MODELING CURRENT PERCOLATION IN GRANULAR SUPERCONDUCTORS

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MODELING CURRENT PERCOLATION IN GRANULAR
SUPERCONDUCTORS Noel Rutter Amit Goyal Oak
Ridge National Laboratory J.H. Durrell, J.L.
Driscoll J.E. Evetts University Of Cambridge
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Aims of modeling coated conductors

• How do the reduced critical currents of grain
  boundaries affect the overall Jc of the tape ?
• What would be the effect of the imperfect
  crystallographic orientations on conductor
  parameters such as
• Grain size
• Conductor dimensions
• Grain shape
• Quality of crystallographic texture
• When is intra-granular dissipation important ?
• How do the properties change in magnetic field ?

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The square and hexagonal grain models

• A 2-dimensional array of identical grains
• Square model
• Only 4 neighbors per grain
• Shortest boundary path across tape is equal to
  tape width
• Hexagonal model
• 6 neighbors
• 2 possible orientations of hexagons

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Using the Potts model for grain growth
The structure is initially represented by an
array of pixels each having a unique label
A pixel is selected randomly and may be assigned
the index (with probability P) of one of its
neighbors depending on its energy within the
structure.
Consider the shaded central grain. Its energy is
defined as the number of its neighbors in
different grains
Change to A No change (DG 0) Change to B
(DG -1) Change to C (DG 2)
If DG is zero or negative the change happens, if
it is positive, it will happen with probability P
given by
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Grain structures generated by the Potts Model
When the number of pixels considered is equal to
the total number of pixels in the array, one
Monte-Carlo Step (MCS) of grain growth has
occurred.
The figure below shows how the grains grow as MCS
increases
MCS 10
MCS 100
MCS 1000
As in a real structure, the grains have a range
of sizes
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Grain Shape and Size

• The conductor is modeled as a 2-D array of grains
  which are continuous throughout the film
  thickness.

• Each grain is made up of 1 or more pixels (p).

p1

• These grain structures both represent a tape with
  Nl25 and Nw10.

• When p1, the structure is identical to that used
  in early work such as Nakamura et al.

p5

• For p5, a more realistic structure is generated.

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Grain misorientation

• Bulk X-Ray data represents the global
  distribution of grain orientations
• Such curves may be fitted by a Gaussian function

• The Grain Boundary Misorientation Distribution is
  the most important factor
• A single misorientation (i.e in-plane only)
  predicts a different result for the grain
  boundary misorientation distribution

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Grain Orientations

• Each grain has a crystallographic orientation
  close to 100lt001gt.

• The misorientations are based on X-Ray
  measurements of global texture, well simulated by
  Gaussian curves.

FWHM10
In-plane orientation

• Individual grains are assigned an orientation
  based on the gaussian(s), assuming no
  correlations between neighbors.

10
0
-10

• The misorientation angles of the pixel boundaries
  are calculated and a grain boundary map generated.

G.B. mis- orientation
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The jc vs q relationship
In general jc(q)jc(0) exp -((q-b)/a) for
q gt b jc(q)jc(0) for q lt b

• The slope (a) has been measured by various
  authors to lie between 2.4 and 4.4
• The value of b probably depends on intra-granular
  dissipation. Verebelyi et al. have proposed that
  twin boundaries within grains will cause them to
  have similar properties to 1.8? grain boundaries

Values used in previous modeling work are
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Calculating the critical current

• Each pixel boundary is then assigned a critical
  current, based on the misorientation angle (q)

• The maximum flow through the pixel network (Ic)
  is equal to the value of the minimum cut across
  it.

• A limiting path is found. Physically, this
  corresponds to the primary flux flow channel and
  is the region in which dissipation is initiated.

• The total critical current of the sample is
  calculated by summing the individual critical
  currents of the pixel edges comprising the
  limiting path.

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Summary of the improved percolation model

• Advantages of the new model
• The grain structure is more realistic, having
  grains of various shapes and sizes
• The algorithm used executes the Ic calculation
  exactly
• There are pixel boundaries which fall within the
  grains - these can be attributed values which
  represent the intra-granular jc.
• The model is easily adaptable to consider
  different grain structures (i.e. aspected grains)
• In theory it may be scaled indefinitely, though
  computing resources limit the size of tapes
  considered so far.

• Remaining limitations
• Boundaries are still pixelated, hence their total
  lengths are overestimated
• It is still unclear whether neighboring grains
  have correlated orientations

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The effect of conductor dimensions
The graph shows how for a given number of grains
across the width, log (Jc) varies linearly with
log (1/Nl) and the slope increases as the width
decreases.
A reasonable approximation of this relationship is
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The effect of conductor dimensions
Plotting
1/Nw
Jc vs
(1/Nl)
The approximation is most accurate when Nl gtgt Nw
and both Nw and Nl are large. Fortunately this is
the case for most practical conductors that one
might consider.
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The effect of conductor dimensions

• It is possible to extrapolate the results to
  tapes with very long length (i.e. 1 km).
• The average grain size in RABiTS samples is
  around 30 mm, hence a tape 1 km long has
  Nl?3x107.
• The figure on the right shows that the drop in Jc
  due to percolation around badly aligned grains is
  not so bad if Nw is greater than around 100.

• In order to obtain high critical current in
  coated conductors, one should
• Produce a well textured superconducting layer
• Ensure that the grain size is very small compared
  with the conductor dimensions
• Make the tape as wide as possible to avoid a
  significant dependence of Jc on the conductor
  length

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Effect of finite intra-granular critical currents

• The plot shows how Jc varies with the in-plane
  FWHM if the grains are equivalent to 0, 1.8 or
  4 grain boundaries. The dotted line represents a
  model in which the grains are not considered.

• If e4 for example, the texture of the tape has
  little effect on Jc until the FWHM exceeds
  approximately 7 .

• The dotted line indicates a limitation of
  previous models. For well aligned samples, Jc
  exceeds jc(0). This is because the length of the
  limiting path is greater than the sample width by
  about 30.

e0
e1.8
e4
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Effect of finite intra-granular critical currents

• Texture (represented by the FWHM) and grain
  quality (represented by e) are both important
  factors to consider.
• The best route to improve the overall Jc of a
  process depends upon where on the 2-D surface the
  tapes lie.
• If the grains limit Jc, there is little gained
  from improving the grain boundary jc (by either
  reducing the misalignments or Ca doping).

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Effect of finite intra-granular critical currents

• It is possible to construct a schematic phase
  diagram for RABiTS tapes at 77 K in self-field
  based on previous figures.

• IG region grains have low jc and sample is well
  textured. Dissipation is entirely within the
  grains.

• GB region grains have high jc but are less well
  textured. Dissipation is entirely at the grain
  boundaries.

• Mixed region A combination of intra- and
  inter-granular dissipation occurs.

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Effect of c-axis field on grain boundaries

• The properties of YBCO grain boundaries on
  bicrystal substrates in magnetic field

0 T
3 T
5 T

• At high magnetic fields, the grains have lower jc
  than the low-angle boundaries.

Verebelyi et al, APL 76 13 (2000)
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Effect of c-axis field on grain boundaries

• The shape of the modeled curve is similar to that
  measured on real tapes

• At around 6 T, there is a crossover from grain
  boundary dissipation to grain-dominated jc.

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Effect of c-axis field on coated conductors
Fernández et al, PRB 67 (2003) 052503
Granular dissipation is favoured if The grains
have low pinning Texture is good Field and
temperature are high
When granular dissipation is dominant, there is
less dependence on sample dimensions. Also, the
tape uniformity should improve.
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Effect of a large applied in-plane field
J. Durrell et al. Phys. Rev. Lett. 90 24 247006
(2003)

• Jc is determined by intra-granular dissipation
  unless the grain boundary and the magnetic field
  are aligned within an angle fk

fk
B GB
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Effect of an in-plane field on coated conductors
9
6 10
Track width 400 mm 10 grains
3 T
1 T
2.5 T
0.75 T
2 T
0.5 T
9
1.5 T
0.25 T
5 10
)
9
4 10
2
(A/m
c
J
9
3 10
9
2 10
T80 K
9
1 10
0
90
180
f
(degrees)
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Effect of an in-plane field on coated conductors
Track width 400 mm 10 grains
2
0)
f
(
c
/J
1
c
J
B1T
T80 K
0
0
90
180
f
(degrees)

• Preliminary measurements on coated conductors
  show a less significant force-free effect than
  single crystals. This may be due to non-uniform
  current flow (i.e. percolation).

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Effect of texture on the in-grain critical current

• If Jc is determined by in-grain dissipation, does
  the crystallographic texture still have an effect
  ?
• Yes, because out-of-plane tilts reduce the
  in-grain critical current.

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Modeling the effect of vicinality

• A FWHM spread of misalignments of 10 about the
  TD causes a 50 reduction in Jc.
• The FWHM of the rocking curve about the RD has
  almost no effect.
• In-plane rotations of grains have no effect.

FWHM (RD) 6?

• Whilst the reduction of Jc due to the vicinal
  effect is important, it is not as severe as the
  effect of grain boundaries.
• The FWHM about the TD is the important parameter.
  In RABiTS tapes, the grains are better aligned in
  this direction due to the substrate rolling
  process.

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Conclusions

• A new current percolation model has been
  developed incorporating a more realistic grain
  structure and the effects of intra-granular
  dissipation.
• The model produces similar results to previous
  models in describing the effects of conductor
  dimensions, grain size and texture.
• Consideration of intra-granular dissipation
  indicates that present RABiTS tapes in self field
  at 77K are likely in a regime of mixed IG and GB
  dissipation, and hence that granular pinning
  needs to be increased also.
• In high-field applications, granular dissipation
  becomes more important. For in-plane fields,
  coated conductors are more isotropic than single
  crystals.
• The in-grain Jc can be increased both by
  increasing the amount of pinning and by keeping
  the misorientation of grains about the transverse
  direction of the tape to a minimum.