Theory of thermomagnetic instability in superconductors

1
Theory of thermomagnetic instability in
superconductors D.V. Denisov1, D.V. Shantsev1,
Y.M. Galperin1,2, Eun-Mi Choi3, Hyun-Sook Lee3,
Sung-Ik Lee3, A.V. Bobyl1,2, P.E. Goa1, A.A.F.
Olsen1 and T.H. Johansen1 1Department of Physics
and Center for Advanced Materials and
Nanotechnology, University of Oslo, P. O. Box
1048 Blindern, 0316 Oslo, Norway 2A.F. Ioffe
Physico-Technical Institute, Polytekhnicheskaya
26, St. Petersburg 194021, Russia
3National Creative Research Initiative Center
for Superconductivity, Department of Physics,
Pohang University of Science and Technology,
Pohang 790-784, Korea e-mail t.h.johansen_at_fys.ui
o.no, homepage http//www.fys.uio.no/super/
dend
Motivation
Goal

• Features of the model
• In the our theoretical study we restrict
  ourselves to a conventional linear analysis of
  the instability and consider the spatio-temporal
  development of small perturbations in the
  electric field, E, and temperature, T.
• The distributions of the current density, j, and
  magnetic induction, B, in the flux penetrated
  region 0ltxltl are determined by the Maxwell
  equation curl Hj
• To find the electric field distribution another
  Maxwell equation and the equation for thermal
  diffusion is used
• curl E-?B/?t ,
• Any B-dependence of jc is neglected, i. e., the
  Bean model is adopted. But E is proportional to
  jn.
• The heat transfer from the superconductor to a
  substrate are taken into account. Consequently,
  the results depend significantly on the heat
  transfer rate, h0, as well as on the film
  thickness, d (dltltw).
• The stationary current and field distributions in
  a thin strip under such conditions is well known
  and the flux penetration depth, l, is related to
  the applied field by the expression
• Within this model, the threshold flux penetration
  depth, l, when the superconducting strip first
  becomes unstable, is given by formula
• Several temperature dependencies were taken into
  account
• ??0(T/Tc)3, h0h0 (T/Tc)3, jcjc0(1-T/Tc),
  U1-T/Tc

The thermomagnetic instability or flux jumping is
commonly observed at low temperatures in type-II
superconductors with strong pinning. These
instabilities can be observed magneto-optically
and often they lead to a highly-non-uniform,
dendritic flux patterns. However such flux
jumping leads to a suppression of transportation
current and in some cases can jeopardize device
itself.
The goal of this work was to develop a model
which can predicts the thermomagnetic
instabilities in various superconducting samples.
This goal was achieved and the model was
published in PRB 73, 014512 (2006).
A superconductor strip on a substrate. The dark
gray area is the flux-penetrated region.
Results
Results
Theoretical stability diagram predicting the
threshold temperature Tth for different film
width. The curve is plotted for parameter values
corresponding to MgB2 films. If the sample width
is smaller then 82µm then no instability will
occur for temperatures higher 4K. From the other
hand for samples wider than several millimeters
the threshold temperature comes to saturation
around its maximum value 10K (for MgB2).
Temperature dependence of the threshold magnetic
field. Experimental data obtained for the 5 mm
wide MgB2 sample and for a 1.8 mm wide Nb film
are plotted as ? and ?, respectively. The full
lines are theoretical fits. The dashed lines show
the limiting temperature above which the
instability vanishes.
There can be two types of instabilities Uniform
Instability (one big flux jump) and fingering
instability (dendrites). Here stability diagram
is shown in the electric field - heat transfer
coefficient plane. The curved line marks the
critical electric field Ec(h0) that separates
fingering (EgtEc) and uniform (EltEc)
instabilities.
Summary

• Dendritic instability occurs much easier in thin
  films than in bulk samples. The corresponding
  threshold fields are proportional to the film
  thickness, d.
• For MgB2 temperature range of the instability
  increases monotonously up to 10K with the strip
  width.
• Instability is suppressed for sufficiently
  narrow strips, is of particular importance for
  design of superconducting electronic devices or
  other applications making use of thin film
  superconductors operating at temperatures below
  the instability threshold value.

Stability diagram in the H-E plane. If heat
transfer coefficient is higher than hcrit then
only dendritic instability can arise in
superconducting sample.
Threshold magnetic field for onset of the
dendritic instability in MgB2 strips of different
width (symbols) plotted together with a fitted
theoretical curve (full line), which diverges at
a finite w indicated by the dashed
asymptote. For the wider samples instability
occurs more easily!

• For more information see
• D.V. Denisov et al., Phys. Rev. Lett. 97,
  077002 (2006).
• D.V. Denisov, A.L. Rakhmanov, D.V. Shantsev,
  Y.M. Galperin and T.H. Johansen. PRB 73, 014512
  (2006).
• A.L.Rakhmanov, D.V.Shantsev, Y.M.Galperin,
  T.H.Johansen
• Phys. Rev. B 70, 224502 (2004)