Mello
Instituto de F́ısica, Universidade Federal Fluminense,
Niterói, R. J., 24210-340, Brazil
J. L. Gonzalez and H. A. Borges
Dept. de F́ısica, Pontif́ıcia Universidade Católica de Rio de Janeiro,
R. Marques de São Vicente 225, R. J., 22453-900 Brasil.
A. Gomes
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil.
A. Loose
FU-Berlin. Berlin. Germany
A. Lopez
Universidade do Estado de Rio de Janeiro, Rio de Janeiro, Brazil.
F. Vieira and E. Baggio-Saitovitch
Centro Brasileiro de Pesquisas F́ısicas, Rio de Janeiro, Brazil.
(Dated: November 21, 2018)
The upper critical field Hc2 is one of the many non conventional properties of high-Tc cuprates.
It is possible that the Hc2(T ) anomalies are due to the presence of inhomogeneities in the local
charge carrier density ρ of the CuO2 planes. In order to study this point, we have prepared good
quality samples of polycrystalline La2−xSrxCuO4 using the wet-chemical method, which has been
demonstrated to produce samples with a good cation distribution. In particular, we have studied the
temperature dependence of the upper critical field, Hc2(T ), through magnetization measurements
on two samples with opposite average carrier concentration (ρm = x) and nearly equal critical
temperatures, namely, ρm = 0.08 (underdoped) and ρm = 0.25 (overdoped). The results close to Tc
do not follow the usual Ginzburg-Landau theory and are interpreted by a theory which takes into
account the influence of the inhomogeneities.
PACS numbers: 74.72.-h, 74.80.-g, 74.20.De, 02.70.Bf
I. INTRODUCTION
High critical temperature superconductors (HTSC)
display many non-conventional properties which remain
to be explained by a concise physical picture[1, 2, 3].
This state of affairs might be due to the fact that, differ-
ently from low temperature superconductors, these ma-
terials have a large degree of intrinsic inhomogeneities.
Although its origin is unknown, there are many evidences
from different experiments that they do not have a homo-
geneous doping level[4, 5]. For instance, recent Scanning
Tunneling Microscopy (STM)[6, 7, 8, 9] measurements
have detected strong variations in the density of states as
the tip travels over a clean and sharp surface. Also, neu-
tron diffraction data have revealed a complex structure of
the charge distribution that has become known as ”stripe
structure”[10]. Nuclear Quadruple Resonance (NQR)[11]
and Angle Resolve Photo-Emission (ARPES)[12] have
also detected inhomogeneities in the local environment
and in the charge distribution.
Based on these experimental evidences we argue that
this behavior and the inhomogeneities are possibly due
to a phase separation transition (PST) connected with
the anomalies seen at the upper pseudogap temperature.
Such PST would bring the system to a disordered charge
distribution state, with the formation of islands or re-
gions of distinct doping levels[13]. Since any PST de-
pends on the chemical mobility, this approach may shed
some light on the reason why some compounds appear
to be more homogeneous or, at least, do not display
any gross inhomogeneity[14, 15], although the phase di-
agrams of cuprates seem to be universal. Therefore, we
think it is possible to formulate a unified theory for the
HTSC despite of their different degrees of disorder.
In this paper we explore this possibility by showing
that the anomalies of the upper critical field Hc2(T ) as
function of the temperature T are in agreement with
charge carriers inhomogeneities in two samples with com-
pletely different average doping levels: one underdoped
and other in the far overdoped region of the super-
conducting phase diagram. In order to achieve this
goal this work is threefold: i) we prepared samples of
http://arxiv.org/abs/0704.1425v2
La2−xSrxCuO4 (LSCO), ii) we made several sets of
Hc2(T ) measurements and iii) perform a theoretical in-
terpretation of the data. The non conventional fea-
tures of Hc2(T ), like a positive curvature and absence
of saturation at low temperatures, are well known from
many previous experiments[16, 17, 18]. We concentrat
in two samples in the underdoped and overdoped dop-
ing levels, and measured Hc2(T ) to detect qualitative
doping dependent behavior. We interpreted the results
through a theory that takes into account the different
contribution from stripes of different local charge den-
sity. These calculations are based on the Cahn-Hilliard
(CH) theory of phase separation[19, 20] for compounds
with ρm ≤ 0.20[2, 13] and on a Gaussian charge fluc-
tuation around the average for ρm ≥ 0.20. In both
cases the resulting inhomogeneous systems are studied
with a Bogoliubov-deGennes approach to a disordered
superconductor [13]. Indeed there are several different
approaches to deal with the inhomogeneities in HTSC,
like the method of Ghosal at al[21] of a local disorder in
the chemical potential, Nunner et al[22] that deals with
out of plane chemical disorder, and Cabo et al[23] that
introduced an in plane Gaussian disorder around de av-
erage doping, to mention just a few of what can be found
in the literature. So far, all these approaches succeeded
in explain some HTSC features, what shows that the in-
homogeneities are important, but only new and refined
experiments will be able to determine the correct way to
deal with them.
II. SAMPLE PREPARATION
Several polycrystalline samples belonging to the
La2−xSrxCuO4 system were prepared by the wet-
chemical method according to reference[24, 25]. Pure
(99,99%) oxide and carbonate compounds, namely
La2O3, CuO, and SrCO3 were dried at 150
0C and
weighted with adequate stoichiometrical proportions.
The powders were dissolved into 50 ml of ultra pure acetic
acid (CH3COOH) and the final solution was dried and
after heated at 9000C during 24 hours in flowing oxygen.
After that the powders were quenched at room temper-
ature, then the mixtures were reground and pressed into
pellets. Finally the pre-sintered samples were sintered at
10500C during 50 hours.
Fig.(1) shows the x-rays diffraction performed in both
samples prepared according the wet-chemical method.
The x-ray diffraction (XRD) patterns were obtained in a
(XPert PRO PANalytical) powder diffractometer using
CuKα radiation (λ = 1.5418Å). Data were collected by
step-scanning mode (200 ≤ 2θ ≤ 900) and 2 s counting
time in each step at room temperature. Orthorhombic
(Bmab) and tetragonal (F4mmm) structure space groups
were assumed in the Rietveld analysis for the 0.08 and
0.25 strontium percent samples respectively.
Once we have characterized the sample, we have per-
formed magnetic measurements by a SQUID magnetome-
20 30 40 50 60 70 80 90
20 30 40 50 60 70 80 90
25 % Sr
8 % Sr
FIG. 1: X-rays diffraction spectra at room temperature for
both samples used in the experiment. The principal peaks
were identified according to Rietveld analysis.
ter, as described in the next section.
III. EXPERIMENTAL PROCEDURE
5 10 15 20 25
-8,0x10-5
-6,0x10-5
-4,0x10-5
-2,0x10-5
18 19 20 21 22 23 24 25 26 27 28
Tc(3kOe)=21.15
Tc(1kOe)=22.6 K 1 kOe
 3 kOe
 7 kOe
 10 kOe
T (K)
25 % Sr
FIG. 2: (Color on line) The zero-field cooling magnetic mo-
ment at some applied magnetic fields as a function of tem-
perature for the sample with 25% of strontium. The insert
shows how the values of Tc were determined for two values of
the applied field.
Magnetization curves as a function of temperature
were obtained with a SQUID magnetometer in a conven-
tional DC mode. The measurements were performed in
zero field-cooling conditions with moderate applied fields
ranging between zero and two Tesla. Fig.(2) contains sets
of M(T) curves for the sample La1.75Sr0.25CuO4 (25 %
of strontium) is presented. In the inset we show details of
how Tc(H) were obtained for two selected fields, namely
3kG and 1 kG. By taking the measured values of Tc(H),
we obtain the plot shown in Fig.(3), from which we can
extrapolate the value of Tc(Hc2 = 0). For the present
case, we obtain Tc(Hc2 = 0) = 24.1K and for the under-
doped sample, we get Tc(Hc2 = 0) = 31.1K. A similar
set of curves was obtained for the 8% strontium sample.
In this manner we got the value of Tc(Hc2 = 0) for both
samples.
18 19 20 21 22 23 24 25 26 27
12000
16000
20000
T (K)
FIG. 3: (Color on line) Values of Tc(H) calculated as shown
in inset of Fig.(2) ( for La1.75Sr0.25CuO4). The value of
Tc(Hc2 = 0) is taken where the curve extrapolates to zero.
In this case we get Tc(Hc2 = 0) = 24.1K, value that will be
used further on (Fig.(7)).
On the other hand, the critical temperatures Tc(H =
0) for both samples were determined from many sets
of resistivity data, by taking the maximum of the
first derivative of the resistivity vs temperature curves,
namely Tc(8%Sr) = 22.8K and Tc(25%Sr) = 19.9K.
From these data, the widths of the superconducting
transitions were estimated at half-maximum of the first
derivative, as displayed in Fig.(4). For the sample with
25% of strontium the ∆Tc was about 4.5 K while in the
0.08% sample the transition was wider pointing out to
the presence of strong inhomogeneities in the sample.
The presence of these disorder will be important in the
discussions presented in the next sections.
10 20 30 40
0 10 20 30 40
∆ΤC=4.5 K
FIG. 4: Resistivity as function of temperature. The insert
shows the first derivative of the data and the maximum is
taken as the superconductivity transition temperature for this
sample at Tc(25%Sr) = 19.9K
IV. PST RESULTS AND DISCUSSION
As discussed in the introduction, there are many exper-
imental evidences showing that some cuprates are highly
inhomogeneous in their charge distribution while others
are not, but all of them display the same phase diagram.
To deal with this non trivial problem, we have intro-
duced the idea of a PST that, depending on the mobility
of the ions, can generates various degrees of decomposi-
tion. This phase segregation process can form patterns
on the sample, as the stripes[10] or patchwork[6], gener-
ating islands with different values of the charge density,
or can merely form small fluctuations around an average
doping level. Applying a Bogoliubov-deGennes (BdG)
theory to these systems we were able to calculate the
local superconducting pairing amplitude at a given site
”i” in a cluster ”l”. Thus, in our calculations, a given
sample with average or mean charge density ρm may be
composed of local regions with local densities ρ(l). Here
we show results on a 14× 14 matrix, that is, 14 clusters
on a stripe form (each stripe has 14 sites) (l = 1to14)
and 196 sites (i runs from 1 to 196).
In general, regions with larger charge densities or dop-
ing levels usually may become superconducting, and
those with very low doping level are insulators and never
become superconducting. This anomalous behavior is de-
tected by the superconducting pairing amplitude ∆(l, T )
which, as the temperature is decreased, starts at Tc(l),
increases and saturates at low temperatures. This allows
us to define a local superconducting temperature Tc(l).
Here we exhibit simulations on a square mesh 14× 14
that display stripe inhomogeneities similar to the exper-
imental results[10], but derived from a CH phase separa-
FIG. 5: (Color on line) Temperature evolution of the local
pairing amplitude ∆(i, T ) at each stripe on a square of 14×14,
that is, with 196 sites ”i”, for different samples. Because of
the inhomogeneities, the sites in the left have ρ(i) ≈ 0 and the
ones in the right ρ(i) ≈ 2ρm. As the systems are cooled down,
more regions become superconducting and they percolate at
Tc(ρm). These percolation threshold temperatures for each
compound are indicated in their respective panel. The panel
with ρ = 0.22 does not have the stripe structure, because it
has a random distribution of doping values, which are sorted
following a Gaussian distribution.
tion theory applied to the LSCO series[13]. According to
the CH results[19, 20], the square mesh phase separates
into a bimodal distribution of charge. For underdoped
samples the phase separation is essentially total and leads
to two halves where the 7 stripes at the left are character-
ized by local doping ρ(l) ≈ 0 and the 7 at the right side
have ρ(l) ≈ 2ρm as showed in the top panel of Fig.(5)
(for ρ = 0.05). As ρm increases, the charge fluctuations
also increases, changing the properties of a compound
into metallic, and superconducting at low temperatures.
Thus, for compounds with 0.06 < ρm < 0.20, at low tem-
peratures, as one can see in Fig.(5), there are also stripes
with non-vanishing densities at low density regions. No-
tice that the values of ∆ are constant in a given stripe,
in all its sites ”i”, and that is why we may plot ∆(i, T )
at each site, although that is meaningful only at a given
stripe ∆(l, T ). As the density of charge carriers increases,
it produces the ∆(i, T ) at the very low doping sites, i.e.,
in the left region of Fig.(5). When this occurs at low
temperatures, it is possible that the system becomes su-
perconductor at Tc(ρm) by the percolation of all the local
superconducting regions[26, 27] and it can hold a dissipa-
tionless current. The percolation among the local regions
where ∆(l, T ) in non-zero may occur either directly or by
Josephson coupling and the associated temperature, or
the superconducting temperature Tc(ρm), is also shown
in the panels of Fig.(5). Notice that the compound with
ρ = 0.22 does not have the stripe structure, it has just a
Gaussian distribution of doping values, since it is larger
than the PST threshold of ρ ≈ 0.2.
Above Tc(ρm), depending on the value of ρm, the com-
pounds may be formed by mixtures of superconducting,
insulator and normal domains and above the pairing for-
mation temperature (the onset temperature which some-
times is called the lower pseudogap), they are a disor-
dered metal with mixtures of normal (ρ(l) > 0.05) and
insulator (ρ(l) ≤ 0.05) regions. For ρm ≥ 0.20 the
charge disorder is practically zero, with a small fluctu-
ation around ρm. From these calculations, we identify
Tonset(ρm) as the highest temperature (Tc(l)) which in-
duces a ∆(l, T ) in a given compound which is easily seen
in the panels of Fig.(5). Tonset may be also identified
with the onset of Nernst signal[13]. To make clear how
the values of Tonset of a given compound are obtained,
we show the larger values of Tc(i) and Tc(l) in Fig.(6) for
the sample with ρ = 0.15.
FIG. 6: (Color on line) To explain the concept of local super-
conducting temperature and specially how certain regions de-
velops a non-vanishing pairing amplitude, we plot the ∆(i, T )
or ∆(l, T ) for each site ”i” of our two dimensional 14 × 14
array. The onset temperatures values for which these pairing
amplitudes develop for each stripe are clearly indicated in the
figure, and the highest value, namely, T = 90K, is taken as
the lower pseudogap temperature of this compound.
V. Hc2 RESULTS AND CALCULATIONS
It is well known that the Ginzburg-Landau (GL) up-
per critical field of a homogeneous superconductor is a
linear function of the temperature near Tc and falls to
zero at this temperature[28]. This behavior is not ob-
served by our measurements, showing another departure
from conventional properties. As we can see in Fig.(7),
the Hc2(T ) experimental points for both samples are lin-
ear only near and below T/Tc ≤ 0.9. As the tempera-
ture increases it performs an upturn curvature falling to
zero further beyond Tc. Consequently, we need some new
ideas or theories to interpret these results.
From these Hc2(T ) curves, we can estimate that the
upper critical field goes to zero at about 31.1 K, which is
substantially higher than the value Tc = 22.8K of the 8
%-Sr sample (from resistivity measurements as discussed
in section II). Similarly, Hc2(T ) falls to zero at 24.1K for
the 25 %-Sr sample which has a Tc = 19.9K. Thus we
see that Hc2(T ) vanishes at temperatures 8-22% larger
than Tc. This effect, the nonzero value of Hc2 above
Tc, is quite unusual for a normal superconductor but it
was also observed in LSCO and Bi-2201 cuprates by the
group of Wang et al[29]. We will show below that this
result may be explained as a consequence of the intrinsic
disorder in HTSC, namely the presence of regions with
different local dopings and distinct superconducting local
temperatures Tc(l).
In order to provide an interpretation to these results,
we applied a generalization of the GL Hc2(T ) expression
following along the lines described by Caixeiro et al[28]:
The GL upper critical field near Tc of a homogeneous
superconductor may be written as
Hc2(T ) =
Tc − T
. (T < Tc) (1)
At a temperature T , we take each superconducting re-
gion in the sample characterized by a ρ(l) as the source
that generates a ∆(l) to produce a magnetic response
and displays a local Hc2, provided that T ≤ Tc(l). As
discussed, these regions can be in stripe or others forms,
but the important point is that they are characterized by
a region of fairly constant density ρ(l) that at T ≤ Tc(l)
may shield the applied magnetic field. Thus each of such
given local region has a local superconducting tempera-
ture Tc(l) and will contribute to the upper critical field
with a local linear upper critical field H l
(T ) near Tc(l)
according to the usual GL approach. This is justified be-
cause each region has a constant density, like a type II
low temperature superconductor, and should posses its
own Hc2 that is expected to vanish linearly at Tc(l).
Consequently the total contribution of the local su-
perconducting regions to the whole sample upper critical
field is the sum of all the H l
(T )’s,
Hc2(T ) =
Tc(l)− T
Tc(l)
(T ). (T < Tc(l) ≤ Tonset(ρm)) (2)
WhereW is the total number of superconducting regions,
stripes or islands with its local Tc(l) ≥ T . The maxi-
mum value of Tc(l) is the pseudogap temperature iden-
tified above as the Tonset(ρm). For the LSCO series a
coherence length of ξab(0) ≈ 22Å is used, in accordance
with the measurements[28]. This value of ξab(0) leads to
Hc2(0)=Φ/2πξ
(0)=64T. Due to the limitations of the
GL approach, we expect the result of this equation to be
accurate only near and above the system Tc.
Fig.(7a) shows both the Hc2 results of the generalized
GL calculations together with the experimental values
for underdoped La1.92Sr0.08CuO4 compound. In the cal-
culations on this compound, we used a maximum tem-
perature of superconducting formation, Tonset ≈ 90K
FIG. 7: Experimental points (connected by a thin line)
against the reduced temperature t = T/Tc and the calcu-
lated curve (thick line) of Hc2 considering inhomogeneous
samples with a stripe distribution of local superconducting
temperature Tc(l) for ρ = 0.08 and a similar calculation with
a Gaussian distribution for ρ = 0.25.
(the maximum Tc(l) for this sample) from pseudogap
estimates[5] and from our calculations shown in Fig.(5).
This is the reason why the calculated curve falls to zero
at large values of the reduced temperature t = T/Tc. The
measured critical temperature is, by the first derivative
of the resistivity, Tc(ρm) = 22.8K. Thus, using no ad-
justable parameters, only values taken from experiments,
we are able to obtain very reasonable agreement with
the Hc2 experimental values and, more importantly, a
clear explanation why it does not vanish at Tc: at tem-
peratures just above Tc there are some superconducting
islands that do not percolate, leading to a finite resis-
tivity, but they are still large enough to produce a clear
magnetic response. The sum of such local magnetic re-
sponses is clearly seen in our experiments and in other
Hc2 measurements[29]. The magnetic contributions from
non percolated islands above Tc was also measured in the
form of an anomalous magnetization[23, 30] for under-
doped compounds. Such results were interpreted within
the framework of the critical state model on a charge
disordered superconductor made up of islands[31], very
close to the above approach.
Fig.(7b) also shows the data and the calculations on
the overdoped La1.75Sr0.25CuO4. Perhaps the PST line
or upper pseudogap vanishes at ρm = 0.20[2], what
is in agreement with many experiments that indicate
more ordered behavior to overdoped than the under-
doped compounds[13, 32]. Thus, we considered just a
small Gaussian variation in the local charge density which
yielded also a small variation in ∆Tc(ρm) ≈ 18%. This
calculation, with small fluctuations instead of large stripe
like variations, is in agreement with the Fermi liquid
behavior of overdoped samples. Accordingly, the vari-
ations of ∆Tc(l) are very similar to the compound with
ρm = 0.22, showed in the last panel of Fig.(5). As a con-
sequence, the calculated Hc2(T ) curve of the overdoped
sample falls to zero just 18% above Tc, while the under-
doped vanishes at a much larger temperature.
VI. CONCLUSION
We observed that the measuredHc2(T ) curves for both
underdoped and overdoped La2−xSrxCuO4 compounds
display several non-conventional features like the positive
curvature and non-vanishing values above Tc.
The original GL approach to Hc2 near Tc fails to repro-
duce these behaviors. However it is possible to describe
qualitatively well the observed behavior by a generaliza-
tion of the GL theory that takes into account the intrinsic
charge inhomogeneities. As an additional step towards
an unified description, we assumed that such disorder
was originated from a phase separation transition, possi-
bly near the upper pseudogap temperature. The calcula-
tions were done in connection with the BdG formalism to
obtain the distribution of local superconducting temper-
ature Tc(l), that is the onset of local pairing amplitude,
on clusters with local charge density ρ(l).
The measurements yield stronger non-conventional be-
havior for the underdoped sample which may be an indi-
cation of a larger degree of inhomogeneity, in agreement
with other experiments[8, 32]. The different degrees of
disorder were taken into account by the phase separa-
tion and this unified approach reproduced well the Hc2
results. Thus we conclude that the observed unusual fea-
tures associated with Hc2 for both samples are consis-
tent with the presence of charge inhomogeneities in the
La2−xSrxCuO4, which depending on the value of x or
ρm, appear either in the form of stripes or in the form of
small fluctuation around the average doping level.
J. L. González and A. Lopez acknowledge financial sup-
port from Faperj. E. V. L. de Mello is grateful to CNPq
for partial financial support.
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http://arxiv.org/abs/cond-mat/0107347
