Current Percolation and the V-I Transition in YBa2Cu3O7 Bicrystals and Granular
Coated Conductors
J. E. Evetts, M. J. Hogg, B. A. Glowacki, N. A. Rutter and V. N. Tsaneva
Department of Materials Science and IRC in Superconductivity, University of Cambridge, Cambridge CB2 3QZ UK
Abstract—There is considerable interest in the dynamics of
vortices in granular ‘coated conductors’ consisting of a 2-D
network of low angle grain boundaries (LAGB). The V-I
characteristic of the conductor is determined by a combination
of flux vortex channelling along the grain boundaries and
current percolation within the grain network. In this work it is
shown that measurements of viscous flow for a YBa2Cu3O7
bicrystal LAGB can be applied in a statistical model that
predicts the characteristic V-I response for a particular grainto-grain dispersion of grain boundary angles.
I. INTRODUCTION
Prototype biaxially textured granular YBa2Cu3O7 coated
conductor tapes are now available with critical current
6
-2
densities, Jc (77 K , 0 T ), of 1-2.10 Acm [1]. The critical
current is limited by a 2-D network of low angle grain
boundaries (LAGB) with typical grain boundary dispersion
determined by electron back scattered diffraction (EBSD) in
the range 1° to 12°. Since Jc of a LAGB decreases
approximately linearly with boundary angle up to 12° current
transport in these conductors is a complex percolation process
that depends on grain size (typically 30-60 µ m) and
distribution as well as grain to grain misalignment. It is of
importance for conductor design and specification to be able
to predict the dynamics of vortices in these materials and the
resultant V-I characteristic [2]. A recent study of the angular
variation of Jc(B) for an isolated LAGB in a bicrystal thin
film suggests that the critical pinning force for vortices at a
0.5
LAGB is one dimensional, (i.e. JcB~B ), this suggests that a
single row of vortices mediates pinning and flux flow [3].
This is the starting point for an analysis of flux flow
measured for an isolated bicrystal LAGB, which then
provides a basis for the treatment of vortex percolation and
the V-I response of a general 2-D LAGB distribution.
Fig. 1. Voltage–current transition for an isolated 4° grain boundary (GB) and
for intragrain (IG) YBa2Cu3O7.
the relation, ρf = (B/Bc2)ρn , where ρn is the normal state
resistivity, corresponds closely to values for Bc2 for
YBa2Cu3O7 in the literature [4]. The E-J characteristic is
linear because the effective sample gauge length is just one
vortex spacing. As a consequence the local electric field E
-1
can be as high as 3 Vcm for a voltage drop of a few µV
between voltage contacts, depending on the applied field B.
In the model presented below the E-J characteristic is
assumed to be linear with slope equal to the intrinsic ρf for
YBa2Cu3O7, although there are deviations at very low E,
where flux creep is dominant, and at high B where there is
some evidence that multiple rows of vortices move along a
LAGB [3].
III. A SIMPLE MODEL FOR CURRENT AND FLUX PERCOLATION
A granular ‘coated conductor’ is represented by a 2-D
network of LAGB (Fig. 2). The dispersion of LAGB graingrain misalignment angles may either be defined by a suitable
statistical distribution function or taken directly from EBSD
measurements on actual coated conductor samples. The
Support is acknowledged from the Engineering and Physical Sciences
Research Council and the European Community under TMR Network No.
CT98-0189 SUPERCURRENT.
FL
vf
e
d
b
II. E-J FOR AN ISOLATED LAGB
We have measured the V-I response for an isolated LAGB
by patterning critical current tracks across a YBa2Cu3O7 thin
film deposited on a bicrystal SrTiO3 substrate [4]. Except at
very low voltages the E-J characteristic is linear (Fig. 1),
indicating clearly that flux flow is dominated by viscous
dissipation, with a slope determined by the intrinsic flux flow
resistivity ρf. A similar characteristic has been obtained for a
number of bicrystal LAGB samples. The corresponding flux
flow resistivity increases linearly with applied field except at
high applied fields (B > 6 T). If it is assumed that flux flow
occurs by the motion of a single line of vortices along the
LAGB the value of the upper critical field Bc2 calculated from
(1,7)
J
B
(1,3)
50 µm
a
c
Fig. 2. Schematic of a tape conductor, the 2-D distribution of LAGB is taken
from the EBSD image (with misorientation information) for a biaxially
textured coated conductor. Vortex flow channels are made up from LAGB
segments with differing Jc values. The conductor is of width W and thickness
t and the applied field B is normal to the conductor plane. The indices (k,u)
define the LAGB segments. For channel a→b k=1, u=1-10. Segments (1,7)
and (1,3) are indicated. We assume that self-field effects are negligible.
model depends on two coupled ‘percolation’ phenomena, (1)
the formation of vortex flow channels spanning the sample
along a connected set of LAGB segments (‘vortex
percolation’), and (2) the distortion of the current distribution
within the sample section so that all the LAGB segments on a
vortex flow channel enter the flow state simultaneously and
maintain a constant voltage drop ∆Vk across the kth vortex
flow channel (‘current percolation’). The conservation of
moving vortices within a channel defines ∆Vk through the
relation for the electric field E=-vf xB. In Fig. 2 simple vortex
flow channels of the type a→b will form when the critical
current is first exceeded; at much higher currents the flow
channels will become more complex as indicated by the
divided channel c→d,e.
A. Relation of dV/dI to Number of active Channels
The invariance of Ek and ρf for all LAGB segments
comprising the kth vortex channel greatly simplifies the
expression for the voltage drop ∆Vk. Fig, 3. shows schematic
E-J characteristics for a set of LAGB segments.
The current density in the uth LAGB segment of the kth
channel is J(k,u) = Jc(k,u) + ∆J(k) where ∆J(k) = Ek/ρf. If the
uth segment has length w(k,u) and thickness t the critical
current of the LAGB segment is Ic(k,u) = t w(k,u) Jc(k,u).
Summing we obtain, I(k) = I c(k) + t w(k)Ek/ρf where w(k) ~ W
is the integrated length of the channel. If the width of a vortex
channel is one vortex spacing ao then ∆Vk= a o Ek and the
expression for the V-I characteristic for the conductor
summed over m active vortex channels becomes,
k =m
ρ a 
V = ∑  f o  [I − I c (k )]
k =1  tW 
where
ρ a 
g f =  f o 
 tW 
∆Ic3
∆Ic1
dV
= g f h∆I c
dI
= constant
<Ic>-∆Ic3
<Ic>
<Ic>+∆Ic3
I
Fig. 4. Schematic of predicted V-I transitions for distributions of GB critical
currents with FWHM ∆Ic3>∆Ic2>∆Ic1. When I = <Ic>, the average GB
critical current, dV/dI(<Ic>) = constant and V(<Ic>) ∝ ∆Ic.
(2)
The number of active vortex channels is a function of the
total current density, m = m(I), and determining m(I) will
determine the gradient of the current-voltage characteristic as
an integral multiple of gf. For a film 1 µm thick and 10mm
-7
wide gf~10 Ω for B = 1 T.
E
V
∆Ic2
(1)
Whence the slope of the characteristic becomes simply,
dV
= g f m(I )
dI
B. The Derivation of dV/dI for a 2-D Dispersion of LAGB
Two approaches are being explored to predict dV/dI for a
2-D dispersion of LAGB. The first method depends on
detailed mapping of the grain texture for prototype coated
conductors using EBSD techniques. Current percolation can
then be modelled directly for a progressively increasing
current to determine the location and distribution of flux flow
channels.
In the second approach expressions for the statistical
distribution of grain-to-grain LAGB angles are applied to
predict the likely form of the V-I characteristic. Model
distribution functions can be used or actual distributions may
be deduced from X-ray rocking curves and phi scans on
prototype conductors. The real form of such distributions will
consist of a mean misorientation angle and an associated
spread. Using a simple triangular distribution of grain
boundary critical currents with height h, mean <Ic(k)> and
FWHM of ∆Ic (where h∆Ic is constant) leads to a distribution
for dm/dI, the rate of formation of vortex channels. If all the
LAGBs are of the same misorientation angle ∆I c = 0 and
dm/dI will become discontinuous when I=<Ic> corresponding
to all possible channels starting to flow at once. The results
are shown schematically in Fig. 4 for three cases
∆Ic3>∆Ic2>∆Ic1.
ρf=dE/dJ
IV. SUMMARY
A framework has been presented for the modelling of the
V-I characteristic for biaxially textured coated conductors. A
number of simplifying assumptions have been made, in
particular the equations presented apply for I close to Ic when
the vortex flow channels are undivided. The regime of
interest for applications depends on both the voltage criterion
for Ic and the ratio of the grain size to the vortex spacing. This
relates directly to the assumption that the V-I characteristics
for LAGBs are linear in the regime of interest. More
extensive measurements on isolated boundaries are necessary
to clarify the occurrence and range of the linear characteristic.
Ek=1
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(1,7) (1,3)
[2]
J(1,u)
∆J(1)
Fig. 3. Schematic E-J curves for the ten LAGB segments in the k =1 vortex
channel (a→b in Fig. 2) each with a different grain boundary angle resulting
in different Jc values. In the flux flow state the value of the electric field Ek=1
is the same for all segments of the vortex channel.
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