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\begin{center}
{\bf PHYSICAL MECHANISMS OF REDUCTION OF CRITICAL CURRENT DENSITY
ACROSS GRAIN BOUNDARIES
 IN HIGH-$T_c$ SUPERCONDUCTORS}
\vspace*{1.5cm}

I.A.Ovid'ko
\vspace*{1cm}


Institute of Problems of Mechanical Engineering,

Russian Academy of Sciences, Bolshoj 61, Vas.Ostrov,

St.Petersburg 199178, Russia; E-mail: ovidko@def.ipme.ru

\end{center}

Received: November 11, 2000 \vspace*{3.5cm}

\noindent The combined effects of dilatation stresses,
stoichiometric and electric-charge inhomogeneities, $d_{x^2-y^2}$
symmetry of the superconducting order parameter and the faceted
microstructure of grain boundaries are theoretically examined here
as those causing the experimentally observed  (see, e.g., Dimos et
al, Phys.Rev. B 41  (1990) 4038) reduction of the critical current
density $J_c$ across grain boundaries in high-$T_c$ cuprates. The
enhancement of $J_c$  across high-quality twist and doped grain
boundaries in high-$T_c$ cuprates is briefly discussed. \vfil


\hrule \vspace*{0.5cm}

This paper is based on presentation given at the International
Workshop on Interface Controlled Materials (ICMRD), St.
Petersburg, Russia, June 7-9, 2000.
 \\ Corresponding author: I.A. Ovid'ko, e-mail:
 ovidko@def.ipme.ru\\
 \copyright~2001 Advanced Study Center Co. Ltd.
 \pagebreak

Grain boundaries (GBs) in polycrystalline high-$T_c$
superconductors are the subject of intensive studies motivated by
unremitting attention to both the fundamentals of high-$T_c$
superconductivity and its applications. Polycrystalline high-$T_c$
superconductors are characterized by the transport
critical-current density ($J_c$) whose values are essentially
lower (commonly by orders) than those of their single crystalline
counterparts, see, e.g., [1-4]. This property plaques high-current
applications of high-$T_c$ superconductors, that potentially are
capable of being extremely wide in electrical engineering. Though
there are many experimentally documented facts concerning the
suppression of $J_c$ across GBs in high-$T_c$ cuprates, its
physical mechanism(s) is (are) not unambiguously recognized yet
[5-7].

Several mechanisms have been proposed relating the GB-induced
reduction of $J_c$ in high-$T_c$ cuprates to
the following factors: (i) stress fields of GBs [8-10];
(ii) deviations from bulk stoichiometry in vicinities of
GBs [11,12]; (ii) the combined effects of
d-wave type symmetry of the superconducting
order parameter and the faceted
microstructure of GBs [13]; (iv) electric-charge inhomogeneities
(band banding)
[9,14-16].
However, each of these mechanisms explains
only a  part of experimental data, if it is treated
as the sole physical mechanism responsible for suppression of high-$T_c$
superconductivity.
The main aims
of this paper are to theoretically describe
the effect of GBs on high-$T_c$ superconductivity with
all factors, (i) to (iv), being taken into consideration,
and to analyse the $J_c$ enhancement
across high-quality twist and doped GBs in cuprates.

The key points of the suggested model are as follows: (a) GB cores
(layer-like cores of high-angle boundaries and cylinder-like cores
of lattice dislocations composing low-angle boundaries (Fig.~1))
are characterized by a low atomic density compared to the bulk
[17,18] and, therefore, by high positive values of dilatation.
Tensile dilatation stresses associated with GB cores induce weak
stoichiometric inhomogeneities within GB cores, because atoms of
different types in polyatomic high-$T_c$ cuprates exhibit
different behaviors in response to the dilatation stresses. More
precisely, relatively large cations (e.g., Y, Ba, Cu in YBaCuO and
Bi, Sr, Ca, Cu in BiSrCaCuO superconductors) substitute small
anions O$^{-2}$ in strained GB cores. The substitution processes
are mediated by enhanced GB diffusion and give rise to creation of
an excess positive charge density $Q_{\rm gb}$ ($>0$) of GB cores.
The concentration of cations within GB cores, resulted from the
substitution processes in question, weakly deviates from the bulk
concentration. Actually, if the substitution does not occur, the
concentration of cations within GB cores is lower than that in the
bulk phase due to a low atomic density inherent to the GB phase.
The subsitution of small oxygen anions by relatively large cations
of other elements of a cuprate results in an increase of the
initially low concentration of cations up to value close to the
bulk concentration of cations. This is why experiments often do
not reveal deviations from the bulk concentration of cations
within GB cores. At the same time, in accordance with our model,
GB cores should be deficient in oxygen, giving rise to the excess
positive charge density $Q_{\rm gb}$. The oxygen concentration is
experimentally measured with large errors (due to low atomic
weight of oxygen) [5], in which case deviations from bulk
concentration of oxygen within GB cores can be not detectable in
experiments.

In order to screen the excess positive charge density $Q_{\rm gb}$,
hole depletion zones characterized by an
excess negative charge density $Q_{\rm z}$ ($<0$) are formed in vicinities
of GB cores (Fig.~1).
Such hole depletion zones have been detected
in electron energy loss spectroscopy experiments [7,19,20]. Due to
high sensitivity of high-$T_{\rm C}$ superconductivity to stoichiometry and
hole concentration [5,7], it is natural to treat the GB
cores and hole depletion zones in their vicinities as non-superconducting
regions responsible for suppression of critical current density $J_{\rm c}$ across
GBs in cuprates.




According to the theory of electron pairs tunneling in superconductors (see,
e.g.~[21]), reduction of the critical-current-density across a non-superconducting
layer is approximately described by factor $exp\{-h/\xi\}$. Here $h$
denotes the layer thickness and $\xi$ the characteristic decay length
which can be the tunneling length for insulating GBs or the
proximity length for metallic GBs (which is close to the coherence
length).

Parallel with stoichiometric and electric-charge inhomogeneities,
additional geometric
factors are capable of affecting the critical current density across tilt boundaries.
Thus, recently, the symmetry of the order parameter in many high-$T_{\rm c}$
superconductors has been experimentally recognized as being
$d_{x^2-y^2}$ [22-26] or $d_{x^2-y^2}$ mixed with an s-wave component [27].
The $d_{x^2-y^2}$ symmetry causes a dependence of the critical-current
density $J_{\rm c}(\alpha)$ across a GB on boundary misorientation
$\alpha$ and orientation of boundary plane relative
to adjacent grains [13,28].
With this taken into consideration, the faceted microstructure
of tilt boundaries, that is often detected experimentally
(see, for instance, [29-52]), also influences the transport properties
of tilt boundaries in high-$T_{\rm C}$ superconductors.



With description [13] of the influence of $d$-symmetry
and GB faceting on $J_c$ taken into account, we find that the combined
effects of
dilatation-induced stoichiometric and electric-charge inhomogeneities, $d_{x^2-y^2}$ symmetry
and faceting cause the following angular dependence
of $J_{\rm c}$ in the case of [001] tilt boundaries:
\begin{equation}
J_{\rm c}(\alpha)/J_{\rm c}(0^\circ)\approx\,\, \frac{1}{S}\,<\!\!\!\! \prod_{m=1,2}[(\sin\alpha_m)^2-(\cos\alpha_m)^2]>_{_F}\,
\int_S\exp\{-h(y,z)/\xi\}\,dy\,dz
\end{equation}
Here $\alpha_m$ ($m=1,2$) is the smallest angle between the grain
boundary plane and a principal crystallographic axis ($\vec{a}$ or $\vec{b}$)
of adjacent grain $m$ ($\alpha_1+\alpha_2=\alpha$), and $< ... >_F$ denotes
the averaging that takes into account the faceted boundary microstructure
(for details, see paper [13]). $h(y)$ denotes the thickness of the regions
with stoichiometric and electric-charge inhomogeneities (Fig.~1),
$y$ denotes the coordinate along GB plane.

We have calculated with the help of formula (1) the angular
dependence of $J_{\rm c}$ -- curve~1 in Fig.~2 -- attributed to
the combined effects in question. In doing so, for high-angle
($\alpha\ge20^\circ$) boundaries, $h$ is taken as $H\approx3$~nm.
For low-angle ($\alpha\le19^\circ$) boundaries, the scale $h$ is
choosen, according to Fig.~1~a,  as $y-$dependent thickness of
boundaries composed of lattice dislocation cores and their
vicinities with "non-ideal" stoichiometry and charge density, that
are characterized by diameters $2R\approx3$~nm. (In this case
interspacing between periodically arranged lattice dislocations
composing low-angle boundaries (Fig.~1~a) depends on $\alpha$ as
[17]: $2d=b\sin(\alpha/2)$, where $b$ is the magnitude of the
dislocation Burgers vector.) In the intermediate range of $\alpha$
from $19^\circ$ to $20^\circ$, $h$ is taken as interpolation of
corresponding values of $h$ for low- and high-angle boundaries.
Results of the averaging $< ... >_F$ are taken from paper [13]
where the averaging procedure is discussed in detail. $\xi$ is
taken as 1.5~nm. (This corresponds to the coherence length in
[001] planes that carry the current [5].)


The calculated dependence $J_{\rm c}(\alpha)$ (curve~1 in Fig.~2)
is in a satisfactory agreement with experimental data~[1-4]
(curve~2 in Fig.~2) for YBaCuO superconductors. This allows us to
think that the idea on the combined effects of dilatation-induced
stoichiometric and electric-charge inhomogeneities, $d_{x^2-y^2}$
symmetry of the superconducting order parameter and the faceted
microstructure of GBs is effective in description of the dramatic
distinction between the transport properties of low- and
high-angle boundaries.

Now let us discuss the doping of polycrystalline high-$T_{\rm C}$
cuprates, which  is capable of strongly influencing the critical
current densities $J_{\rm c}$ across GBs. The remarkable
experimental fact in this area is a record enhancement of $J_{\rm
c}$ (by about a factor of 8) in Ca-doped YBaCuO superconductors
[16].

In the framework of the suggested model, doping can decrease value
of $Q_{\rm gb}$ and, therefore, enhance values of $J_{\rm c}$
across GBs in the following cases: (A) valency $\tilde{v}$ of
dopant cations is lower than that ($v_k$) of host cations; (B)
radius $\tilde{r}$ of dopant cations is larger than that ($r$) of
host cations. In the case (A) the excess charge $Q_{\rm gb}$
decreases directly due to the doping-induced decrease of the sum
charge of cations within GB core. In the case (B) the doping
decreases the effect of dilatation stresses. That is, substitution
of relatively small host cations by relatively large subsitute
cations results in an increase of the atomic density within GB
cores. As a corollary, dilatation stresses associated with GB
cores are decreased, in which case so does the driving force for
diffusional exchange of cations and oxygen anions within boundary
cores, thus decreasing $Q_{\rm gb}$. In context of the former
mechanism (A) of the influence of doping on the critical current
density $J_{\rm c}$ across GBs, the Ca-doping of
YBa$_2$Cu$_3$O$_{7-\delta}$ bi-crystals results in the
experimentally detected [16] $J_{\rm c}$ enhancement due to the
following. Commonly Ca$^{2+}$ cations substitute Y$^{3+}$ cations
in YBaCuO cuprates [33]. This gives rise to a decrease of the
excess positive charge density $Q_{\rm gb}$, associated with
Ca$^{2+}\rightarrow$Y$^{3+}$ substitution in GB cores, and, as a
corollary, to the $J_{\rm c}$ enhancement.

In context of point (A), it is potentially interesting
to experimentally test the effect of Na-doping on $J_{\rm c}$ in bi-crystalline
YBaCuO. Actually, Na$^{1+}$ cations substitute Y$^{3+}$ cations
in YBa$_2$Cu$_3$O$_{7-\delta}$ cuprates [34]. This non-isovalent
doping can substantially decrease $Q_{\rm gb}$ and, therefore increase
$J_{\rm c}$ in bi-crystalline Y$_{1-x}$Na$_{x}$Ba$_2$Cu$_3$O$_{7-\delta}$
superconducting cuprates.

Recently, high-quality $c$-axis twist boundaries have been
fabricated in BiSrCaCuO superconductors, that exhibit the enhanced
transport properties [35-37]. These twist boundaries carried
critical current as high as their constituent single crystals
(adjacent grains), regardless of the misorientation angle [35-37].
The specific structural feature of high-quality twist boundaries
is the zero thickness of boundary core. That is, the twist
boundaries are atomically intact and clean, with no detectable
$c$-axis spacing increase or chemical changes between the BiO
double layers [35,36]. A direct consequence of the zero thickness
of high-quality twist boundary cores is the fact that there are no
dilatation stresses associated with such cores. In context of our
previous analysis, it means that high-quality twist boundaries
should not affect the critical current density. The aforesaid is
in correspondence with experimental data [35-37].

In general, GBs in high-$T_c$ superconductors can undergo
structural transformations induced by their stress fields and/or
their elastic interaction with other defects (see experimental
data [38-40] and theoretical models [41-43]). Such transformations
dramatically change GB core structures and stress fields in
vicinities of GBs, in which case, following the model suggested in
this paper, they are capable of strongly influencing the critical
current density across GBs. As a corollary, the transformations in
question should be definitely taken into account in interpretation
of experimental data on measurements of the critical currents in
polycrystalline high-$T_c$ superconductors.

In conclusion, the combined effects of dilatation-induced
stoichiometric and electric-charge inhomogeneities at GBs,
$d_{x^2-y^2}$ symmetry and GB faceting account for the
experimentally observed [1-4] reduction of the critical current
density $J_c$ with increasing GB misorientation angle in
high-$T_c$ superconductors. The representations on these effects
can be used in understanding the nature of the enhancement of
$J_c$ across high-quality twist and doped GBs in cuprates. A more
detailed report of this study will be published in paper [44].



\vspace*{2ex}

Many thanks are due to R.Masumura and C.Pande for valuable
discussions. This work was supported, in part, by the Office of US
Naval Research (grant N00014-99-1-0896) and INTAS (grant
99-01216).
\newpage

{\bf \large Figure Captions}

\bigskip
Fig. 1. Stoichiometric and electric-charge inhomogeneties within and near tilt boundaries.
(a) Cores of lattice dislocations (composing low-angle boundaries) and
(b) cores of high-angle boundaries are characterized by deviations
from bulk stoichiometry and an excess positive charge density $Q_{\rm gb}$.
Their vicinities (dotted regions) are
hole depletion zones.\\

Fig. 2. Angular dependences of the critical current density
$J_{\rm c}$ across [001] tilt boundaries. Solid curve~1, according
to formula (1), is attributed to the combined effects of
stoichiometric and electric-charge inhomogeneities, $d_{x^2-y^2}$
symmetry and the faceted microstructure of tilt boundaries. The
experimentally detected~[1-4] angular dependence of $J_{\rm c}$ is
shown, according to [13], as dashed curve~2.
\\
\newpage
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\end{document}
(In papers [14-16] the excess electric charge within high-angle
boundary cores has been treated as that responsible for the
formation of hole-depletion zones in vicinities of GBs. According
to [16], however, the excess electric charge can be either
positive or negative, in which case hole-enhancement zones in
vicinities of GBs could be expected to form as likely as
hole-depletion zones do. At the same time, only hole-depletion
layers have been detected in electron energy loss spectroscopy
experiments [7,19,20]. The model [9] focused on the range of
$\alpha$ from 5$^\circ$ to tentatively 25$^\circ$ leaving the
dependence $J_{\rm c}(\alpha)$ unclear, for  $\alpha<5^\circ$ and
$\alpha>25^\circ$. In addition, the electric-charge
inhomogeneities have been described in paper [9] as those
associated with stress-induced inhomogeneities of the averaged
(over an elementary cell) ion density, in which case the
polyatomic structure of cuprates has been ignored. The
phenomenological model [12] uses the existence of hole depletion
zones and their experimentally measured characteristics as an
experimentally documented input of the
Ginzburg-Landau-formalism-based description of the transport
properties of tilt boundaries in YBaCuO superconductors.)
