Commun. Theor. Phys. (Beijing, China) 49 (2008) pp. 1631–1634
c Chinese Physical Society
Vol. 49, No. 6, June 15, 2008
An Extended Model for Coexistence of Superconductivity and Paramagnetism in
High-Tc Superconductors
F. Inanir
Department of Physics, Rize University, Rize, 53100, Turkey
(Received November 7, 2007)
Abstract In the present work, the total magnetization in superconducting state is separated into critical state and
paramagnetic components in terms of an H(x)-dependent magnetic flux density. Utilizing this model, we reproduce
successfully M -H curves measured by Sandu et al. [Phys. Rev. B 74 (2006) 184511] and Sandu et al. [J. Supercond.
Incorp. Novel Magn. 17 (2004) 701] for different forms of Jc .
PACS numbers: 74.72.-h, 74.25.Ha, 74.25.Qt
Key words: superconductivity, paramagnetism, high-Tc superconductor, critical-state model
1 Introduction
2 Result and Discussion
Important characteristics of hard type-II superconducting materials such as critical fields, Meissner expulsion, trapped flux, hysteresis losses, interplay of superconductivity, and magnetism, etc., can be determined
from their magnetization curves. The critical state model
(CSM) has extensively been employed to analyze magnetization data on these materials. It gives the relationship
between the measured irreversible magnetization Mirr and
the bulk supercurrent density J introduced by Bean.[1,2]
This model is based on two simple assumptions: (i) the supercurrent density is defined by a critical current density
Jc , and (ii) any changes in the flux distribution are introduced at the sample surface. The Bean model takes Jc
to be independent of local flux density B. Later, various
forms of Jc , which is dependent on B have been proposed
by others.[3−10]
Many workers have reported that the magnetic moment of the various type-II superconductors especially
containing paramagnetic ions changes its sign at some
value of the increasing magnetic field after zero-field cool
(ZFC) processes.[11−27] This unusual behavior has generally been attributed to the coexistence of superconductivity and paramagnetism. Common Bean’s CSM[1,2] is
insufficient to explain these observations and to reproduce experimental M -H curves with this unusual magnetic crossing. To the best of our knowledge, some theoretical calculations have been presented in literature to
reproduce this abnormality based on CSM. For instance,
Qin et al.[23] and Hari Babu et al.[24] employed the critical
state model in combination with the standard theory of
paramagnetism with the extended Brillouin function and
Fisher et al.[25] have developed two velocity hydrodynamic
model of the CSM.
In this paper, we extend our approach[28] to calculate
the families of magnetization curves by considering the influence of volume currents on local paramagnetism based
on critical state model and compare our values with the
available magnetization data in the literature. The experimental results have been justified quite well in the frame
work of our model by taking into account proper field dependence of Jc (B).
2.1 Outline of the Model
We have considered a type-II superconducting slab
with thickness 2D (−D < x < D), which is infinite in the
y- and z-directions, initially cooled to low temperature in
the absence of a magnetic field. The thickness of the slab
is assumed to be much larger than London penetration
depth λ. The influence of the demagnetization effect and
surface barrier effects are neglected in the present treatment since surface barrier effect does not essentially cause
an explicit deformation to the M (H) curves.[29−31] The
magnetic field is applied parallel to the wide faces of the
slab. The magnetic field penetrates into the superconductor and induces shielding current in the regions of flux
penetration. The flux penetrated regions carry a shielding
current density Jc determined by Ampére’s law,
∂B
= ±µJc .
(1)
∂x
Here, µ = µ0 (1 + χ) is permeability of material, where µ0
is permeability of vacuum and χ is magnetic susceptibility. χ > 0 corresponds to paramagnetism and χ < 0 for
diamagnetism. Jc is a function of B(x) = µ0 (1 + χ)H(x)
and thus, paramagnetism and supercurrents can interact
with each other. The sign ± is determined by the slope
of the flux density in the specimen, i.e., the direction to
which the critical current flows. The boundary condition
for Eq. (1) is B(x = 0) = B(x = ±D) = µHa . From the
symmetry, it is sufficient to consider only the half-part of
the slab.
Following critical state approach,[1,2] magnetization M
for an idealized slab arising from the flux produced by the
external magnetic field is calculated by
Z D
1
M=
B(x) dx − Ba ,
(2)
µ0 D 0
where D is the thickness of the slab, B(x) is the local flux
density permeating the slab transverse to the x axis and
Ba is the applied field with Ba = µ0 Ha .
Substituting the definition B(x) = µ0 (1 + χ)H(x) into
Eq. (2), we have
Z
Z D
1 D
1
M=
H(x) dx − Ha + χ
H(x) dx . (3)
D 0
D
0
RD
The part (1/D) 0 H(x) dx − Ha in Eq. (3) defines the
superconducting critical state magnetization. The second
integral on the right-hand side of Eq. (3) is the measure
of a paramagnetic or strong diamagnetic contribution to
the superconducting magnetization. In this picture, total magnetization contains both irreversible critical state
1632
F. Inanir
magnetization Mcrt and normal state-like magnetization
(paramagnetism and diamagnetism) Mnorm .
Vol. 49
The coexistence of superconductivity and paramagnetism of single crystal Y0.47 Pr0.53 Ba2 Cu3 07−δ specimen
was investigated by Sandu et al.[26] Figure 1 displays the
magnetization versus applied magnetic field up to 1 kOe
at 2 K reproduced from Sandu et al.[26] The most prominent feature of the data of Sandu et al. is the migration
of the magnetization curves from the negative (diamagnetic) part to the positive (paramagnetic) part at certain
value of the increasing magnetic fields. Our primary interest in this analysis is to reproduce the change of this
sign as a function of applied field. The hysteretic loop is
typical for the superconducting systems coexisting with
local paramagnetic moments. They have stated that total
magnetization loop is the superposition of superconducting magnetization and paramagnetic moment due to Pr
ions. Experimental (circle) and the best fitted curve (full
line) are displayed in Fig. 2.
The experimental data in Fig. 2 were carefully extracted from Ref. [26]. The hysteresis loop shows a little indication of reversible component in the magnetization. The experimental data could be fitted directly by
the model. Evidently, the model reproduces successfully
the entire hysteretic behaviour and the crossing from diamagnetism to paramagnetism. The experimental values
can be generated theoretically quite well by employing
exponential critical state model,[8,32]
|B| Jc (B) = Jc0 exp −
,
(4)
B0
which leads to the description of many observations of hysteretic magnetization curves in high Tc superconductor.
Substituting Eq. (4) into Eq. (1), for the given boundary
condition, we derive the flux profile equation as follows:
x i
1 h
,
(5)
H± (x) = ln exp(p Ha ) ∓ p 1 −
p
D
where p is called the pinning parameter which determines the order of field dependence of the critical current,
B = µ H, and Ha denotes the magnetic flux density at the
specimen surface. The parameters Jc0 and H0 are related
to Hp and p by Hp = (1/p) ln(1 + p) and p = Jc0 D/H0 ,
in which Jc0 and H0 are temperature-dependent. Here,
Hp is the full penetration field, i.e. the field required to
penetrate the sample up to the center. Chen et al.[32] have
given expressions for M -H curves of type II superconductors based on exponential model. The geometry for our
calculation is considered as an infinite slab and therefore
the correlation of the demagnetization field was omitted.
Theoretical contribution of χhH(x)i-dependent paramagnetic magnetization to the critical state magnetization can
be visually determined by comparing each pair of adjacent
curves displayed in Fig. 3.
Fig. 2 Observed magnetization data extracted from Ref. [26]
(open circle) together with a fitted model behaviour (full
line). The fitting curve was calculated employing Eq. (3) and
exponential CSM (Eqs. (4) and (5)). The parameters are:
p = Jc0 D/H0 = 0.2, χ = 0.075.
Fig. 3 Effect of paramagnetic magnetization on superconducting one. The upper magnetization curve (black line) of
Fig. 2 and the curve (gray line) calculated by the same input as the former except that χ = 0, hence the paramagnetic
magnetization component of Eq. (3) is absent.
2.2 Comparisons with Experimental Results
(i) Coexistence of superconductivity and paramagnetism[26]
Fig. 1 Magnetization M versus applied field H curve in
a single crystal Y0.47 Pr0.53 Ba2 Cu3 O7−δ reproduced from
Sandu et al.[26] The measurement was carried out at 2 K
in increasing and decreasing magnetic field up to 1 kOe,
perpendicular to the ab plane of the specimen.
(ii) Coexistence of superconductivity and paramagnetism with Meissner diamagnetism: observations of Sandu et
al.[27]
No. 6
An Extended Model for Coexistence of Superconductivity and Paramagnetism in High-Tc Superconductors
1633
Figure 4 reproduces the measurements of M -H loop at 65 K (Tc = 90 K on a polycrystalline Eu0.7 Sm0.3 Ba2 Cu3 O7−δ
reported by Sandu et al. (Fig. 1 of Ref. [27]). The irreversible magnetization chances sign (negative to positive) at
the certain value of the increasing magnetic field after ZFC and recovers the negative values for the reverse cycle.
We have confined ourselves to generate theoretically the anomalous sign alternations in the experiment employing our
approach. From the behavior of the curve it is deduced that total magnetization comprises two contributions to the
superconducting magnetization. One of them is paramagnetic magnetization which arises from paramagnetic Eu ions,
that is responsible for the sign alternation (negative to positive) in the ascending branch. Another one is the reversible
or equilibrium magnetization which causes the second sign change in the descending branch (positive to negative),
some downward displacement and some reversible part in curves (see the inset Fig. 4).
Fig. 4 Magnetization curve measured by Sandu et al.[27] on
a Eu0.7 Sm0.3 Ba2 Cu3 O7−δ at 65 K. The dotted line represents
equilibrium magnetization Meq calculated by Eq. (3) in that
reference. This displays their observations of sign alternations
in magnetic moment in forward and reverse field cycle. Inset:
M -H loop at 80 K (lower curve), and at 86 K (upper curve).
Fig. 5 Corresponding curve of Fig. 4 calculated via Eqs. (3),
(6), and (7), introducing contribution of Meissner current.
The parameters are: n = 0.4, χ = 0.000 95, Hc1 = 0.5 H∗ ,
H0 = H∗ , s = 0.2.
We have adopted the generalized Kim’s critical state
model[3,4] to fit experimental data.[27] In this model, Jc (B)
is given by
Jc (B) = Jc0 /[(1 + B/B0 )n ] ,
(6)
where B0 and n are phenomenological parameters and Jc0
is the critical current density at zero magnetic field. Chen
and Goldfarb[33] calculated the magnetization curves of
an infinitely long orthorhombic specimen with finite rectangular cross section for n = 1. Figure 5 shows the
corresponding curve combining Eq. (5) and the Kim’s
critical state model (Eq. (6)) including Meissner current,
IM = Mrev = (µ0 Ha − µ Hs ). Here, IM denotes the diamagnetic Meissner surface current flowing perpendicularly
to Ha per unit dimension along Ha , and Mrev denotes the
reversible Abrikosov diamagnetism. Again, the magnetic
inductions are normalized with respect to H∗ = Jc0 D and
the calculated magnetization is normalized accordingly.
Over the range 0 ≤ Ha ≤ Hc1 , IM = µ0 Ha , and
Hs = 0 for all previous histories of µ0 Ha , hence for all
configurations of H(x). For the case Hc1 < Ha < Hc2 , we
employ the simple expression,[34−36]
s+1
IM = Hc1
/Has ,
(7)
where s is Meissner exponent, Hc1 is the lower critical
fields and both are regarded as temperature-dependent
parameters. The influence of paramagnetic and Meissner
diamagnetism contribution to the shape of CSM magnetization of the specimen can be visually assessed comparing each pair of adjacent curves displayed in Fig. 6. In
these calculations, the adjustable parameters are n, s, and
Hc1 . The parameters χ and Hc1 /H∗ are adopted in order to produce the observed magnetic behaviors of the
specimen. The parameter n for bulk current and exponent s for Meissner current are assessed by various trials
to obtain the best fit of the calculated and experimental
curves. We find that the calculated magnetizations are
nearly equal to the experimental values. But the pattern
of the calculated M -H loop exhibits some little deviation
from the experimental data at the lower values of the reverse field cycle. We adopted Kim-type dependence of the
critical current density to fit experimental data since it
provides smooth descending function with a wide change
in range. However, both Kim’s and exponential models
are not perfect to fit experimental data. Therefore, more
complex Jc (B) functions are required to improve the results. In addition, we note that it is complicate to analyze
the magnetic behaviour of the granular superconductor
samples since there are many other factors effecting material data, such as size, shape and orientation of the grains,
the current circulating inside and between the grains, flux
pinning properties of grain and matrix etc. Overall, it is
remarkable that theoretical curves employing a simple expression for the dependence of the critical current density
Jc (B) on the magnetic field and temperature reproduce
striking features of the experimental data pretty well as
seen from Figs. 4 and 5.
1634
F. Inanir
Vol. 49
Our treatment especially chances the results at low fields when it is compared to the previous works that the
paramagnetism is directly calculated from the applied field (see Refs. [23] and [24]). At the initial increase of field, a
low-field diamagnetic peak appears in all measurements (see Figs. 1 and 4), which is generally typical of magnetization
of type-II superconductors. This low field peak can be reproduced quite well by our treatment, besides migration from
negative magnetic moment to positive magnetic moment can be generated. The influence of the surface barrier effects
are neglected in the present treatment, since surface barrier effect does not cause an explicit deformation to the M (H)
curves, but merely expands them up and down.[31]
Fig. 6 Influence of Meissner surface current (b) and paramagnetic magnetization (c) to the calculated M -H curves.
The parameters are (a) χ = 0, Hc1 = 0 (b) χ = 0, Hc1 = 0.5 H∗ , and (c) χ = 0.000 95, Hc1 = 0.5 H∗ . Other parameters
are the same as adopted in Fig. 5.
3 Conclusions
The sign change of magnetic moment in the M -H loop of several high-Tc superconductors owing to the coexistence
of superconductivity and paramagnetism could be described quite well by critical state combined with a normal state
χH(x)-dependent paramagnetic contribution to the magnetization. Based on the treatment presented here, we have
successfully reproduced magnetization curves measured by Sandu et al.[26,27] exploiting a simple expression for the
dependence of the critical current density Jc (H) for the magnetic field and temperature.
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