PHYSICA
Physica C 219 (1994) 465470 NorthHolland
Charged grain boundaries as pinning centers in highTc superconductors B.Ya. Shapiro a n d I.B. Khalfin Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Barllan University,RamatGan 52900, Israel Received 7 July 1993 Revised manuscript received 16 October 1993
The theoryof the Abnkosovvortexpinningby chargedgrain boundaries in hightemperature superconductorsis presented. We show that due to the small ratio ~/lo in hightemperature superconductors (where ( is the coherence length and lo is the Debye screening length) the attraction of the Abrikosov vortices to the grain boundary is significantly greater than in conventional superconductors.
I. Introduction The influence of the external electric field on superconductivity is at present a subject of intensive investigations [ 18 ]. The electric field, directed perpendicular to the surface of the sample, increases the carrier density n in the thin layer of the width of the Debye screening length, lo = (4n ve2/E)  ~/2, where v is the density of states at the Fermi level and e is the dielectric constant. As a result, the surface density of states and, in turn, the superconducting coupling constant, g, grows too. This leads to the appearance of a surface layer of width ID where the superconducting characteristics are improved. Note that this effect is asymmetric and depends on the polarity of the electric field, i.e., there appears a layer with suppressed superconductivity at the surface if the electric field is inverted. The electricfield effects in superconductors (such as the depletion of superconducting properties at the surface and the shift of the critical temperature) are determined by the ratio Io/~ [4,7,8]. Though ID is small (of order of 15 A) and ~~ 1001000 A in conventional superconductors, these effects can be neglected because of the small ratio ID/~<< 1. AS has already been found [79], these effects are more pronounced in the superconductors with l o l l ~ 1. Thus, the hightemperature superconductors
(HTSC's) are favorable to study such a phenomenon because these systems are characterized by both small ~ and by large ID owing tO the small density of states and large dielectric constant ~~2030. The main efforts have been focussed on the increase of the local critical temperature and formation of a localized state at the surface due to the external electric field [ 46 ]. The recent measurements [ 4 ] of the critical current, Jc, have attracted great interest. It was shown experimentally [ 3,4 ] that the 10% growth of the carrier density due to the electric field (for a YBa2Cu307 film at the electric field E,~ 107 V / c m ) leads to a 50% growth in Jc. On the other hand it is well known that in reality many of the novel superconductors are not ideal; they are characterized by a granular structure and contain shortrange dislocations between the grains. These structure defects are known to be pinning centers and they restrict the Abrikosov vortex (AV) motion in the superconductor [1013]. Because of the local tension and of the broken atomic bonds such defects can be a source of an electric field. Usually this fact may be neglected because of the charge carriers screening in the grains. If the thickness of these intergrain defects, d, is small in comparison with the screening length, i.e., if d < lo, the effect of the charged plane should be taken into account. In this case the additional interaction of AV's with the grain bound
09214534/94/$07.00 © 1994Elsevier ScienceB.V. All rights reserved. SSDI 09214534(93)E09779
B. Ya. Shapiro, I.B. Khaifin / Charged grain /)oundartc~
466
ary will be determined by the overlap of the AV core of the size ~ with the Debye layer lD (because lr,/ ~>d/~). In a strong magnetic field AV's from the chains stretched along the pinning center [14,15]. The stability of these multivortex chains as a function of temperature, electric field and density of vortices is one more subject of our consideration.
2. Vortex energy and pinning force The GinzburgLandau free energy, G. in a inhomogeneous medium can be expressed as [ 79.16,17 ]
super x ortex

~(BH) 2 G= f ( 8~
Fig. 1. G e o m e l r y of the problem.
8n/no = ( E / E * ) e x p ( .v/l~>) . (1) where A is the superconducting energy gap (which has the meaning of the order parameter), B and H are the magnetic induction and external magnetic field, respectively, and j = x , y, z. The electric field, E, being perpendicular to the y  z plane, can be accounted for by an expansion ofc~ in eq. ( 1 ) over 8n/ no, where no is the carrier density far from the plane and 8n is the deviation of the charge density at the charged plane due to E: 8n c~=z+g ,
where E*=4~elDno/e, E(,vc)=0. In order to calculate the correction to the vortex energy caused by its interaction with the surface layer, it is convenient to write J = J o + S A , where Ao is the order parameter in the absence of the electric field. Near the vortex core Ao is spacedependent, vanishing at the center of the vortex. If a vortex is located at (vo, 0), we can use the approximation
IAo(X, v)t 2=
((xx°)/~±)2+(Y/~ll)" r _ 1+ ((.VXo)/~)2+(y/~11)22 B
where r= ( T ~  T ) / T o T~ is the critical temperature at E = 0 and g = (0 In Tc)/(O In no) is the superconducting coupling constant. The form of the expansion (2) is justified by the small amplitude of the order parameter A(r) (see for example ref. [17] ). According to the considered model (fig. 1), the function 8n in the interval xe [0, d] can be approximated by a constant (in this region pinning is determined by the properties of the intergrain substance), in contrast the interval x > d, where the local electric field changes the properties of superconductor inside the grain. In the framework of the ThomasFermi approximation we get for the halfspace (for d
"
(4)
(2)
?/0
(31
where we account for the anisotropy by introducing ¢± and ¢ll as the coherence lengths perpendicular and parallel to the surface of the grain boundary. Taking the variation of eq. ( 1 ), we get the correction to the GinzburgLandau free energy due to the electric field E: AG(xo) =  vgE~
ckv (7
dr ×
×exp(x/lD)(lAo(x,y)12ldo(OO){2),
(5t
where A 2 ( o o ) = r / ( 2 f l ) is the order parameter far from the vortex. Performing the integration in eq. (4), we get
AG=
vr~g E ~ ~ ~,,,,
!
exp(X/lD)dAw / l + ( ( X   X o ) / ~ j )2'
(6)
B. Ya. Shapiro, LB. Khalfin / Charged grain boundaries
The energy correction AG as a function of Xo/(± forms the energy profile well. Its minimum is located at Xo=0. Note that at large ~./lD~ 100, which is relevant for conventional superconductors, the well AG(0) is negligible. At smaller ~±/lD~1, relevant for HTSC's, AG becomes significant. The maximum force of the vortex interaction with the charged grain boundary, F = OAG/Oxo,can be expressed easily from eq. (6). Considering the finite thickness of the plane one has to replace the lower limit of these integrals by d and to take into account that Xo> d. The pinning force as a function of the distance from the charged plane is presented in fig. 2.
0.2
467
From fig. 2(b) is clear that if d < ID, the dependence of the thickness is weak.
3. Energy of vortex chain Attraction of AV's to the planes leads to the creation of vortex chains stretched along them (these chains are similar to the multivortex states considered in refs. [ 14 ] and [ 15 ] ). The stability of the vortex chains in the grain boundaries is determined by the competition of two processes, namely, the attraction to the plane and the intervortex electromagnetic repulsion. The full energy of such a chain (per length 2 II) can be expressed as
¢~ I I,L hr m=z,_~ ~ Ko(m~/r/NL) E(NL)=NLAG+ 8rt22~ (7)
0.18 0.16 0.14 0.12 =~
0.1 t
~
0.08
~ 0.061 ~. 0.04 ~ 0.02
1
2
3
4
5
6
7
8
9
10
DISTANCE
(a)
where Ko is the Hankel function of the secondorder integer index (McDonald function), NL is the concentration of AV's per length 2 iio (NL = 2 jio/ro, where ro is the distance between the nearest vortices) and 0o is the flux quantum. The first term in eq. (7) is the pinning energy of the chain, and the second term is the energy of intervortex interaction in the chain. Using the integral representation of Ko and performing the summation we get for the second term of eq.
(7) <3o
0.2
1
0.18" 0.16 ! 0,14 
d=0
~
0.1
0.08
(8)
d = 0.05
0,12d = 0.125
d = 0.25
///
",:
.
0.060.04
#
d=0.5
0.02 0 0
/ [
I
I
I
I
I
2
3
4
5
From eqs. (7) and (8) we see that the energy of the chain has a minimum for certain concentration of AV's. This dependence for different ratios ID/~±is shown in fig. 3. Clearly for conventional superconductors the pinning energy is negligible in comparison with HTSC's.
DISTANCE
(b) Fig. 2. Pinning force of a vortex, F, in terms of (vzxE~u)/ (2flgE*~±) vs. distance from the plane, Xo/~±, for different ratios e = ~± ~lo, when d = 0 (a), and for different thicknesses of the intergrain boundaries, d, when ~../lD= 1 (b). Here E/E*~0.25.
4. Stability of the chain The vortex chain stretched along the plane is a onedimensional system with a shortrange intervortex interaction. That is why the conditions of the chain stability are so important in order to reach a great
B. Ya. Shapiro, I.B. Khalfin / Chargedgrain boundaries
468
f
+ ,:~: V(R/  R i' + z ,  = i )
Ou: Ouj~ Oz, Oz: dz, dz:. (IL))
3
where V ( R /  R 1:~ + : ,  : : ) is the energy of the interaction between the elements of vortices with coordinates ( R / , :,) and ( R / . :j): AG" and U" have been derived from eqs. (6) and (7):
z
2
z ~2
AG"
1
• ..
0.5
.r
CONCENTRATION
A~[
i
exp(lxl//D)Ck~
i
1.5
[l+(.v/~,
)~]3,'2 .j,
(ll>
REI,. [:NITS )
(12)
Fig. 3. Pinning energy, in terms of (u~xE)/(2flgE*)~ll vs. the concentration of the vortices, at ~ / I o ~ 1 (dashed line) and at ~±/ID~ 100 (solid line).
(i,.: for the coefficients we have
critical current. It is obvious, that its stability strongly depends on the AV attraction to the plane. The destruction of the vortex chain can be caused by therm o d y n a m i c fluctuations. The most dangerous fluctuations are the long wave (in the zdirection) transverse ones. Namely, these fluctuations break the chain of AV's. The probability of the thermodynamic fluctuations, IV, can be expressed as [ 17 ] Woc exp (

Rmin/ k B T) ,
(9 )
where Rm~, is the m i n i m u m energy of the fluctuating configuration. Expanding the energy (7) by powers of small vortex displacements, u, from the equilibrium state and taking into account the tilt of vortices (following, for example ref. [ 1 8 ] ) we get the expression
vr~g E
.t1= 2~ E*~ll and

8~/, i
Performing a Fourier transformation 1
Ltl
f~i 2 14kkkz el[/'±R!~+/'~z~l ~/ ]~ k t kz
as a result we get for Rmm
Rmin=EE(uO) =NL ~
[AG"U',I,~,(1cos(kL2L/N~))
klk:
+c44k~]luk~, 12 ,
(13)
where c44 is a tilt modulus. In the longwave approximation it has a well known form [ 18 ]: C44 = 0 2 / 4 7 I (
1 nt'~'2k~ ),
"~~11=2~ .
(14)
For the mean square (u~i ~=) we obtain the resuh
B. Ya. Shapiro, I.B. Khalfin / Chargedgrain boundaries
469
in the plane and the critical temperature for such a center exist. It should be noted that in HTSC's the conditions of the onedimensional vortex chain stability are more favorable than in conventional ones. The reason of this fact is a large ratio ~/ID in usual superconductors in comparison with HTSC's. In fact, the pinning forces stabilizing the vortex chain in HTSC systems are greater.
0.6 ¸ 0.5
OA
2 0.3 0.2 0.I
0 0.I
0.2
0.3
0.4
0.6
0.5
E/E*
5. Conclusions (a) 6
5
..~3
"~
2
I
I
I
I
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(b)
Fig. 4. Criterion of the chain stability, C*= C(~2/22) vs. electric field (a), and critical concentrationof vortices, NLcm(per length 2o), vs. reduced temperature z for E/E* ~ 0.25 (b).
(uLk,) = C , { 1  C ( ~ / X ~ ) [ 1  c o s ( k± 2 . /NL) ]} + ¢1t2£44k z2 , (15) where c44 is the tilt modulus, Ci = A G " ~ and C = U"2~/C1. It is clear that under the condition C ( ~ / 2 ~ ) > ½ the mean square displacement diverges and the chain cannot be stable (at least in the point kz=0). The parameter C depends on the external electric field E, the ratio ~_L/lo, and the concentration of vortices Art.. The diagrams of the existence of stableunstable states of the vortex chain in a charged plane are presented in figs. 4(a) and (b). From these plots it is clear that the critical values of the electric field, the concentration of vortices
The existence of the charged grain boundaries leads to the appearance of regions (with thickness of the order of lD) with depleted superconducting properties. The result of such a depletion can be a broadening of the pinning centers and, in turn an improvement of the pinning properties of the grain boundaries and growth of the critical current due to the electric field. Therefore, the samples with a high number of grain boundaries may be very useful in reaching a large critical current. From this point of view, HTSC's are the best substances. Indeed, due to the small ratio ~/lD~ 1, the contribution of the electricfield effect to the enhancement of the pinning force and, in turn, the critical current, J¢=Fm~x/Oo, are two orders of magnitude larger in comparison with the conventional superconductors (see fig. 5). In a strong magnetic field (when the intervortex interaction becomes significant) chains of vortices can appear at the grain boundaries. For a fixed value of the electric field, the critical temperature of the chain instability, T * ( H ) , can be determined. As one can find from fig. 4(b), this temperature, T * ( H ) , drops down with an increase in the density of vortices in the twin boundary. We can estimate the effect for YBaCuO: for E/E*=O.1, H = l O 0 G, 20= 1000 A, and NLt =250 A, we get T*=0.92 T¢. It should be noted that in a sufficiently large magnetic field the influence of the bulk vortices must be taken into account. This leads to a stabilization of the vortex chain under the condition d  1~ ArE,where d is the intervortex distance in the bulk (this phenomenon will be considered in detail separately). In distinction from single vortex pinning the intervortex interaction in the planar pinning center
B. Ya. Shapiro. I.B. Khalfin / Charged grain boundarie,~
470
changes the pinning lbrce and, as a result, thc crttical current, J~. This means that the critical current drops to zero at the temperature T * < Y\. because the lherm o d y n a m i c fluctuations destroy the pinning center
0.25
0.2
(fig. 6). References
0.15
Z ~2
\
0.05
0
2O
40
60
80
I00
Fig. 5. Critical current, J~, in units of Fm~,/Oo vs. {,/ID for E~ E*~0.25.
Jc
xx~Nx \
T*
x
\
h Tc
TEMPERATURE Fig. 6. Critical current J~ vs. temperature (qualitatively')
J 1 ] R.E. Glover and I).M. Sherill, Phys. Rev. Lctl. 5 t lU00 ! 248. [2] H i . Stadler, Phys. Re','. Left. 14 11960) 979 [3]J, Mannharl, D.G Scholm, J.G. B e d n o r z a n d K . ,~, Miillm, Phys. Rev. Letl. 67 ( 1991 ) 2009. [4] J. Mannhart, Mod, Phys. Len. B 6 (1992) 555 [5] Z. Trybula, J. Stankowski and J. Baszynski, Physica (" l~m (1988) 485. J 6 ] X.X. XL Q. ki, C. Doughty, ('. Kwon, S. Bhaltacharya, A . I Findikoglu and T, Venkatesan, Appl. Phys. Left. 59 I I'*~1 i 3470. [7] B. Ya. Shapiro, Phys. Lelt. ~105 (1984) 344. [8] k. Burlachkov, [.B. Khalfin and B.Ya. Shapiro, Phys. Re~ B 48 (1993). [9]B.Ya. Shapiro and I.B. Khalfin, Phys. kctt ', (IL)~3~ submitted. [ 10] J. Mannhart, D. Anselmetti, J.G. Bednorz, A. ( atana, (11. Gerber, K.A. Mailer and D.G. Scholm, Z. Phys. B 86 ( 1992 ) 177. [11 ] L.J. Swartzendruber, A. Roitburd, D.L. Kaiser, F.W. (iayle and L.H. Bennett, Phys. Rev. Left. 64 (1990) 483: ~. Roitburd, L.J. Swartzendruber, D.L. Kaiser, F.W. Gaylc and L H . Bennett, Phys. Rev. Left. 64 ( 1990 ) 2962. [ 12] W.K. Kwok, U. Welp, G.W. Crabtree, K.G. Vandervoort, R. Hulscher and J.Z. Liu, Phys. Rev. Lett. 64 (1990) 966, A. Umezawa, G.W. Crabtree. W.K. Kwok, U. Welp, K . G Vandervoorl and J . Z Liu, Phys. Rev. B 42 ( 1990 ) 8/44. [13]I.N. Khlyustikov and Aft. Buzdin, Adv. Phys 36 (19871 271. [ 14 ] I.B. Khalfin and B.Ya. Shapiro, Physica C 202 ( 1992 ) 31~3. [ 15] I.B. Khalfin, V.A. Larkin, B.Ya. Shapiro and V. Obozno~. Phys. Rev. B 46 (1992) 11243. [ 16 ] M. Tinkham, Introduction to Superconductivity ( McGra~Hill, New York, 1974). [17]L.D. Landau and E.M. Lifshits, Statistical Physics ( Pergamon, Oxford, 1969 ). [ 18 ] E.H. Brandt and A. Sudbo, Physica C 180 ( 199 l ) 426: A. Sudbo and E.H. Brandt. Phys. Rev. Leu. 66 ( 1991 ) 1781.