PHYSICAL REVIEW B
VOLUME 57, NUMBER 21
1 JUNE 1998-I
Current transport through low-angle grain boundaries in high-temperature superconductors
A. Gurevich
Applied Superconductivity Center, University of Wisconsin, Madison, Wisconsin 53706
E. A. Pashitskii
Institute of Physics, National Academy of Sciences of Ukraine, Kiev, 252650, Ukraine
~Received 4 November 1997!
We consider mechanisms which can account for the observed rapid decrease of the critical current density
J c ( u ) with the misorientation angle u through grain boundaries ~GB’s! in high-T c superconductors ~HTS’s!.
We show that the J c ( u ) dependence is mostly determined by the decrease of the current-carrying cross section
by insulating dislocation cores and by progressive local suppression of the superconducting order parameter c
near GB’s as u increases. The insulating regions near the dislocation cores result from a strain-induced local
transition to the insulating antiferromagnetic phase of HTS’s. The structure of the nonsuperconducting core
regions and current channels in GB’s is strongly affected by the anisotropy of the strain dependence of T c
which is essentially different for YBa2Cu3O7 and Bi-based HTS’s. We propose a mechanism of the progressive
superconductivity suppression on GB’s with u due to an excess ion charge on the GB’s which shifts the
chemical potential in the layer of the order of screening length l D near the GB’s. The local suppression of c
is amplified by the proximity of all HTS’s to a metal-insulator transition, by their low carrier density and
extended saddle point singularities in the electron density of states near the Fermi surface. Taking into account
these mechanisms, we calculated J c ( u ) analytically by solving the Ginzburg-Landau equation. The model well
describes the observed quasiexponential decrease of J c ( u ) with u for many HTS’s. The d-wave symmetry of
the order parameter weakly affects J c ( u ) in the region of small u and cannot account for the observed drop of
J c ( u ) by several orders of magnitude as u increases from 0 to u .20° –40°. @S0163-1829~98!06121-9#
I. INTRODUCTION
Mechanisms of current transport through grain boundaries
~GB’s! in high-temperature superconductors ~HTS’s! have
attracted much attention. The critical current density J c ( u )
through GB’s is very sensitive to the orientation of the adjacent crystallites and strongly decreases with the misorientation angle u in the range from u .3° –5° to u .30° –40°.1–7
The d-wave symmetry of the order parameter in HTS’s also
contributes to the decrease of J c ( u ) with u and gives rise to
novel effects, such as p junctions and fractional vortices on
GB’s.8–12 The rapid decrease of J c ( u ) with u essentially
limits the current-carrying capability of HTS materials which
inevitably contain GB networks or colonies of misoriented
grains.13–18 For instance, the decrease of the fraction of highangle GB’s in biaxially textured HTS’s substantially increased the critical current to a higher level determined by
vortex dynamics and pinning, rather than by the GB
transparency.19–21
A conventional model22,23 describes a symmetric lowangle GB’s as a chain of edge dislocations with the Burgers
vector b perpendicular to the GB plane ~Fig. 1!. The structure of GB’s in HTS’s can be more complicated than the
idealized one shown in Fig. 1 due to partial GB dislocations,
faceting, long-range strain fields, and compositional variations near GB’s.24–31 These factors give rise to structural and
chemical inhomogeneities along GB’s on a broad variety of
scales, from the nanoscale of individual dislocation cores to
the macroscale of GB facets. Yet even in those rare cases
when the atomic structure of a GB is known, the effect of
this structure on superconducting properties remains unclear,
0163-1829/98/57~21!/13878~16!/$15.00
57
since the correlation between atomic displacements and a
microscopic mechanism of high-T c superconductivity is still
uncertain. In general, the material inhomogeneities along
GB’s cause modulations of the superconducting coupling
across GB’s which determines its global critical current density J c after averaging the microscopic supercurrents j(x)
over the relevant spatial scales.
The GB dislocation structure naturally accounts for the
decrease of J c ( u ), if one assumes that there are regions of a
suppressed superconducting order parameter D near the dislocation cores which block the supercurrent through GB’s.1
These nonsuperconducting core ~NC! regions of radius r i
;b can result from local compositional and hole concentration variations, additional electron scattering, and significant
FIG. 1. Chain of edge dislocations which form a symmetric
low-angle grain boundary in the y-z plane. The nonsuperconducting
core regions are shadowed.
13 878
© 1998 The American Physical Society
57
CURRENT TRANSPORT THROUGH LOW-ANGLE GRAIN . . .
strains near the dislocation cores. High-resolution electron
microscopy has shown that the low-angle GB’s do consist of
chains of edge dislocations separated by regions of a weakly
distorted crystalline lattice and also exhibit compositional
variations and strain fields near the Cu-rich dislocation cores
on the scale .10–300 Å both along and across GB’s.24–31
The dislocation model predicts the critical misorientation
angle u c 52sin21(b/4r i ).20° –40°, above which the NC regions overlap, and GB’s become a continuous insulating or
normal Josephson contact with J c much smaller than the
intragrain critical current density J g . For u , u c , this model
gives an approximately linear dependence J c ( u ).(1
2 u / u c )J g , if one assumes that J c ( u ) is determined by the
area of current channels in GB’s between the dislocation
cores.4 However this linear dependence J c ( u ) is much
weaker than J c ( u )}exp(2u/u0) with u 0 .4° –5°, which is
usually observed experimentally.1–7 Models have been proposed in which the current channels in GB’s are described as
an array of parallel point contacts32–34 which exhibit weaklink behavior, if their width becomes smaller than the superconducting coherence length j ,35 that is, d22r i , j , or u
, u 1 .b/ @ j (T)12r i # . At 77 K, the in-plane j (T) is about
35 Å, whence u 1 '5° for r i 5b53.8 Å, which correlates
with the observed sharp decrease of J c ( u ) above u
.3° –7°.1,7
The theoretical description of the observed dependence
J c ( u ) remains incomplete, not least because of a variety of
relevant physical mechanisms and the multiscale heterogeneity of GB’s. The rapid decrease of J c ( u ) is usually ascribed
to the strain-induced, or compositional suppression of D near
dislocation cores and in the layer of thickness l( u ) near
GB’s.24–41 Then the array of parallel current channels in
GB’s is regarded as an effective Josephson contact, for
which
J c }exp@ 2l ~ u ! / j N ~ u !# ,
l@ j N ,
~1!
where j N is the characteristic decay length which can be the
proximity length for metallic GB’s or the tunneling length
for insulating GB’s.5 Thus, basically any increase of l( u )
with the dislocation density b/d. u or decrease of j N due to
additional electron scattering on GB’s dislocations and compositional variations near GB’s could account for the observed rapid decrease of J c ( u ) with u . However, this phenomenological approach has several inconsistencies. First,
Eq. ~1! can be used for a clean metallic GB, provided that
l.2 j N .2 j (0)T c /T. This implies a fairly wide layer of suppressed order parameter near the GB (l.600 Å at 4.2 K!,
which would result in negligible J c at 77 K. Yet, the weaklink behavior of GB’s and strong decrease of J c ( u ) have
been observed both at 4.2 and 77 K, with J c at 77 K exhibiting rather high values ;105 A/cm2 in thin film
YBa2Cu3O7 bicrystals.3,5–7 Second, the strain fields of a symmetric GB exponentially decay over the length l5d/2p ,22
which decreases as u increases. To account for the increase
of l( u ) with u , one has to assume nonperiodicity of the GB
dislocation structure and distribution of d,41 compositional
variations near GB’s, macroscopic strain fields produced by
the GB facets,31 etc. The periodic long-range strains near
low-angle GB’s can even locally increase T c for small u and
depress T c at larger u due to proximity effect coupling of the
13 879
NC core regions.42 For the d-wave symmetry of the order
parameter, faceting can also cause decreasing J c ( u ),10 but
this mechanism gives the relatively weak dependence J c
}cos2 2u which is not sufficient to explain the observed drop
in J c ( u ) by two to three orders of magnitude without the
assumption of the local depression of D on the GB’s. The
gap suppression due to local nonstoichiometry near the GB’s
is mostly determined by materials factors, such as diffusion
of oxygen in remanent strain fields around GB’s which may
contribute to the significant scatter in J c ( u ) usually observed
on HTS bicrystals of the same misorientation u .5,7 However,
the nearly ‘‘universal’’ dependence J c ( u )}exp(2u/u0) observed on many HTS’s ~Refs. 5 and 6! seems to indicate a
fundamental intrinsic mechanism of the GB weak-link behavior common for all HTS’s.
In this paper we propose a model which describes the
observed J c ( u ), taking into account the GB dislocation structure and the fact that a comparatively small shift of the
chemical potential m near GB’s can strongly decrease T c , or
even turn HTS’s into an insulating antiferromagnet.50 Because of the proximity of HTS’s to the metal-insulator transition, the strains and excess ion charge of the GB dislocation structure can locally induce a dielectric phase near
dislocation cores and cause progressive overall suppression
of the superconducting order parameter with u in a narrow
layer of the order of the screening length near GB’s. The
model provides an intrinsic mechanism for the rapid decrease of J c ( u ) with u and describes well the observed J c ( u )
dependence in HTS bicrystals, even without invoking the
local nonstoichiomentry and heterogeneity of GB’s. For
small u , the local suppression of superconductivity at GB’s
is shown to affect J c ( u ) much more strongly than the symmetry of the order parameter which manifests itself at higher
u . We obtain J c ( u ) by solving the Ginzburg-Landau ~GL!
equation which describes well the practically important temperature range T.77 K for HTS’s.
The paper is organized as follows. In Sec. II we discuss
mechanisms which determine the structure of the NC regions
and the gap suppression near GB’s. We first consider a strain
mechanism which gives rise to a composite structure of the
NC regions consisting of a dielectric core surrounded by a
normal shell. The shape of the NC regions is strongly affected by the in-plane anisotropy of the strain dependence of
T c . Then we consider the electron screening of the excess
ion charge of GB’s which results in superconducting gap
suppression near the GB’s amplified by the proximity of
HTS’s to the metal-insulator transition, low carrier density,
and extended saddle point singularities near the Fermi surface. In Sec. III we propose a model described by the GL
equation which takes into account the current channel structure of a low-angle GB’s and the local gap suppression near
GB’s. This exactly solvable nonlinear model enabled us to
obtain an analytical formula for J c ( u ) which gives a strong
~though nonexponential! decrease of J c ( u ) with u . In Sec.
IV we compare the model with experiment and show that the
theoretical J c ( u ) dependence describes well the observed
J c ( u ) in YBa2Cu3O7 and Bi-based HTS’s.
II. CURRENT CHANNELS IN GB’s
A. Strain mechanism
A qualitative description of NC regions can be made, assuming that the superconducting coupling constant l(r) near
A. GUREVICH AND E. A. PASHITSKII
13 880
57
the dislocation core becomes spatially inhomogeneous due to
its dependence on the local strain tensor e ik . The suppression of D near the cores can be quantified in terms of the
tensor C ik which determines the shift of T c in a weakly deformed superconductor:
T c 5T c0 2C ik e ik .
~2!
The strain dependence of T c in HTS’s can be highly
anisotropic.43–47 For instance, for optimally doped
YBa2Cu3O7 single crystals, it was found that dT c /d p a 5
2(1.9–2) K/GPa, dT c /dp b 51.9–2.2 K/GPa, and dT c /d p c
52(0 –0.3) K/GPa,45 where p i is the stress along the ith
axes. The hydrostatic pressure derivative dT c /d p
5 ( i dT c /dp i yields a comparatively small value dT c /d p
.0.3 K/GPa which results from a cancellation of large, almost equal, and opposite in-plane effects of different signs.
This is usually ascribed to the influence of Cu-O chains,
orthorhombic distortions of the Cu-O planes, charge transfer
effects, etc.43–46 By contrast, Bi2Sr2CaCu2Ox exhibits nearly
isotropic in-plane pressure derivatives dT c /dp a .1.6 K/GPa,
dT c /dp b .2 K/GPa, but large negative dT c /dp c .22.8 K/
GPa along the c axis, so that dT c /dp5 ( i dT c /d p i again
largely cancels under hydrostatic pressure. Doping can substantially affect dT c /dp i ; for example, the dT c /d p i values
for underdoped YBa2Cu3O7 with T c '40 K are about 2–10
times larger than those for the optimally doped
YBa2Cu3O7.48
For planar deformations, Eq. ~2! becomes
d T c 52C @ e 1p ~ e xx 2 e y y ! cos 2w 12p e xy sgn~ x ! sin 2w # ,
~3!
where p5(C a 2C b )/(C a 1C b ), e 5 e xx 1 e y y , C5(C a
1C b )/2, and w is the angle between the x axis taken along
the normal to the GB’s and the a axis in Fig. 1. The constants C a 52 ] T c / ] e aa and C b 52 ] T c / ] e bb determine the
change of T c under uniaxial compression ( e i ,0); for example, C a 52217 K and C b 5316 K for optimally doped
YBa2Cu3O72 d single crystals.47 For C a 5C b , which corresponds to the nearly isotropic in-plane dependence T c ( e ik )
found in Bi2Sr2CaCu2Ox ~Bi-2212!, Eq. ~3! reduces to d T c
52C e . Assuming isotropic elastic constants in the ab
plane,49 the shape of the NC region for a single edge dislocation can be evaluated from the condition T c (r)50, using
Eq. ~3! with w 50 and e ik from Appendix A. For the dislocation at x5y50, the boundary of the NC region is described by
r ~ f ! 52r 0 sin f ~ 11p 0 cos2 f ! ,
~4!
r 05
bC ~ 122 s !
,
4 p T c0 ~ 12 s !
p 05
C a 2C b
,
~ 122 s !~ C a 1C b !
~5!
where s is the Poisson ratio and f is the polar angle. For the
isotropic strain dependence of T c , Eq. ~4! describes a circle
of radius r 0 centered at x50,y52r 0 @Fig. 2~a!#. For anisotropic T c ( e ik ) with u p 0 u @1, the characteristic size of the NC
region becomes r 1 5r 0 u p 0 u , and its shape changes as shown
in Figs. 2~b! and 2~c!.
Figure 2 shows the NC regions in a grain boundary calculated from the condition T c (r)50, where T c is given by
Eq. ~3!, 2 w 5 p 2 u , and e ik for the periodic chain of GB
dislocations is taken from Appendix A. For isotropic T c ( e ik ),
FIG. 2. Nonsuperconducting core regions in a symmetric GB
calculated from Eq. ~3! for u 515° and s 50.28. ~a! shows the
isotropic case p50 for C/T c0 520. ~b! and ~c! show the NC regions
for the anisotropic case: C/T c0 510, p55.3 ~b! and C/T c0 510, p
525.3 ~c!.
57
CURRENT TRANSPORT THROUGH LOW-ANGLE GRAIN . . .
13 881
FIG. 3. Phase diagram T(c) of high-T c superconductors ~a!,
where I, N, and S correspond to the insulating, normal, and superconducting states, respectively. ~b! shows ~a! replotted in the T-e
coordinates.
the NC regions shrink and flatten along GB’s as u increases,
but they do not touch for any d. The latter is due to partial
compensation of the dipole strain fields of GB dislocations,
resulting in the exponential decay of e ik }exp(22puxu/d)
away from GB’s. For anisotropic T c ( e ik ), the NC regions do
not overlap as well, though their shape depends on the sign
of p and can significantly differ from the isotropic case. As
an illustration, Figs. 2~b! and 2~c! show NC regions for typical YBa2Cu3O72 d values p565.3, where the plus sign corresponds to the case shown in Fig. 1, and the minus sign
corresponds to the transposition of the a and b axes. For the
anisotropic strain dependence of T c , the current-carrying
cross section of GB’s can be strongly reduced by offdiagonal components of e ik . This may contribute to the
weak-link behavior of @001# tilt GB’s in the basal plane of
YBa2Cu3O72 d .
The above description, based on the linear strain dependence ~2!, can be used for NC regions much larger than b.
This implies C@T c0 in Eq. ~5!, which corresponds to underdoped HTS,43–46 or significant local nonstoichiometry near
GB’s. However, for optimally doped HTS’s ( s 50.28, C
.300 K, and T c0 .100 K!, Eq. ~4! yields a fairly small
diameter of the NC region, 2r 0 .0.3b.1 Å. This indicates
that the linear approximation ~2! of T c ( e ik ) is not selfconsistent, since the elastic strains e i j }b/2p r 0 near the dislocation cores become so strong that the linear expansion ~3!
of T c in e i j is hardly adequate. Thus, we have to take account
of the actual nonlinear dependence of T c on e i j shown in Fig.
3 with a characteristic maximum at the optimum strain e m ,
which reflects the nonmonotonic dependence of T c of HTS’s
on the hole concentration c on the Cu-O planes.50,52 In the
absence of structural transitions, T c ( e ) may be approximated
by a conventional parabolic function, which describes well
the observed dependence of T c on hydrostatic pressure p in
HTS’s.43–46 The general quadratic dependence T c ( e i j ) in anisotropic HTS’s can be written in the form T c 5T c0 2C i j e i j
2Q i jkl e i j e kl , where the tensor Q i jkl is symmetric with respect to the transposition i↔ j and k↔l, and the principal
values C a , C b , and C c of C i j determine the observed
dT c /d e i for e i j →0. For orthorhombic symmetry, Q i jkl has
nine independent components,23 but for planar deformations
Q i jkl has four independent components; so
FIG. 4. Composite structure of the nonsuperconducting core regions in a GB for the isotropic case calculated from Eq. ~7! for u
515°, T577 K, DT m 51 K, C5300 K, T c0 590 K, and s 50.28.
The inner region ~I! is in an insulating state surrounded by the
normal regions.
T c 5T c0 2C a e aa 2C b e bb 2Q a e 2aa 2Q b e 2bb 2Q 1 e aa e bb
2Q 2 e 2ab .
~6!
The simplest version of Eq. ~6! with C a 5C b , Q a 5Q b
5Q 1 /2, and Q 2 50 corresponds to a purely isotropic T c ( e )
which depends only on the dilatation e ,
T c 5T c0 2C e 2 ~ C e ! 2 /4DT m .
~7!
Here DT m 5T cm 2T c0 , and the coefficients in Eq. ~7! are
chosen such that the maximum value of T c equals T cm at
e m 522DT m /C, and dT c /d e gives the observed C at e
→0. The pressure experiments have shown that DT m !T c0 ,
at least for optimally doped HTS’s.43–46 In this case T c ( e )
vanishes at comparatively weak strain e i . A2DT m T c0 /C
!1, for which the linear elasticity theory is still applicable,
since C.T c0 .
Now we can estimate the size of the NC region for the
nonlinear T c ( e ). For the isotropic T c ( e ), the condition T c
50
yields a quadratic equation for e 5b(1
22 s )sin f/2p (12 s )r. Its solution gives two circular NC
regions described by Eqs. ~4! and ~5! with p 0 50 and r 0
6
replaced by r 6
i ~Fig. 4!. The radii r i are
r6
i 5
r0
~ A11 a 0 61 ! ,
2
~8!
where a 0 5T c0 /DT m . For DT m !T c , the nonlinearity of
T c ( e ) increases r i by the factor (T c /4DT m ) 1/2 as compared to
r 0 . For optimally doped Bi-2212, s 50.28, C.300 K, T c0
.100 K, and DT m 54 K; we obtain from Eq. ~8! that 2r 1
i
'b. The local nonstoichiometry near the Cu-rich dislocation
cores24 may considerably increase the ratio C/T c0 in Eq. ~8!,
thus further increasing r i .
For the anisotropic T c ( e ik ) dependence ~6!, the solution
of the quadratic equation T c @ e ik „r( f )…# 50 for r( f ) yields
13 882
A. GUREVICH AND E. A. PASHITSKII
r ~ f ! 5r 0 sin f @ A~ 112p 0 cos2 f ! 2 1g ~ f !
6 ~ 112p 0 cos2 f !# ,
~9!
g ~ f ! 5g 0 1g 1 ~ cos2 f 2 s sin2 f !
1g 2 @~ 122 s ! 2 24 cos2 f # 1g 3 ctg 2 f cos2 f ,
~10!
where g 0 54QT c0 /C 2 , g 1 54(Q a 2Q b )T c0 /(122 s )C 2 , g 2
5(Q 1 22Q)T c0 /4(122 s ) 2 C 2 , g 3 5Q 2 T c0 /4(122 s ) 2 C 2 ,
and
Q5(Q a 1Q b )/2.
For
the
isotropic
case,
b 0 5g 1 5g 2 5g 3 50, Q5C 2 /4DT m , Eq. ~9! reduces to Eq.
~8!. Unlike the isotropic case, the shape of r( f ) for the anisotropic case can be changed by the nonlinear terms in Eq.
~6!. We do not discuss here the anisotropic case in more
detail because of a lack of experimental data on Q i for
YBa2Cu3O7.
The structure of the NC regions may be further clarified
by the phase diagram of HTS’s T c (c), shown in Fig. 3. Indeed, using the electroneutrality condition, the isotropic
strain dependence of T c ( e ) can be qualitatively mapped onto
the domelike dependence of T c (c) on the hole concentration
c, if e (r) varies weakly on microscopic scales, such as the
screening length ~see below! and j . Figure 3 clearly shows
that the slope C5dT c /d e increases as T c decreases, giving
the observed strong sensitivity of the ratio C/T c0 to the local
doping level.43–46,51 Under these assumptions, we can replot
the phase diagram in T-e coordinates as shown in Fig. 3~b!
and conclude that for T50 the NC regions are partly in an
insulating ~I! state,5,24 since the point T c 50 is close to the
region of the HTS phase diagram which corresponds to an
insulating antiferromagnet. Recent measurements of resistivity r (T) of La22x Srx CuO4 in high pulsed magnetic fields52
revealed an insulating behavior of r (T) even in the superconducting region of the phase diagram after suppression of
the order parameter by a magnetic field. Therefore, the
strains might cause the local S-I transition near the dislocation cores in larger domains determined by T c ( e )5T i ,
where T i corresponds to the intersection of the superconducting and insulating regions on the phase diagram in Fig. 3.
The nonzero T i can be taken into account by replacing T c0
with T c0 2T i in Eq. ~8!. Notice that Winkler et al.53 reported
on a significant dielectric fraction in GB’s for a 32° @ 001# tilt
YBa2Cu3O7 bicrystal which manifests itself in Fiske resonances on I-V characteristics. A large T c depression was also
observed on a 7° Bi-2212 tilt bicrystals.54
The NC regions therefore have a composite structure consisting of insulating and normal domains, whose boundaries
are defined by the conditions T c ( e ).T i and T c ( e ).T, as
shown in Fig. 4. For a single dislocation, the dielectric do1
main is a circle of radius r 1
i centered at x50, y52r i ~as
2
seen from Fig. 3, the second solution r i for underdoped
HTS’s corresponds to a normal domain!. The outer shape of
the normal domain is two circles which touch at x5y50,
their radii r 6
n being described by Eqs. ~4! and ~8! in which
T c0 should be replaced by T c0 2T:
r6
n 5
r0
~ A11 a 0 t 61 ! ,
2t
~11!
57
where t 5(T c0 2T)/T c0 . The size of the normal NC region
2
1/2
along GB’s, L n 52(r 1
n 1r n )52r 0 (11 a 0 t ) / t , increases
with temperature. For optimally doped Bi-2212,43–46 we
have T c0 .100 K, DT m .1 –2 K, a 0 5T c0 /DT m 545–90, T
577 K, and t 51/7; we obtain L n .(100–140)r 0 '50–70 Å.
As above, L n (T) may increase due to the local nonstoichiometry near GB’s.
Using Eq. ~11!, we can estimate the critical angle u n
52 sin21(b/2L n ), above which the normal parts of the NC
core regions of neighboring dislocations start overlapping:
u n 52 sin21
p ~ 12 s !~ T c0 2T !
C ~ 122 s ! A11 a 0 t
~12!
.
For 77 K and the numbers used above, this yields
u n (77 K).3° –5°, in agreement with the observed onset of
the sharp drop in J c ( u ).1–7 By contrast, the dielectric parts of
the NC core regions may start overlapping at a much higher
angle u c than u n . Using Eq. ~8!, we can estimate the critical
angle u c 52 sin21 (b/4r 1
i ) as follows:
u c 52 sin21
2 p ~ 12 s ! T c0
C ~ 122 s !~ 11 A11 a 0 !
.
~13!
For C.300 K, T c0 .90 K, and DT.1 –2 K, we obtain from
Eq. ~13! that u c .30°–40°.
B. Electron screening in GB’s
The strain decay length d/2p .b/2p u becomes smaller
than b'4 Å at comparatively small angles u .9°, for which
additional mechanisms can contribute to the suppression of
superconductivity near GB’s. We consider here a local redistribution of the carrier density n(x,y) and the shift of the
chemical potential m at the GB’s which both strongly affect
T c in HTS’s. If e (x,y) varies over scales shorter than the
Debye screening length l D 5 @ k ` /4p e 2 N(E F ) # 1/2, the local
electroneutrality assumed in the previous section becomes
invalid. Here k ` .20–30 is the dielectric constant of the
ionic lattice of HTS’s,58 N(E F ) is the density of states on the
Fermi surface, and 2e is the electron charge. Unlike low-T c
superconductors, HTS’s have l D comparable to the coherence length j 0 ,55,56 and so the charge effects are essential, if
the width of current channels in GB’s becomes smaller than
2l D .
For an ideal electron gas in layered materials, the in-plane
screening length l D is independent of the electron density,57
with l D 5( k ` sr B ) 1/2/2 for s! k ` r B and l D 5 k ` r B /2 for s
@ k ` r B . Here r B 5\ 2 /me 2 is the Bohr radius, m is the inplane effective mass, and s is the interlayer spacing. For
HTS’s, the characteristic values of l D 55 –10 Å ~Ref. 56!
become larger than the strain decay length b/2p u for u
.4° –7°. For higher u , the sizes of the NC core regions are
smaller than the thickness 2l D of the space charge layer near
GB’s; and thus the suppression of D on GB’s is mostly due
to the electrostatic potential F(x) near GB’s.
We describe screening around the GB dislocations in the
normal state,63 using the Thomas-Fermi equation
¹ 2 F2F/l 2D 524 p Ze d N ~ x,y ! / k ` ,
~14!
57
CURRENT TRANSPORT THROUGH LOW-ANGLE GRAIN . . .
where Ze is the ion charge and d N(x,y) is the perturbation
of the total ion density N 0 caused by the GB strains. Notice
that for a two-dimensional ~2D! ideal electron gas, the nonlinear corrections to Eq. ~14! are absent.57
The atomic structure of the GB’s in layered HTS’s can be
very complicated, since it depends not only on u , but also on
the way the different ions in adjacent crystalline lattices are
matched on GB’s.29 However, the important qualitative feature of the GB structure is that it has an excess ion charge Q
which is screened by electrons ~holes!. We calculate Q for
low-angle GB’s, for which we can use the continuous elasticity theory for the GB dislocations, and thus express the
excess ion density at the GB’s in terms of measured macroscopic parameters. To do so, we consider the simplest isotropic expansion of d N(x,y) in elastic strain e (x,y),
d N52N 0 ~ e 1 z e 2 ! .
~15!
Here the linear term in e describes the periodic ion density
variation in elasticity theory which gives zero charge after
averaging along GB’s. The quadratic term describes the first
anharmonic correction, where z is the Grüneisen parameter
which determines the thermal expansivity and is usually
about 2 above 20–30 K.59,60 Although small, as compared to
the linear term in Eq. ~15!, the quadratic term is important,
since it has a nonzero mean value Q52N 0 Ze z * ^ e 2 & dx
which gives the excess ion charge per unit area of GB’s due
to reduced ion density in the layer of thickness d/2p near
GB’s:
Q52 z eZN 0
52
E
`
dk
( u e ~ k,G ! u 2 2 p
2` G
z ZeN 0 ~ 122 s ! 2 b 2
16p d ~ 12 s !
2
d
ln ,
ri
F 1 ~ x,y ! 5F a
(
G.0
bolic dependence T c 5T cm 2A(n2n m ) 2 , the uniform shift of
the critical temperature d T c (x)5T c (x)2T c0 along GB’s is
given by d T c 52A ^ F 1 /4p l 2D e & 2 , where ^ & stands for spatial averaging over y. In GL theory the suppression of D near
GB’s is determined by the dimensionless parameter G 1 5
2 * d T c (x)dx/ j 0 T c0 ~see below! which can be obtained from
Eq. ~17! in the form
G 15
AF 2a
@ e 2 u x u G 2Ge 2 u x u G 0 /G 0 # sin Gy, ~17!
where
F a 522 p N 0 Zeb(122 s )l 2D /d(12 s ) k ` ,
G
1/2
52 p n/d, G 0 5(G 2 1l 22
D ) , and n50,61,62, . . . . Here
F 1 (x,y) oscillates along GB’s and exponentially decays
across GB’s on the lengths d/2p n. Therefore, the screening
effects in the first order in e only renormalize the size of the
NC regions calculated in the previous section, since the fluctuation of the carrier concentration d n(x,y)5
2F 1 (x,y)/4p l 2D e varies on the same spatial scales as the
lattice strains e (x,y).
The oscillations of d n(x,y), along GB’s also result in a
uniform suppression of the local T c near GB’s. For the para-
(
~ 21g ! A11g1g 2 22g22
G ~ 11g ! 3/2~ 11 A11g !
2T c0 j 0 G.0
,
~18!
where g5(l D G) 22 . For u .b/2p l D , that is, g!1, the expression under the sum in Eq. ~18! reduces to 5/8l 4D G 5 . Since
F a in Eq. ~18! is proportional to 1/d, the parameter G 1 }d 3
}1/u 3 rapidly decreases with u . Therefore, the first linear
term in Eq. ~15! due to elastic strains cannot give rise to any
progressive depression of D on GB’s with u .
By contrast, the anharmonic contribution F 2 (x,y) contains a nonoscillating term which decays over l D across GB’s
and describes a shift of the mean electrostatic potential
F(x)5 ^ F 2 (x,y) & due to the excess ion charge on GB’s. As
shown in Appendix B, F(x) is given by
F ~ x ! 52Fl 2D
Gm
2Gl D e 2 u x u /l D 2e 22 u x u G
G.0
~ 2l D G ! 2 21
(
,
~19!
where F5 p eZN 0 z b 2 (122 s ) 2 /2d 2 (12 s ) 2 k ` . Since the
sum ~19! diverges logarithmically at large G, we introduced
a cutoff G m ;2 p /r i determined by the size r i of the dielectric core region, where Eq. ~14! becomes invalid. For u
.b/2p l D , the second term in the numerator of Eq. ~19! can
be neglected, and F(x) takes the form
F ~ x ! 5F 0 exp~ 2 u x u /l D ! ,
~16!
where the Fourier component e (k,G) was taken from Appendix B. The total ion charge Q( u ) increases approximately
linear with the GB dislocation density, b/d}sin(u/2). The
nonzero Q results in a uniform shift of the mean F(x) which
decays across GB’s on the screening length l D which is
larger than d/2p in the crucial region of intermediate u .
It is convenient to write F as F5F 1 1F 2 , where F 1 and
F 2 are determined by the linear and quadratic terms in
d N( e ), respectively. The solution of Eq. ~14! for F 1 (x,y)
obtained in Appendix B has the form
13 883
~20!
where F 0 to logarithmic accuracy is given by
F 0 .2
eZN 0 z b ~ 122 s ! 2 l D ln~ d/r i !
4 ~ 12 s ! 2 k `
u
sin .
2
~21!
The amplitude F 0 increases approximately linear with u for
u !1. Taking for YBa2Cu3O7,56,62 s 50.25, ZN 0 5n 0 55
31021 cm23 , and k ` 520, we obtain eF 0 ;6 meV for d
54b54r i , u 520°, l D 58 Å, z 52.
The 2D isotropic equation ~14! gives the simplest selfconsistent description of charge effects near GB’s, although
it does not take into account all characteristic features of the
electron band structure of HTS’s and nonlocal electron
screening. The nonlocality increases the effective screening
length for charge fluctuations with the wave vector k comparable to the 2D Fermi wave vector k F .(2 p n 0 s) 1/2.57 Due
to the low carrier density n 0 , the condition k;2 p /d;k F is
characteristic of GB’s in HTS’s, resulting in the effective
screening length l D which can substantially exceed its
Thomas-Fermi value. Thus, the electric potential F(x) near
GB’s becomes more long range, varying on lengths of the
order of the mean spacing between carriers ;n 21/3
which is
0
about 6 Å for optimally doped YBa2Cu3O7 (n 0 .5
31022 cm23 ). We do not consider here the more complicated nonlocal screening near GB’s, restricting ourselves to a
qualitative description in the framework of the Thomas-
A. GUREVICH AND E. A. PASHITSKII
13 884
57
Fermi approach with an effective l D extracted from observed
strong electric field effects in HTS’s which indicate that l D
.5 –10 Å.56
The nonlocality also makes F(x) dependent on details of
the electron band structure of HTS’s, such as the extended
saddle point singularities near the Fermi level and planar
portions on the Fermi surface along the @100# and @010# directions in the Brillouin zone.64–67 For s@ k ` r B , the
screened potential F(x,y) is long range57 (F}1/r 3 ) and also
exhibits Friedel’s oscillations,63,59
F~ x !}
c 0 cos Qx 2 u x u / j
1,
e
uxu
Q u x u .1,
~22!
along the a or b axes. Here Q is the nesting wave vector
between the neighboring planar portions of the Fermi surface, which is Q'0.3 Å21 for optimally doped HTS’s.64–66
This gives the period of the oscillations, l f 52 p /Q.20 Å.
The factor exp(2uxu/j1) accounts for the damping of the Friedel oscillations in the superconducting state, where j 1 (T)
weakly depends on temperature and approximately equals
the superconducting coherence length j (0) at T50.68,69 Impurities do not affect the Friedel oscillations, if the electron
mean free path l i is much larger than j 0 . The case l i . j 0 is
characteristic of HTS materials which usually correspond to
the clean limit j 0 !l i . Therefore, the length l D ; j 1
.10–15 Å can be regarded as an intrinsic effective thickness over which the order parameter is depressed near the
GB’s.
C. Heterogeneity scales of the T c suppression on GB’s
Now we consider how the heterogeneity of the GB’s can
affect superconducting properties. The NC regions, spaced
by d, consist of an insulating domain of size 2r i surrounded
by metallic shell in which the local superconducting coupling constant l is suppressed on the scale of a few atomic
spacings. Below T c , these small metallic core regions (r 1
, j ) become superconducting due to the proximity effect
and therefore transparent to the transport current. However,
the dielectric core regions are not affected by the proximity
effect and remain stronger barriers for the current flow. For
the ideal GB’s shown in Fig. 1, the size r i calculated above
in the linear elasticity theory decreases with u due to the
compensation of strain fields in GB’s as d decreases ~see
Appendix A!. However, if r i ( u ) becomes comparable with
b, the decrease of r i is limited by plastic effects near dislocation cores.22,72 The local suppression of D on GB’s due to
the NC regions is determined by the parameter42
G 1;
L 2n
dj0
u
}sin ,
2
~23!
which is a fraction occupied by the normal NC regions in the
layer of thickness j 0 near GB’s. These normal regions give
rise to the proximity effect suppression of D linear in the GB
dislocation density.
For u .b/2p l D , the NC regions become smaller than the
screening length l D ; thus the superconductivity suppression
on GB’s is mostly due to the shift F 0 of the averaged electrical potential on GB’s. This results in the opposite shift of
the chemical potential m by m 0 52eF 0 to provide a con-
FIG. 5. Suppression of the order parameter near GB’s due to the
variation of the chemical potential in the layer of thickness 2l D
, j (T). The dashed curve depicts the energy of the extended saddle
point singularity in N(E).
stant electrochemical potential m e 5 m 1eF. However, T c of
HTS’s is generally quite sensitive to small shifts of m , since
comparatively small changes of the in-plane hole concentration c can cause the transition into an insulating
antiferromagnet50 ~see Fig. 3!. Due to the small Fermi energy
in HTS’s, E F ;0.1 eV, the above-estimated shift of m on
GB’s by eF;5 –10 meV;T c appears to be of the order of
the change of m which can cause strong local suppression of
D.50,67,70,71
These qualitative estimates indicate that the charge effects
can provide a universal mechanism of the local superconductivity suppression on GB’s. This mechanism is amplified by
the extended saddle point singularities in the HTS electron
density of states N(E) observed in photoemission experiments at energies E52 m 1 '20–30 meV below the Fermi
level.64–67 The resulting singularity in N(E) near the Fermi
surface not only can enhance the bulk T c , 70,71 but also can
make T c sensitive to any shifts of m of order m 1 . Since eF 0
on GB’s turns out to be comparable to the m 1 , the shift of the
peak in N(E) away from its presumably optimum position at
m 1 can further suppress D in the layer of thickness 2l D
around GB’s, as shown in Fig. 5.
We estimate the suppression of D(x) due to the local
variations of F(x) in GL theory which has been used to
describe D(x,y) near various structure defects for both sand d-wave symmetries of the order parameter74 and in the
presence of extended saddle point singularities.71 We illustrate here the essential physics by the simplest s-wave GL
equation, since J c ( u ) calculated below turns out to be only
weakly sensitive to the symmetry of D in the region of small u .
The GL equation with a locally nonuniform superconducting
coupling constant l(r) has the form73
F
G
T c ~ r! D 2
2 2 D50.
T
D0
j 20 ¹ 2 D1 ln
~24!
Here j 0 and D 0 are of the order of the zero-temperature
coherence length j (T)5 j 0 / At and the gap D(T)5 At D 0 ,
respectively, and t 5(T c0 2T)/T c0 !1 near the bulk T c0 .
The value j 0 5( f 0 /2p H 8c2 T c ) 1/2 can be extracted from the
observed temperature derivative of the upper critical field
H 8c2 at T c , giving j 0 '13 Å for YBa2Cu3O7 (T c 590 K,
H 8c2 '2 T/K!. Taking the BSC-type dependence T c
57
CURRENT TRANSPORT THROUGH LOW-ANGLE GRAIN . . .
5v0exp(21/l) and assuming that T c is mostly determined
by the local depression of l near GB’s, we can rewrite Eq.
~24! in the form
F G F
j 20 ¹ 2 D1 t 2
D2
D 20
D5
G
1
1
2
D,
l ~ r! l `
~25!
where l ` is the coupling constant away from GB’s. The
right-hand side of Eq. ~25! describes the localized perturbation V(r)D due to the suppression of l(F) near GB’s. Assuming a smooth dependence of l on m , we get V(x)5
2eF(x)( ] l ` / ] m )/l 2` , if eF 0 ( u ) is smaller than the critical shift of the chemical potential m c which causes the
superconductor-insulator transition. Since F 0 on GB’s increases with the dislocation density, this mechanism provides the progressive decrease of D on GB’s with u , which,
in turn, gives rise to the rapid decrease of J c ( u ) considered
in the next section. The angular dependence J c ( u ) is thus
determined by the domelike dependence l( m ) characteristic
of HTS’s.
Therefore, besides the spacing between the GB dislocations, there are two additional intrinsic length scales associated with GB’s. The first one is the size r i .b of the insulating core regions which provide strong barriers for current
flow. The second scale is the width 2l D of a layer of suppressed D across GB’s which is determined by the charge
effects.
III. CRITICAL CURRENT
Now we calculate J c for a uniform current flowing perpendicular to GB’s, by solving the GL equation
j ¹ c1c2c 2
2
2
3
i2
c3
5V ~ c ,r! ,
~26!
where c 5D/D b is the order parameter normalized to its bulk
value D b 5D 0 At , i5J/J 0 is the dimensionless transport current density away from GB’s, J 0 5c f 0 /16p 2 l 2L j is of the
order of the depairing current density, and l L (T)5l L0 / At
is the London penetration depth.
The left-hand side of Eq. ~26! is the usual GL equation in
the presence of uniform current,73 and the right-hand side
describes the perturbation caused by GB’s,
V5
F
G
i 20 ~ r! 2i 2
1
1 1
2
c1
,
t l ~ r! l `
tc 3
~27!
where i 0 (r) is the normalized current density near GB’s,
where both i 0 (r) and l(r) are strongly inhomogeneous due
to the local T c suppression and the channel structure of
GB’s. The first term on the right-hand side of Eq. ~27! describes the local T c suppression across the GB’s and the
modulation of D(x,y) along GB’s caused by the NC regions.
The second term in Eq. ~27! results from the decrease of the
GB current-carrying cross section by the insulating parts of
the NC regions. This gives rise to a local concentration of
j(x,y) in the current channels, which further depresses D.
Equation ~26! describes the distribution of D(x,y) for the
proximity coupled NC regions. The general case of d
. j (T) is very complicated; however, the situation simplifies
13 885
if the effective GB thickness l 0 52l D and the dislocation
spacing d are much smaller than j (T) @which corresponds to
u .b/ j (T);5° for YBa2Cu3O7 at 77 K#. In this case the
particular shape of the NC regions becomes less essential,
because D(x,y) is mostly determined by V(x,y) averaged
along GB’s.42 Since l D ! j (T) near T c , the localized potential V(x,y) can be written as
V5V 0 d ~ x ! ,
V 05
E
`
2`
^ V ~ x,y ! & dx,
~28!
where ^ ••• & stands for the spatial averaging along GB’s, and
d (x) is the delta function. It is convenient to rewrite Eq. ~26!
in the following dimensionless form:
c 9 1 c 2 c 3 2i 2 / c 3 2G ~ c ,i ! d ~ h ! 50,
~29!
where the prime denotes differentiation with respect to h
5x/ j (T) and G( c ,i)5V 0 ( c ,i)/ j (T). Equation ~29! can be
solved exactly for any dependence G( c ).75 The solution
c ( h ) obtained in Appendix C has the form
c 2 ~ h ! 5 c 2m 1 ~ c 2` 2 c 2m ! tanh2
F
~ u h u 1 h 0 ! Ac 2` 2 c 2m
A2
G
.
~30!
Here c ` and c m are given by
i 2 5 c 4` ~ 12 c 2` ! ,
c 2m 52i 2 / c 4` .
~31!
The first equation in Eq. ~31! describes the suppression of c
by uniform current,73 and h 0 is defined by c (0)5 c 0 . The
order parameter c 0 on GB’s satisfies the following equation:
2E2 c 20 1
c 40 i 2 G 2 ~ c 0 !
2 25
,
2
4
c0
~32!
where E5 c 2` 23 c 4` /4. For i50, Eqs. ~27!, ~31!, and ~32!
yield G( c )5G 1 c , c ` 51, c m 50, and
h 0 5 A2 tanh
21
SA
11
G 21
8
2
G1
2 A2
D
.
~33!
The distribution c (x) described by Eqs. ~30! and ~33! is
shown in Fig. 6, where c 0 512G 1 /2A2 at G 1 !1, and c 0
5A2/G 1 for G 1 @1.
For a qualitative analysis of c (x,i) at i.0, it is convenient to use an analogy of Eq. ~29! with the equation that
describes the classical motion of a particle of the unit mass
and energy E in the potential
U~ c !5
i2
c2 c4
2 1
,
2
4
2c2
~34!
where c and h play the role of the particle coordinate and
time, respectively. The term G( c ) in Eq. ~29! describes an
elastic reflection of the particle at c 5 c 0 . We seek a symmetric trajectory c ( h ) of the particle which starts moving
from point b with an infinitesimal velocity and c 5 c ` , then
gets reflected at c 5 c 0 , and comes back to b with zero velocity. The graphic solution of Eq. ~32! for c 0 is shown in
Fig. 7, where c 0 corresponds to the intersection points of the
A. GUREVICH AND E. A. PASHITSKII
13 886
57
FIG. 6. Distribution c ( h ) near GB’s described by Eqs. ~30! and
~33! for i50 and G 1 510. The coordinate x is normalized by j (T).
curves 2G 2 ( c )/4 and U( c )2E. As seen from Fig. 7, Eq.
~32! has several roots, of which only those with d c 0 /di
,0 are stable.75 The only stable solution s in Fig. 7 disappears at i.i c , where i c determines the critical current density through GB’s which is basically the depairing current
density for the self-consistent distribution ~30!.
Now we consider G( c ) which follows from Eq. ~27!:
G ~ c ! 5G 1 c 1G 2 i 2 / c 3 .
~35!
The parameter G 1 , which describes the local suppression of
T c on GB’s, increases with u due to the domelike dependence of the superconducting coupling constant l @ m
1eF( u ) # in HTS’s on the chemical potential. We do not
discuss here specific dependences l( m ) for various microscopic models published in the literature, focusing instead on
a phenomenological description of J c ( u ) which can be obtained by expanding l.l ` 1( ] l ` / ] m )eF 0 in Eq. ~27! for
small eF 0 . This gives G 1 which increases approximately linearly with the GB dislocation density:
G 15
u
sin .
l ` m c j 0 At 2
21
2l D eF m
FIG. 7. Graphic solution of Eq. ~32! for J,J c ~a! and J5J c ~b!.
The particle of energy E moves in the potential U( c ), reflecting in
the intersection points of U( c )2E ~solid curve! and 2G 2 ( c )/4
~dashed curve!.
cores and in the layer of thickness .4r i around GB’s, where
r i is an effective radius of the NC regions. Then G 2 can be
written in the form
~36!
Here m c 5 u ] lnl ` / ] m u
is of the order of the shift of m
which causes the S-I transition, and the amplitude F m is
defined by u F 0 ( u ) u 5F m sin(u/2), where F 0 ( u ) is given by
Eq. ~21!. The nonlinearity of the function T c ( m ) could be
taken into account by assuming the conventional parabolic
dependence T c (x)5T c0 @ 12a 1 F(x)2a 2 F 2 (x) # in Eq. ~24!,
where a 1,2 are constants. This results in a nonlinear and fairly
cumbersome dependence of G 1 } * ln(Tc /Tc0)dx on sinu which
we do not consider here, since the much simpler Eq. ~36!
based on the linear expansion of l( m ) already provides a
good description of the observed J c ( u ) ~see below!.
The parameter G 2 in Eq. ~35! accounts for the decrease of
the current-carrying cross section of GB’s by the insulating
NC regions. In this case the local current density is enhanced
in the superconducting channels between the dislocation
G 25
4r i
F
d2
j 0 At ~ d22r i !
2
G
21 5
4r i n ~ 22 n !
j 0 At ~ 12 n ! 2
.
~37!
Here the term d 2 /(d22r i ) 2 describes the local enhancement
of i 2 in the current channels, and
n5
sin~ u /2!
,
sin~ u c /2!
~38!
where u c is the critical angle, u c 52 sin21(b/4r i ) at which the
insulating NC regions overlap.
For arbitrary G 1 , G 2 , the critical current i c is determined
by two cumbersome algebraic equations for c 0 (i) and i c . A
simpler case G 2 50, G 1 .0 corresponds to a uniform Josephson contact with suppressed order parameter, for which
J c is determined by a single algebraic equation given in Appendix C @numerical simulation of Eqs. ~29! and ~35! for
CURRENT TRANSPORT THROUGH LOW-ANGLE GRAIN . . .
57
13 887
G 2 50 was performed by Campbell36#. Hereafter we focus on
the most interesting case G 1 @1, G 2 .0, for which a GB
exhibits weak-link behavior. As follows from Eq. ~36!, the
condition G 1 @1 does hold, since 2l D . j 0 , t !1 and
eF 0 /l ` m c .1 for weak coupling. If G 1 @1, then i 2 !1, and
so we can put c 2` 512i 2 and tanh2(h0 /A2)5 c 20 22i 2 in
Eqs. ~30! and ~31!. In this case Eq. ~32! reduces to ~see
Appendix C!:
a5
u 4 22u 3 12qu 2 1 a 2 q 2 50,
~39!
G 1G 2
,
21G 1 G 2
~40!
q5 ~ 21G 1 G 2 ! G 21 i 2 ,
where u5 c 20 . A graphic analysis of Eq. ~39! similar to that
in Fig. 7 shows that the stable solution u s (q) exists only
below the critical value q,q c which determines J c . At q
5q c points s and u in Fig. 7 merge at u5u c , and the stable
solution disappears, if q.q c . The value u c can be found by
differentiating Eq. ~39! with respect to u, giving q c
5u c (3/22u c ). Substituting this q c into Eq. ~39! and solving
a quadratic equation for u c ( a ), we calculate q c 5u c (3/2
2u c ) and finally obtain J c in the following scaling form:
J c5
F
J m 129 a 2 1 ~ 113 a 2 ! 3/2
n
~ 11 a !~ 12 a 2 !
J m5
a5
J 0 b At
2 A2 b j 0 sin~ u c /2!
G
1/2
,
~42!
,
b n 2 ~ 22 n !
2 t ~ 12 n ! 2 1 b n 2 ~ 22 n !
~41!
~43!
.
Here the dependence of a on n and t was found, using Eqs.
~36!, ~37!, and ~40!. The control parameter b
52l D beF m / j 20 m c l ` , which determines how fast J c ( u ) decreases with u , can be obtained from Eq. ~21!:
b5
F
G
e 2 n 0 z ln~ b/r i u ! bl D ~ 122 s ! 2
.
2 k `m cl `
j 0 ~ 12 s !
~44!
Notice that for G 1 @1, the parameter G 2 enters Eq. ~41! only
in the combination G 1 G 2 , and so even a weak blockage of
current by the NC core regions (G 2 !1) can markedly suppress J c , if G 1 G 2 .1. For G 1 G 2 @1, Eq. ~41! yields
J c5
3 A3J 0
8G 1 AG 1 G 2
J 15
5J 1
t 5/2~ 12 n !
n 2 ~ 22 n ! 1/2
3 A3bJ 0 ~ 0 !
8 b 3/2j 0 sin~ u c /2!
,
,
~45!
~46!
where J 0 (T)5J 0 (0) t 3/2. The suppression of J c due to the
current channels manifests itself in the additional factor
AG 1 G 2 .1 in Eq. ~45!, as compared to J c ;J 0 /G 1 for G 2
50 and G 1 @1 ~see Appendix C!. Equation ~45! corresponds
to the GL temperature region T.T c in which J c } t 5/2 rapidly
decreases with T and u . The temperature dependence ~45!
differs from the J c } t for S-I-S Josephson contacts and is
rather
consistent
with
the
observed
quadratic
FIG. 8. Fit with the experimental data on thin film @001# tilt
YBa2Cu3O7 bicrystals with b 52.7 and u c 530°. Curves s and d
correspond to the s and d pairings, respectively. They are related by
Eq. ~47!, where J c ( u ) for the s pairing is given by Eq. ~41!. The
dashed line shows the function exp(2u/u0) with u 0 54°.
dependence,5,76,77 J c } t 2 characteristic of S-N-S contacts.78
In this model which neglects the Josephson tunneling
through insulating dislocation cores, J c ( u ) vanishes at u
5 u c . When taking into account the tunneling through the
insulating NC regions at u . u c , the critical current remains
finite, though its angular dependence J c ( u ) might change as
compared to u , u c .
IV. COMPARISON WITH EXPERIMENT
AND DISCUSSION
Equation ~41!, which gives the explicit dependence of J c
on u and T, contains three control parameters b , u c , and J m .
When comparing the calculated J c ( u ,T) with experiment,
the amplitude J m can be eliminated by normalizing J c ( u ) to
its value J c ( u 0 ) for a certain misorientation angle u 0
.3° –5° at which J c ( u 0 ) equals J g in the grains. Small u
, u 0 correspond to a plateau on the observed J c ( u ) curves in
Fig. 8 in the region between u 0 and the base data point
logi50 at u 50, where J c ( u ) values through GB’s are hardly
accessible in resistive experiments. Thus, we are left with
two parameters b ;1 and u c .20° –40° which can be extracted from the best fit with experimental data, which are
usually well described by the exponential dependence J c
}exp(2u/u0) with u 0 .4° –5°. The fact that all uncertain
microscopic parameters near GB’s, l D , m c , z k ` , and l ` ,
can be combined in a single parameter b markedly simplifies
the comparison of our model with experiment.
As an illustration, Fig. 8 shows the observed J c ( u ) for
YBa2Cu3O7 bicrystals7 at 77 K similar to analogous J c ( u )
dependences for other thin film and bulk HTS
bicrystals.1–3,5,6 Given the typical scatter of experimental
data, the fit, which corresponds to b 52.7 and u c 530°, turns
out to be quite good. The algebraic function ~41! describes
well the apparent quasiexponential decrease of J c ( u ) by two
to three orders of magnitude in the region 0, u ,30°. The
calculated J c ( u ) also appears to be rather close to the
13 888
A. GUREVICH AND E. A. PASHITSKII
straight dashed line in Fig. 8 which corresponds to J c ( u )
5J 0 exp(2u/u0) with u 0 54°.
So far we have not specified the symmetry of the order
parameter, since the final J c ( u ) dependence is in fact only
weakly sensitive to the symmetry of D at u , u c . For instance, the J (d)
c ( u ) for d-wave pairing is related to the
(
u
)
as follows:
s-wave J (s)
c
J ~cd ! ~ u ! 5cos2 2u J ~cs ! ~ u ! ,
~47!
where the factor cos2 2u accounts for the matching conditions of the superconducting d-wave gap function on
GB’s.9,80 Equation ~47! also implies that the gap suppression
on GB’s is weakly sensitive to the symmetry of the order
parameter, as was assumed by Hilgenkamp et al.10 As follows from Fig. 8, the factor cos2(2u) only gives a relatively
small correction to the much stronger decrease of J c ( u )
caused by the local superconductivity suppression near
GB’s.
Since Eq. ~41! describes well the observed angular dependence J c ( u ), we can independently check the selfconsistency of the model by comparing the absolute value of
J c given by Eq. ~45! with experiment. Likewise, the parameter b extracted from the fit can be compared with what
follows from Eq. ~44!. For b 52.7, u c 530°, b54 Å, j 0
513 Å, l L0 51500 Å, t 51/7, and u 515°, we obtain, from
Eqs. ~45! and ~46!, that J c ;23105 A/cm2 , in agreement
with typical observed J c values for @001# 15° tilt YBa2Cu3O7
thin film bicrystals at 77 K.3,6,7 In order to evaluate b , we
take d54b54r i , u 55°, l D 58 Å, j 0 513 Å, s 50.25, z
52, and n 0 5531021 cm23 . Then Eq. ~44! gives b 52.7 for
l ` m c .9 meV.T c . For the 15° GB, the d-wave correction
in Eq. ~47! reduces J c by 25%.
The critical angle u c lies in the region 20° –40°, if r i .b
@see Eq. ~13!#. A more accurate estimate for u c is hard to
obtain, not only because the detailed shape of the NC core
regions is sensitive to the local parameters near GB’s which
are not well known, but also because u c depends on the
atomic structure of the plastically deformed and compositionally different dislocation cores in HTS’s.24–29 However,
the dependence ~41! is only weakly affected by the uncertainty in u c , if u is not too close to u c . The parameter b
which determines the superconductivity suppression on
GB’s, increases for layered HTS materials which have a low
carrier density, short coherence length j 0 , and large screening length l D . Notice that the local superconducting properties on GB’s are determined not only by atomic displacements near GB’s seen by electron microscopy,24–29 but rather
by the resulting variations of the local density of states
N(E F ,r) along the GB’s. The variations of N(E F ,r) could
be revealed by scanning tunnel microscopy, thus providing
very important information on the electron structure of the
insulating NC regions and the space charge layer near GB’s.
In this paper we considered the region of high temperatures T c 2T!T c , in which many uncertainties of microscopic mechanism of high-T c superconductivity can be effectively treated by the universal GL equations. The
description of GB’s at lower temperatures is more model
dependent and requires much more complicated Eilenberger’s or Bogolubov–de Gennes equations.79–81 However, the
similarity in J c ( u ,T) dependences observed on GB’s at high
57
and low temperatures allows us to assume that our model
may give a qualitative description of the low-T region as
well. Yet there are new features characteristic of the d-wave
symmetry of D, such as localized zero-energy electron states
at the S-I interface, which may contribute to the tunneling J c
of Josephson contacts in anisotropic HTS’s.79
When calculating J c , we assumed uniform current flow
through GB’s which is characteristic of thin film bicrystals of
thickness L z much smaller than the Josephson penetration
depth l J 5(c f 0 /16p 2 l L J c ) 1/2, where c is the speed of light
and f 0 is the flux quantum. For bulk bicrystals, with L z
@l J , the transport current only flows in the peripheral layer
of thickness l J , and so the total critical current through
GB’s equals I c 5 Pl J J c , where P is the perimeter of the
GB’s.82 Thus the mean critical current density J b 5J c /A,
normalized to the cross-sectional area A, is given by
J b5
~ c f 0 J c ! 1/2P
4 p A Al L
~48!
.
Equation ~48! corresponds to an isotropic superconductor.
For an anisotropic slab with A5L y L z and London penetration depths l Ly and l Lz along the y and z axes, respectively,
the factor P/A Al L should be replaced by
P
A Al L
→
2
L y Al Lz
1
2
L z Al Ly
.
~49!
The apparent J b in bulk bicrystals depends on the sample
geometry and is strongly reduced by the factor Pl J /A!1,
as compared to the local J c . At the same time, J b ( u )
}J 1/2
c ( u ) exhibits weaker dependence J b }exp(2u/2u 0 ) than
J c ( u ) at u , u b , where u b is defined by l J ( u b )
.min(Lz ,Ly). At u b . u b , the slope of lnJc(u) increases by 2
times, since, for u . u b , current flows uniformly through
GB’s; and thus J b ( u ).J c ( u ).
In our model, the strong decrease of J c ( u ) with u is due
to the excess ion charge Q( u ) of the GB dislocation structure. This was shown for the simplest anharmonic correction
z e 2 in an isotropic approximation characterized by the single
constant z . For orthorhombic crystalline symmetry, the term
z e 2 turns into the quadratic invariant z a e 2aa 1 z b e 2bb
1 z 1 e aa e bb 1 z 2 e 2ab which is proportional to the elastic energy density in the Grüneisen approximation. The crystalline
anisotropy somewhat complicates the above analysis, but it
does not change the key point that Q( u ) increases with the
GB dislocation density b/d}sin(u/2), resulting in a shift of
the chemical potential and the superconductivity suppression
on GB’s. This was obtained in the framework of the continuous elasticity theory which can be used for low-angle GB’s
with d@b. The discreteness of the crystalline lattice and broken atomic bonds along GB’s can affect Q for higher u and,
in principle, could result in dips in Q( u ) for certain symmetric misorientations. This might pertain to the non-weak-link
behavior observed on some high-angle GB’s.83,84 The atomic
displacements on GB’s which give rise to the nonzero Q( u )
could also be independently obtained by scanning electron
tunnel microscopy,24,29 which would enable one to calculate
Q( u ) and J c ( u ) for high-angle GB’s as well.
57
CURRENT TRANSPORT THROUGH LOW-ANGLE GRAIN . . .
Charge effects in HTS’s have recently attracted much attention due to a possibility to affect T c by applying strong
electric field E, which can have various applications in superconducting electronics.56 As was shown above, the significant suppression of J c ( u ) can be due to localized E(x) of
the charged GB dislocation structure which shifts the chemical potential on GB’s as u increases. This is similar to the
suppression of the order parameter on GB’s by external gate
voltage in HTS transistors, resulting in a significant change
of J c of GB’s observed in experiment.56 Another indication
of the importance of the charge effects on GB’s follows from
electromigration experiments33 which showed a noticeable
change of J c across GB’s after applying pulse electric fields
at T.T c . This is consistent with our model in which even
subtle changes in the ionic structure on GB’s due to electromigration can significantly affect Q and thus J c .
Summarizing the obtained results, we can point out the
following main factors which can contribute to the low J c
values of GB’s in HTS materials.
~1! The proximity of the HTS transition to the metalinsulator transition makes the order parameter on the GB’s
sensitive to small shifts of the chemical potential caused by
local excess ion charge on structural defects screened by
electrons ~holes!. This can result in the strong superconductivity suppression near the GB amplified by the extended
saddle point singularities in the electron density of states of
HTS’s.
~2! The antiferromagnetic insulating phase in the HTS
phase diagram manifests itself in the insulating core regions
caused by large strains near the GB dislocation cores and
charge effects on the GB’s. These insulating core regions are
strong barriers for current flow which also significantly enhance the effect of the local superconductivity suppression in
the current channels at J.0.
~3! The suppression of the superconducting coupling constant occurs in the double-charge layer near GB’s comparable to the coherence length j 0 . This results in a large effective thickness of the GB’s, which is also specific to HTS’s
due to their short j 0 and large l D . By contrast, GB’s in lowT c superconductors with l D ! j 0 are not strong barriers for
current flow.
can hardly change qualitatively the general quasiexponential
dependence of J c ( u ) on u which is mostly determined by the
segments of GB’s with maximum J c values.
Another important issue concerns the dependence
J c ( u ,H) in the magnetic field H, which is determined not
only by the local current transport through GB’s, but also by
depinning of vortices localized on GB’s. The latter depends
on the change of the core structure of vortices on GB’s and
their magnetic interaction with strongly pinned bulk
vortices.86 Here the charge effects on GB’s can also essentially contribute to the vortex pinning.55
ACKNOWLEDGMENTS
This work was supported by NSF MRSEC Program No.
DRM 9214707. The authors are grateful to D.C. Larbalestier
for his stimulating interest and helpful discussions. A.G. also
thanks D.K. Christen and D.O. Welch for useful comments.
APPENDIX A
Here we give necessary formulas22 for the elastic strain
tensor e ik . For a single dislocation, e ik is given by
e xx 5
e yy5
by @~ 322 s ! x 2 1 ~ 122 s ! y 2 #
4 p ~ 12 s !~ x 2 1y 2 ! 2
by @~ 122 s ! y 2 2 ~ 112 s ! x 2 #
4 p ~ 12 s !~ x 2 1y 2 ! 2
e xy 52
bx ~ x 2 2y 2 !
4 p ~ 12 s !~ x 2 1y 2 ! 2
,
~A1!
,
~A2!
.
~A3!
For a symmetric GB, e ik can be obtained by replacing y by
y2nd and summing up over the integer n. This yields
e xx 1 e y y 5
e xx 2 e y y 5
V. CONCLUDING REMARKS
In this paper we mostly focused on those general features
of the current transport through low-angle GB’s which are
rather insensitive to the microscopic mechanism of superconductivity, the symmetry of the order parameter, and detailed
structure of the dislocation core regions. Our model, which
links the dependence J c ( u ) with normal properties and the
phase diagram of HTS’s, describes well the observed strong
dependence of J c ( u ) on misorientation angle even for the
simplest ideal GB dislocation structure shown in Fig. 1. The
partial dislocation structure of GB’s in HTS’s ~Ref. 31! increases the effective thickness of GB’s, thus making the
charge effects and the suppression of D on GB’s more pronounced. Macroscopic inhomogeneites of GB’s are due to
local nonstoichiometry, faceting, and resulting long-range
strain fields which can subdivide GB’s into weakly coupled
segments connected in parallel.85 Though important for current percolation in polycrystalline HTS’s,13,16 the faceting
13 889
e xy 52
e b ~ 122 s ! sinq
,
coshp2cosq
e b psinqsinhp
~ coshp2cosq ! 2
~A5!
,
e 1 ~ cosqcoshp21 ! p
2 ~ coshp2cosq ! 2
~A4!
,
~A6!
where e 1 5b/2(12 s )d, p52 p x/d, and q52 p y/d.
Using Eqs. ~3! and ~A4!–~A6!, we can write the equation
T c (x,y)50 for the NC core regions in the form
~ cosh p2cos q ! 2 h5 ~ cosh p2cos q ! sin q
1p 0 @ psin qsinh pcos 2w
2 u p u ~ cosh pcos q21 ! sin 2w # ,
~A7!
and
p 0 5(C a
where
h52d(12 s )T c0 /Cb(122 s )
2C b )/(122 s )(C a 1C b ). Equation ~A7! was used to calculate the shapes of the NC core regions in Fig. 2.
13 890
A. GUREVICH AND E. A. PASHITSKII
For the isotropic strain dependence of T c ( e ik ) (p 0 50),
Eq. ~A7! gives the explicit dependence x(y) in the form
x5
F
G
d
2py 1 2py
1 sin
cosh21 cos
.
2p
d
h
d
x6
n 5
F
G
2py
d
2py 2pr6
n
1
sin
cosh21 cos
.
2p
d
d
d
F
2pr6
d
n
1
ln
2p
d
AS
2pr6
n
d
D G
d 21 2 p r 6
n
.
y6
5
tan
m
p
d
~A10!
~A11!
The NC regions shrink as d decreases. In the linear elasticity
theory the NC regions touch only if d→0 when y 6
m →d/2.
However, for small d the elastic approximation becomes invalid, and the shape of the NC regions is determined by
plastic and charge effects near GB’s.
¹ 2 F 1 2F 1 /l 2D 54 p ZeN 0 e / k ` ,
~B1!
E
`
dk
F ~ k,G ! e ikx1iGy ,
2` 2 p
~B2!
where G52 p n/d is the reciprocal lattice vector along GB’s.
The Fourier component e (k,G) equals
ibG ~ 122 s !
2d ~ 12 s !~ k 2 1G 2 !
.
~B3!
Therefore, F 1 (k,G) is given by
F 1 ~ k,G ! 52
2 p iN 0 Zeb ~ 122 s ! G
d ~ 12 s !~ k 2 1G 2 !~ k 2 1G 2 11/l 2D ! k `
.
~B4!
Substituting Eq. ~B4! into Eq. ~B2! and integrating over k,
we arrive at Eq. ~17!.
Now we consider the equation for F 2 ,
¹ 2 F 2 2F 2 /l 2D 54 p ZeN 0 z e 2 / k ` ,
from which we obtain
~B6!
e z ZN 0
p k ` G,G 8
(
3
E E
`
2`
dk
`
2`
dk 8
e ikx1iGy e ~ k 8 ,G 8 ! e ~ k 8 2k,G 8 2G !
k 2 1G 2 11/l 2D
.
~B7!
The averaged in y potential F(x)5 ^ F 2 (x,y) & is given by
the term with G50 in Eq. ~B7!, whence
F~ x !5
z ZeN 0 b 2 ~ 122 s ! 2
4 p d ~ 12 s ! k `
2
3
2
(
G
8
E E
`
2`
dk
`
2`
dk 8
e ikx G 8 2
~ k 2 11/l 2D !~ G 8 2 1k 8 2 !@ G 8 2 1 ~ k 8 2k ! 2 #
.
~B8!
Performing integrations in k and k 8 , we arrive at Eq. ~19!.
APPENDIX C
c 8 ~ 6` ! 50,
c 8 ~ 10 ! 52 c 8 ~ 20 ! 5G ~ c 0 ! /2.
~C1!
~C2!
From Eq. ~29!, we obtain the integral of ‘‘energy:’’
can be solved by the Fourier transformation,
e ~ k,G ! 5
F 2 ~ x,y ! 5
c ~ 6` ! 5 c ` ,
The equation for F 1 ,
(G
~ k 8 ,G 8 ! e *
We seek for the solutions of Eq. ~29! with the following
boundary conditions:
APPENDIX B
F ~ x,y ! 5
Ee
(
~ k 1G 2 11/l 2D ! k ` G 8
Here an asterisk implies the complex conjugate, and e (k,G)
is given by Eq. ~B3!. In the coordinate representation we get
2
11 ,
2e z ZN 0
2
3 ~ k 8 2k,G 8 2G ! dk 8 .
~A9!
The boundary of the dielectric NC region can be obtained by
1
replacing r 6
n by r i . The maximum sizes x m and y m of the
NC regions across and along GB’s are given by
x6
m5
F 2 ~ k,G ! 52
~A8!
For the isotropic case, we can also obtain the shape of the
normal NC regions x(y) for the nonlinear dependence T c ( e ).
Substituting Eqs. ~A4!–~A6! into Eq. ~7! and solving the
quadratic equation T c ( e )5T for e , we get
57
~B5!
1 2
c 8 1U ~ c ! 5E,
2
U~ c !5
i2
c2
c4
1
,
2
2
4
2c2
~C3!
~C4!
where E5U( c ` ). The boundary condition ~C2! implies that
the particle moving in the potential U( c ) undergoes an elastic reflection at c 5 c 0 . Equation ~32! for c 0 can be obtained
for any G( c ) by substituting Eq. ~C2! into Eq. ~C3! at c
5 c 0.
Equation ~32! has several roots which correspond to
stable (s) and unstable (u) distributions c ( h , c 0 ) in Fig. 7.
Due to the symmetry c ( h )5 c (2 h ), the solution c ( h ,G)
for G.0 can be obtained from the solution c ( h ,0) of Eq.
~29! with G50 and the same boundary conditions at h 6`
by the following rule:
c ~ h ,G ! 5 c ~ u h u 1 h 0 ,0! .
~C5!
Here the constant h 0 is chosen to satisfy
c 0 5 c ~ h 0 ,0! ,
~C6!
CURRENT TRANSPORT THROUGH LOW-ANGLE GRAIN . . .
57
where c 0 is determined by Eq. ~32!.
The value c m is determined by the condition of energy
conservation, U( c ` )5U( c m ), which enables us to factorize
Eq. ~C3! in the form
c8
2
5 ~ c 2` 2 c 2 ! 2 ~ c 2 2 c 2m ! /2c 2 .
~C7!
Equation ~C7! automatically provides the correct boundary
conditions c 8 (0)5 c 8 (6`)50 for c ( h ,0). Here h 50 corresponds to the value c 5 c m which can be found by comparing Eqs. ~C3! and ~C7!. This gives a simple relationship
between c m and c ` :
c 2m 52i 2 / c 4` 5222 c 2` .
S
i2
1
G 1 c 0 1G 2 3
2
c0
D
2
5
1
c 20
~ c 2` 2 c 20 ! 2 ~ c 20 2 c 2m ! .
v2 w
1 5e,
2
v
~C10!
where v 5 c 20 /(11G 21 /4), w5i 2 /(11G 21 /4) 3 , and e52E/(1
1G 21 /4) 2 . The cubic equation ~C10! has three roots of which
only one with d v /di,0 corresponds to the stable c ( h ). As
seen from Fig. 7, the stable solution exists only below the
critical current i,i c at which the points s and u merge.
Differentiating Eq. ~C9! with respect to u, we obtain
12 v 2w/ v 2 50.
1
2
v c5 2
3
~C11!
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4 2e
2 .
9
3
~C12!
Substituting Eq. ~C12! back into Eq. ~C11!, we arrive after
some algebra at the following equation for i c :
i c 5 ~ 12S !~ 112S ! 1/2,
~C13!
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~ 41G 21 ! 2
~C14!
.
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~C9!
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cases G 2 50 and G 1 @1. For G 2 50, we have
v2
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~C8!
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13 891
J c5
J0
,
2G 1
G 1 @1.
~C15!
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contacts.78
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c 0 }1/G 1 !1, c ` 51, c 2m 52i 2 , and i 2c !1. Then we can retain only the term c 0 G 1 and all inverse powers of c 0 in Eq.
~C9! which thus becomes
S
1
i2
G 1 c 0 1G 2 3
2
c0
D
2
512
2i 2
c 20
.
~C16!
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by Eqs. ~40! into Eq. ~C16!, we arrive at Eq. ~39!.
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