Supercond. Sci. Technol. 11 (1998) 1181–1185. Printed in the UK
PII: S0953-2048(98)93386-3
The range of giant flux instabilities in
the H –T plane in hard
superconductors: calculations and
experiment
V V Chabanenko†, A I D’yachenko†, A V Chabanenko†,
M V Zalutsky†, H Szymczak‡, S Piechota‡ and A Nabialek‡
† Physico-Technical Institute, National Academy of Sciences,
ul. R. Luxembourg 72, 340114 Donetsk, Ukraine
‡ Institute of Physics, Polish Academy of Science, Al. Lotnikow 32/46,
02-668 Warsaw, Poland
Received 2 February 1998
Abstract. We have studied magnetothermal instabilities, giant flux jumps, both
theoretically and experimentally. Magnetostriction and magnetization hysteresis
loops with flux jumps were calculated over a wide range of experimental parameters
employing two critical-state models: the Kim–Anderson model and the exponential
model. The influence of the magnetic history on the flux jumps’ magnetostriction
and magnetization were investigated for the LaSrCuO single crystal. The shape of
the unstable region of the critical state in the temperature–magnetic field plane was
constructed from calculations and experimental results. In the Kim–Anderson
model H –T diagrams of flux instabilities agree well with experimental results on
HTSC single-crystal LaSrCuO and experimental results on LTSC niobium.
1. Introduction
When the magnetic field H is ramping, the appearance of
a small increase of the temperature leads to a decrease of
the value of the critical current density of the hard type
II superconductors. In adiabatic conditions this increases
dissipation and can lead to ‘thermomagnetic catastrophe’.
As a result, giant flux jumps appear and the sample goes
to the resistive or normal state. The investigation of
flux instabilities in HTSCs is interesting because of the
potential applications as well as the importance of gaining
a fundamental understanding of the phenomenon. We have
studied magnetothermal instabilities both theoretically and
experimentally in LaSrCuO.
There are a lot of papers devoted to giant flux jumps
attributed to the avalanche effect in the flux movement. For
our main aim, the build-up of diagrams of the instability
range (or flux jump range) of the critical state of type II
superconductors (SCs) in the temperature–magnetic field
plane and comparison of them with the experimental
data, three papers proved to be of importance [1–3].
In [1] Kim–Anderson model analytical expressions for
flux jumps in magnetization are given. Another paper
[2] is devoted to computation of field dependences of
magnetostriction of SCs, but without consideration of flux
jumps. The present paper on the basis of a combination
c 1998 IOP Publishing Ltd
0953-2048/98/101181+05$19.50 of these approaches to the problem gives the computation
of both magnetization M(H ) and magnetostriction 1L(H )
loops with flux jumps. We apply the same criterion of
instability as in [1]; however, for our calculations we
used numerical computation of the magnetic properties of
type II SCs. The algorithm used by us makes it possible
to derive these for any analytical dependences of critical
current versus magnetic field. In particular we select
two very widespread types of j (H ) dependences, the socalled Kim–Anderson model and the exponential model
(according to [2] the exponential model is most favourable
for describing the measured magnetostriction curves). The
analysis of the width of the region of the magnetic fields
where the critical state of SCs is unstable, carried out for
different temperatures, allowed us to derive calculated H –T
diagrams for flux jumps. In the Kim–Anderson model
the shape of the unstable region of the critical state in
the temperature–magnetic field plane agrees well with our
experimental results on HTSC single-crystal LaSrCuO and
experimental results [3] on LTSC niobium.
2. H –T diagram of flux instabilities: theory and
experiment
The influence of the magnetic history on the flux jumps
was investigated in the LaSrCuO crystal. Magnetostriction
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V V Chabanenko et al
and magnetization hysteresis loops with flux jumps were
calculated employing two critical-state models: the Kim–
Anderson model and the exponential model.
2.1. Critical state
Let us consider the pinning-induced magnetization and
magnetostriction loops with flux jumps for a specimen
having a slab geometry with thickness of L = 2d (−d ≤
x ≤ d). Suppose that the external magnetic field He is
applied parallel to the slab face. The magnetic field profile
H (x) inside the slab is given by the solutions of the critical
state equation
dB/dx = −µ0 J (x)
(1)
where current density |J (x)| ≤ |Jc (x)| and µ0 is the
permeability of vacuum.
This critical-state equation
can be used to determine the magnetization and the
magnetostriction when H Hc1 (for YBCO the lower
critical field H is of the order of several hundred gauss at
T = 0 while the flux jumps are observed at fields µ0 H of
the order of several tesla). According to the Kim–Anderson
model [4]
Jc (B) = ±J0 (T )B0 /(B0 + |B|).
(2)
Here B0 is a phenomenological parameter and J0 is the
critical current density at zero magnetic field B; in HTS
materials J0 (T ) ≈ J0 (0)(1 − T /Tc ). Integration of
equations (1) and (2) gives
B± (x) = B0 − [(B0 − B ∗ )2 ± 2µ0 J0 (x − x ∗ )B0 ]1/2
B(x) < 0
B± (x) = −B0 + [(B0 + B ∗ )2 ± 2µ0 J0 (x − x ∗ )B0 ]1/2
B(x) > 0.
(3)
The +(−) sign corresponds to domains of vortices where
dB/dx > 0 (dB/dx < 0) and the field B ∗ = B(x ∗ ). In
the exponential model [5] the critical current density is
expressed as follows:
Jc (B) = ±J0 exp(−B/B0 ).
(4)
The sign indicates the flow direction of the current.
Integration of equations (1) and (4) provides the local field
distribution:
B± (x) = B0 ln[exp(B ∗ /B0 ) ± µ0 J0 (x − x ∗ )/B0 ]
B(x) > 0
B± (x) = −B0 ln[exp(−B ∗ /B0 ) ± µ0 J0 (x − x ∗ )/B0 ]
B(x) < 0.
(5)
Here the +(−) sign again corresponds to the local field
distribution where dB/dx > 0 (dB/dx < 0) and the
boundary field B ∗ = B(x ∗ ). The full magnetic field profile
in the sample is given by the combination of solutions (3)
or (5). The parameters x ∗ and B ∗ are chosen so that the
condition µ0 He = B(x = 0) holds and the field B(x)
distribution is continuous. The corresponding formulae for
x ∗ in the Kim–Anderson model are given in [1]. For the
exponential model as well as for any form of j (H ) when the
number of jumps becomes very large the use of analytical
expressions for x ∗ is not very sensible.
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Figure 1. Magnetization and magnetostriction loops with
flux jumps; calculation in the Kim–Anderson model.
Comparison of the solutions (3) and (5) shows that
they are close to each other if B ∗ /B0 1. From the
four solutions (3) and (5), all the necessary magnetic field
profiles can be constructed [1, 2]. When the value of
the magnetic field B(x) is known, the magnetization M
and magnetostriction 1L/L0 can be calculated by simple
integration:
Z
d
1L(B)/L0 = −1/(µ0 EL0 )
[Be2 − B 2 (x)] dx
(6)
0
Z
M(B) = 1/(µ0 d)
d
B(x) dx − He .
(7)
0
Here E is the elastic constant of the material along the x
axis and L = 2d.
Flux instabilities in hard superconductors
Figure 3. Magnetization loops (calculation) which illustrate
the influence of the maximal temperature T ∗ reached by
the sample during flux jump; T1∗ > T2∗ .
|He |). Because of the moving of flux lines, the magnetic
field, B(x), in the sample increases by 1B(x). The 1B(x)
value has to be calculated from equation (1) in following
way. Let xi be the points at which the B(xi ) = Bi values
are calculated with the index l ≤ i ≤ n, xl = 0, xn = d and
0
− Bi ,
B 0 (x) as the initial local field. We put 1B = Bi+1
where i > 1, Bi = B(x = 0) = µ0 He . The critical-state
equation (1) gives the conditions:

|1B| < µ0 Jc (Bi )

 B(xi ) + 1B
(8)
B(xi+1 ) = B(xi ) + sign(1B)µ0 Jc (Bi ) dx


|1B| > µ0 Jc (Bi )
Figure 2. H –T ranges of the flux instability: (a ) experiment
on LaSrCuO; (b ) calculation (d = 5 mm, jc = 109 A m−2 ,
B0 = 2 T; Bp , full penetration field); (c ) experiment [8] on
Nb.
where dx = xi+1 − xi .
Because of exchange with the magnetic field, B(x), the
energy per unit volume Q is dissipated in the vicinity of x
where
Z d
[B(x 0 ) − B 0 (x 0 )] dx 0 .
(9)
Q(x) = Jc (x)
x
Under local adiabatic conditions one obtains
1T (x) = Q(x)/CV
2.2. Algorithm for numerical calculation of
magnetization and magnetostriction loops with flux
jumps, instability criterion
The instability criterion (under adiabatic conditions) in
the case of the Kim–Anderson type of critical current
density Jc (B) can be derived analytically [1]. For the
exponential model (4) this is not possible and we have
used in this case the calculations based on the adiabatic
flux–B jump instability criterion. Let the applied field
He increase (decrease) by a small amount 1He (|1He | (10)
where CV is the specific heat of the superconductor. For
YBCO [6], CV (1–26 K) (J cm−3 ) = 2.68 × 10−4 T −2 +
1.01 × 10−4 T + 2.53 × 10−6 T 3 + 7.17 × 10−9 T 5 and CV
(26–100 K) (J cm−3 ) = 2.02 × 10−4 T 2 − 7.82 × 10−9 T 4 .
The increase of temperature 1T reduces the critical current
Jc :
1Jc (x) = (∂Jc /∂T )1T (x).
(11)
According to the critical-state equations (8), the reduction
in Jc leads to a decrease in the shielding ability of the
1183
V V Chabanenko et al
conditions of the sample. The critical state conditions (8)
for type II SCs allow one to calculate magnetic field
profiles in a slab and for any totality of the flux jumps,
independently of the magnetic history.
2.3. Results of calculation and comparison with
experiment
Figure 4. Magnetostriction loops for different temperatures,
LaSrCuO.
superconductor by 1HS , where
Z
d
1HS = −
1Jc (x 0 ) dx 0 .
(12)
x
The magnetic configuration is unstable with respect to a
flux jump if 1HS > He [7, 8].
During a flux jump, the temperature inside the sample
increases and reaches a maximum value T ∗ < Tc , which
is assumed to be independent of the position x inside the
sample. The critical current density at T ∗ is Jc (T ∗ ) <
Jc (T0 ), where T0 is the temperature of the sample before
the flux jump. Therefore, according to the critical-state
conditions, the magnetic field configuration is given by the
solution of the equations (8) for Jc = Jc (T ∗ ). After the
flux jump the temperature inside the sample is returned
to the starting value T0 , but the magnetic field profiles
in the slab are unchanged, i.e. they are the same as for
T = T ∗ . An increase (decrease) of the external field, He ,
leads to the new magnetic field distributions. It is assumed
that the maximum temperature, T ∗ , reached during a jump,
which is less than Tc , is the same for all jumps. Generally
speaking the T ∗ value is unknown because it depends on
microstructure of the sample, sweep rate and the adiabatic
1184
These results and data [1, 2] are used for computer
simulations. Magnetostriction and magnetization loops are
constructed over a wide range of experimental parameters.
In figure 1, one can see an example of theoretically
calculated magnetostriction and magnetization hysteresis
loops with magnetostriction and magnetization jumps on
the basis of the Kim–Anderson model. 1Hfj is the flux
instability range of the magnetic field. Similar calculations
were performed for different temperatures and were used
for the construction of the H –T phase diagram of flux
instability (figure 2(b)).
The calculations done make it possible to follow the
change in the pattern of flux jumps (figure 3) when varying
the maximal temperature T ∗ reached by the sample during
a jump (or when varying the final depth of flux penetration
into the sample). During the experiment a similar result
can be, for example, reached by changing the velocity of
heat removal from sample surface. However, it is rather
difficult to control this parameter in the experiment.
Experiments reported here have been performed on
an La1.85 Sr0.15 CuO4 single crystal with the fields oriented
along the c-axis. The sample had dimensions of 2.15 ×
2.24 × 5.42 mm3 , with the c-axis the shortest test axis. The
change of length 1L of the sample was measured in the
a–b-plane along the longest dimension, using a strain gauge
technique. The resistance change of strain gauges was
measured by an AC bridge. The magnetic field was swept
at a constant cyclic rate of 0.5 T min−1 after the sample was
zero-field cooled to the measuring temperature. An onset
temperature for superconductivity of 35.5 K was observed
in a field of 10 G. Figure 4 shows the manifestation of
flux jump instabilities in the virgin magnetization curve for
LaSrCuO single crystal. Hirr is the field where M becomes
reversible.
As can be seen from figures 2(a) and 2(c), the overall
pictures of the H –T ranges of flux instability are similar
in shape for Nb [3] and LaSrCuO. One can also see
a good qualitative agreement between the experimental
and the calculated phase diagrams. This shows that the
Kim–Anderson model is applicable at large distances away
from the irreversibility line for HTSC materials whereas
the exponential model is applicable in the vicinity of the
irreversibility line.
As is clear from both calculation and experiment, the
first and third quadrants have significantly fewer flux jumps
than the second and fourth quadrants. For example, for
the T = 4 K curve (figure 1(a)) and experiment [9],
flux jumps are completely absent in the first and third
quadrants. According to our calculation, the difference
in the frequency of occurrence of the flux jumps in the
different quadrants is related to the shape of the flux profiles
in the various quadrants.
Flux instabilities in hard superconductors
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