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Pinning properties of a disordered planar defect
PhysicaC210(1993) North-Holland
AE
114-126
Pinning properties of a disordered planar defect D. Agassi and R.D. Bardo Naval Surface Weapons Center, Dahlgren Division, White Oak Detachment, Silver Spring, MD 20903-5000, USA Received 19 November 1992 Revised manuscript received 19 January
1993
The effects of penetration depth fluctuations in a disordered-regular superconducting junction are evaluated in the London limit. This configuration demonstrates a disorder-proximity effect: the disorder on the one side of the junction induces pinning potential fluctuations on the other side which penetrate a distance of the order of the penetration depth into the regular side. The associated correlation functions and critical currents are calculated and discussed.
1. Introduction Strong flux pinning at elevated temperatures and fields is a key prerequisite for magnetic applications of high critical temperature superconductors (HTSC) [ 11. The associated flux pinning potentials are specific to the type and configuration of the defects [ 21, yet the measured critical current is determined by the combined effect of all inhomogeneities in the sample. Consequently, it is experimentally difficult to assess the various pinning contributions in order to improve the materials processing. This difficulty motivated studies of engineered grain boundaries (GBs) [ 31 and textured specimens [ 41, to identify the factors controlling the electromagnetic properties of GBs. By the same token, theoretical modeling of specific pinning sources provides a valuable tool for quantitative evaluation of pinning mechanisms [ 5,6]. A case in point is a recent pinning model of an isolated twin boundary, where a S-S’-S superconducting heterojunction with a slab geometry has been assumed [ 7 1. The objective of this paper is to extend the latter model by adding disorder of the superconducting parameters [ 8 1. This extension is motivated by several reasons. ( 1) Disorder is probably a feature of twins [ 9 ] and GBs. Recent EELS [IO] and EDS [ 111 data demonstrated oxygen deficiency distribution around GBs 0921-4534/93/$06.00
0 1993 Elsevier Science Publishers
and twin boundaries. This distribution is likely to be disordered given that the basal-plane oxygen atoms in YBCO are notoriously mobile. (2) To accommodate critical currents normal to the slab interface, the translational invariance along the slab interface must be broken. Disorder is a relevant mechanism for breaking this invariance symmetry. ( 3 ) The extended model manifests a disorder-proximity effect which is now introduced in an heuristic way. Consider a type II regular-disordered planar superconducting junction (fig. 1). For a vortex located on the regular side of the junction within less than a penetration depth distance from the interface, the swirling shielding currents straddle the regular and disordered sides of the junction. Starting on the regular side, consider the circular motion of a shielding current volume element along a smooth trajectory until it crosses into the disordered side of the junction. At that point, and along the entire passage on the disordered side, the motion will follow a jagged trajectory given the spatial fluctuations of the superconducting properties on that side. Therefore, upon reentry to the regular side of the junction, its trajectory and volume have been altered in a random manner, depending on the specific entry-exit points across the interface. The randomly altered, shielding current volume element smoothly flows through the regular side until crossing back into the disordered side, where the process repeats. The net effect is that,
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D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
115
traduced. In section 3, the model of a single vortex in a disordered superconductor is analyzed for weak and short correlation length disorder. The disordered-regular superconducting junction is treated in section 4. This model yields expressions for the pinning potential correlation functions and renormalization. Discussion of the results and the model domain of validity are included in section 5.
A
-
x
2. Basic equations hi
Fig. 1. Schematic plot of the disordered-ordered superconducting junction and definition of the nomenclature and coordinate system used in the text. The hatched domain represents the disordered domain.
e.g., on the regular side of the junction two volume elements associated with two adjacent trajectories will not be “similar”, reflecting the different scrambling they experience while on the disordered side. Therefore, on the regular side of the junction a spatially random shielding current is created. This “random” current, and the associated magnetic field, represents a proximity effect in the sense that disorder characteristics are carried over into the regular superconductor side [ 121. We refer to this effect as the disorder-proximity effect. It should be realized, however, that unlike the superconductivity-proximity effect which pertains to electrons pairing in proximity to a superconductor, the present “proximity”effect represents electromagnetic coupling. In order to quantify these features two contigurations are studied: an isolated vortex in an infinite disordered superconductor and in a disordered-regular superconducting junction. The evaluation of a disordered slab, as a model for a grain boundary, is taken up in a subsequent publication [ 131. The models yield the pinning potentials and analytic estimates of the associated critical currents. A bonus of the present study is the establishment of a framework for treating weak disorder. Beginning in section 2, the basic equations and the statistical assumptions defining the disorder are in-
Our starting assumptions are the validity of the London limit and a continuous superconducting medium. The former follows since, for HTSC, rc=n( 0) / r(O) rO( 100) [ 141. The layered structure of the HTSC [ 151 and the a-b plane anisotropy are considered inessential complications in the present context of focus on disorder effects. In the London limit and the absence of time varying fields, the free energy F is expressible in terms of the local magnetic induction, hereafter denoted by B (or “h” in some notations [ 161): F= &
I
d~[B-B+~2(R)(VxB)*(VxB)],
(1)
where R= (x, y, z), and the local penetration depth, n(R), is the only phenomenological parameter. The cgs unit system is used throughout. The previous analysis of a piecewise constant penetration length [ $71 is extended here to a non-constant n(R), where the coordinate dependence reflects random spatial structural and composition variations. In general vortex lines will have an undulating shape, and the associated B field depends on the three coordinates [ 171. Here we limit ourselves to a simpler class of models, where A2(R) +A2 (r) and r= (x, y), see section 5. It implies perfectly rigid vortex lines aligned in the z-direction (parallel to the junction interface, fig. 1 ), and the corresponding London equation is [ 5 ] B-
&(12(r) g)- $(A2(r) $)=@,&r,) , (2)
where the vortex location is at t-r,, the flux quantum is e0 = xfic/ I e 1,c is the speed of light and B= (0,
D. Agassi, R.D. Bardo
116
/Pinning properties of disordered planar defect
0, B). For a planar boundary at rr= (xi, y), the boundary conditions are [ 7,18 ] (3) where
the
interface
discontinuity
is defined
as
A[(~(~(r)),=,(,,l=y(x,+~, Y)-V~I--C Y) for e+O. Equation (2) is further simplified provided the non-constant part of 2 ( r) oscillates “rapidly” and is “weak”. In order to elucidate this statement and introduce the basic parameters, suppose that A’(r) NA~+A$ sin(x/o), where o is a typical oscillation length of the penetration depth, ;i,_ is its “mean”, and AF is its oscillation amplitude. Hereafter we use interchangingly the terms “disorder”, “fluctuations” and “oscillation” in reference to quantities such as the penetration depth, the magnetic field and the pinning potential. “Rapid” fluctuations imply cs
~cos(xi~)[a(B)iax+as[B],ax] -n2(~)[a2(~)/ax~+a2s[~]/ax2]
=@&(x-XCJ)
.
(4)
Consider now the various terms in eq. (4). Since (B) -A~a2(B)/ax2x@, d(x-x0), the remaining terms in eq. (4) constitute a differential equation for 6 [B] with derivatives of (B) as source terms. The various terms are estimated by replacing derivatives with typical ratios, e.g., a;12(r)/axNA:/a and aB/ ax-d(B)/ax+f6[B]I/o. The dominating 6[B] derivativetermisI~a26[B]/ax2xnt8[B]/a2,which is considerably larger than the other 6 [B] derivative term (~~/a)cos(x/a)a6[B]/axx~:G[B]/02inthe assumed regime of AL>> Izr. This second derivative term is part of the fourth term on the left hand side of eq. (4) and must be retained. Next, the important (B)-derivatives “source” terms in eq. (4) are S, = (2$/a) cos(x/a)a(B)/ax and S,=nc sin(x/ a)a2( B) /ax2. Their relative magnitude depends on whether x is “near” or “far” from the vortex position, x0. Assuming (B) z Ko( Ix-x0 I /A,) as a rough estimate for the average field (K. is the modified Bessel function of order zero), it follows that for 1x--x0 1I (T,where (B) varies rapidly, S2 2 S,. Far from the vortex position, i.e. AL> Ix-x0 I > 0, the reverse is true. Now, the quantity of physical interest is the induced field at the position of the vortex (in the present notation this quantity is approximately 6 [B]; see eq. (9) ). In analogy to the case in electrostatics, the induced field results from the “bare” vortex field reflections off inhomogeneities such as interfaces and disorder. Consequently, for conligurations where a vortex is placed within a distance 0 from an inhomogeneity, or when inhomogeneities at distances larger than 0 contribute little for statistical reasons (section 3 ), it is a good approximation to retain just the S2 source term. This term also originates from the fourth term on the 1.h.s. of eq. (4). The above considerations justify retaining only the fourth term on the 1.h.s. of eq. (4), i.e., the London equation (2 ) is approximated by B-i’(r)($
+ $)=qbo6(r-rv),
(5)
which forms the basis for the subsequent analysis. The approximate equation (5 ) is deficient “far” ( Z+ a) from the vortex position. Its solution, however, is still qualitatively similar to the exact solution since the London equation is elliptic, i.e., it yields waves decaying off the vortex position. Conse-
D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
quently, the effect of the neglected source term S, is to modify the decay rate of the solution away from the vortex position, but not change its decaying characteristics. Nor will the neglect of S, change the order of magnitude of the average-field decay length (A,), since this length scale is determined by the extension of the supercurrents around the vortex as discussed in the Introduction. Thus, for the applications discussed in this and the accompanying paper, where the correct description of the induced field at close proximity to an inhomogeneity and an estimated decay length capture the essential physics, the approximate equation (5 ) is adequate (see also section 5 ). In order to simulate structural and composition disorder we assume that the penetration depth is a two-dimensional gaussian, white noise random variable. The corresponding statistical assumptions are (A2(r)) (p(r)
=A2 )
5(0)
(8)
Note that the &functions in eqs. (2) and (6) represent variations over different length scales, namely cr and r(O), respectively. Notwithstanding this distinction, the mathematical idealization (6) is convenient and does not lead to unphysical results. Having articulated the basic equation and characterized the disorder, we conclude this section by establishing the connection between the London equation solution and the pinning potential. An important ramification of eq. (8) is that, on a length scale of <( 0)) the penetration depth fluctuations are approximately constant. Consequently [ 7,18 1, for a vortex of length L, the pinning potential per unit vortex length in the presence of a planar defect (PD) and penetration depth fluctuations (fl), u PD+fl(r=rv)/L, is given by
uPD+fl(rv) = &D+fl+V(rV) -Ffl+dr,) L
> =O,
tv(r)p(r’)>=
117
2nezrr2 74*-r’),
L = g PND(rv; rv) .
(6)
(9)
0
where the convenient, fined as
random
variable
p(r) is de-
(7) As above, the 0 parameter in eq. (6) signifies the correlation length associated with the penetration length fluctuations and the parameter e2=( (8[12(r)])2)/(A2)2 measures the amplitude of relative fluctuations. These parameters characterize the disorder. The particular parameter combination in eq. (6) is motivated by substituting for the &function a gaussian normalized to unit area, exp( - (r-r’)2/202)/(2m2). The rationale for the white noise statistical assumption (6) is based on our assumption that awl,, describing a regime, where the length scale associated with composition inhomogeneities is considerably shorter than the penetration depth. Consequently, variations over a length G are set to zero. In addition, it is unrealistic to expect g<<(O), since the coherence length is already of atomic dimensions in HTSC. Thus the subsequent analysis (in particular eq. (9), below) applies in the regime
In eq. (9), the free energy of a vortex (V) at r, in the absence of a planar defect and the presence of fluctuations is denoted by F’a+,( r), the free energy of a vortex at r, in the presence of a planar defect and fluctuations is denoted &,+fl+v(r), and the induced field (the B field excess over the vortex’s field) at r= r,, emanating from a vortex at r,,, is denoted by biND( r= r,; rv). The latter is the calculated quantity. Note that the second r.h.s. of eq. (9) yields only the excess free energy due to a planar defect [ 7 1. The free energy F,,, (rv) is evaluated separately in section 3. As a consequence of the randomness of A2(r) the ensuing pinning potential is spatially fluctuating. Consequently, a generic pinning potential, denoted hereafter by U,(r), has the decomposition into a mean and fluctuations
u,(r,)=(up(r,))+G[up(r,)l,
(10)
where ( 6[ U,(r) ] ) =O. The pinning potential fluctuations represent a dense set of pinning potential peaks and troughs, superimposed on a smoothly varying mean ( U,(r) ) _Statistical properties of such a function are characterized by the correlation func-
118
D. Agassi. R.D. Bardo
/Pinning properties of disordered planar defect
tion (6[ U,(r)]6[ U,(r’)]). Its diagonal part, in particular, yields the mean potential barrier between adjacent peaks and troughs. Denoting the latter by wfl(r),
W”(r,)=[((~[~~(r,)l)2)l”2,
(11)
the pinning potential due to fluctuations and the critical current normal to the interface are associated with W,(r).
3. Disordered superconductor The simplest manifestation of flux pinning due to disorder is in a model consisting of a single vortex in an infinite superconductor with a randomly-varying penetration depth. Consequently, the important boundary conditions eq. (3) do not enter the present discussion. The analysis serves the dual purpose of introducing a framework for treating disorder, and a necessary preparation for the subsequent calculations in the presence of boundaries. For a constant penetration depth [ 18 1, A2(r) =A2, the free energy of a vortex in an infinite superconductor, Fv( r), is constant due to translational invariance. This implies no pinning since the vortex is unable to lower its free energy at any point. In the presence of A2( r) fluctuations, however, the free energy is spatially fluctuating and pinning is due to the local, randomly-distributed minima of FV+fl (r). Thus the entire pinning in this case is due to fluctuations and the quantity of interest is W”(r), eq. ( 11). To solve eq. ( 5 ) with a fluctuating A2(r), the equation is first cast in the form
= 40(r(r-rv), A2(rv)
(12)
where V= V(r; rv) denotes the total vortex field at point r, emanating from a vortex located at r=r,. Equation ( 5 ) is obtained by expanding ( A2(r) ) - ‘: 1 ;120=
1 A2+6[12(r)]
1 @O (r)+... = 2 - TV
,
(13)
which is valid in the assumed weak disorder limit, i.e., E< 1. The notation in eq. ( 12) and below is that
capital letters denote total fields, while lower case letters, as in eq. (9), denote particular field components. In eq. ( 12) the stochastic term is multiplicative, a form rendered solvable by standard scattering theory methods. Specifically, the d-function on the r.h.s. of eq. (12) implies that V(r; r,,) is the Green function of a “hamiltonian” [ 201, where ( ( @o/A2)& r) ) plays the role of a potential, and the terms inside the square brackets define an unperturbed “hamiltonian”. The corresponding Born expansion of the full Green function in terms of the “unperturbed” Green function provides, therefore, an explicit expansion of v(r; rV)
1201,
Vu; rv)
=N(r,(rIv+v~v+~v~v+...Ir,).
(14)
In eq. ( 14), u is the “unperturbed” Green function, corresponding to a vortex in a constant penetration depth (A) superconductor, and N(r,) is a spatially random normalization. The corresponding expressions are [ 19,2 1] u(r-r,)=[jli
-($
+ $)r’$d(r-rv)
dkew[ik*(r-rv)l
$0
=gzy2 12
N(G) =
n2(r”)
s
=l-@()l+?(rv)+...
P+(l/IZ)2
’
(15)
Expansion ( 14) is the central result of this section. This series provides an explicit separation between the fluctuation and non-fluctuating factors, thereby allowing for a systematic perturbative evaluation of disorder effects. In particular, we demonstrate now that in the parameter regime (8) and weak disorder ( E< 1)) it suffices to consider the lowest-order terms in p(r). The corresponding pinning potential is evaluated according to eq. (9). Since the fluctuations are approximately constant over a scale of r( 0) (eq. (8) ), and in eq. (14) only the zero-order term (in q(r)) diverges logarithmically at r+rv, the pinning potential per unit vortex-length, 6[ U,(r) l/L, is
D. Agassi. R.D. Bardo /Pinning properties
F,+,(r,) -&(r”)
Ufl(r,) -=
L
L
=$
(16)
[V(r;r,)-v(r;r,)ll.,,“,
where Ffl+,(rv) and F,(r,) denote the free energy of a vortex in an infinite superconductor in the presence and absence of fluctuations, respectively. In general, fluctuations have two effects, namely to renormalize the mean of the pinning potential ( U,( rv) ) and give rise to spatial pinning potential fluctuations 6 [ U,( r,) 1. The statistical assumptions (6) imply that terms in eq. ( 14) containing an odd number of p(r)-factors, which vanish upon taking the ensemble average [ 221, constitute the pinning potential fluctuations. The contributions to the mean pinning potential arise from terms with an even number of p(r). The lowest order of such contributions are
+...lr”>
Ir+,
(17)
where an overline connecting a pair of p(r) factors i.e. the application of eq. denotes a “contraction”, (6) to that particular pair of p(r). The corresponding expressions, using eqs. (6) and ( 15 ), yield
119
of disordered planar defect
factor [ 2 11. The second factor, fi = (es/l)‘, arises from intermediate integrations in a contraction over a “u” factor. Since for a HTSC A( O)/r( 0) N 0( 100) and G/J. - 5( 0) /A - 0( 0.0 1)) typically f, 2: 30 while for an assumed E= 0.1, one obtains f2= (ta/ 1)2- 10P6. It follows that the lowest-order corrections to the mean pinning potential are proportional to a “small” parameter fi f2 u 0( 0.00003 ). This justifies a perturbative approach to eq. ( 17 ), and similarly for the pinning potential correlation function (below). Note that the smallness of&, the prime reason for the excellent convergence of the Born series, is a direct consequence of the assumed short correlation length 0 (eq. 8) ). Another important feature to be noted from the analysis of eq. (18) is that the last line, which involves the least number of intermediate summations, gives the largest contribution. This rule is substantiated below and used in section 4. We turn now to the pinning potential fluctuations. From eqs. ( 14) and ( 15), the lowest-order contributions are 6[G(rv)l
$0
= G
L
(rlv~v-~O~(rV)v+...Ir,>I,-,,,
(19) the
and
quantity
of
interest
( 6 [ U, ( rv) ] 6 [ U,( r: ) ] ) . Straightforward
.
algeb::
using eqs. (6) and ( 15 ) yields v(r--rV)
I14rv
=
$5
ln(UkXo))
C,(IArVI)=~~~(r,)~(r:)(r,I~lr,)(r:I~lr:)
,
2
(rlwv4rv)
( > @O
I-
=(2xn)Z
~2[2~ln(~/5(o))12~1(l~~l),
(18)
= &
y
’ [27tln(1/<(0))12,
0 where the first line in eq. ( 18 ), pertaining to Fv ( rv), is added here for reference. Expressions ( 18 ) highlight two recurring dimenfirst, f,=2rtln(ll/ factors. The sionless <( 0) ) - v( ( r-r, ( u <( 0) ) originates from a contraction of the two arguments in a v( r- rv) factor. The cutoff at the coherence length momentum in integrals like ( 15 ) produces this well-known logarithmic
(20) where we introduced F,(z)=e-Z2/2”2,
the functions F2(z)=Ko(z)2,
D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
120
F,(z)=2zK,(2z),
(21)
and p=r/l, K,,(z) and K,(z)are the Kelvin functions [ 231, and AI/A= 1r-r' 1/A=Ap. Thus the pinning potential correlation function to lowest order is
<~[~fl(rv)l~[~fl(r~)l> L2
(22)
+C3(Irv--rLI)l.
The correlation function, eq. (22), is comprised of terms with disparate ranges and strengths. The strongest contribution, C, ( )Ar, I), has also the shortest range, namely IJ. The other terms have a range of the order I and are smaller by orders of magnitude. This outcome may be understood by
4. Disordered-regular
noting that since u is the fluctuation “range”, correlations over that range are expected to be of “full strength” ( N t’). The other terms in eq. (22) represent processes combining propagation in a random medium and fluctuations, where the former substantially dampens the contributions ( thef2 factor in conjunction of eq. ( 17 ) ). It is for this reason that the dominating term involves the least intermediate summations. In summary, we have established a perturbative expansion of the mean and correlation function of the pinning potential that is consistent with eq. (8 ) and weak disorder. The analysis yields the leading contributions and a rule for identifying the dominant terms. It further indicates that fluctuations contribute primarily to the pinning potential fluctuations, while the corrections to the corresponding mean are negligible.
superconducting junction
In this section we consider a model of a disordered-regular superconducting junction with a planar interface (fig. 1). In comparison to section 3, the new elements here are the boundary conditions (3) which link the regular and disordered sides of the junction. As argued in the Introduction, this linkage endows the regular side with a feature of disorder, i.e. pinning potential fluctuations. Notwithstanding the different underlying physics, this effect is termed the disorder-proximity effect, in analogy to the situation for superconductor-nonsuperconducting junctions [ 12 1. As a reference case, consider first a junction where both sides are regular superconductors [ 18 1. As mentioned earlier, the translational invariance parallel to the junction interface implies pinning only normal to the interface. In the presence of disorder on one side of the junction, the above is qualitatively modified. First and foremost, the above translational invariance is broken, giving rise to pinning in that direction. The other difference is that fluctuations quantitatively modify the surface barrier [ 18 1. Consider first the case of a vortex located on the regular side of the junction. The corresponding London equations are
(23)
where the “A” and “T ” labels refer to the corresponding homogeneous, its solution has the decomposition
domains,
see fig. 1. Since on side A, eq. (23 ) is in-
(24)
BA(~rv)=U~(r;rv)+baND(r;r,), where uA(r; rv)=uA(r-r,) is given in eq. (15) with 1=A H, and by” eq. (24). Both terms have the representation [ 71
(r; rv), the
induced
field, is defined
by
D. Agassi, R. D. Bardo /Pinning properties of disordered planar defect
1
O”
(25)
where the symbols
in eq. (25) denote
x=L(k,
P=
ky)
9
121
tr,v) = (4 Y)llH
Dx(xy)
9
=,/x:+t1/Rx)* ,
Rx=;IxlA~t (26a)
forX=A,T, and evt-i[xyv*i~X(xy)tl)
$0
Dx(xy)
t26b)
.
The + and - signs in eqs. (25) and (26b) apply when c> & and <<&, respectively. On the disordered side (T), eq. (23 ) has the structure of a scattering wave equation [ 13 1, with the dop( r) / 1% factor playing the role of a potential. Consequently the solution has the Born series expansion where the “unperturbed” Green function is z+ (r- r’ ), and the “scattering” solution of the corresponding homogeneous equation has the representation
Expressions (25) and (27) define the unknowns, bA(xy) and &(x,,)_ The latter is not a fluctuating quantity since the fluctuating factors are explicitly separated out in eq. (27). Matching eqs. (25) and (27) according to (3), and following the paradigm in section 3, the mean induced field, in lowest order, is 00
= &
H
I
)
Y
=
corrected
to first order, r(l), is given by
PA --R~&(x~)/Q&~) n,txy) +R:&(xJl&(xy) ’
72
T
0
.2
.4
.6
A
(28)
’
DA(&)
--co
where the “reflectivity” t(~)tX
e--2D4~~%rW(xy)
dx,
.*
t4(Xy) =
I
1.2
i$
1+NGtXy) 1-N&(Xy) ’
Fig. 2. The mean pinning potential of a disordered-regular junction in the &=A,, limit. The quantity I(<) is the integral in eq. (30) and C=x/&.
122
D. Agassi, R.D. Bardo /Pinning
FT(<&
h)
Fn(-h
properties of disordered planar defect
t’l)
_._.-. t
_._.-. id; A) 1j&t(o); A)
-
i&i T)/ i&(o); T)
10-i
a
t,
Fig. 3. The pinning potential correlation functions and associated critical currents. In fig. 3(a) the integrals in eqs. (32) and (39) are plotted for the cases of RT= 1.5 (broken line) and R,= 5 (solid line) on the respective sides of the junction. The coordinate system is defined in fig. 1 and &x/L,. Figure 3 (b) shows the estimated critical currents perpendicular to the interface, normalized to their values at the interface as given by eq. (40). For positive <, the broken and solid curves correspond to RT= 1.5 and R,=5, respectively, for the disorder parameters of section 5. The curves represent j,( I; 5; A; PD+fl)/j,( I; t(O); A; PD+fl). For negative 5, only the results for R,=5 are given: the dot-dash curve represents j,( I; CT; PD+fl)/j,( I; c(O); T; PD+fl) and the solid curve represents j,( I; e T; fl)/ j,( I; r(O); A, PD+fl).
An important feature of the mean field (28) is that there is no qV-dependence, i.e., the fluctuations do not generate an average pinning potential parallel to the interface. Hence all y-pinning (see fig. 1) is associated with the pinning potential fluctuations. The fluctuations correction to eq. (28) enters via the dimensionless parameter A, eq. (29 ), which consists of factors already encountered in eq. ( 18 ). Consequently, the fluctuation correction to the mean pinning potential is negligible, as for the model of section 3. An exception is the limit As is evident from eqs. (28) and (29) in the absence of fluctuations (A =O) when (L:(r)) =n:=A&. T(‘) (x,,) =O, i.e., the induced mean field vanishes as it should. However, in the presence of fluctuations, eq. (29) yields T(‘) (x,,) -A/D% (x,,) and (30)
The positive pinning potential, eq. (30), plotted in fig. 2, represents repulsion between the vortex on the regular side and the interface. It extends over a distance AH, which is considerably longer than the fluctuations correlation length. This is a first manifestation of the disorder-proximity effect. We turn now to the calculation of the pinning potential correlation function, eq. (1 1 ), following the paradigm of section 3. The result for the leading term, in lowest order, is
D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
123
(31) where co co ca FA(P”;P:)
=
SSI
-cc -co -cc
dqdQdQ’exp[iq(rl,-rl:)-(D,(Q+q/2)+D,(Q-q/2))t;,
(32) and the unperturbed
7(X)=7’o’(X)=
(see eq. (29) ) is [ 71
reflectivity
1-GDT(X)IDA(X) 1+GD,(x)lD,(x)
(33)
.
All other terms in the lowest order are smaller by at least a factor (a/n.). The important q,-dependence of eq. (32) is a manifestation of the disorder-proximity effect, i.e., the creation of pinning parallel to the defect interface, outside the disordered domain. Note that the &-dependence in eq. (32) is invariant under v,-translation, as it should be. The calculated examples in fig. 3 (a) show an approximately exponential falloff with distance from the junction interface, with an approximate decay constant AH. is approximately A,/ The logarithmic divergence of eq. (32) for t,, cV+O, at a cutoff distance -r(O), <( 0) (ln(n,/c( 0) )2. It is a common artifact of the delta function source term in the London equation [ 2 11. From eqs. (30) and (32) the relative magnitude of the mean and fluctuating pinning potential at the interface (<;,=~~=<(O)) can be derived. Denotingp,=(t(O), q), we obtain (34) This estimate indicates that the pinning fluctuation contribution may become comparable to that of the mean pinning potential. In particular, the surface barrier [ 7,181 is qualitatively modified. Note also that in the limit of a disordered “weak” superconductor, where I, z+ AH, then 7% - 1 and the r-factors in eq. (32) contribute an enhancement of a factor -4. By the same token, in a disordered “strong” superconductor, 7- 1, and the pinning fluctuations are suppressed on the ordered side. The latter result is consistent with the physical picture that the shielding currents penetration into the disordered side is suppressed in this case, and hence the proximity effect is attenuated. We turn now to the case of a vortex on the disordered side (T) of the junction, fig. 1. In this case, the solution is the sum of a single vortex solution in a disordered superconductor, VT(r; rV) (eq. ( 16) ), and the general solution of the homogeneous equation, &(r,
rV) = vT(r, rV) +GND(r,
rV) ,
(35)
where byD (r) has the Born series expansion (27). On the A side the solution of the homogeneous has the form (24). Similar algebra yields the leading contribution to the mean pinning potential:
equation
co
(b:NDkPv)) = where
&
T
s
emh(xY)h
dxyD,(x
--m
7(Xyh(Xy) Y
>
(36)
124
D. Agassi, R.D. Bardo
l-
/Pinning properties of disordered planar defect
(~~lh)21n(Wt(0) )
nr(Xy)= 1 +0.25(D,(X,)R,)-2(ta/~T)21n(~T/r(0)
(37)
)r(X,) ’
As for eq. (28), the mean pinning potential (36) does not depend on the qy coordinate, i.e., no pinning in that direction even on the disordered side. The renormalization of the mean pinning potential due to fluctuations is again a small correction due to the typical factor (~a/1)~ << 1. It is interesting to consider again the limit where (A2(r)) =A$ =A&. In this case 7(x,,) =O and eq. (36) implies no pinning on the T side even with disorder. The marked asymmetry between the two sides of the junction, fig. 2, can be heuristically understood as follows: for a vortex located on the disordered side, the shielding currents incurring into the regular side of the junction are not “scrambled” any further, hence there are no additional induced fields due to the boundary. On the other hand, for a vortex located on the regular side of the junction, the shielding currents are always “scrambled” upon incursion into the disordered side, and hence the ensuing self-field generates pinning. In order to evaluate the pinning potential fluctuations on the T side, only the leading term in the lowest order, is considered. This yields (38) where the lengthy expression
MP,;P,)=
for FT(pv; p:) is given in the Appendix.
Its diagonal
part is
7 7 7 dqdQdQ’exp[(DT(Q-q4/2)+DT(Q’+q/2))r,l -03 -ccl--m
(39) The correlation functions (3 1) for the regular side and eq. (38) for the disordered side are the central results of this work. In fig. 3(a) two representative cases are shown. Note again the sharp falloff with distance from the interface, typical of the induced fields. The RT~ and (o/AT)* prefactors in eq. (38) strongly suppress the pinning potential fluctuations on the disordered side for RT > 1. Comparing the prefactors in eq. ( 38 ) to those of the C, term in eq. (22), the latter is the dominating term on the disordered side (see section 5). This is a plausible result. It reflects the fact that pinning potential fluctuations correlate poorly in a disordered domain since it involves a field propagating from the vortex to the interface and back in a random domain. 5. Discussion
and summary
The S’-S disordered-regular junction considered here yields an estimate of the critical currents normal and parallel to the junction interface. The connection between the calculated pinning potential and critical current involves the derivative of the pinning potential in the appropriate direction and at the appropriate evaluation point [ 7 1. A pinning force nor-
ma1 to the junction interface is already present in the regular-regular superconducting junction and, therefore, is not discussed here. The new feature in the disordered-regular junction is the pinning force parallel to the junction interface, associated with the critical current normal to it. For the purpose of estimating derivatives a typical range over which the pinning potential fluctuations
D. Agassi, R. D. Bardo /Pinning properties of disordered planar defect
vary is o. On the regular side, the only contribution to the critical current is through the disorder-proximity effect, U,,,, (r), see eq. (9), whereas on the disordered side there are two contributions, i.e. u PD+B(r)+ Q,(r), see eqs. (9) and (16). Expressions(31),(38),(22),(11),and(9)and(l6)yield the following critical currents (j,) normal to the interface ( I ), on sides X= A, T, evaluated at r= <( 0): j,( 1, r=<(O);
A; PD+fl)
(40) Inserting the parameter (T= 15 A, e=O.l, a/ r(O)= 1, AH=900 A, @oo=1633 (eV A)‘.5 we obtain ontheAsidej,(_L;r=<(O);A;PD+fl)-4X108A/ cm2. This estimate is subject to uncertainties in the e and CJparameters which can vary the results by two orders of magnitude. On the T side, expressions (40) show that the critical current associated with “PD+fl” scales as RT~ (o/n,), while the one associated with “fl” scales as Ry’. Thus the latter is dominant for the measured RT [ 241, and is smaller than the critical current on the A side. Typical critical currents are calculated and compared in fig. 3 (b ). Note the approximate exponential falloff on the regular side, with an approximate decay length ?An, while on the disordered side the decay length AT= RTLu > An. This type of exponential falloff is in keeping with the remarks following the introduction of the basic equation (5). The foregoing analysis is based on a number of simplifying assumptions. The first is a rigid vortex line, corresponding to the glass phase regime [ 25,26 1. For undulations over a length scale small in comparison to the penetration length, the present analysis provides a local approximation of the pinning potential in the vortex line direction. This interpre-
125
tation is consistent with the absence of a pancakes chain flux line in the present analysis [ 15 1. The single-vortex feature of the model corresponds to a low magnetic field regime where collective pinning does not enter [ 271. The low temperature and low field validity ranges of the model are in keeping with the high critical current estimates obtained above. The exponential distance dependence of the critical currents implies that pinning may be enhanced provided the distance between boundaries is smaller than the penetration depth. Twin boundaries form such a dense set of planar defects; however, they may lack other necessary attributes. This observation suggests that provided the twin boundaries are modified to conform with a superconducting junction, the corresponding critical intragrain critical current is enhanced. To our knowledge, no systematic studies of jc and twin boundary density exist [ 91. In summary, we have considered penetration depth fluctuations in the context of a vortex in an infinite superconductor and in a regular-disordered superconducting junction. This particular focus is motivated by the probable structural and composition disorder inside a grain and twin boundaries. The analysis is limited to a “short” correlation length and “weak” disorder, the parameter regime applicable to HTSC ceramics. An important qualitative result is that in the presence of spatial disorder the pinning potential is not spatially smooth in an adjacent regular domain. In other terms, there is a disorder-proximity effect whereby fluctuation effects extend beyond the original domain of disorder over a distance of the penetration length. This effect may be of relevance in the context of flux creep. The corrections to the mean pinning potential are proportional to the small parameter (es/l)’ N 1O-‘j. The fluctuations, however, give rise to a substantial critical current normal to the interface. The implications of this analysis for finite width grain boundaries are discussed in a subsequent paper [ 13 1.
Acknowledgement We would like to acknowledge ulating discussions.
C. Pande for stim-
D. Agassi, R.D. Bardo
126
/Pinning properties of disordered planar defect
Appendix The general expression for the kernel in eq. (38), valid for 0 > & > - 03 is given below. Using eq. (26a) and defining D*=D3+D4, where
&=&(Q'-q/2), (A.11 D3=DT(Q-q/2),Dq=DT(Qf+q/2) > LA=&-(Q+q/2),
and
K,(q,Q,Q+X,=j& 1
2
x(exp(-DD,r,-D2P;)P(Dl+D2+D*;r:,0) +ew( -DLL +D&)P(D~
-D2 +D*; L G)
+exp(Dt~v+D2~)p(-D,-D2+D*;0,~v)), (A.21 eD&_eDCL
~(D;tv,r:)= the correlation
(‘4.3)
9
A
function
in eq. (38) is given by
&(Pv; PI)
dQ'K,(q, Q,Q'; L r:) -cm
= 7 dq 1 dQ ,=/
-cc -m
x
T(Q--d2)T(Q’+d2) DID~&D~
.
(A.4)
References [ 1] D. Larbalestier,
Phys. Today June ( 199 1) 24. [2] For instance: P.H. Kes, PhysicaC 185-191 (1991) 288; C.J. Van der Beek and P.H. Kes, Phys. Rev. B 43 ( 1991) 13032; E.V. Thuneberg, Cryogenics 25 (1989) 236; Y. Zhu, J. Tafto and M. Suenaga, Mater. Res. Bull. November ( 199 1) 54. [3] D. Dimos, P. Chaudhari, J. Mannhart and F.K. LeGoues, Phys. Rev. Lett. 61 (1988) 219; D. Dimos, P. Chaudhari and J. Mannhart, Phys. Rev. B 41 (1990) 4038; D.C. Larbalestier et al., Physica C 185- 189 ( 199 1) 3 15; SE. Babcock, X.Y. Cai, D.L. Kaiser and D.C. Larbalestier, Nature (London) 347 ( 1990) 167; C.B. Eom, A.F. Marshall, Y. Suzuki, B. Boyer, R.F.W. Pease and T.H. Geballe, Nature (London) 353 (1991) 544. [ 41 Y. Zhu, M. Zhang, H. Wang and M. Suenaga, J. Mater. Res. 6 (1991) 2507. [ 51 L.L. Daemen and J.E. Gubematis, Phys. Rev. B 43 ( 199 1) 413;
K.K. Likharev, Rev. Mod. Phys. 5 1 ( 1979) 101. [6] H. Svensmark and L.M. Falicov, Phys. Rev. B 42 ( 1990) 9957. [7] D. Agassi and J.R. Cullen, Physica C 195 ( 1992) 277; D. Agassi and J.R. Cullen, in: Physics and Materials Science of High Temperature Superconductors, II, eds. R. Kossowsky, B. Raveau, D. Wohlleben and S.K. Patapis, (KIuwer, Dordrecht, 1992) p. 139. [ 81 For instance, A. Barone, A.I. Larkin and Y.N. Ovchinnikov, J. Supercond. 3 ( 1990) 155. [9] C.J. Jou and J. Washburn, J. Mater. Res. 4 (1989) 795; M.J. Kramer and R.W. McCallum (1992) preprint; Y. Zhu and M. Suenaga, in: Interfaces in High Temperature Superconducting Systems, eds. D.A. Rudman and S.L. Shinde (Springer, New York, 1992). [lo] Y. Zhu, Z.L. Wang and M. Suenaga, Philos. Mag. A ( 1992), in press; K.B. Alexander, D.M. Kroeger, J. Bentley and J. Brynestad, Physica C 180 ( 1991) 337; M.J. Kramer and R.W. McCallum (1992) preprint. [ 111 S.E. Babcock and D.C. Larbalestier, Appl. Phys. Lett. 55 (1989) 393. [ 121 H. Totsuji, Phys. Rev. B 43 (1991) 5287; G. Deutcher and P.G. De Gennes, in: Superconductivity, ed. R.D. Parks (Marcel Dekker, New York, 1969) chapter 17. [ 131 D. Agassi and R.D. Bardo, Physica C 209 ( 1993) 127. [ 141 B. Batlogg, Phys. Today, June ( 199 1) 44; G.W. Crabtree, W.K. Kwok and A. Umezawa, in: Quantum Field Theory as an Interdisciplinary Basis, eds. F.C. Khanna, H. Umezawa, G. Kunstatter and H.C. Lee (World Scientific, Singapore, 1988). [ 151 J.R. Clem and M.W. Coffey, Phys. Rev. B 43 ( 1991) 7837. [ 161 P.G. DeGennes, Superconductivity of Metals and Alloys, (Benjamin, New York, 1966) p. 3. [ 171 D. Ertas and M. Kardar, Phys. Rev. Lett. 69 (1992) 929; S. Chakravarty, B.I. Ivlev and Y.N. Ovchinnikov, Phys. Rev. Lett. 64 (1990) 3187. [ 181 A.M. Grishin, Sov. J. Low Temp. 9 (1983) 138. [ 191 For instance, T. Van Duser and C.W. Turner, Principles of Superconductive Devices (Elsevier, New York, 198 1). [20] P. Roman, Advanced Quantum Theory (Addison-Wesley, Reading, 1965) chapter 4. [ 2 I] D. Belitz, in: High Temperature Superconductivity, ed. J.W. Lynn (Springer, New York, 1990). [22] For instance, T. Lukes, in: Solid State Theory, ed. P.T. Landsberg (Wiley, London, 1969) chapter 13. [ 23 ] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC, 1964) chapter 9. [ 241 A. Duran, P.L. Gammel, R. Wolfe, V.J. Fratello, D.J. Bishop, J.P. Rice and D.M. Ginsberg ( 1992) preprint. [25] D.R. Nelson and V.M. Vinokur, Phys. Rev. Lett. 68 (1992) 2398. [26] L. Krusin-Elbaum, L. Civale, V.M. Vinokur and F. Holtzberg, Phys. Rev. Lett. 69 (1992) 2280. [27] For example, E.H. Brandt, lnt. J. Mod. Phys. B 5 (1991) 751.
AE
114-126
Pinning properties of a disordered planar defect D. Agassi and R.D. Bardo Naval Surface Weapons Center, Dahlgren Division, White Oak Detachment, Silver Spring, MD 20903-5000, USA Received 19 November 1992 Revised manuscript received 19 January
1993
The effects of penetration depth fluctuations in a disordered-regular superconducting junction are evaluated in the London limit. This configuration demonstrates a disorder-proximity effect: the disorder on the one side of the junction induces pinning potential fluctuations on the other side which penetrate a distance of the order of the penetration depth into the regular side. The associated correlation functions and critical currents are calculated and discussed.
1. Introduction Strong flux pinning at elevated temperatures and fields is a key prerequisite for magnetic applications of high critical temperature superconductors (HTSC) [ 11. The associated flux pinning potentials are specific to the type and configuration of the defects [ 21, yet the measured critical current is determined by the combined effect of all inhomogeneities in the sample. Consequently, it is experimentally difficult to assess the various pinning contributions in order to improve the materials processing. This difficulty motivated studies of engineered grain boundaries (GBs) [ 31 and textured specimens [ 41, to identify the factors controlling the electromagnetic properties of GBs. By the same token, theoretical modeling of specific pinning sources provides a valuable tool for quantitative evaluation of pinning mechanisms [ 5,6]. A case in point is a recent pinning model of an isolated twin boundary, where a S-S’-S superconducting heterojunction with a slab geometry has been assumed [ 7 1. The objective of this paper is to extend the latter model by adding disorder of the superconducting parameters [ 8 1. This extension is motivated by several reasons. ( 1) Disorder is probably a feature of twins [ 9 ] and GBs. Recent EELS [IO] and EDS [ 111 data demonstrated oxygen deficiency distribution around GBs 0921-4534/93/$06.00
0 1993 Elsevier Science Publishers
and twin boundaries. This distribution is likely to be disordered given that the basal-plane oxygen atoms in YBCO are notoriously mobile. (2) To accommodate critical currents normal to the slab interface, the translational invariance along the slab interface must be broken. Disorder is a relevant mechanism for breaking this invariance symmetry. ( 3 ) The extended model manifests a disorder-proximity effect which is now introduced in an heuristic way. Consider a type II regular-disordered planar superconducting junction (fig. 1). For a vortex located on the regular side of the junction within less than a penetration depth distance from the interface, the swirling shielding currents straddle the regular and disordered sides of the junction. Starting on the regular side, consider the circular motion of a shielding current volume element along a smooth trajectory until it crosses into the disordered side of the junction. At that point, and along the entire passage on the disordered side, the motion will follow a jagged trajectory given the spatial fluctuations of the superconducting properties on that side. Therefore, upon reentry to the regular side of the junction, its trajectory and volume have been altered in a random manner, depending on the specific entry-exit points across the interface. The randomly altered, shielding current volume element smoothly flows through the regular side until crossing back into the disordered side, where the process repeats. The net effect is that,
B.V. All rights reserved.
D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
115
traduced. In section 3, the model of a single vortex in a disordered superconductor is analyzed for weak and short correlation length disorder. The disordered-regular superconducting junction is treated in section 4. This model yields expressions for the pinning potential correlation functions and renormalization. Discussion of the results and the model domain of validity are included in section 5.
A
-
x
2. Basic equations hi
Fig. 1. Schematic plot of the disordered-ordered superconducting junction and definition of the nomenclature and coordinate system used in the text. The hatched domain represents the disordered domain.
e.g., on the regular side of the junction two volume elements associated with two adjacent trajectories will not be “similar”, reflecting the different scrambling they experience while on the disordered side. Therefore, on the regular side of the junction a spatially random shielding current is created. This “random” current, and the associated magnetic field, represents a proximity effect in the sense that disorder characteristics are carried over into the regular superconductor side [ 121. We refer to this effect as the disorder-proximity effect. It should be realized, however, that unlike the superconductivity-proximity effect which pertains to electrons pairing in proximity to a superconductor, the present “proximity”effect represents electromagnetic coupling. In order to quantify these features two contigurations are studied: an isolated vortex in an infinite disordered superconductor and in a disordered-regular superconducting junction. The evaluation of a disordered slab, as a model for a grain boundary, is taken up in a subsequent publication [ 131. The models yield the pinning potentials and analytic estimates of the associated critical currents. A bonus of the present study is the establishment of a framework for treating weak disorder. Beginning in section 2, the basic equations and the statistical assumptions defining the disorder are in-
Our starting assumptions are the validity of the London limit and a continuous superconducting medium. The former follows since, for HTSC, rc=n( 0) / r(O) rO( 100) [ 141. The layered structure of the HTSC [ 151 and the a-b plane anisotropy are considered inessential complications in the present context of focus on disorder effects. In the London limit and the absence of time varying fields, the free energy F is expressible in terms of the local magnetic induction, hereafter denoted by B (or “h” in some notations [ 161): F= &
I
d~[B-B+~2(R)(VxB)*(VxB)],
(1)
where R= (x, y, z), and the local penetration depth, n(R), is the only phenomenological parameter. The cgs unit system is used throughout. The previous analysis of a piecewise constant penetration length [ $71 is extended here to a non-constant n(R), where the coordinate dependence reflects random spatial structural and composition variations. In general vortex lines will have an undulating shape, and the associated B field depends on the three coordinates [ 171. Here we limit ourselves to a simpler class of models, where A2(R) +A2 (r) and r= (x, y), see section 5. It implies perfectly rigid vortex lines aligned in the z-direction (parallel to the junction interface, fig. 1 ), and the corresponding London equation is [ 5 ] B-
&(12(r) g)- $(A2(r) $)=@,&r,) , (2)
where the vortex location is at t-r,, the flux quantum is e0 = xfic/ I e 1,c is the speed of light and B= (0,
D. Agassi, R.D. Bardo
116
/Pinning properties of disordered planar defect
0, B). For a planar boundary at rr= (xi, y), the boundary conditions are [ 7,18 ] (3) where
the
interface
discontinuity
is defined
as
A[(~(~(r)),=,(,,l=y(x,+~, Y)-V~I--C Y) for e+O. Equation (2) is further simplified provided the non-constant part of 2 ( r) oscillates “rapidly” and is “weak”. In order to elucidate this statement and introduce the basic parameters, suppose that A’(r) NA~+A$ sin(x/o), where o is a typical oscillation length of the penetration depth, ;i,_ is its “mean”, and AF is its oscillation amplitude. Hereafter we use interchangingly the terms “disorder”, “fluctuations” and “oscillation” in reference to quantities such as the penetration depth, the magnetic field and the pinning potential. “Rapid” fluctuations imply cs
~cos(xi~)[a(B)iax+as[B],ax] -n2(~)[a2(~)/ax~+a2s[~]/ax2]
=@&(x-XCJ)
.
(4)
Consider now the various terms in eq. (4). Since (B) -A~a2(B)/ax2x@, d(x-x0), the remaining terms in eq. (4) constitute a differential equation for 6 [B] with derivatives of (B) as source terms. The various terms are estimated by replacing derivatives with typical ratios, e.g., a;12(r)/axNA:/a and aB/ ax-d(B)/ax+f6[B]I/o. The dominating 6[B] derivativetermisI~a26[B]/ax2xnt8[B]/a2,which is considerably larger than the other 6 [B] derivative term (~~/a)cos(x/a)a6[B]/axx~:G[B]/02inthe assumed regime of AL>> Izr. This second derivative term is part of the fourth term on the left hand side of eq. (4) and must be retained. Next, the important (B)-derivatives “source” terms in eq. (4) are S, = (2$/a) cos(x/a)a(B)/ax and S,=nc sin(x/ a)a2( B) /ax2. Their relative magnitude depends on whether x is “near” or “far” from the vortex position, x0. Assuming (B) z Ko( Ix-x0 I /A,) as a rough estimate for the average field (K. is the modified Bessel function of order zero), it follows that for 1x--x0 1I (T,where (B) varies rapidly, S2 2 S,. Far from the vortex position, i.e. AL> Ix-x0 I > 0, the reverse is true. Now, the quantity of physical interest is the induced field at the position of the vortex (in the present notation this quantity is approximately 6 [B]; see eq. (9) ). In analogy to the case in electrostatics, the induced field results from the “bare” vortex field reflections off inhomogeneities such as interfaces and disorder. Consequently, for conligurations where a vortex is placed within a distance 0 from an inhomogeneity, or when inhomogeneities at distances larger than 0 contribute little for statistical reasons (section 3 ), it is a good approximation to retain just the S2 source term. This term also originates from the fourth term on the 1.h.s. of eq. (4). The above considerations justify retaining only the fourth term on the 1.h.s. of eq. (4), i.e., the London equation (2 ) is approximated by B-i’(r)($
+ $)=qbo6(r-rv),
(5)
which forms the basis for the subsequent analysis. The approximate equation (5 ) is deficient “far” ( Z+ a) from the vortex position. Its solution, however, is still qualitatively similar to the exact solution since the London equation is elliptic, i.e., it yields waves decaying off the vortex position. Conse-
D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
quently, the effect of the neglected source term S, is to modify the decay rate of the solution away from the vortex position, but not change its decaying characteristics. Nor will the neglect of S, change the order of magnitude of the average-field decay length (A,), since this length scale is determined by the extension of the supercurrents around the vortex as discussed in the Introduction. Thus, for the applications discussed in this and the accompanying paper, where the correct description of the induced field at close proximity to an inhomogeneity and an estimated decay length capture the essential physics, the approximate equation (5 ) is adequate (see also section 5 ). In order to simulate structural and composition disorder we assume that the penetration depth is a two-dimensional gaussian, white noise random variable. The corresponding statistical assumptions are (A2(r)) (p(r)
=A2 )
5(0)
(8)
Note that the &functions in eqs. (2) and (6) represent variations over different length scales, namely cr and r(O), respectively. Notwithstanding this distinction, the mathematical idealization (6) is convenient and does not lead to unphysical results. Having articulated the basic equation and characterized the disorder, we conclude this section by establishing the connection between the London equation solution and the pinning potential. An important ramification of eq. (8) is that, on a length scale of <( 0)) the penetration depth fluctuations are approximately constant. Consequently [ 7,18 1, for a vortex of length L, the pinning potential per unit vortex length in the presence of a planar defect (PD) and penetration depth fluctuations (fl), u PD+fl(r=rv)/L, is given by
uPD+fl(rv) = &D+fl+V(rV) -Ffl+dr,) L
> =O,
tv(r)p(r’)>=
117
2nezrr2 74*-r’),
L = g PND(rv; rv) .
(6)
(9)
0
where the convenient, fined as
random
variable
p(r) is de-
(7) As above, the 0 parameter in eq. (6) signifies the correlation length associated with the penetration length fluctuations and the parameter e2=( (8[12(r)])2)/(A2)2 measures the amplitude of relative fluctuations. These parameters characterize the disorder. The particular parameter combination in eq. (6) is motivated by substituting for the &function a gaussian normalized to unit area, exp( - (r-r’)2/202)/(2m2). The rationale for the white noise statistical assumption (6) is based on our assumption that awl,, describing a regime, where the length scale associated with composition inhomogeneities is considerably shorter than the penetration depth. Consequently, variations over a length G are set to zero. In addition, it is unrealistic to expect g<<(O), since the coherence length is already of atomic dimensions in HTSC. Thus the subsequent analysis (in particular eq. (9), below) applies in the regime
In eq. (9), the free energy of a vortex (V) at r, in the absence of a planar defect and the presence of fluctuations is denoted by F’a+,( r), the free energy of a vortex at r, in the presence of a planar defect and fluctuations is denoted &,+fl+v(r), and the induced field (the B field excess over the vortex’s field) at r= r,, emanating from a vortex at r,,, is denoted by biND( r= r,; rv). The latter is the calculated quantity. Note that the second r.h.s. of eq. (9) yields only the excess free energy due to a planar defect [ 7 1. The free energy F,,, (rv) is evaluated separately in section 3. As a consequence of the randomness of A2(r) the ensuing pinning potential is spatially fluctuating. Consequently, a generic pinning potential, denoted hereafter by U,(r), has the decomposition into a mean and fluctuations
u,(r,)=(up(r,))+G[up(r,)l,
(10)
where ( 6[ U,(r) ] ) =O. The pinning potential fluctuations represent a dense set of pinning potential peaks and troughs, superimposed on a smoothly varying mean ( U,(r) ) _Statistical properties of such a function are characterized by the correlation func-
118
D. Agassi. R.D. Bardo
/Pinning properties of disordered planar defect
tion (6[ U,(r)]6[ U,(r’)]). Its diagonal part, in particular, yields the mean potential barrier between adjacent peaks and troughs. Denoting the latter by wfl(r),
W”(r,)=[((~[~~(r,)l)2)l”2,
(11)
the pinning potential due to fluctuations and the critical current normal to the interface are associated with W,(r).
3. Disordered superconductor The simplest manifestation of flux pinning due to disorder is in a model consisting of a single vortex in an infinite superconductor with a randomly-varying penetration depth. Consequently, the important boundary conditions eq. (3) do not enter the present discussion. The analysis serves the dual purpose of introducing a framework for treating disorder, and a necessary preparation for the subsequent calculations in the presence of boundaries. For a constant penetration depth [ 18 1, A2(r) =A2, the free energy of a vortex in an infinite superconductor, Fv( r), is constant due to translational invariance. This implies no pinning since the vortex is unable to lower its free energy at any point. In the presence of A2( r) fluctuations, however, the free energy is spatially fluctuating and pinning is due to the local, randomly-distributed minima of FV+fl (r). Thus the entire pinning in this case is due to fluctuations and the quantity of interest is W”(r), eq. ( 11). To solve eq. ( 5 ) with a fluctuating A2(r), the equation is first cast in the form
= 40(r(r-rv), A2(rv)
(12)
where V= V(r; rv) denotes the total vortex field at point r, emanating from a vortex located at r=r,. Equation ( 5 ) is obtained by expanding ( A2(r) ) - ‘: 1 ;120=
1 A2+6[12(r)]
1 @O (r)+... = 2 - TV
,
(13)
which is valid in the assumed weak disorder limit, i.e., E< 1. The notation in eq. ( 12) and below is that
capital letters denote total fields, while lower case letters, as in eq. (9), denote particular field components. In eq. ( 12) the stochastic term is multiplicative, a form rendered solvable by standard scattering theory methods. Specifically, the d-function on the r.h.s. of eq. (12) implies that V(r; r,,) is the Green function of a “hamiltonian” [ 201, where ( ( @o/A2)& r) ) plays the role of a potential, and the terms inside the square brackets define an unperturbed “hamiltonian”. The corresponding Born expansion of the full Green function in terms of the “unperturbed” Green function provides, therefore, an explicit expansion of v(r; rV)
1201,
Vu; rv)
=N(r,(rIv+v~v+~v~v+...Ir,).
(14)
In eq. ( 14), u is the “unperturbed” Green function, corresponding to a vortex in a constant penetration depth (A) superconductor, and N(r,) is a spatially random normalization. The corresponding expressions are [ 19,2 1] u(r-r,)=[jli
-($
+ $)r’$d(r-rv)
dkew[ik*(r-rv)l
$0
=gzy2 12
N(G) =
n2(r”)
s
=l-@()l+?(rv)+...
P+(l/IZ)2
’
(15)
Expansion ( 14) is the central result of this section. This series provides an explicit separation between the fluctuation and non-fluctuating factors, thereby allowing for a systematic perturbative evaluation of disorder effects. In particular, we demonstrate now that in the parameter regime (8) and weak disorder ( E< 1)) it suffices to consider the lowest-order terms in p(r). The corresponding pinning potential is evaluated according to eq. (9). Since the fluctuations are approximately constant over a scale of r( 0) (eq. (8) ), and in eq. (14) only the zero-order term (in q(r)) diverges logarithmically at r+rv, the pinning potential per unit vortex-length, 6[ U,(r) l/L, is
D. Agassi. R.D. Bardo /Pinning properties
F,+,(r,) -&(r”)
Ufl(r,) -=
L
L
=$
(16)
[V(r;r,)-v(r;r,)ll.,,“,
where Ffl+,(rv) and F,(r,) denote the free energy of a vortex in an infinite superconductor in the presence and absence of fluctuations, respectively. In general, fluctuations have two effects, namely to renormalize the mean of the pinning potential ( U,( rv) ) and give rise to spatial pinning potential fluctuations 6 [ U,( r,) 1. The statistical assumptions (6) imply that terms in eq. ( 14) containing an odd number of p(r)-factors, which vanish upon taking the ensemble average [ 221, constitute the pinning potential fluctuations. The contributions to the mean pinning potential arise from terms with an even number of p(r). The lowest order of such contributions are
+...lr”>
Ir+,
(17)
where an overline connecting a pair of p(r) factors i.e. the application of eq. denotes a “contraction”, (6) to that particular pair of p(r). The corresponding expressions, using eqs. (6) and ( 15 ), yield
119
of disordered planar defect
factor [ 2 11. The second factor, fi = (es/l)‘, arises from intermediate integrations in a contraction over a “u” factor. Since for a HTSC A( O)/r( 0) N 0( 100) and G/J. - 5( 0) /A - 0( 0.0 1)) typically f, 2: 30 while for an assumed E= 0.1, one obtains f2= (ta/ 1)2- 10P6. It follows that the lowest-order corrections to the mean pinning potential are proportional to a “small” parameter fi f2 u 0( 0.00003 ). This justifies a perturbative approach to eq. ( 17 ), and similarly for the pinning potential correlation function (below). Note that the smallness of&, the prime reason for the excellent convergence of the Born series, is a direct consequence of the assumed short correlation length 0 (eq. 8) ). Another important feature to be noted from the analysis of eq. (18) is that the last line, which involves the least number of intermediate summations, gives the largest contribution. This rule is substantiated below and used in section 4. We turn now to the pinning potential fluctuations. From eqs. ( 14) and ( 15), the lowest-order contributions are 6[G(rv)l
$0
= G
L
(rlv~v-~O~(rV)v+...Ir,>I,-,,,
(19) the
and
quantity
of
interest
( 6 [ U, ( rv) ] 6 [ U,( r: ) ] ) . Straightforward
.
algeb::
using eqs. (6) and ( 15 ) yields v(r--rV)
I14rv
=
$5
ln(UkXo))
C,(IArVI)=~~~(r,)~(r:)(r,I~lr,)(r:I~lr:)
,
2
(rlwv4rv)
( > @O
I-
=(2xn)Z
~2[2~ln(~/5(o))12~1(l~~l),
(18)
= &
y
’ [27tln(1/<(0))12,
0 where the first line in eq. ( 18 ), pertaining to Fv ( rv), is added here for reference. Expressions ( 18 ) highlight two recurring dimenfirst, f,=2rtln(ll/ factors. The sionless <( 0) ) - v( ( r-r, ( u <( 0) ) originates from a contraction of the two arguments in a v( r- rv) factor. The cutoff at the coherence length momentum in integrals like ( 15 ) produces this well-known logarithmic
(20) where we introduced F,(z)=e-Z2/2”2,
the functions F2(z)=Ko(z)2,
D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
120
F,(z)=2zK,(2z),
(21)
and p=r/l, K,,(z) and K,(z)are the Kelvin functions [ 231, and AI/A= 1r-r' 1/A=Ap. Thus the pinning potential correlation function to lowest order is
<~[~fl(rv)l~[~fl(r~)l> L2
(22)
+C3(Irv--rLI)l.
The correlation function, eq. (22), is comprised of terms with disparate ranges and strengths. The strongest contribution, C, ( )Ar, I), has also the shortest range, namely IJ. The other terms have a range of the order I and are smaller by orders of magnitude. This outcome may be understood by
4. Disordered-regular
noting that since u is the fluctuation “range”, correlations over that range are expected to be of “full strength” ( N t’). The other terms in eq. (22) represent processes combining propagation in a random medium and fluctuations, where the former substantially dampens the contributions ( thef2 factor in conjunction of eq. ( 17 ) ). It is for this reason that the dominating term involves the least intermediate summations. In summary, we have established a perturbative expansion of the mean and correlation function of the pinning potential that is consistent with eq. (8 ) and weak disorder. The analysis yields the leading contributions and a rule for identifying the dominant terms. It further indicates that fluctuations contribute primarily to the pinning potential fluctuations, while the corrections to the corresponding mean are negligible.
superconducting junction
In this section we consider a model of a disordered-regular superconducting junction with a planar interface (fig. 1). In comparison to section 3, the new elements here are the boundary conditions (3) which link the regular and disordered sides of the junction. As argued in the Introduction, this linkage endows the regular side with a feature of disorder, i.e. pinning potential fluctuations. Notwithstanding the different underlying physics, this effect is termed the disorder-proximity effect, in analogy to the situation for superconductor-nonsuperconducting junctions [ 12 1. As a reference case, consider first a junction where both sides are regular superconductors [ 18 1. As mentioned earlier, the translational invariance parallel to the junction interface implies pinning only normal to the interface. In the presence of disorder on one side of the junction, the above is qualitatively modified. First and foremost, the above translational invariance is broken, giving rise to pinning in that direction. The other difference is that fluctuations quantitatively modify the surface barrier [ 18 1. Consider first the case of a vortex located on the regular side of the junction. The corresponding London equations are
(23)
where the “A” and “T ” labels refer to the corresponding homogeneous, its solution has the decomposition
domains,
see fig. 1. Since on side A, eq. (23 ) is in-
(24)
BA(~rv)=U~(r;rv)+baND(r;r,), where uA(r; rv)=uA(r-r,) is given in eq. (15) with 1=A H, and by” eq. (24). Both terms have the representation [ 71
(r; rv), the
induced
field, is defined
by
D. Agassi, R. D. Bardo /Pinning properties of disordered planar defect
1
O”
(25)
where the symbols
in eq. (25) denote
x=L(k,
P=
ky)
9
121
tr,v) = (4 Y)llH
Dx(xy)
9
=,/x:+t1/Rx)* ,
Rx=;IxlA~t (26a)
forX=A,T, and evt-i[xyv*i~X(xy)tl)
$0
Dx(xy)
t26b)
.
The + and - signs in eqs. (25) and (26b) apply when c> & and <<&, respectively. On the disordered side (T), eq. (23 ) has the structure of a scattering wave equation [ 13 1, with the dop( r) / 1% factor playing the role of a potential. Consequently the solution has the Born series expansion where the “unperturbed” Green function is z+ (r- r’ ), and the “scattering” solution of the corresponding homogeneous equation has the representation
Expressions (25) and (27) define the unknowns, bA(xy) and &(x,,)_ The latter is not a fluctuating quantity since the fluctuating factors are explicitly separated out in eq. (27). Matching eqs. (25) and (27) according to (3), and following the paradigm in section 3, the mean induced field, in lowest order, is 00
H
I
)
Y
=
corrected
to first order, r(l), is given by
PA --R~&(x~)/Q&~) n,txy) +R:&(xJl&(xy) ’
72
T
0
.2
.4
.6
A
(28)
’
DA(&)
--co
where the “reflectivity” t(~)tX
e--2D4~~%rW(xy)
dx,
.*
t4(Xy) =
I
1.2
i$
1+NGtXy) 1-N&(Xy) ’
Fig. 2. The mean pinning potential of a disordered-regular junction in the &=A,, limit. The quantity I(<) is the integral in eq. (30) and C=x/&.
122
D. Agassi, R.D. Bardo /Pinning
FT(<&
h)
Fn(-h
properties of disordered planar defect
t’l)
_._.-. t
_._.-. id; A) 1j&t(o); A)
-
i&i T)/ i&(o); T)
10-i
a
t,
Fig. 3. The pinning potential correlation functions and associated critical currents. In fig. 3(a) the integrals in eqs. (32) and (39) are plotted for the cases of RT= 1.5 (broken line) and R,= 5 (solid line) on the respective sides of the junction. The coordinate system is defined in fig. 1 and &x/L,. Figure 3 (b) shows the estimated critical currents perpendicular to the interface, normalized to their values at the interface as given by eq. (40). For positive <, the broken and solid curves correspond to RT= 1.5 and R,=5, respectively, for the disorder parameters of section 5. The curves represent j,( I; 5; A; PD+fl)/j,( I; t(O); A; PD+fl). For negative 5, only the results for R,=5 are given: the dot-dash curve represents j,( I; CT; PD+fl)/j,( I; c(O); T; PD+fl) and the solid curve represents j,( I; e T; fl)/ j,( I; r(O); A, PD+fl).
An important feature of the mean field (28) is that there is no qV-dependence, i.e., the fluctuations do not generate an average pinning potential parallel to the interface. Hence all y-pinning (see fig. 1) is associated with the pinning potential fluctuations. The fluctuations correction to eq. (28) enters via the dimensionless parameter A, eq. (29 ), which consists of factors already encountered in eq. ( 18 ). Consequently, the fluctuation correction to the mean pinning potential is negligible, as for the model of section 3. An exception is the limit As is evident from eqs. (28) and (29) in the absence of fluctuations (A =O) when (L:(r)) =n:=A&. T(‘) (x,,) =O, i.e., the induced mean field vanishes as it should. However, in the presence of fluctuations, eq. (29) yields T(‘) (x,,) -A/D% (x,,) and (30)
The positive pinning potential, eq. (30), plotted in fig. 2, represents repulsion between the vortex on the regular side and the interface. It extends over a distance AH, which is considerably longer than the fluctuations correlation length. This is a first manifestation of the disorder-proximity effect. We turn now to the calculation of the pinning potential correlation function, eq. (1 1 ), following the paradigm of section 3. The result for the leading term, in lowest order, is
D. Agassi, R.D. Bardo /Pinning properties of disordered planar defect
123
(31) where co co ca FA(P”;P:)
=
SSI
-cc -co -cc
dqdQdQ’exp[iq(rl,-rl:)-(D,(Q+q/2)+D,(Q-q/2))t;,
(32) and the unperturbed
7(X)=7’o’(X)=
(see eq. (29) ) is [ 71
reflectivity
1-GDT(X)IDA(X) 1+GD,(x)lD,(x)
(33)
.
All other terms in the lowest order are smaller by at least a factor (a/n.). The important q,-dependence of eq. (32) is a manifestation of the disorder-proximity effect, i.e., the creation of pinning parallel to the defect interface, outside the disordered domain. Note that the &-dependence in eq. (32) is invariant under v,-translation, as it should be. The calculated examples in fig. 3 (a) show an approximately exponential falloff with distance from the junction interface, with an approximate decay constant AH. is approximately A,/ The logarithmic divergence of eq. (32) for t,, cV+O, at a cutoff distance -r(O), <( 0) (ln(n,/c( 0) )2. It is a common artifact of the delta function source term in the London equation [ 2 11. From eqs. (30) and (32) the relative magnitude of the mean and fluctuating pinning potential at the interface (<;,=~~=<(O)) can be derived. Denotingp,=(t(O), q), we obtain (34) This estimate indicates that the pinning fluctuation contribution may become comparable to that of the mean pinning potential. In particular, the surface barrier [ 7,181 is qualitatively modified. Note also that in the limit of a disordered “weak” superconductor, where I, z+ AH, then 7% - 1 and the r-factors in eq. (32) contribute an enhancement of a factor -4. By the same token, in a disordered “strong” superconductor, 7- 1, and the pinning fluctuations are suppressed on the ordered side. The latter result is consistent with the physical picture that the shielding currents penetration into the disordered side is suppressed in this case, and hence the proximity effect is attenuated. We turn now to the case of a vortex on the disordered side (T) of the junction, fig. 1. In this case, the solution is the sum of a single vortex solution in a disordered superconductor, VT(r; rV) (eq. ( 16) ), and the general solution of the homogeneous equation, &(r,
rV) = vT(r, rV) +GND(r,
rV) ,
(35)
where byD (r) has the Born series expansion (27). On the A side the solution of the homogeneous has the form (24). Similar algebra yields the leading contribution to the mean pinning potential:
equation
co
(b:NDkPv)) = where
&
T
s
emh(xY)h
dxyD,(x
--m
7(Xyh(Xy) Y
>
(36)
124
D. Agassi, R.D. Bardo
l-
/Pinning properties of disordered planar defect
(~~lh)21n(Wt(0) )
nr(Xy)= 1 +0.25(D,(X,)R,)-2(ta/~T)21n(~T/r(0)
(37)
)r(X,) ’
As for eq. (28), the mean pinning potential (36) does not depend on the qy coordinate, i.e., no pinning in that direction even on the disordered side. The renormalization of the mean pinning potential due to fluctuations is again a small correction due to the typical factor (~a/1)~ << 1. It is interesting to consider again the limit where (A2(r)) =A$ =A&. In this case 7(x,,) =O and eq. (36) implies no pinning on the T side even with disorder. The marked asymmetry between the two sides of the junction, fig. 2, can be heuristically understood as follows: for a vortex located on the disordered side, the shielding currents incurring into the regular side of the junction are not “scrambled” any further, hence there are no additional induced fields due to the boundary. On the other hand, for a vortex located on the regular side of the junction, the shielding currents are always “scrambled” upon incursion into the disordered side, and hence the ensuing self-field generates pinning. In order to evaluate the pinning potential fluctuations on the T side, only the leading term in the lowest order, is considered. This yields (38) where the lengthy expression
MP,;P,)=
for FT(pv; p:) is given in the Appendix.
Its diagonal
part is
7 7 7 dqdQdQ’exp[(DT(Q-q4/2)+DT(Q’+q/2))r,l -03 -ccl--m
(39) The correlation functions (3 1) for the regular side and eq. (38) for the disordered side are the central results of this work. In fig. 3(a) two representative cases are shown. Note again the sharp falloff with distance from the interface, typical of the induced fields. The RT~ and (o/AT)* prefactors in eq. (38) strongly suppress the pinning potential fluctuations on the disordered side for RT > 1. Comparing the prefactors in eq. ( 38 ) to those of the C, term in eq. (22), the latter is the dominating term on the disordered side (see section 5). This is a plausible result. It reflects the fact that pinning potential fluctuations correlate poorly in a disordered domain since it involves a field propagating from the vortex to the interface and back in a random domain. 5. Discussion
and summary
The S’-S disordered-regular junction considered here yields an estimate of the critical currents normal and parallel to the junction interface. The connection between the calculated pinning potential and critical current involves the derivative of the pinning potential in the appropriate direction and at the appropriate evaluation point [ 7 1. A pinning force nor-
ma1 to the junction interface is already present in the regular-regular superconducting junction and, therefore, is not discussed here. The new feature in the disordered-regular junction is the pinning force parallel to the junction interface, associated with the critical current normal to it. For the purpose of estimating derivatives a typical range over which the pinning potential fluctuations
D. Agassi, R. D. Bardo /Pinning properties of disordered planar defect
vary is o. On the regular side, the only contribution to the critical current is through the disorder-proximity effect, U,,,, (r), see eq. (9), whereas on the disordered side there are two contributions, i.e. u PD+B(r)+ Q,(r), see eqs. (9) and (16). Expressions(31),(38),(22),(11),and(9)and(l6)yield the following critical currents (j,) normal to the interface ( I ), on sides X= A, T, evaluated at r= <( 0): j,( 1, r=<(O);
A; PD+fl)
(40) Inserting the parameter (T= 15 A, e=O.l, a/ r(O)= 1, AH=900 A, @oo=1633 (eV A)‘.5 we obtain ontheAsidej,(_L;r=<(O);A;PD+fl)-4X108A/ cm2. This estimate is subject to uncertainties in the e and CJparameters which can vary the results by two orders of magnitude. On the T side, expressions (40) show that the critical current associated with “PD+fl” scales as RT~ (o/n,), while the one associated with “fl” scales as Ry’. Thus the latter is dominant for the measured RT [ 241, and is smaller than the critical current on the A side. Typical critical currents are calculated and compared in fig. 3 (b ). Note the approximate exponential falloff on the regular side, with an approximate decay length ?An, while on the disordered side the decay length AT= RTLu > An. This type of exponential falloff is in keeping with the remarks following the introduction of the basic equation (5). The foregoing analysis is based on a number of simplifying assumptions. The first is a rigid vortex line, corresponding to the glass phase regime [ 25,26 1. For undulations over a length scale small in comparison to the penetration length, the present analysis provides a local approximation of the pinning potential in the vortex line direction. This interpre-
125
tation is consistent with the absence of a pancakes chain flux line in the present analysis [ 15 1. The single-vortex feature of the model corresponds to a low magnetic field regime where collective pinning does not enter [ 271. The low temperature and low field validity ranges of the model are in keeping with the high critical current estimates obtained above. The exponential distance dependence of the critical currents implies that pinning may be enhanced provided the distance between boundaries is smaller than the penetration depth. Twin boundaries form such a dense set of planar defects; however, they may lack other necessary attributes. This observation suggests that provided the twin boundaries are modified to conform with a superconducting junction, the corresponding critical intragrain critical current is enhanced. To our knowledge, no systematic studies of jc and twin boundary density exist [ 91. In summary, we have considered penetration depth fluctuations in the context of a vortex in an infinite superconductor and in a regular-disordered superconducting junction. This particular focus is motivated by the probable structural and composition disorder inside a grain and twin boundaries. The analysis is limited to a “short” correlation length and “weak” disorder, the parameter regime applicable to HTSC ceramics. An important qualitative result is that in the presence of spatial disorder the pinning potential is not spatially smooth in an adjacent regular domain. In other terms, there is a disorder-proximity effect whereby fluctuation effects extend beyond the original domain of disorder over a distance of the penetration length. This effect may be of relevance in the context of flux creep. The corrections to the mean pinning potential are proportional to the small parameter (es/l)’ N 1O-‘j. The fluctuations, however, give rise to a substantial critical current normal to the interface. The implications of this analysis for finite width grain boundaries are discussed in a subsequent paper [ 13 1.
Acknowledgement We would like to acknowledge ulating discussions.
C. Pande for stim-
D. Agassi, R.D. Bardo
126
/Pinning properties of disordered planar defect
Appendix The general expression for the kernel in eq. (38), valid for 0 > & >
&=&(Q'-q/2), (A.11 D3=DT(Q-q/2),Dq=DT(Qf+q/2) > LA=&-(Q+q/2),
and
K,(q,Q,Q+X,=j& 1
2
x(exp(-DD,r,-D2P;)P(Dl+D2+D*;r:,0) +ew( -DLL +D&)P(D~
-D2 +D*; L G)
+exp(Dt~v+D2~)p(-D,-D2+D*;0,~v)), (A.21 eD&_eDCL
~(D;tv,r:)= the correlation
(‘4.3)
9
A
function
in eq. (38) is given by
&(Pv; PI)
dQ'K,(q, Q,Q'; L r:) -cm
= 7 dq 1 dQ ,=/
-cc -m
x
T(Q--d2)T(Q’+d2) DID~&D~
.
(A.4)
References [ 1] D. Larbalestier,
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