Physica C 369 (2002) 82–86
www.elsevier.com/locate/physc
The structure of vortex matter avalanches in a niobium plate
V.V. Chabanenko a,*, V.F. Rusakov b, S. Piechota c, A. Nabialek c,
S. Vasiliev c, H. Szymczak c
a
Physical and Technical Institute, National Academy of Sciences, Ulitsa R. Luxembourg, 72 83114 Donetsk, Ukraine
b
National University, 83055 Donetsk, Ukraine
c
Institute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland
Abstract
We observed a threshold for enter of the huge flux avalanches in the shielding experiments (in other words, for the
increasing of the screening properties of superconductor before flux jump). We also found a threshold for exit of residual flux in the trap experiments. In any case, the value of threshold achieved up to 15% of the full volume of the flux
jumps. The results of our experiment allow to analyse quantitatively inertial properties of the vortex matter using the
balance of energy. Ó 2001 Published by Elsevier Science B.V.
PACS: 74.60 Ge
Keywords: Mixed state; Flux lattice; Lattice dynamics
1. Introduction
Our experiment identifies avalanches as nonlocal events propagating rapidly over the distance
comparable with the size of the sample. Earlier
thermal effects in massive rods of niobium as a
result of giant flux jump were observed [1]. Some
observations of the behaviour and structure of flux
jumps were presented in [2]. While recording the
voltage pulses produced by the magnetothermal
instability, we observed a threshold for enter of the
huge flux avalanches in the shielding experiments
(in other words, for the increasing of the screening
properties of superconductor before flux jump) [3].
We also found a threshold for exit of residual flux
in the trap experiments. In one of the last reports
[4] in the mixed state authors revealed ‘‘negative
vortices’’, whose penetration leads to the expulsion
of magnetic field. The existence of the above
mentioned thresholds has been established experimentally by recording the voltage pulses produced
by abrupt changes of the flux density resulting in a
huge flux avalanche in the superconductor sample
during a slow ramp of the magnetic field. The
details of the flux jump structure were discussed.
We made an attempt to estimate the value of the
vortex matter mass using our results of experimental investigation.
2. Experimental
*
Corresponding author. Fax: +380-622-521-074.
E-mail address: chaban@host.dipt.donetsk.ua (V.V. Chabanenko).
With the aid of the Hall probe we monitored
the dynamics of catastrophic avalanches of the
0921-4534/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V.
PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 2 2 4 - 2
V.V. Chabanenko et al. / Physica C 369 (2002) 82–86
magnetic flux in superconducting Nb plate during
a slow sweep of an external magnetic field. The
probe was placed in the centre of the sample (Fig.
1a) and it measured the surface induction Bsurf ¼
l0 H þ M, where H is the external magnetic field,
M is the magnetization supported by the circulating supercurrent and l0 is the permeability of
vacuum. A sensitive crossover area of the sensor
chip (n-InSb thin film layers doped with Sn) was as
low as 20 50 lm2 . The high electron mobility
(104 cm2 /V s) gives the high capability at the
frequencies <109 Hz. Consequently, in our experiment a flux avalanches are detected in real time
(treal < 106 s) without distortion. In our experiment we examine the Hall sensor voltage simul-
83
taneously by the following two ways: directly in
the transient recorder (model TCC-1000, Riken
Denshi Co., Ltd.) with memory (each jump was
registered by 1020 dots) and after amplifying its
voltage with the aid of a Keithley 182 voltmeter. A
semiconductor thermometer attached to the sample monitored its temperature. In the shielding
experiments, zero field cooling mode (ZFC) was
used. The field trapping mode is realized by an
increase of an external magnetic field above the
second critical magnetic field Hc2 at the temperature of experiment. The data presented in this
study come from polycrystalline Nb sample. The
dimensions of the sample are following (see Fig.
1a): L ¼ 11 mm, l ¼ 3:5 mm, and 2R ¼ 5 mm.
3. Results
Fig. 1. (a) Geometry used for experimental observation of the
structure of magnetic instabilities; (b) the hysteretic magnetization loop of Nb sample. Insert: Hall probe voltage vs. external magnetic field Hext .
Fig. 2 shows the temperature and temporal
evolution of the surface magnetic induction Bsurf in
shielding and trapping modes. The structure of the
jump principally differs from a simple step. It has
been shown experimentally (see Fig. 2a–c) that the
flux avalanche is preceded by certain phenomenon
under which the field on superconductor surface
decreases prior to an abrupt increase resulting
from the development of the instability. The value
of negative peak riches even 16% of the total value
of flux jump. Energetically, this is a significant
value enabling us to speak of a potential barrier,
which prevents flux entering. The duration of
negative peak is of 0:2–12 ms. Now let us consider
peculiarities of the structure of the thermomagnetic avalanche for the case of magnetic flux
trapping. In this mode (Fig. 2e,f), a positive peak is
observed that precedes the development of the
instability, similar as in the case of shielding (Fig.
2a–c). The magnitude of this peak is approximately 5–6% of the main jump with 0:2–0:3 ms
duration of the process. After the development of
instability, in the mode of flux trapping (Fig. 2d–f)
no peculiarities are observed since the state of the
sample becomes normal.
The above-described peculiarities of the thermomagnetic instability process are not sensitive to
the external magnetic field sweep rate dH/dt in the
range 0.05–2 T/s (Fig. 3).
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V.V. Chabanenko et al. / Physica C 369 (2002) 82–86
Fig. 2. Successive investigation of the avalanche jumps structure for (a–c)––shielding and (c, d)––trapping regimes for Nb. Inserts show the local magnetic induction on the surface of the sample as function of increasing and decreasing external magnetic field;
T ¼ 4:5 K.
Until now there is no precise explanation of the
above mentioned phenomena. However, these peculiarities in the flux jump structure resemble the
case of Faraday’s and Lenz’s law of electromagnetic
induction: the direction of an induced emf is always
such that the resulting magnetic flux of the induced
current will oppose the change in flux, giving rise
to the emf. Hence, such phenomenon could be
explained by the Le Chatelier’s thermodynamic
principle, whose particular case is the Faraday’s
law. This principle, however, is not valid in the case
of a mixed state of the hard superconductor, because this system is metastable. Hence, the observed
phenomenon needs a special theoretical approach.
On the basis of the experiment with thermomagnetic instability we made an attempt to estimate the mass of the vortex matter. Estimations of
an effective mass of the Abrikosov vortex were
performed earlier on the basis of microwave experiments [5]. There is a critical frequency above
which the impedance for subcritical currents becomes the impedance of the ‘‘ideal’’ (without pinning) vortex matter. As a result the microwave
field causes energy dissipation putting the vortex
lattice into motion. Experimental data on microwave field attenuation contain information on inertial properties of a vortex. The effective mass per
flux tube were determined in this manner: it is
V.V. Chabanenko et al. / Physica C 369 (2002) 82–86
85
t2 dq mt t2 dn
¼
:
ð1Þ
2
2
Here dq ¼ mt dn, dn is a change in concentration of
vortices, mt is a mass of unit length of the vortex.
In Eq. (1) we neglect the contribution connected
with the change of vortex velocity, because this
velocity change gives contribution of higher order
of smallness.
The work done by forces of internal friction
under infinitely small displacement of vortex, we
estimate to be
dEc ¼
dA0 ¼ gt dx;
Fig. 3. The structure of flux jump in the trapping mode for
different rate of the magnetic field ramping; T ¼ 4:5 K.
where g is coefficient of viscosity per unit length of
vortex. In Eq. (2) we neglect the contribution
connected with change in velocity, as well. Hence,
the density of dissipating energy associated with
the increment of vortices density may be represented as
dA ¼ dA0 dn ¼ gtdx dn:
8
ð2Þ
ð3Þ
20
approximately 10 electron masses/cm (10
kg/
m) of flux tube.
In our experiments pinning force disappears as
a result of thermomagnetic instability and this
fact enables the Lorentz force to accelerate the
vortex matter. Parameters of vortex matter motion during a flux jump could be analyzed by investigation of a magnetic induction jump. This
was done in our experiment and used for estimation of the inertial properties of the vortex
matter.
On the basis of the jumps’ structure found in
our experiment we made an attempt to estimate
the mass of the vortex matter. To do this we will
use the balance of energy. We will equate the increment of kinetic energy of a vortex to the work
of all forces acting on it. Let us consider a vortex
of the mixed state from the screening layer of superconductors. Vortex moves with velocity v under
Lorentz force FL as a result of thermomagnetic
avalanche. We suppose that Lorentz force is much
greater than pinning force. Consequently, a contribution to the Lorentz force reaction comes only
from the flux flow resistance, which is characterized by a viscosity.
An increment of kinetic energy of unit volume
(kinetic energy density) is
Lorentz force acting on vortex is the main cause
of its motion. The work done by Lorentz force
may be estimated by the same way it has been
done for force of viscous friction
dA ¼ fL dx dn:
ð4Þ
Using for Lorentz force expression fL ¼ jU0
(where j is the screening current density) and
combination of Eqs. (1), (3) and (4), it can be
easily obtained the energy balance:
mt dnt2
¼ jU0 dx dn gtdn dx:
2
For mass of unit length of vortex we get
mt ¼ 2j
U0
dx
dx 2g ;
t
t2
ð5Þ
assuming dx ¼ vDt, we can obtain
mt ¼ 2j
U0
Dt 2gDt:
t
ð6Þ
Here Dt is duration of flux jump. Its value, as well
as the flux velocity, can be estimated using either
diffusion coefficient of magnetic flux or experimental flux jump data.
We estimate this time from our experimental
dependence B(t) (Fig. 2). The plot shows that
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V.V. Chabanenko et al. / Physica C 369 (2002) 82–86
Dt 5 104 s. Average velocity is found as
t ¼ S=Dt, where S is the path traveled by a vortex
during jump time. As the path S we assume d/3,
where d is characteristic dimension of the sample.
In this assumption we take into account, that field
has already partially penetrate into sample by time
of evolution of flux jump, i.e. vortex has already
been in some distance from boundary of sample,
and in some distance from of vortex structures
collide with one another in a center of sample.
The coefficient of viscosity we estimate according to Ref. [6]: g ¼ 1:5 108 NS=m2 .
Using the expression for velocity, Eq. (6) may
be rewritten in the form
of the magnetic flux had passed. More exact
evaluation for vortex mass may be obtained in the
quantitative theory of the thermomagnetic instability where the inertial properties of the vortex
matter would be taken into account.
Acknowledgements
We are grateful to J. Kolacek and V. Vinokur
for helpful discussions and hints. The European
ESF program VORTEX is also acknowledged.
The Polish Government Agency KBN under contract no. 8 T11B 038 17 supported this work.
2
mt ¼ 6
jU0 ðDtÞ
2gDt:
d
ð7Þ
Assuming that the transport current density is
equal to the critical one for our sample (is from
magnetic measurement it results that jc ¼ 3:8 107 A/m2 , Fig. 1b), one can easily obtain for inertial mass of unit length of vortex the value mt 4 1012 kg/m.
It is clear that our evaluation for the vortex
mass is overestimated. Indeed, we have used the
value of the viscosity coefficient corresponding to
the state of the superconductor after the avalanche
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