Physica C 349 (2001) 125±138
www.elsevier.nl/locate/physc
Distribution of the magnetic ®eld and current density in
superconducting ®lms of ®nite thickness
D.Yu. Vodolazov, I.L. Maksimov *
Nizhny Novgorod University, 23 Gagarin Avenue, Nizhny Novgorod 603600, Russian Federation
Received 20 January 2000; accepted 24 April 2000
Abstract
An 1D equation describing the distribution of the e€ective vector potential A…y† across a ®lm width, which holds for
thin (d < k) and thick (d > k) ®lms alike, is derived based on the analysis of a 2D Maxwell±London equation for
superconducting ®lms in a perpendicular magnetic ®eld. For a ®nite k case, the distributions of the local magnetic ®eld
and current density are found both inside and outside superconductors. An approximation dependence A…y†, ®nite (with
all of its derivatives) across the entire ®lm width, is found for ®lms in the Meissner state. The ¯ux-entry ®eld is evaluated
for a ®lm of arbitrary thickness. An approximation expression is obtained for the distribution of the sheet current
density in the mixed state of a pin-free superconducting ®lm with an edge barrier. The latter approximation allows one
to estimate magnetic ®eld concentration factor at the ®lm edge as a function of external magnetic ®eld and geometrical
parameters of the sample. Ó 2001 Elsevier Science B.V. All rights reserved.
PACS: 74.60.Ec; 74.76.-w
Keywords: Meissner state; Mixed state; Superconducting ®lms; Surface barrier
1. Introduction
Of active interest recently is the theoretical [1±
14] and experimental [15±22] investigation of
mixed static and dynamic states in superconducting ®lms in a perpendicular magnetic ®eld.
Theoretical calculations of various magnetic characteristics of ®lms in such a geometry, using
the microscopic theory or the Ginzburg±Landau
*
Corresponding author. Tel.: +7-831-2656255; fax: +7-8312658592.
E-mail address: ilmaks@phys.unn.runnet.ru (I.L. Maksimov).
equations does not seem possible because of the
mathematical intricacies involved in their solution.
Therefore, in practical calculations, the following
two approaches are mostly employed: In the ®rst
method the Maxwell equation is analysed jointly
with the London equation, which yields an integral±di€erential equation [1,2,8±13] for distribution of the current density integrated over the ®lm
thickness, d. This equation was derived on the
assumption that the ®lm is thin (d k, k is the
London penetration depth), which naturally limits
its applicability range.
The other approach is based on the theory
of complex variable functions (TCVFs) [6,7,14],
used in transformation of integral equations [3±5].
In this case, the phenomenological dependences
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 3 4 ( 0 0 ) 0 1 5 2 2 - 7
126
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
B…H† and J…E† are employed as an additional
condition instead of the London equation. It
should be noted that neither of these methods has
ever focussed on investigating the e€ects related to
®niteness of the London penetration depth, k.
Besides, the equality of the results obtained
through solving of the integral equation for thin
®lms and by the use of the TCVF methods for
thick ®lms leads us to believe that for ®lms of arbitrary thickness there should exist one equation
describing distribution of the current density and
the magnetic ®eld across the ®lm width.
The present paper deals with a study on the
distribution of the magnetic ®eld and current
density in the Meissner and mixed states for ®lms
placed in a perpendicular magnetic ®eld. It is
shown that for the ®nite-thickness ®lms, the
Maxwell±London equation [1,2,8] describes distribution of the vector potential (current density)
averaged over the ®lm thickness, d, provided the
latter, is much smaller than the ®lm width, W:
d=W 1. For thin ®lms, d k, this equation is
valid practically over the entire width of the ®lm,
while in the case d k, it holds everywhere in the
®lm, except for the areas near the edges, which
measure as W =2 ÿ d=2 6 jY j < W =2.
The paper is organized as follows: Section 2
describes derivation of an 1D equation for distribution of the ®lm-thickness-averaged vector potential across a sample width, based on the
analysis of 2D Maxwell±London equation. In
Sections 2.1 and 2.2, the 1D and 2D equations are
numerically studied and compared for thin and
thick ®lms, respectively. An approximation dependence for A…y† is obtained through numerical
solution of these equations. The distribution of the
vector potential (or the local current density) and
of the local magnetic ®eld over a superconductor
cross-section is described. Section 3 deals with estimation of the ®eld for the ®rst vortex entry into
thin and thick superconducting ®lms (Sections 3.1
and 3.2, respectively). In Section 3.3, we discuss
the in¯uence of surface defects and a layered
structure of superconductors on a barrier suppression ®eld value. Section 4 is concerned with
the structure of a mixed state in thin and thick
superconducting ®lms of an arbitrary width in the
absence of bulk pinning. An approximation formula is found for the integral (over thickness)
current density, which is then used as a basis for
constructing the magnetization curves for these
superconductors. Section 5 sums up the main results obtained in this work.
2. The structure of the Meissner state
Assume an in®nite (in the X direction) superconducting ®lm of width, W and thickness, d in a
perpendicular magnetic ®eld; the geometry is
shown in Fig. 1. Let us ®rst consider the Meissner
state. The Maxwell equation has the form (in a
gauge r A ˆ 0),
DA ˆ ÿ
4p
j;
c
1†
Fig. 1. Geometry of the problem: Points with coordinates (y ˆ 1, z ˆ 1) and (y ˆ 1, z ˆ 0) determine side edges and equator line of
the ®lm, respectively.
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
where D is the 2D Laplacian operator. It also
follows from the symmetry of the problem that
only the x components of the vector potential
A ˆ …Ax ; 0; 0† and of the current density j ˆ
jx ; 0; 0† are not zero. The boundary conditions are
oAx =oY †jY !1 ˆ ÿH1 , …oAx =oZ†jZ!1 ˆ 0, where
H1 is the ®eld far from ®lm H ˆ …0; 0; H1 †. It
should be noted that by the local magnetic ®eld h
in a ®lm, here we imply a microscopic ®eld averaged over scales much larger than the atomic one
but much smaller than k.
In this case, it is convenient to change over from
the di€erential Eq. (1) to its integral analog. Using
the Green function of the 2D Laplacian operator,
we rewrite Eq. (1) as
Ax …Y ; Z† ˆ A0x …Y † ÿ
2
c
Z
Z
W =2
ÿW =2
d=2
ÿd=2
ln jR ÿ R0 j ‡ C†
jx …Y 0 ; Z 0 †dY 0 dZ 0 ;
2†
where C is the constant generic for the 2D Green
function, A0x …Y † is the vector potential of an external ®eld: A0x …Y † ˆ ÿH1 Y and the origin of coordinates is chosen in the ®lm centre.
Employing the London equation j ˆ ÿcA=4pk2
and introducing dimensionless coordinates y ˆ
2Y =W , z ˆ 2Z=d, Eq. (2) reads
Ax …y; z† ˆ A0x …y† ‡
‡
Wd
16pk2
d
W
Z
ÿ1
ÿ1
0 2
z ÿ z †
Z
1
ln …y ÿ y 0 †
2
!
2
Wd ~
C
‡
16pk2
1
Z
1
ÿ1
Z
1
ÿ1
Ax …y 0 ; z0 † dy 0 dz0
Ax …y 0 ; z0 † dy 0 dz0 ;
3†
where C~ ˆ C ‡ 2 ln…W =2†. The latter integral in
Eq. (3) is directly proportional to the total current.
In a magnetic ®eld (without a transport current)
the total current is equal to zero due to the symmetry, so the last term in the right-hand side of Eq.
(3) vanishes. We now average Eq. (3) over the ®lm
thickness, which yields the following
expression
R1
(hereafter by A…y† we imply 0:5 ÿ1 Ax …y; z†dz):
127
Wd
4pk2
Z 1 Z
Z
A…y† ˆ ÿH1 yW =2 ‡
‡
Wd
32pk2
0
Z
1
ÿ1
0
ÿ1
1
ln jy ÿ y 0 jA…y 0 † dy 0
!
b2
ln 1 ‡
2
y ÿ y 0 †
ÿ1
1
ÿ1
0
Ax …y ; z † dy dz dz0 ;
4†
where the integral kernel of Eq. (3) is written in the
form
!
2
d
0 2
0 2
z ÿ z †
ln …y ÿ y † ‡
W
!
b2
0
;
ˆ 2 ln jy ÿ y j ‡ ln 1 ‡
2
y ÿ y 0 †
and the designation, b ˆ …z ÿ z0 †d=W 1 is introduced (as in this case of superconducting ®lms
d=W 1).
2
The function ln…1 ‡ b2 =…y ÿ y 0 † † is not zero
only in a small region around point y 0 : jy ÿ y 0 j
6 jbj. In this case, integration over y 0 in the second
integral of Eq. (4) can be done only over this small
region. For the same reason, we can expand the
function Ax …y 0 ; z0 † into the Taylor series in terms of
y 0 near point y:
Ax …y 0 ; z0 † ˆ Ax …y; z0 † ‡
oAx …y; z0 † 0
y ÿ y† ‡ :
oy
5†
Note that expansion (5) (in the above-speci®ed
limit) is valid for thin (d < k) ®lms over the entire
sample width. In thick (d > k) ®lms, the validity of
this expansion is violated in the near-edge regions
with dimensions of order d=W . More details on the
applicability of Eq. (5) are provided at the end of
this section.
After the series expansion of function Ax …y 0 ; z0 †
in Eq. (5), it is now possible to calculate the last
term of Eq. (4):
Z
y‡b
yÿb
Z
1
ÿ1
Z
1
ÿ1
ln 1 ‡
d
W
2
z ÿ z0 †2
y ÿ y 0 †
!
2
oAx …y; z0 † 0
y ÿ y† ‡ dy 0 dz dz0 ;
Ax …y; z0 † ‡
oy
6†
128
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
(note, that the upper (lower) limit of the integration over y 0 will change by 1…ÿ1†, when point
y becomes close to the ®lm edge, i.e., when
1 ÿ jbj 6 jyj < 1; this is because the integration in
Eq. (4) is carried out only over a sample volume)
and to show that integral (6) is equal to zero in a
wide parameters range. Indeed, integration of Eq.
(6), ®rst in terms of y 0 and then in z and z0 , provides
a direct evidence (bearing in mind that function
Ax …y 0 ; z0 † is even in z0 ) that integral equation (6) is
zero for all values of y satisfying the inequality
jyj 6 1 ÿ jbj. In the region 1 < jyj 6 1 ÿ jbj, integral
equation (6) leads to appearance of nonzero terms
that for thin ®lms are small due to the presence of
n
the corrections of …d=k† (n > 1) type. They have
to be taken into account when we are interested in,
for example, the distribution of the derivative
dA=dy near the ®lm edges (since disregard for these
terms will cause a logarithmic divergence of the
®rst derivative). For thick ®lms, allowance for the
near-edge regions of a superconductor in integral
Eq. (6) cannot largely a€ect the A…y† distribution
o€ the ®lm edges because of smallness of jbj.
Thus, the 2D equation (3) is reduced to a 1D
equation for the ®lm-thickness-averaged vector
potential A…y†, which is valid in the region
jyj 6 1 ÿ jbj
Wd
A…y† ˆ ÿH1 yW =2 ‡
4pk2
Z
1
ÿ1
Besides, the vector potential in this case is
practically independent of z (but not the derivative
oAx …y; z†=oz). Thus, at d ˆ k, the relation Ax …y; 1†=
Ax …y; 0† 1:07, i.e., the di€erence is about 7%.
With a lower d=k ratio, the relation Ax …y; 1†=
Ax …y; 0† tends to unity.
We have derived an asymptotic expression for
the vector potential distribution, satisfying Eq. (7)
(and, hence, Eq. (3) averaged over z) with a suciently high accuracy (not less than 3% at a ®lm
edge and far from the edge region, and not less
than 6% in the near-edge region, Fig. 2):
keff H1 y
A…y† ˆ ÿ p ;
a…1 ÿ y 2 † ‡ b
8†
2
where keff: ˆ k2 =d, b ˆ 2keff =pW ‡ 4…keff =W † , and
the dependence a…W =keff † is shown in Fig. 3 together with approximation (9)
a 0:25 ÿ
0:63
0:5
W =keff †
‡
1:2
W =keff †0:8
:
9†
At W < keff , the dependence A…y† becomes almost
linear. Formula (8) with a ˆ 0:25, b ˆ 0 was derived analytically in Refs. [1,2] by solving Eq. (7)
(to be more exact, a simpli®ed version of Eq. (7) in
which the left-hand part is omitted, which corre-
ln jy ÿ y 0 jA…y 0 † dy 0 :
7†
In the following sections, the results of a numerical solution of Eqs. (7) and (3) are provided
for thin and thick ®lms.
2.1. Thin ®lms (d < k)
It turns out that in ®lms satisfying the condition
d=k 6 1=4, the di€erence between the solutions of
Eqs. (7) and (3) (averaged over thickness) is about
1% far from the ®lm edge and less than 4% in a
narrow near-edge region. An appreciable error in
the near-edge region (which is slightly growing
towards the ®lm edge with a larger numerical step)
depends on the presence of the small corrections in
Eq. (7) that were ignored.
Fig. 2. Distribution of averaged vector potential, for di€erent
ratios W =keff : curves 1±5 for W =keff ˆ 1, 5, 10, 50, 200, respectively. The dotted lines represent numerical results, and the
solid lines represent approximation (8).
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
Fig. 3. Dependence of the parameter a on W =keff . Circles represent numerical results and, solid curve 1 represent approximation (9).
sponds to the condition W =keff 1). Moreover, in
the wide-®lm limit, the expression for A…y† (see Eq.
(8)) at y ˆ 1 coincides with the value A…1† obtained
analytically in Refs. [8,9].
The resultant dependence A…y† allows one to
calculate the ®eld concentration parameter c ˆ
edge
edge
hz =H1 (where hz
is the edge ®eld averaged
over superconductor thickness) for ®lms of such a
type:
edge
h
2keff
a
:
10†
c ˆ z ˆ p 1 ‡
b
H1
W b
At W keff , Eq. (10) transforms into
p r
p 2p W
;
cˆ
10
keff
p
where the coecient preceding
W =keff is a
quantity of the order of unity. The di€erence between Eq. (10) and the numerically obtained expression for c may reach 30%: for wide ®lms
(W keff ), Eq. (10) yields an overestimated result
(see insert in Fig. 4). This is because, unlike the
A…y† function itself, the ®rst derivative equation (8)
with respect to y (magnetic ®eld) adequately satis®es the numerical solution everywhere except for
the narrow region near a ®lm edge (Fig. 4). Due to
the logarithmic divergence of the magnetic ®eld at
129
Fig. 4. Distribution of hz inside the ®lm for W =keff ˆ 200. Dots
represent numerical result, and solid line represent expression
obtained from approximation (8). Insert shows dependence
edge
hz …W =keff †: circles represent numerical result, dashed line
represent the expression
p (10), solid line represent the ®tting
function 1 ‡ 0:66 W =keff .
a ®lm edge, which follows from the solution of Eq.
edge
(7), the approximation expression for hz
was
compared with the numerical solution of Eq. (3).
Fig. 4 p
also
shows
the interpolation function


1 ‡ 0:66 W =keff for the numerical analysis data
(see solid line in the insert). Thus, the di€erence
between Eq. (10) and the numerical result in the
wide ®lm limit W =keff 1 is about 17%.
2.2. Strip of ®nite thickness (d k)
A comparative numerical analysis of the solutions to Eq. (3) integrated over a superconductor
thickness and Eq. (7) was also carried out for the
case k d W . It was found out that the solutions coincide (to the accuracy of about 3%) in the
region jyj < 1 ÿ d=W and (quite surprisingly) in
points jyj ˆ 1. In the near-edge region, we observed a di€erence in the solutions of Eqs. (3) and
(7).
An approximation equation has also been derived for the dependence A…y†. It turned out to be
exactly the same as the dependence equation (8)
with the selection a 0:25, b 0:64k2 =dW . One
can easily see that the values obtained for a and b
130
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
Fig. 5. Distribution of averaged vector potential at thick ®lm
(d ˆ 10k† for various widths: (1) W ˆ 50k, (2) W ˆ 100k, (3)
W ˆ 200k. Dots represent numerical result, solid lines represent
expression (8) with a ˆ 0:25, b ˆ 2k2 =p dW .
practically coincide, to a numerical error, with
those obtained for thin ®lms at k2 =Wd 1. Fig. 5
shows the dependence A…y† derived from the solution to Eq. (3), and also to Eq. (8). It is seen from
the latter that maximum deviation of the solution
to Eq. (3) from Eq. (8) (and, hence, from the solution to Eq. (7)) occurs in the region y >
1 ÿ d=W , but in point y ˆ 1, however, both solutions coincide again. Thus, expression (8) provides
an adequate description of the A…y† distribution
(the di€erence from the numerical solution of Eq.
(3) does not exceed 3%) in the region jyj 6 1 ÿ d=W
and at jyj ˆ 1. This con®rms the above statement
that Eq. (7) describes the distribution of the A…y†
dependence in a ®lm depth well. Besides, Eq. (7) is
apparently valid immediately at the edge of a superconductor as well, which, in our opinion, is a
quite surprising fact.
Fig. 6 shows the distribution of the current
density over a ®lm cross-section and the distribution of the absolute value of local magnetic ®eld
1=2
(jhj ˆ …h2z ‡ h2y † ) both inside and outside a ®lm
with dimensions W ˆ 100k, d ˆ 10k. As seen from
Fig. 6a and c, the magnetic ®eld reaches its maximum at the corners (side edges) of a superconductor. At the equator (y ˆ 1, z ˆ 0) the magnetic
®eld is less intensive, but not appreciably smaller
Fig. 6. Contour lines of the intensity of magnetic ®eld (a, c) and
current density (b, d) of thick (W ˆ 100k, d ˆ 10k) ®lm in applied perpendicular magnetic ®eld. The step for magnetic ®eld is
0:41H1 , for current density, it is 0:1j…1; 1†. Maximum values of
magnetic ®eld …jhj ˆ 4:1H1 † and current density …j ˆ 1† are
reached at the corners of the ®lm.
than the ®eld at the corners (for comparison, at the
corners jhj 4:1H1 , while in the middle of a side
face jhj 2:9H1 for the given parameters of ®lm).
It is easily seen that towards the ®lm interior
(along the y-axis), the magnetic ®eld is practically
uniform through the entire sample thickness, except for the near-surface areas with the thickness
of order, k, where magnetic lines abruptly change
direction. A similar behaviour is demonstrated by
a current density (Fig. 6b and d). It is proved
numerically that both the current density and the
magnetic ®eld fall o€ towards a sample centre by a
law similar to the exponential one, not only from
the side faces but also from the top and bottom
ones. So, in thick ®lms k d W , screening
currents ¯ow only in the near-surface layers of
thickness about k. On the same scale, there is a
decrease of a local magnetic ®eld in superconducting samples of such a type.
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
The numerical solution of Eq. (3) also provides
a possibility to determine the ®eld at a ®lm edge.
Unfortunately, unlike with thin ®lms, Eq. (8) differs largely from the numerical result for the nearedge region (this discrepancy may reach 30%).
Therefore, the use of Eq. (8) in calculations of a
thickness-averaged z-component of the magnetic
®eld at a ®lm edge certainly leads to a considerable
error.
In Fig. 7, the obtained numerical dependences
edge
hz =H1 (the z-component of magnetic ®eld, averaged over thickness) and hz …1; 0†=H1 (the zcomponent
of magnetic ®eld on equator) on
p
W =d are presented. It is clearly seen that with a
good accuracy, the dependences are linear even for
the W =d values which are close to unity. Note that
edge
the coecient of proportionality between hz =H1
1=2
and …W =d† is equal to 1.03, which practically
is the same as its estimate ( 1) found in Ref. [14].
Generally speaking, the value of this coecient
depends on a ®lm thickness (or, rather, the ratio
d=k). Thus, the insert in Fig. 7 illustrates ratios
edge
hz =H1 , hz …1; 0†=H1 for various values of ®lm
thickness and shape parameter, W =d ˆ 5. It is seen
that only the quantity hz …1; 0† is practically inde-
131
pendent of the ratio d=k. The strongest dependence on ®lm thickness is exhibited by hz …1; 1† (it
grows with the increase of the ratio d=k). This
edge
results in a slight increase of the average ®eld hz
with the growth of ®lm thickness (given the same
W =d ratio). However, for very thick ®lms, d k,
edge
hz =H1 is supposed to practically cease to be
dependent on d=k. Indeed, in the limit of interest,
the equator ®eld is independentpof
d=k,
 while the
corner ®eld hz …1; 1† increases as W =k (which was
derived from expression (11c) and (11d)). The
sharpest variation of the magnetic ®eld intensity
occurs at a distance of order k from the top/bottom surfaces. Correspondingly, the contribution of
edge
those regions in hz =H1 will be of the order
1=2
k…W =k† =d ˆ …W =d†1=2 …k=d†1=2 and will become
negligible with the increase of the ratio d=k.
Besides the approximation expression for A…y†,
we have found numerical estimates for the vector
potential at …1; 1† (on a corner); (1; 0) (on the
equator), and also the distribution of the vector
potential (current density) over the upper/lower
surfaces:
r
W
k;
11a†
Ax …1; 0† ' ÿH1
d
1=2
1 d
;
Ax …1; 1† ' Ax …1; 0† 1 ‡ p
16p k
11b†
keff H1 y
Ax …y; 1† ˆ ÿ p ;
a…1 ÿ y 2 † ‡ b
11c†
where
0:25
;
2
1 ‡ …d=k† =2p†
2keff
p :
bˆ
pW …1 ‡ …d=k†= 16p†
aˆ
edge
Fig. 7. Dependences hz
() and hz …1; 0† ( ) on the parameter …W =d†1=2 . Solid line 1 is the ®tting function 1=3 ‡
1:03…W =d†1=2 , dotted line 2 is the ®tting function 1=3 ‡
edge
0:92…W =d†1=2 . Insert shows the dependences of hz (),
hz …1; 0†( ) and hz …1; 1†() on the ®lm thickness for W =d ˆ 5.
11d†
It is easily seen that at d k, the vector potential
in points …1; 1† will largely exceed its value in
points …1; 0†. Using expression (11c) which is
similar to Eq. (8) with renormalized parameters a,
b, we can ®nd the points (lines) on the upper and
lower surfaces, where the vector potential will coincide in absolute value with that on the equator.
Simple calculations show that these lines are at
132
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
a distance d=p from the side surfaces of the strip
having k d W .
These results allow us to con®rm the assumption (Section 2.1) on the possibility of expanding
Ax …y 0 ; z0 † in the limits (y ÿ b, y ‡ b). Indeed, in the
case of thin ®lms, the vector potential (or current
density in a mixed state; Section 4) varies on scales
much larger than k and, hence, than d (see Eq. (8)).
For thick ®lms, the vector potential (current density) far from edges is ®nite only in the surface
layers of thickness of order k. At the same time,
the scale of variation for Ax …y; z† along y o€ sample
edges is macroscopic (see expression (11)). Therefore, expansion (6) is also possible o€ the edges.
Near the edges, however, Ax …y; z† 6ˆ 0 over the
entire thickness, and the scale of A…y; z† variation
(Fig. 6b and d) in this region is k (in the y direction). Hence, expansion (6) is not valid in the limits
y ÿ b 6 y 0 6 y ‡ b near the edges (jyj ! 1) of thick
superconducting ®lm.
3. The conditions for vortex entry in superconducting ®lms
Using expressions (8) and (11), it is possible to
estimate the edge barrier suppression ®eld, Hs or
the ®rst-vortex entry ®eld into superconducting
®lms.
3.1. Thin ®lm
The vortices may enter into a thin ®lm in the
Meissner state provided the condition jA…1†j ˆ
Acrit is met; here Acrit U0 =2pn [22,23] (U0 is the
quantum of a magnetic ¯ux, n is the coherence
length). The resultant expression for Hs is
U0 p
Hs b;
12†
2pnkeff
with b being the same as in expression (8). Dependence (12) in the limit W keff and W keff
coincides to a factor of order one with the expression for the Meissner state breakdown ®eld
obtained in the limiting cases of wide and narrow
thin ®lms in Refs. [1,2]. From Eqs. (10) and (12),
we can easily ®nd the value of the magnetic ®eld
(or, rather, the thickness-averaged z-component)
edge
at a ®lm edge hz , when vortices start penetrating
edge
in it. For example, for wide ®lms hz is
edge
hz
U0
:
10nkeff
To an accuracy of the factor of order one the
above expression coincides with the ®eld in the
core of a Pearl vortex which is equal to U0 =4pnkeff
[24].
3.2. Thick ®lm
The main di€erence between thick and thin
®lms is that the vector potential for the former
largely depends on z. However, we should apparently anticipate that the conditions of vortex entry
here will be qualitatively similar to those for thin
®lms. Indeed, as shown in Refs. [22,23], after the
vector potential has reached its critical value at a
superconductor edge in the Meissner state (in the
mixed state it is the gauge-invariant potential
P ˆ U0 ru=2p ÿ A that should reach a critical
value), the order parameter, W ˆ jWjeiu is strongly
suppressed and vortex formation begins. The
above papers dealt with bulk superconductors and
thin-®lm samples, which, due to the symmetry of
the problem or problem geometry, could be assumed homogeneous along the z-axis.
It should be expected that in thick ®lms, the
order parameter will be suppressed in the regions
where the vector potential Ax …y; z† reaches its critical value.
First, the condition jAx …y; z†j ˆ Acrit will be satis®ed at the side edges …1; 1† of a superconductor (as the magnetic ®eld increases from zero).
This means that the order parameter will be ®rst
suppressed in these points ®rst. With a further
increase of the magnetic ®eld, this situation may
develop by two scenarios:
(1) Suppression of the order parameter results
in tilted vortices that start to form at the corners of
a superconductor cross-section. When the vector
potential at the equator reaches the critical value,
two tilted vortices fragments (from top and from
bottom) will join each other to form one rectilinear
vortex. In the absence of pinning, the latter is able
to penetrate into the sample centre driven by the
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
Lorentz force. A similar vortex entry scheme was
discussed in Ref. [14].
(2) In the course of further magnetic ®eld increase, the order parameter becomes suppressed in
the region of side edges. This causes areas with a
suppressed order parameter to appear near the
side edges, which would allow one to regard a ®lm
cross-section as a rectangular with rounded-o€
edges. It should be emphasized that the geometrical sizes of sample remain unchanged in this situation, only its physical properties vary in the
regions near side edges. This scenario allows us to
explain the physical mechanism behind the formation of the ``geometrical rounds-o€'' near the
corners of a rectangular cross-section sample,
which were considered in Refs. [6,7]. However,
unlike Refs. [6,7,14], our approach is based on
the assumption that vortices will start entering
deep inside a superconductor when the condition
jAx …y; z†j ˆ Acrit is ful®lled at the equator. By that
moment, the e€ective ``round-o€'' radius will reach
a value of order d=2.
The practically feasible scenario can be found
out only through experiment. Theoretically, this
question can be answered by numerical solution
of a problem on a vortex entry in a 3D sample
within the time-dependent Ginzburg±Landau theory.
The feature common for the above two schemes
is, actually, the assumption that vortices enter
deeply a thick ®lm when the condition jAx …1; 0†j ˆ
Acrit is met. This allows one to estimate the ®eld of
vortex entry inside a sample. Using Eq. (11a) (regardless of possible variation due to the penetration of tilted vortices or the existence of areas with
a suppressed order parameter), we now derive the
expression for ®eld, Hs , which is equivalent (12)
(with b ˆ 2keff =pW ). This similarity is due to the
fact that the Ax …1; 0† value is de®ned practically
by the same expression for both thin and thick
®lms, provided parameter k2 =dW is the same.
By analogy with thin ®lms, one can ®nd a value
of the ®eld at a ®lm edge when the vortices start to
penetrate deep into a sample. Yet, as opposed to
largely depends on the z cothin ®lms, ®eld hedge
z
ordinate in the sample region. Estimations of the
equator ®eld yields
heq
z ˆ hz …1; 0† 133
U0
;
2pnk
edge
which is practically equivalent to the hz (Fig. 7).
Note also that heq
z …Hs † to the factor of the order of
unity is equal to the thermodynamical ®eld, Hc , or
the surface barrier suppression ®eld for bulk superconductors in the absence of edge defects.
3.3. The in¯uence of surface defects and anisotropy
The resultant values for ®eld, Hs , are characteristics of isotropic superconductors with ideal
surface. As was established in Refs. [23,25], surface
defects can considerably decrease the value of Hs .
For example, in Ref. [25] for the case j 1 (j is
the Ginzburg±Landau parameter) maximum suppression of the entry ®eld, g ˆ Hs =Hen (where Hen
is the ®eld of vortices entry in a superconductor
p
with surface defects) was estimated as g j=p.
Thus, for j ˆ 100 we will have g ' 5:6.
A strong in¯uence on the value of Hs may be
also produced by anisotropy or, rather, layered
structure of superconductors. If the layers are not
Josephson coupled (or are weakly coupled), a superconductor should be regarded as a stack of
superconducting layers. This geometry can be
simulated, if we multiply the integrand in Eq. (3)
(and, hence, in integral (6)) by the step periodic
function, z, which is equal to one in the superconducting layer and is zero in the interlayer
space.
Let a period in such a structure be much smaller
than k (which is practically always ful®lled for
HTSC), a layer thickness be l and an interlayer
separation be m. Then, we can assume the distribution of Ax …z† to be a smooth function z, similar
to the dependence Ax …z† for a homogeneous superconductor. In this case, the integral in Eq. (3)
for a layered superconductor will be …l ‡ m†=l
times smaller than that for an isotropic superconductor. In other words, we can replace this integral
for a layered superconductor by the integral for an
isotropic case by introducing an e€ective penetra1=2
tion depth k0 ˆ k……l ‡ m†=l† . In this way, we
immediately obtain the distributions of A…y†,
Ax …y; 1†, and also the values for Ax …y; z† at the
134
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
side edges and the equator with allowance for the
layered structure of superconductor.
One particular e€ect of the anisotropy is that
1=2
times
the value of Ax …1; 0† will be ……l ‡ m†=l†
larger than that for an isotropic superconductor,
all other conditions being equal. This, in its turn,
will lead to a ……l ‡ m†=l†1=2 times smaller ®eld of
vortex entry in a superconductor. For example, at
_ typical for BiBaCaCuO, one
l ˆ 3A_ and m ˆ 12A,
1=2
®nds ……l ‡ m†=l† 2:2.
Thus, the two above-mentioned factors, i.e.,
surface defects and layered structure of superconductors may cause a considerable (10-fold and
more) decrease of the vortex entry ®eld in layered
superconductors with surface defects, as compared
to the vortex entry ®eld in isotropic perfect-surface
superconducting ®lms.
Another conclusion following from the fact of a
layered structure in such systems is that the scale
of a local magnetic ®eld penetration, for example,
for thick ®lms, will be k0 ˆ k……l ‡ m†=l†1=2 , and for
thin ®lms, the parameter keff has to be changed
for k0eff ˆ k0 2 =d. At the same time, the thicknessaveraged current density, unlike the thicknessaveraged vector potential, will practically remain
unchanged across the entire ®lm width, except for
the regions lying close to sample edges. This can be
accounted for by the fact that the expression for
current density, j ˆ ÿcA=4pk2 will include k only
at a superconductor edge (see Eq. (8)). Likewise,
all other quantities that obviously do not include k
(for example, a degree of the magnetic ®eld concentration at the equator of thick ®lm) will remain
the same.
4. The structure of a mixed state
Let us now discuss the parameters of a mixed
state arising in a superconducting ®lm in ®elds
larger than Hs . Here, we neglect the presence of
bulk pinning, which is justi®ed for soft superconductors. This problem was studied earlier in Refs.
[6,10±12,14,26]. In [12,26] the authors considered a
case of narrow thin ®lms Wd=k2 1; Ref. [10]
deals with wide thin ®lms Wd=k2 1, and Refs.
[6,14] is a study on thick ®lms that formally obey
the condition Wd=k2 1. We will analyse a gen-
eral case to show that it embraces either of the
above two situations. Besides, the resultant distribution of current density will be ®nite across the
entire ®lm width, as opposed to the results of the
above cited works.
Consider a superconducting ®lm in a mixed
state, placed in a perpendicular magnetic ®eld. The
current density and the vector potential in the
London limit are related as
j ˆ ÿ…A ÿ U0 ru=2p†c=4pk2 ;
where u is the order parameter phase. The Maxwell equation will have form (1), in which D is now
a 3D Laplacian operator. Using the Green function for D, we can write this equation in an integral
form:
Z Z Z
1
j…r0 †
dx0 dy 0 dz0 :
13†
A…r† ˆ A0 …r† ‡
c
jr ÿ r0 j
We now subtract function ru…r† from the left- and
right-hand sides of Eq. (17) and take curl from
these parts (using the property r ru…r† ˆ
2pd…r ÿ r0 †, where r0 ˆ …x0 ; y 0 † are the vortex coordinates). Next, we do the averaging over coordinates x; y on scales much larger than an intervortex
distance. After these operations, the distribution of
a current density becomes uniform along the xaxis and we can perform integration over x0 in Eq.
(13). We then average the obtained equation over
a ®lm thickness and use the results of the integral
(6) calculations. This will yield
R d=2an equation for the
sheet current density i…y† ˆ ÿd=2 jx …y; z†dz,
8pkeff di…y† 2
‡
cW
dy
c
Z
1
ÿ1
i…y 0 †
dy 0 ˆ ÿH1 ‡ n…y†U0 ;
y0 ÿ y
14†
where n…y† is the density of vortices, the distance
being measured in units of W =2. For the ®rst time
this equation was derived in Ref. [8] for thin ®lms
d k. Just like Eq. (7), Eq. (14) is valid at
jyj 6 1 ÿ jbj for thick ®lms, and across the width
for thin ®lms (excluding an extremely narrow nearedge region of size d k). Besides, it should be
expected by analogy with the Meissner state that
Eq. (14) for thick ®lms will also be valid directly at
a sample edge.
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
135
We analysed Eq. (14) numerically for di€erent
values of parameter W =keff , using the condition
that current density is zero in the region where
vortices exist, and takes ®nite value in vortex-free
regions [6,7,10,11,14]. Besides, we set a boundary
condition j…1† ˆ js on a current density (in
increasing magnetic ®elds), which allows for an
edge barrier. The value of the current density, js , of
order of the Ginzburg±Landau depairing current
density for ideal-surface superconductors [22,23].
In result, we have obtained the approximation
expression for i…y†
i…y† ˆ
8
<0
: 4p
cH1 …z‡1†
q
 sign…y†
jyj‡a†2
1‡a†2
a…1ÿz2 †‡b
0 < jyj < a;
a < jyj < 1;
15†
where
zˆ
2…y 2 ÿ a2 †
ÿ 1;
1 ÿ a2 †
8 1 keff
16
‡
bˆ
p 1 ÿ a2 W
1 ÿ a†2
keff
W
2
;
a…H1 † is the half-width of the vortex-®lled region
and parameter, a, is de®ned by expression (9) in
which W has been replaced by W …1 ÿ a†.
Expressions (15), being not derived, represents a
rather useful interpolation for the distribution of
the sheet current density for a ®lm in a mixed state.
We would like to emphasize again that, in the thin®lm case, W =keff can be both smaller (narrow
®lms) and larger (wide ®lms) than unit, whereas
for thick ®lms this ratio is always much greater
than 1.
Fig. 8 shows the dependence i…y† for di€erent
values of a magnetic ®eld. The di€erence of approximation (15) from the numerical result does
not exceed 4% in the vortex-free zone (a < y < 1).
Note, that in the near-edge region and close to the
boundary of the vortex-®lled region deviation may
come to about 9%.
The dependence a…H1 † (in increasing magnetic
®eld) is to be found from the following equation:
Fig. 8. Distribution of the sheet current i…y† for ®lm with parameter W =keff ˆ 200 and for di€erent values of a: (1) 0.0, (2)
0.4, and (3) 0.8. Dashed lines represent numerical results and
solid lines represent expression (15).
2
8 1 keff
16
keff
‡
2
W
p 1 ÿ a2 W
1 ÿ a†
2 !
2
H1
8 keff
keff
ˆ
‡ 16
:
Hs
W
p W
For W =keff 1, we have
a…H1 † ˆ
q
2
1 ÿ …Hs =H1 † ;
H1 ' Hs ;
or
r
2keff Hs
;
a…H1 † ˆ 1 ÿ
pW H1
H1 Hs ;
while for W =keff 1, we have
a…H1 † ˆ 1 ÿ Hs =H1
for all values of H1 .
Using dependence (15), we can ®nd distribution
of the z-averaged magnetic ®eld across a ®lm
width:
hz …y† ˆ H1 ‡
2
c
Z
1
ÿ1
i…y 0 †
dy 0 :
y ÿ y0
16†
136
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
Using Eq. (15), we can estimate the dependence
edge
of hz =H1 on the parameters of a ®lm and an
increasing external magnetic ®eld (for thin ®lms;
see Section 2.2)
edge
hz
2keff 1
1
b…1 ÿ a†
4
p 1 ‡
2a ÿ
ˆ
:
H1
W
b
2
1 ÿ a2
b
17†
Fig. 9. Distribution of the averaged z-component of magnetic
®eld for W =keff ˆ 200 and a ˆ 0:6. Solid line is obtained with
help of expression (15) and (16), dotted line represents the numerical result. Insert shows detailed distribution of the ®eld;
dashed line is the function …a2 ÿ y 2 †1=2 =…1 ÿ y 2 †1=2 from Refs.
[10,14].
Fig. 9 shows the dependences hz …y† for a ®lm
with W =keff ˆ 200 and a ˆ 0:6 (H1 ' 1:3Hs ), obtained numerically and by means of expressions
(15) and (16). It is seen that these dependences
practically coincide across the entire width of a
®lm, except for the near-edge regions and the
boundary of the vortex-®lled zone. The di€erence
in the magnetic ®eld value at y ˆ 0:6 should be
attributed to the inaccuracy of approximation
(more precisely, its ®rst derivative at the boundary
of the vortex-®lled area). Dependence hz …y† shown
in the insert to Fig. 9 was obtained theoretically in
[6,10,14]. It is seen that, as opposed to this analytical dependence, a non-zero magnetic ®eld does
exist outside the vortex-®lled region also, and it is
quite strong (>0.1H1 ) for a ®lm with the given
parameters. Another distinguishing feature is the
occurrence of a jump from zero to some ®nite
value for the dependence n…y† ˆ hz …y†=U0 at y ˆ a.
The reason of the vortex density discontinuity is
explained by a non-zero magnetic ®eld in the region …a; 1† and the condition of a magnetic ®eld
continuity at y ˆ a.
It is clearly seen that in the limit a ! 1(H1 !
edge
1), the ratio hz =H1 tends to 1.
Knowing the dependence i…y†, we can ®nd the
magnetization curves of superconducting ®lms for
di€erent values of W =keff . Fig. 10 illustrates the
results obtained. One can see that with an increase
in the parameter W =keff the magnetization curves
become similar to the dependence, M…H †, derived
theoretically in Ref. [10] for wide thin ®lms. As
W =keff decreases, the magnetization curves tend to
the dependence which is valid for thin narrow ®lms
[12,26]. Thus, even at W =keff ˆ 1, the dependence
M…H † for a narrow ®lm and the M…H † obtained
numerically by the use of Eq. (15) practically coincide. So, expression (15) allows us to obtain
Fig. 10. Magnetization curves of superconducting ®lms for
di€erent ratios W =keff : curve 1 for W =keff ˆ 1, curve 2 for
W =keff ˆ 200 and, curve 3 for W =keff ˆ 1. Curve 3 practically
coincides with the magnetization curve for narrow W keff
®lms [26].
D.Yu. Vodolazov, I.L. Maksimov / Physica C 349 (2001) 125±138
magnetization curves for such superconductors at
an arbitrary ratio W =keff . Note that the magnetization curves in this case, i.e., at any value of parameter W =keff lie between two curves ÿ1 and 3 as
shown in Fig. 10 (in dimensionless units).
5. Conclusion
It is shown in the present paper that the Maxwell±London equation used so far only for thin
®lms is also valid for samples of ®nite thicknesses.
This equation is shown to de®ne the distribution of
a thickness-averaged vector potential and/or current density (in the mixed state case) across a
sample width. For thin ®lms, the equation holds
practically everywhere in a ®lm, whereas in the
thick ®lm case its applicability is restricted only
within a narrow bands near the edges W =2 ÿ d=
2 6 jY j < W =2.
An approximation expression is found, describing distribution of vector potential A (or
current density j…y†) across the width of a ®lm in
the Meissner state, which applies to both thin and
thick ®lms. For thick ®lms, analytical expressions
have been derived, de®ning the value of the vector
potential (local current density) at the equator
(Y ˆ 1, Z ˆ 0), side edges (Y ˆ 1, Z ˆ 1), and also
on the top and bottom surfaces (Y , Z ˆ d=2) of a
sample. Besides, analytical approximation expressions have been found for the magnetic ®eld at
the equator and for the thickness-averaged edge
magnetic ®eld.
The vector potential distribution data were used
to evaluate the ®eld of the the ®rst vortex entry
(barrier suppresion ®eld) for superconductors of
such geometry. It is described by an universal expression (12) valid for both thin and thick ®lms. It
is shown that besides surface defects the layered
structure of superconductor may result in a signi®cant suppression of the vortex entry ®eld. Thus,
mutual in¯uence, both surface defects and layered
structure, may lead to suppression of Hs by a
factor of 10 (and even greater).
The study of a mixed state yields an interpolation expression for distribution of a sheet current
density i…y† in superconducting ®lms without bulk
pinning. This result allows one to, for the ®rst
137
time, estimate the dependence of the magnetic ®eld
edge
concentration, c ˆ hz =H1 on the parameters of
superconductor and external magnetic ®eld, H1 .
Besides, these data were used to calculate the
magnetization curves for ®lm superconductors at
any values of parameter Wd=k2 .
Acknowledgements
The authors are obliged to Prof. J.R. Clem, Dr.
G.M. Maksimova for helpful discussions of the
results obtained. This work is supported by the
Science Ministry of Russia (Project no. 98-012),
and in part Basic Foundation for Fundamental
Research (Project no. 97-02-17437). Partial support of the International Center for Advanced
Studies (INCAS) through Grant no. 99-2-03 is
gratefully acknowledged.
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