Physica C 369 (2002) 227–231
www.elsevier.com/locate/physc
Role of the field dependence of the heat capacity for the flux
jump process in HTSC materials
V.V. Chabanenko a,*, V.F. Rusakov b, A.I. D’yachenko a, E.M. Roizenblat c,
S. Piechota d, S. Vasil’ev d, H. Szymczak d
a
Physical and Technical Institute, National Academy of Sciences, Ulitsa R Luxembour, 72, 83114 Donetsk, Ukraine
b
National University, 83055 Donetsk, Ukraine
c
Research Institute of Materials and Electronic Engineering, 83036 Donetsk, Ukraine
d
Institute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland
Abstract
Magnetization and magnetostriction loops were calculated in terms of the Kim–Anderson model incorporating the
flux jump instability criterion and taking into account magnetic field and temperature dependencies of the heat capacity
of YBaCuO. Regions in the H –T diagram where flux jumps occur were determined. The stabilizing role of the specific
heat for the critical state of the material was observed. Ó 2001 Published by Elsevier Science B.V.
PACS: 74.60 Ge; 74.72 Bk
Keywords: Mixed state; Lattice dynamics; Thermal conductivity
1. Introduction
Quasipermanent magnets of HTSC can now
trap a multi-tesla field [1]. This fact and similar
applications are limited from the low temperature
side by thermomagnetic instabilities (giant flux
jump) and damages [2] or magnetolamination [3]
caused by magnetic pressure. At high temperature
T the trapping flux decreases, which is connected
with a decrease in the critical current density jc .
With the availability of sizable bulk materials of
uniform structure [1] and superconducting prop-
*
Corresponding author. Fax: +38-0622-521-074.
E-mail address: chaban@host.dipt.donetsk.ua (V.V. Chabanenko).
erties, the range of magnetic fields and temperatures has been essentially broadened to 6 T and 40
K, respectively, which is a region were the thermomagnetic instabilities occur. In determining the
boundaries of the instability region for the critical
state in the H –T plane, of primary importance
becomes the allowance for changes in all of the
material properties under field and temperature
effect. The effect of thermal property CðT ; H Þ of
material on the magnetic field of the first jump
HFJ ¼ ð3l0 Cjc =jdjc =dT jÞ1=2 (this expression is valid
in the Bean’s model for the adiabatic conditions of
instability development at plane geometry of the
sample [4]) is qualitatively evident: a growth in
heat capacity will increase the field where the
critical state of the superconductor fails (flux
jump). However, it is impossible analytically to
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V.V. Chabanenko et al. / Physica C 369 (2002) 227–231
estimate the influence of the whole instability region on the H –T plane. This is, first of all, connected with the complexity, multi-stage description
[4] of the flux jump runaway process in which
different physical parameters of the material may
play a defining role at different stages.
In order to calculate the H –T regions of instability a procedure is needed which is sufficiently
general to embrace the whole of the parameters
and dependence in total. We have elaborated such
a program for determining the region of instability
of the critical state of hard superconductors with
different field dependence of the critical current [5].
It includes experimental data on the change of the
thermal properties ðCðT ÞÞ of a material with temperature. Recently thermal properties CðH ; T Þ in
strong magnetic field for YBaCuO superconductor
have been determined [6]. There are two physical
models discussing the influence of the magnetic
field on the thermal properties of the superconductor. Caroli et al. in their calculations [7] assumed a fully gapped superconductor. It was
calculated by Volovik [8] that, for a superconductor with lines of nodes in the gap function, the
dominant contribution to the magnetic field dependence of the specific heat should come from
quasiparticle excitations outside the vortex core.
This result is derived from the Doppler shift of the
quasiparticle excitation spectrum and from the
dependence of the intervortex spacing on the
magnetic field.
The aim of this paper is to calculate the influence of the change in thermal properties of the
material, under the action of a magnetic field, on
the H –T regions of critical state instability. For
one simulation we take typical parameters for the
YBaCuO superconductor.
Suppose that the external magnetic field He is applied parallel to the slab face. Let the external
magnetic field He increase (decrease) by a small
amount DHe (jDHe j jHe j). Because of the motion
of flux lines, the magnetic field, BðxÞ in the sample
increases by DBðxÞ. The DBðxÞ value has to be
calculated from the equation of the critical state
dB=dx ¼ l0 Jc ðxÞ:
Here l0 is the permeability of vacuum and the
current density absolute value at point x is
jJ ðxÞj 6 Jc ðx; BðxÞÞ. Under such conditions everywhere in the superconductor the pinning force
balances the Lorentz force. Let xi is the point at
which the Bðxi Þ Bi values are calculated with the
index 1 6 i 6 n, x1 ¼ 0, xn ¼ d, and B0 ðxi Þ B0i is
the initial field. We put DB B0iþ1 Bi , where
i > 1, dx ¼ xiþ1 xi ; B0 ¼ Bðx1 Þ ¼ He . The critical
state equation (1) gives the condition:
Bðxiþ1 Þ Biþ1
8
< B0iþ1 þ DB; jDBj < l0 Jc ðBi Þ dx;
¼ Bi þ signðDBÞl0 Jc ðBi Þ dx;
:
jDBj > l0 Jc ðBi Þ dx:
The algorithm given in [5] was used for computer simulation of magnetic properties of superconductors in a wide range of experimental
parameters taking into account the magnetic field
and temperature dependences of the specific heat
CðT ; H Þ. Let us consider a sample having a slab
geometry with thickness W ¼ 2d (0 6 x 6 2d).
ð2Þ
When the applied magnetic field Ha increases by a
small amount than DBðxÞ ¼ BðxÞ B0 ðxÞ, where
B0 ðxÞ is the virgin magnetic field profile inside the
slab.
The energy Q per unit volume dissipated in the
vicinity of the coordinate x is
Z
ð3Þ
QðxÞ ¼ Jc ðxÞ DBðx0 Þ dx0 :
The fluctuations cause a sudden rise of temperature DT ðxÞ at point x. Under local adiabatic conditions one obtains
DT ðxÞ ¼ QðxÞ=Cv ;
2. The algorithm of calculation
ð1Þ
ð4Þ
where Cv ðT ; H Þ is the specific heat of the superconductor.
Temperature and magnetic field reduces the
critical current density by
DJc ðxÞ ¼ Jc ðx; T þ DT ; B0 ðxÞ þ DBðxÞÞ Jc
ðx; T ; B0 ðxÞÞ:
ð5Þ
The reduction in Jc weakens the shielding ability of
the superconductor by DHs ,
V.V. Chabanenko et al. / Physica C 369 (2002) 227–231
DHs ¼ Z
229
d
DJc ðx0 Þ dx0 ;
ð6Þ
0
which is obtained directly from Maxwell’s equation (1). The flux jump occurs if DHs > DHe (instability criterion) [9].
During a flux jump the temperature inside the
sample increases and reaches a maximum at
T < Tc , which is assumed to be independent of the
position x inside the sample. The critical current
density at T is Jc ðT Þ < Jc ðT0 Þ, where T0 is the
temperature of the sample before the jump.
Therefore, according to the critical state equation
(1) and the adiabatic condition DHs > DHe the new
magnetic field configuration B ðxÞ is given by solution of the Eq. (2) for new Jc ðBÞ ¼ Jc ðB; T Þ <
Jc ðB; T0 Þ.
Therefore, the critical state algorithm (2) allows
one to calculate the magnetic field profile in the
slab in the presence of jumps, independently of the
magnetic prehistory and forms of the Jc vs. H and
T dependencies.
For calculations of magnetization MðH Þ and
magnetostriction DLðH Þ loops with instability criterion we used function Cv ðT ; H Þ ¼ C0 ðT Þ þ
0:89T ðH Þ1=2 [6]. We took the value of the critical
current J0 in our calculations in the region: 1010 –
1012 A/m2 . Such values are smaller than a maximal
potential critical current; they lie closer to the
maximal potential critical currents observed in
HTSCs. The model of effectiveness of planar surface and boundary pinning structures in films and
multi-layers predicts [10] for YBCO a maximum
potential critical current value of 1012 A/m2 .
3. Results
Fig. 1 shows the calculated magnetization
MðH Þ and magnetostriction DLðH Þ=L0 loops for
the Kim–Anderson model (Jc ðH Þ ¼ Jc0 =ð1 þ H =
H0 k Þ, Jc0 ¼ 1:6 1010 A/m2 , H0 k ¼ 0:3 T, Hc2 ¼
40 T, T ¼ 3:5 K). For comparison we show two
lines. Solid line depicts magnetic properties of the
superconductor taking into account CðH Þ dependence and dash line depicts it without influence of
the magnetic field on the thermal properties. The
stabilizing role of the specific heat in the critical
Fig. 1. Calculated magnetostriction (a) and magnetization (b)
loops for the Kim–Anderson model taking into account the
Cv ðHÞ dependence (––); without Cv ðH Þ dependence (- - -). The
following parameters are used: the critical current
Jc0 ¼ 1:6 1010 A/m2 , Hc2 ¼ 40 T, and H0 k ¼ 0:3 T, T ¼ 3:5 K.
state material is observed for all quadrants. For
example, they are increasing the magnetic field of
the first flux jump and decreasing the number of
jumps in the first quadrant ðDH1inst ðwith CðH ÞÞ <
DH2inst ðwithout CðH ÞÞÞ when taking into account
the specific heat of the system. Fig. 2 shows the
influence of temperature on the remagnetization
loops in these two cases.
In Fig. 3, H –T diagrams of instability ranges for
three values of the critical current Jc0 (6 1010 ,
16 1010 and 80 1010 A/m2 ) are presented. In
each figure, there are calculated two regions of flux
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V.V. Chabanenko et al. / Physica C 369 (2002) 227–231
Fig. 2. Calculated magnetization loops for the Kim–Anderson
model for different temperatures: taking into account the Cv ðH Þ
dependence (right column); without Cv ðH Þ dependence (left
column). The parameters used are: the critical current is
Jc0 ¼ 80 1010 A/m2 , Hc2 ¼ 40 T, d ¼ 5 mm, and H0 k ¼ 0:3 T.
jumps: one in the presence of the field dependence
of heat capacity and second––without it. In the first
case (with CðH Þ dependence) on the lines parallel
to the H-axis, magnetic field values, at which
magnetic flux jumps take place at fixed temperature, are marked by dark squares. Positions of the
flux jumps were determined on the basis of the
calculated magnetization dependences MðH Þ (see
Fig. 2). In the second case, for each number (n) of
the jump a temperature dependence of its position
on the magnetic field axis HnFJ ðT Þ has been marked.
The influence of the change of the thermal properties of the superconductor, under the influence of
the magnetic field on the instability region for its
critical state is clearly shown.
Several essential characteristics of the H –T regions of instability can be understood with reference to Fig. 3. First of all, this is a diminution in
the instability region with heat capacity increase in
the magnetic field typical of all the quadrants and
Fig. 3. H –T -ranges of the instability calculated by taking into
account the Cv ðHÞ dependence and without it for a plate geometry with thickness d ¼ 5 mm for the Kim–Anderson model
of critical state and for different values of the critical current: (a)
Jc0 ¼ 6 1010 A/m2 , (b) Jc0 ¼ 16 1010 A/m2 and (c)
Jc0 ¼ 80 1010 A/m2 .
critical currents values but to a different degree.
The largest effect from of the thermal properties is
observed in the second quadrant, and the smallest––in the third one. It is clear from Fig. 3, that
the influence of the thermal properties of the superconductor on the critical state stability for
strong magnetic fields is larger than for weak ones.
Irrespective of the thermal properties, in the
second quadrant there is a region of magnetic
fields with no magnetic flux jumps. It is between
the instability region for this quadrant and the axis
V.V. Chabanenko et al. / Physica C 369 (2002) 227–231
of temperature. Such a property of flux jumps was
observed experimentally [11]. A full understanding
of this phenomena needs further investigations.
Acknowledgements
The Polish Government Agency KBN under
contract no. 8 T11B 038 17 supported this work.
The European ESF program VORTEX is also
acknowledged.
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