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\begin{document}
\makeatletter
\renewcommand{\@oddhead}{Mater.Phys.Mech. {\bf 1} (2000) 00-00\hfil\thepage}
\renewcommand{\@oddfoot}{\copyright 2000 Advanced Study Center Co. Ltd.\hfil}
\makeatother

\vspace*{2cm}
\begin{center}
{\Huge\sf\bf
Critical Current Density in Polycrystalline High-$T_c$
Superconductors with Disordered\\ Tilt Boundary}
\end{center}
\vspace{0.3cm}
\centerline{\bf S.A. Kukushkin, A.V. Osipov and I.A. Ovid'ko}

\begin{center}
{\small Institute of Problems of Mechanical Engineering,
Russian Academy of Sciences}
\end{center}
\begin{quote}
{\small Received  November 1999}
\end{quote}
\begin{quote}
{\small{\bf Abstract.}
A theoretical model is suggested which describes the
effect of tilt boundaries with chaotically arranged dislocations
on the critical current density in high-transition-temperature
($T_c$) superconductors. Stress fields of such boundaries, that
suppress high-$T_c$ superconductivity, are revealed to be long-range
as compared to those of tilt boundaries with periodically
ordered grain boundary dislocations. With this factor taken into
consideration, the dependence of the critical current density
accross tilt boundaries on boundary misorientation $\theta$ is
calculated and compared with the experimental data [1].}
\end{quote}

\section{Introduction}

The transport properties of grain boundaries in high - $T_c$
superconducting materials are the subject of intensive
theoretical and experimental studies (e.g., see [1-14]) motivated
by extremely large interest to appllications of such materials.
It has been experimentally revealed that grain boundaries
drastically suppress the superconducting critical current [1-3]. In
particular, the critical current density, $j_c$, accross
tilt boundaries in $Y{Ba}_2{Cu}_3{O}_{7 - \delta}$
superconductors decreases with boundary misorientation angle
$\theta$ as follows : $j_c (\theta) \approx
j_c (0^{0}) exp (- \theta/{8}^0)$, for $\theta$ ranging
from ${0}^0$ to ${15}^0$. The critical current density
${\gamma}_c (\theta)$ accross high-angle tilt boundaries with
$\theta > {15}^0$ is tentatively constant and low : $j_c
(\theta > {15}^0) \approx 0.02 j_c (0^{0})$ [1-3].


Microscopic mechanism of the grain boundary effect on high-$T_c$
superconductivity is not understood yet, though several models of
the effect in question have been suggested [5-14]. In
particular, theoretical models have been elaborated describing
the grain boundary effect as that related to the following
factors : ($i$) a decrease of electron free path in vicinities of
grain boundaries [6]; ($ii$) stress-field-induced structural
disorder and pinning of magnetic vortices at grain boundaries
[7-9]; ($iii$) formation of the antiferromagnetic phase
within grain boundary cores [10]; ($iv$) faceting of grain
boundaries and $d$ - symmetry of the superconducting order
parameter [11]; ($v$) deviations from bulk stoichiometry in
vicinities of grain boundaries [12-14]. Most models
discussed treat stress fields of grain boundary dislocations as
those responsible for suppression of the superconducting order
parameter in vicinities of grain boundaries in high-$T_c$
superconductors [7-10]. Though the concrete mechanism of
the suppression is not unambigiously recognized yet, many
researchers assume the existence of a critical stress,
$\sigma_c$, which characterizes
superconductor-to-non-superconductor transition in stressed
regions [7-10]. More precisely, superconductivity is
supposed to be completely suppressed in regions where some
component(s) of stress tensor is (are) larger than $\sigma_c$.
However, this idea is in contradiction with experimental data in
case of low-angle boundaries consisting of periodically arranged
lattice dislocations (Fig.1a).
\begin{figure}
\centerline{\epsfig{file=fig1.eps, height=3.5in,width=3.5in}}
\vspace*{11.5cm}


{\small{\it Figure 1:}
Ordered and disordered, low-angle tilt boundaries are
represented as $(a)$ periodically ordered and $(b)$ disordered
walls of crystal lattice dislocations of the edge type, respectively.
}
\end{figure}
Actually, for any value of
$\sigma_c$, the volume fraction of "critically stressed" regions
(regions where some component(s) of sress tensor is (are) larger
than $\sigma_c$) in vicinities of such low-angle
boundaries (that create short- or intermediate-range stress
fields which drop as $exp (-x/h_0)$, where $x$ denotes the
distance from a boundary plane, and $h_0$ the interspacing
(period) between periodically arranged dislocations, see (Figs.
1a and 2a)) decreases with boundary misorientation $\theta$.
\begin{figure}
\centerline{\epsfig{file=fig2.eps, height=3.5in,width=3.5in}}
\vspace*{8cm}


{\small{\it Figure 2:}
High-angle tilt boundaries with
$(a)$ periodically ordered and $(b)$ disordered
ensembles of grain boundary dislocations.
}
\end{figure}

\makeatletter
\renewcommand{\@evenhead}{\thepage\hfil {\it S.A. Kukushkin, A.V. Osipov and I.A. Ovid'ko}}
\renewcommand{\@oddhead}{\thepage\hfil {\it Critical Current Density}}
\renewcommand{\@oddfoot}{}
\makeatother
This, in context of the idea on existence of the critical stress
$\sigma_c$, results in increase of the critical current density,
$j_c$, with boundary misorientation $\theta$, while experiments
[1-3] are indicative of the fact that $j_c$ drastically
decreases with $\theta$ in the range of $\theta$ from $0^{0}$ to
tentatively $15^{0}$. In order to avoid the contradiction
discussed, Gurevich and Pashitskii [10] assume that the
non-superconducting, critically stressed regions consists of two,
normal metal and antiferromagnetic, phases, in which case the
antiferromagnetic phase effectively destroys the coupling of
Cooper electrons in vicinities of grain boundaries.

In this paper we suggest an alternative model of the grain
boundary effect on high-$T_c$ superconductivity, based on both
the idea on critical stress $\sigma_c$ and that on disorder in
spatial arrangement of grain boundary dislocations in high-$T_c$
superconductors. Actually, high-$T_c$ superconducting materials
commonly are synthesized at non-equilibrium conditions at which
grain boundary dislocations can form "non-equilibrium" structures
(Figs. 1b and 2b) that are different from "equilibrium", periodic
ones (Figs. 1a and 2a). It this situation, grain boundaries in
high-$T_c$ superconducting materials, as with boundaries in
conventional materials synthesized at non-equilibrium conditions
[15,16], create long-range stress fields which drop as
$x^{-1/2}$ or $x^{-3/2}$. The main aim of this paper is to calculate
the angular dependence of the critical current density, $j_c
(\theta)$, in high-$T_c$ superconductors with tilt boundaries
containing disorderedly distributed dislocations.


\section{Models of low- and high-angle tilt boundaries in high-$T_c$
superconductors}

Stress fields of grain boundaries, in fact, are created by
boundary defects (first of all, dislocations) whose parameters
are caused by the structural geometry of boundaries. So,
low-angle tilt boundaries (with misorientation $\theta$ ranging
from $0^{0}$ to $\theta _c \approx 15^O$) are effectively
represented as walls of lattice dislocations (of the edge type)
with Burgers vectors being those of the crystalline lattice
[17,18] (Fig.1). For $\theta > \theta_c$, cores of lattice
dislocations overlap which compose grain boundaries. As a
corollary, high-angle tilt boundaries (with $\theta > \theta_c$)
have layer-like cores (characterized by thickness $\approx 1 nm$)
(Fig.2) with the structure being different from the crystalline
structure of adjacent grains. The layer-like cores of high-angle
boundaries commonly contain grain boundary dislocations (Fig.2)
with Burgers vectors being those of displacement-shift-complete
lattices of high-angle boundaries [18]. Moduli of such Burgers
vectors commonly are lower than moduli of Burgers vectors of
crystal lattice dislocations composing low-angle boundaries.

Grain boundaries at equilibrium conditions form either periodic
(Figs.1a and 2a) [17,18] or quasiperiodic [18-20]
ensembles that correspond to minimum elastic energy density of
such boundaries. Grain boundaries in polycrystalline materials
synthesized at highly non-equilibrium conditions are
non-equilibrium in the sense that they are characterized by
disorder in spatial arrangement of boundary dislocations (see
Figs.1b and 2b). In doing so, ensembles of grain boundary
dislocations at high-angle boundaries (Fig. 2b), in general, are
more disorderedly distributed than lattice dislocations composing
low-angle boundaries (Fig. 1b). The aforesaid is related to the
fact that dislocations at high-angle boundaries are characterized
by small Burgers vectors as compared to those at low-angle
boundaries; in this case displacements of dislocations from their
equilibrium positions at high-angle boundaries are accompanied by
low changes in the elastic energy density as compared to those of
dislocations composing low-angle boundaries.

Now let us consider a low-angle tilt boundary of infinite
extent, which consists of lattice dislocations periodically spaced
in plane $yz$ in coordinate system shown in (Fig. 1a).  Such
dislocations are characterized by ordinates $y_n = n h_0$ and
Burgers vectors $\vec b = (b,0,0)$ with $h_0$ and $b$ being in
the Frank's relationship [18] : $b= 2 h_0  sin ({\theta}/2)$.
Dislocations at "non-equilibrium" tilt boundary (Fig. 1b) are
assumed to be displaced by $h_0 {\delta}_n$ $(-a < {\delta}_n < a)$
in a randomly uniform way from their "equilibrium" positions
corresponding to the periodic arrangement shown in Fig. 1a. In
other words, the probability of finding the $n$th dislocation in
an infinitesimal interval $(y,y +dy)$ belonging to the interval
$y_n - a \le y \le y_n + a$ is equal to $dy/2a$. In the situation
discussed, ordinates of the dislocations composing the
"non-equilibrium" tilt boundary (Fig. 1b) are given as :


\begin{equation}
y_n = b (n + {\delta}_n) / 2 \sin ({\theta}/2)  \, \, ,
\label{1}
\end{equation}
where

\begin{equation}
<{\delta}_n> = 0 \, , <{\delta}_n^2> = a^{2}/3 \, , <{\delta}_m
{\delta}_n> = 0 \, ,
\label{2}
\end{equation}
with $<...>$ being the averaging on the non-equilibrium dislocation
ensemble. The dispersion of grain boundary stress is equal
to the sum of the dispersions of stresses induced by the
boundary dislocations :


\begin{equation}
D \sigma_{\alpha \beta} (x,y) = \sum_{n = - \infty}^{\infty}
\left\{<{[\sigma_{\alpha \beta} (x,y)]}^2> - \,
{<[\sigma_{\alpha \beta} (x,y)]>}^2\right\} \, \, \, \,
(\alpha,\beta = (x,y))
\label{3}
\end{equation}
with $\sigma_{\alpha \beta}^{(n)}$ (on the r.h.s. of formula \refer{3})
being the stress field of the $n$th dislocation, which, following
[17], can be written as:


\begin{equation}
\frac{\sigma_{xx}^{(n)}}{G} = \frac{b}{2 \pi (1 - \mu)}
\frac{(y-y_n) [3 x^2 + (y-y_n)^2]}{{[x^2 + (y-y_n)^2]}^2} \, ,
\label{4}
\end{equation}
\begin{equation}
\frac{\sigma_{yy}^{(n)}}{G} = \frac{b}{2 \pi (1 - \mu)}
\frac{(y-y_n) [x^2 - (y-y_n)^2]}{{[x^2 + (y-y_n)^2]}^2} \, ,
\label{5}
\end{equation}
\begin{equation}
\frac{\sigma_{xy}^{(n)}}{G} = \frac{b}{2 \pi (1 - \mu)} \frac{x
[x^2 - (y-y_n)^2]}{{[x^2 + (y-y_n)^2]}^2} \, , \label{6}
\end{equation}
\begin{equation}
\frac{\sigma_{zz}^{(n)}}{G} = \frac{\mu}{G} (\sigma_{xx}^{(n)} +
\sigma_{yy}^{(n)}) \, , \label{7}
\end{equation}
Here $G$ denotes the shear modulus, and $\mu$ the Poisson ratio.


After some algebra based on the calculation scheme [15], from
\refer{1} - \refer{7} we have found the averaged (on the
ordinate $y$) dispersion, $D \sigma_{\alpha \beta} (x)$, of
stress tensor components in vicinity of a non-equilibrium tilt
boundary. Following the results of our calculations, the largest
dispersion is that of $\sigma_{xx}$ :

\begin{equation}
D \sigma_{xx} (x) = \left(\frac{G b}{2 \pi h_0
(1 - \mu)}\right)^2\frac{5 \pi a^2}{4 (x/h_0) [(x^2/h^2_0) +
a^2]} \, . \label{8}
\end{equation}
For any $x$, values of $D \sigma_{xy}$ and $D \sigma_{yy}$ are
lower by factor $5$, while values of $D\sigma _{zz}$ (for
$\mu=0.2$; see experimental data [21]) are lower by factor
$\approx 4$ than those of $D \sigma_{xx}$. Therefore, in context
of the idea [7-10] on the crucial effect of the largest
stress tensor component on high-$T_c$ superconductivity, here and
in the following we will focus on analysis of the component $D
\sigma_{xx}$. In doing so, we find the mean value of modulus of
$\sigma_{xx}$ in vicinity of a non-equilibrium tilt boundary of
infinite extent to be as follows :


\begin{equation}
<|\sigma_{xx}|> = \sqrt \frac{D \sigma_{xx}}{\pi} \, , \label{9}
\end{equation}
because $<\sigma_{xx}> = 0$.

The dispersion of stress tensor components in the case of grain
boundaries with partly relaxed dislocation distribution can be
calculated by the same methods as with previously examined
non-relaxed disordered dislocation ensembles. So, in paper
[16] it has been shown that the dispersion in question can be
written as ${\Delta}^2 D \sigma_{xx}$ with $\Delta$ $(0 < \Delta
<1)$ being the factor of relaxation. In these circumstances, the
mean value of modulus of $\sigma_{xx}$ depends on the distance,
$x$, from a partly relaxed tilt boundary as follows :


\begin{equation}
\sigma(x)  \equiv <|\sigma_{xx}|> = \Delta \sqrt \frac{D
\sigma_{xx}}{\pi} = \frac{G \Delta}{2 \pi (1 - \mu)} \sqrt \frac{5
\sin ({\theta}/2)}{2 (x/b) [1 + 4 (x^2/a^2 b^2) \sin^2(\theta/2)]} \, .
\label{10}
\end{equation}
According to formula \refer{10}, $\sigma(x)$ is strongly affected
by the two non-dimensional parameters, $a$ and $\Delta$ : $\sigma
\propto x^{-3/2}$, if $x \gg a b/2 \sin (\theta /2)$; and $\sigma
\propto x^{-1/2}$, if $x \ll a b/2 \sin (\theta /2)$ [15].

In spirit of models [7-10], let us assume that high-$T_c$
superconductivity disappears in stressed regions in which $\sigma
(x) > {\sigma}_c = {\alpha}_c G$, where ${\alpha}_c \ll 1$.
Following [8], ${\alpha}_c \sim {10}^{-2}$. In the situation
discussed, we find that the non-superconducting phase in vicinity
of a tilt boundary with a partly relaxed dislocation distribution
lies in the layer : $- x_c < x < x_c$, where $x_c$ is extracted
from equation $\sigma (x = x_c) = {\alpha}_c G$. For
definiteness, let us assume that the non-superconducting phase
exhibits the normal metal properties. Then, according to the
theory of Cooper pairs tunneling [22,23], we find the
critical current density accross a normal metal layer of
thickness $2 x_c$ in vicinity of the tilt boundary to be as
follows :


\begin{equation}
\frac{j_c(\theta )}{j_{c} (0^0)} = \exp(-2x_c(\theta )/\xi ) \, ,
\label{11}
\end{equation}
where $\xi$ is the electron coherence length in the normal metal
phase. Formulae \refer{10} and \refer{11} allow one to calculate
$x_c(\theta )$ and, therefore, the dependence of the critical
current density, $j_c$, on boundary misorientation.

\begin{figure}
\centerline{\epsfig{file=fig3.eps, height=3.5in,width=3.5in}}
\vspace*{10cm}


{\small{\it Figure 3:}
Fig.3. Theoretical (solid curve) and experimental [1] (dots)
dependences of the averaged density ($j_c$) of the
superconducting critical current accross "non-equilibrium" tilt
boundaries on boundary misorientation ($\theta$).
}
\end{figure}
In the framework of the suggested model, the angular dependence
of $j_c$ was calculated as that optimally corresponding to the
experimental data [1], in which case parameters, $a$ and
$\Delta $, was used as adjusting parameters. In doing so, for
characteristic values of $\mu = 0.2$ [21], $\alpha _c =
10^{-2}$ and $\xi /b=5$, we find the optimal correspondence
between our theoretical calculations of $j_c (\theta)$ and the
experiments [1] (Fig.3) to come into play at $a_{opt} \approx
4.13$ and $\Delta _{opt}=0.313$. Fig.3 is indicative of a
satisfactory agreement between the model under consideration and
the experimental data [1].

Analysis of the theoretical dependence $j_c (\theta)$ shows that
this dependence is weakly affected by parameters, $\xi /b$ and
$\alpha _c$, because $a_{opt} \approx 0.83 \xi /b$, and $\Delta
_{opt}$ runs parallel with $\alpha _c$ at constant $a$. When $\xi
/b$ grows, value of $\Delta _{opt}$ weakly increases. With this
taken into consideration, we find that "structural transition"
from low-angle tilt boundaries (weakly disordered walls of
dislocation with large Burgers vectors; see Fig.1b) to high-angle
tilt boundaries (containing highly disordered ensembles of
dislocations with small Burgers vectors; see Fig.2b) occuring at
$\theta \approx {\theta}_c$ does not result in any jump-like
changes of the theoretical dependence $J_c (\theta)$ at $\theta
\approx {\theta}_c$ (see Fig.3).

In conclusion, in this paper we have developed a theoretical
model which describes the tilt boundary effect on high-$T_c$
superconductivity as that related to long-range stresses (which
suppress the superconducting order parameter) created by tilt
boundaries with disorderdly arranged dislocations. In this
context, the disorder of grain boundary dislocation ensembles is
caused by highly non-equilibrium conditions of synthesis of
polycrystalline high-$T_c$ superconductors. The angular
dependence, $j_c (\theta)$, calculated within the framework of
the suggested model, is in a satisfactory agreement with
the experimental data [1] (see Fig.3).

Acknowledgments. This work was supported, in part, by the Office
of US Naval Research (grant N00014-99-1-0896).


\newpage

\begin{center}
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\end{center}
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\end{document}

