PHYSICAL REVIEW B
VOLUME 59, NUMBER 2
1 JANUARY 1999-II
Onset of flux penetration into a thin superconducting film strip
A. V. Kuznetsov and D. V. Eremenko
Department of Quantum Electronics, Moscow State Engineering Physics Institute, 115409 Moscow, Russia
V. N. Trofimov
Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia
~Received 26 June 1998!
Penetration of vortices into a thin superconducting film and the edge barrier arising in a transverse magnetic
field are considered. The vortex Gibbs energy is calculated on the base of the Kogan’s solution for the Pearl’s
vortex situated near the film edge. The penetration field and the magnetization in the critical state are calculated. The influence of the edge barrier on magnetization of high-T c thin films is estimated.
@S0163-1829~99!13901-8#
I. INTRODUCTION
Application of thin films of high-temperature superconductors ~HTSC’s! in superconducting devices requires precise knowledge of the films’ response to external fields. Recently the critical-state problem has been solved for a film
inserted in a magnetic field applied normally to the film.1,2
Penetration of the magnetic flux into a film in the presence of
an edge barrier has been described by the geometrical barrier
model.3–5 Both models give the magnetic response when
spatial distribution of the critical current or its dependence
on the flux density are known. Since they deal with currents
and flux averaged over several intervortex distances, neither
entry of the first vortex into a film nor features of flux penetration following from peculiarities of thin-film vortices are
treated in the frame of these models.
As was shown by Pearl6 thin-film vortices interact with
external and self-induced fields primarily through the free
space adjacent to the film, where no screening currents flow.
As a sequence, the vortical current nonexponentially decreases with a distance from the vortex core and the vortex
self-energy depends on both film dimensions7,8 and a vortex
position in the film.7,9 In addition, vortices interact with
shielding current flowing throughout the film in an external
field. All this leads to strong dependence of the vortex Gibbs
energy on a position in the film.
It is well known that the Bean-Livingston barrier for the
entry of vortices exists at the surface of bulk
superconductors10–12 and at the film edges.7,13 The barrier
was analyzed in a thorough discussion of the Meissner response of pinning-free thin films given by Likharev.7 A set
of integral equations for computing the Gibbs free energy
was obtained. However, generality of the analysis makes it
difficult to use the obtained results. Some simplification was
introduced to account for the edge barrier in the critical state.
It was shown that the vortex self-energy rises towards the
film center so vortices are attracted by the edges.7 The interaction of a vortex with its image outside the edge was used
to describe the attraction,13 but the exact solution by Kogan9
for a vortex situated near the edge contradicts such an approximation. To calculate interaction of vortices with the
shielding current I the distribution I}1/AW2x was used
near the edge where vortices are nucleated.13 This distribution diverges at the edge position W while the edge current is
0163-1829/99/59~2!/1507~7!/$15.00
PRB 59
limited.7,14 To obtain the edge current one should solve the
London equation playing a major role in the physics of thin
superconducting films.4,7,9,14
In the present paper we compute the London equation and
using Kogan’s solution calculate the Bean-Livingston barrier
in a film strip. We treat the case of strong pinning which is
inherent in real films14 because of interaction of vortices with
defects, dislocations and tensions arising at a film-substrate
interface.
The paper is organized as follows. In Sec. II the Meissner
response is analyzed. Using Kogan’s solution we calculate
the vortex Gibbs energy and describe the Bean-Livingston
barrier arising in thin films. In Sec. III influence of the barrier on flux penetration is analyzed for the case of strong
pinning. We account for the vortex self-energy gradient in
the balance of the forces acting upon a vortex and calculate
the magnetization on the base of the critical-state equations
by McDonald and Clem.2 In Sec. IV we discuss suppression
of the barrier by defects at the edge and calculate the penetration field in which vortices begin to enter a strip. Section
V contains the barrier-related estimations made for
YBa2 Cu3 O72 d films. In Sec. VI a brief summary of our results is given. In the Appendix the barrier-related saturated
magnetization is calculated.
II. VIRGIN STATE
Let us start with an infinite superconducting film strip
inserted in a uniform magnetic field H applied normally to
the strip. We choose the origin of the coordinate frame at the
midpoint of the strip with the y axis lengthwise and the x axis
in the lateral direction, so that the edges are located at
(2W,y,0) and (W,y,0) ~see Fig. 1!. The film thickness d is
small in comparison with the penetration depth l. We neglect the change of a current density J across the thickness
and consider shielding current I5 * d/2
2d/2Jdz. The strip is
wide in comparison with the screening length, W@L
52l 2 /d. Because of the symmetry of the current and flux
distribution in the strip we analyze only the x>0 half of the
strip.
In this section we treat the virgin state in which no vortices are in the film in the zero field. Then the field increases
1507
©1999 The American Physical Society
A. V. KUZNETSOV, D. V. EREMENKO, AND V. N. TROFIMOV
1508
FIG. 1. Sketches of the superconducting strip, the shielding current distribution and the vortex Gibbs energy. Vectors in the energy
sketch represent directions of a force f acting upon a vortex.
up to a small value and the shielding Meissner current induces in the strip. The current distribution is calculated from
the London equation14
1
L ]I
5
2 ]x 2p
E
W
I ~ x 8 ! dx
2W
x 8 2x
2H.
~1!
The first term at the right-hand side of Eq. ~1! represents the
demagnetization field produced by the current itself. Integrating over x and calculating the integration constant from
the boundary condition I(0)50 following from the odd current symmetry I(x)52I(2x), one writes
I5
W
pL
E U
1
ln
0
U
2xH
u1x/W
.
I ~ u ! du2
u2x/W
L
~2!
We calculated the current distributions presented in Fig. 2. In
the central part of the strip the current is described by the
well-known dependence7,14
PRB 59
FIG. 3. The vortex Gibbs energy. The energy is presented in
units of the self-energy of a vortex situated at the strip center. The
applied field h5H/H * is taken in units of H * 5 f 0 / m 0 L ALW.
I52
2Hx
AW 2 2x 2
~3!
.
The larger the W/L ratio, the smaller the difference between
Equation ~3! and the calculated distribution. Eq. ~3! diverges
like 1/AW2x at the edge while the calculated current is
limited by the maximal value I m 5H A2 p W/L. The edge
current is close to the dependence14 i511( z / p )ln(gz/4)
where i52I/I m , z 5(W2x)/L, and g 50.577 21 . . . is the
Euler constant. As seen from Fig. 2, independently of the
W/L ratio the deviation of the current distribution from Eq.
~3! occurs at a distance &L from the edge. In the belt z
<1
the
current
is
fitted
by
i5 @ 1
1( z / p )ln(gz/4) # exp(0.357z 1.2) with an accuracy of a few
percent.
Because of demagnetization the local field and current
strongly increase at the edge with applied field. Vortices
spontaneously nucleate at the edge via dynamic suppression
of the order parameter.15 The self-energy of a vortex situated
near the edge of a wide superconducting film is9
«5
F S D S DG
f 20
rc
W2x
F0
2F 0
4 m 0L
2L
L
,
~4!
where f 0 is the flux quantum, the vortex core radius r c is of
the order of the coherence length j ,F 0 5Y 0 2H0 , and
H0 , Y 0 are the zero-order Struve and the second-kind
Bessel functions in the notation of Ref. 16.
The vortices interact with the shielding current. Though a
vortex in a strip carries less than f 0 of a flux, the interaction
of such a vortex with the shielding current is still described
in terms of the common Lorentz force9 f L 5 f 0 I. Integrating
the Lorentz force one obtains the interaction energy and
writes the vortex Gibbs energy
FIG. 2. Distribution of the Meissner shielding current. The inset
represents the current near the strip edge. The curves are calculated
for W/L5100 ~solid! and 2500 ~dashed!. The dotted curves correspond to Eq. ~3! ~the right curve in the inset! and the edge current
calculated in Ref. 14.
g5«1 f 0
E
W
x
I ~ x 8 ! dx 8 .
~5!
The energy calculated from Eqs. ~2!, ~4!, and ~5! is shown
in Fig. 3 for some values of applied field. As seen, the potential barrier preventing an entry of vortices exists in the
PRB 59
ONSET OF FLUX PENETRATION INTO A THIN . . .
strip. The vortices arising at the edge are pushed out of the
strip by the force f52¹g since their energy rises toward the
strip inner part.
There should be no appreciable thermal activation over
the barrier the scale energy of which is large, f 20 /4m 0 L*
63104 K for L&1 m m. Vortices penetrate the strip central part only when the Gibbs energy maximum is located at
the edge. Using the equation ¹g50 relating the position of
the Gibbs energy maximum x m to the applied field we obtained that I m is of the order of the depairing current when
x m ;W2r c . Note that only qualitative estimations can be
made in a belt of a width r c ; j adjacent to the edge since the
London description and, consequently, Eqs. ~2! and ~4! fail
there.
When H exceeds the field at which the Gibbs energy
maximum is located at the edge, vortices enter the strip and
migrate toward its central part. The migration is affected by
pinning. In the next section we consider the arising critical
state accounting for both the pinning and the vortex energy
gradient.
FIG. 4. Magnetization curves calculated for i p 51 in the presence ~top! and absence of the edge barrier.
follows from the critical-state model for a strip with nonuniform critical current.2 Here a is the flux front position related
to the applied field by the equation
III. CRITICAL STATE
In presence of pinning vortices migrate under the influence of the following forces. The self-energy gradient and
the Lorentz force tend to move a vortex, the pinning force fp
tends to prevent its displacement. If fp exceeds 2¹g the
vortex is pinned. In the opposite case it moves to the point
where fp 5¹g. That point is situated far from the edge when
pinning is weak. As a result, a region where vortices entering
the strip are accumulated, is separated from the edge by a
wide (@L) region which is free of flux. Such a flux penetration has been described in the geometrical barrier model.3,4
We assume that pinning is strong enough, so a small
(;L) displacement from the edge leads to stopping of the
vortex. In this case a region of the critical state forms at the
edge and grows with the field. Therefore flux penetration can
be treated in the frame of the critical-state model.
Flowing in the critical-state region the critical current I c
provides a balance of the forces acting upon a vortex. From
the equation fp 5¹g one obtains f 0 I c 5« 8 2 f px . Here « 8
denotes the derivative with respect to x. We take the pinning
force independent of the flux density and express fp in terms
of the pinning current I p 5const.0. Since fp acts oppositely
to the direction of the vortex displacement, f px 5 f 0 I p , when
flux enters the strip and f px 52 f 0 I p when flux exit it. Calculating the self-energy gradient from Eq. ~4! accounting
that16 Y 08 52Y 1 and H08 52/p 2H1 we write the critical current
I c5
f0
S D G
F
2
W2x
2F 1
6i p ,
L
4 m 0L p
2
~6!
where i p 54 m 0 L 2 I p / f 0 , signs plus and minus correspond to
the exit and entry of flux, respectively.
The current distribution throughout the strip
I5
H
I c~ x ! ,
2
p
a<x,
Aa 2 2x 2
E
W
a
I c ~ x 8 ! dx 8
~ x 8 22x 2 !
Ax 8 2a
2
,
2
x,a,
~7!
1509
H5
1
p
E
I c dx
W
a
Ax 2 2a 2
~8!
.
Integrating the current distribution one calculates magnetization of the strip
m5
1
Wd
E
W
~9!
Ixdx.
0
Substituting Eq. ~7! for I, dividing the integral into two parts
a
W
W
*W
0 . . . dx5 * 0 dx * a . . . dx 8 1 * a . . . dx, inverting the integration order in the double integral, integrating * a0 . . . dx and
summing the result with the second part we obtain
m5
1
2Wd
E
W
a
Ic
2x 2 2a 2
Ax 2 2a 2
dx.
~10!
Calculating both H and m from Eqs. ~8! and ~10! for a
set of flux front positions one constructs a magnetization
curve. When the barrier is absent, i.e., I c 5I p 5const,
calculations
give
the
known
dependences1
H
21
5(I p / p ) cosh (W/a), m52(WI p /2d) A12(a/W) 2 . For
strip with W5100L and i p 51 we computed the magnetization
curve
presented
in
Fig.
4
with
m5
2(WIp/2d)tanh(pH/Ip) following from the dependences
mentioned above. As seen, the edge barrier results in an increase of the magnetization.
Using Eq. ~6! in the frame of the critical-state model2 one
calculates the magnetization in a decreasing field as well as
hysteretic magnetization curve. This problem cannot be
solved in the present work because of its complexity. However, we analyzed the saturated remanent magnetization m rs
remaining in the strip when H decreases from a high value
down to zero. We found that the measurement of m rs and the
saturated virgin magnetization m s allows one to separate the
barrier-related and pinning-related contributions to the magnetization.
In the saturated state the critical current flows throughout
the strip. Its distribution is given by Eq. ~6! where the sign of
i p is positive for the remanent state and negative for the
1510
PRB 59
A. V. KUZNETSOV, D. V. EREMENKO, AND V. N. TROFIMOV
FIG. 5. Ratio of the saturated magnetization calculated for i p
51 and L/r c 5250. The top curve represents the ideal barrier, the
other curves correspond to the barrier suppressed by edge imperfection. For a detailed explanation see Sec. IV.
virgin state. From Eqs. ~6! and ~9! one obtains that the integrals containing pinning current annihilate in the sum m s
1m rs ; * « 8 xdx. On the contrary, the integrals containing
the self-energy gradient annihilate in the difference m s
2m rs ; * i p xdx. In the Appendix we calculated the ratio of
the barrier-related and the pinning-related magnetization
S
D
m s 1m rs
4L
L
.
ln 10.81 .
m s 2m rs p i p W r c
Since the barrier results in an increase of the critical current
only near the edge, its influence on the magnetization is independent of a strip width while the pinning-related magnetization is proportional to W. Therefore, as seen from Fig. 5
where dependence of the ratio on W is depicted, the barrier
slightly affects the magnetization of wide strips.
The vortex self-energy and the critical current primarily
change in a narrow belt of a width ;L adjacent to the edge
~see Fig. 3! so a damage of the strip in this region strongly
affects the penetration of vortices through the barrier. In the
next section we discuss the onset of flux penetration accounting for the edge imperfection.
IV. PENETRATION FIELD
This is a general point that the surface roughness must
suppress the Bean-Livingston barrier. Indeed, in the recent
study by Bass et al.17 it was shown that the surface roughness leads to both a decrease of the barrier energy and a shift
of the barrier maximum from the surface, however no dramatic suppression of the barrier has been obtained. For example, for the root-mean-square surface roughness ;0.4l
the decrease of the energy barrier from its value for ideal
surface is only about 10%.17 Note that such a roughness
value is quite large since in the treated case of a type-II
superconductor the barrier is located at a distance ;l from
the surface. For a thin film one also expects no strong suppression of the barrier by the edge unevenness.
As follows from the experiments in which the surface of a
superconductor was damaged by electron irradiation, there is
strong suppression of the surface barrier under the influence
of the surface defects.18 To understand this fact one should
go beyond the Bean-Livingston approximation which deals
with an already nucleated vortex but is unable to analyze the
vortex nucleation.
Transition to the mixed state in a type-II superconductor
occurs as a result of sufficiently large fluctuations of the
order parameter.11 The vortex nuclei in the form of a semicircle, the ends of which touch the superconductor surface,
are produced near the surface. There exists a minimal critical
size ~and energy! of the vortex nucleus for which the nucleus
is not ‘‘closed up,’’ continues to develop further, and penetrates into the superconductor.11,12 The critical energy decreases and the fluctuation frequency increases in the regions
with suppressed order parameter, therefore the surface defects suppress the surface barrier.
In the case of a thin film the numerical simulation of the
time-dependent Ginzburg-Landau equation completed by the
appropriate Maxwell equations gives the following description for the vortices nucleation.15 The flux penetration into a
thin superconducting film occurs via the dynamic suppression of the order parameter on the macroscopic scale and can
be viewed as the nucleation of the extended droplets of the
normal phase in the superconducting sample. The droplets,
emerging at the film edge, stretch for the distance w from the
edge. This distance is of the order of some tens of j for
narrow film strips ( j !W!L) and is expected to be as large
as l for wide ones. The droplets contain multiple topological
charge and therefore they are unstable with respect to splitting into singly charged vortices. The time scale of formation
and splitting of the droplets is ;1029 s. When the droplets
split, the vortices arise inside the film and some of them are
situated at the distance ;w from the edge. In the presence of
the defects the normal phase tries to settle at the defect sites
where the order parameter is already suppressed. The droplets percolate along easy paths connecting the defect sites
and split into the vortices which are pinned at the defects.
So far we treat a strip with ideal edges. Real samples,
however, have the edge defects which are introduced by intergrain boundaries, if the film is polycrystalline, and by lithography because of the decreasing of the film thickness
and damage of the superconductor. Via the dynamic suppression of the order parameter the vortices arise in the defect
regions in a sufficiently low applied field. One estimates w
;d for a lithography-related defects. In the case of the polycrystalline film w is of the order of the grain diameter, as
shown in Fig. 6.
Consider first the zero-pinning behavior. If the maximum
of the vortex Gibbs energy is situated far from the edge,
x m ,W2w, vortices nucleating in the defect regions are
pushed out of the strip. When x m >W2w vortices enter the
strip and migrate under the influence of the force 2¹g acting towards the strip center. The penetration field H p , in
which the first vortex enters the strip, is calculated from the
equation ¹g50 solving for x5W2w.
Pinning tends to hold a vortex at the site of its nucleation.
The first vortex is pinned when ¹g and fp becomes equal at
some field. From the equation ¹g5fp one calculates the penetration field when pinning is present
m 0H p5
f0
F 1 22/p 2i p
L ALW
4 A2 p i
.
~11!
PRB 59
ONSET OF FLUX PENETRATION INTO A THIN . . .
1511
for a vortex situated in the film center.8 Using m
.4 f 0 W(2 ln 211)/p,9 Eq. ~4! and Ref. 16 we write
m 0 H c1 f .
FIG. 6. Sketch of the strip near the edge. The polycrystalline
film ~grey! consists of single-crystal grains of a characteristic diameter D which are separated by the intergrain boundaries ~lines!. The
bottom cross sections indicate that the edge rounding varies along
the strip. The dark grey color marks the region of the edge damage
related to lithography. w is a characteristic width of the edge imperfection due to granularity or lithography.
Here the current i and function F 1 are taken at w/L. Note
that the same dependence of the penetration field on a
sample width H p }1/AW was obtained earlier for a thin flat
type-II superconductor.3,5
The penetration field as a function of w and i p is depicted
in Fig. 7. H p is strongly suppressed by increase of w. The
stronger, the pinning, the lower the penetration field. Moreover, H p is zero for some w. This means that any vortex
arising in the infinitesimal field must be pinned. At the same
time the penetration field is finite.
In a weak field the minimum of the sample free energy is
achieved when the film is in the Meissner state. The Meissner state becomes metastable with respect to formation of the
mixed state when energies of these states become equal with
field increase. The corresponding field, named the lower
critical field of film H c1 f , is calculated from the equation
H c1 f 5«/ m where the moment m and the energy are taken
FIG. 7. The penetration field vs w and i p ~inset!. The field h p
5H p /H * is presented in units of H * 5 f 0 / m 0 L ALW.
S
D
L
f0
ln 10.81 .
16~ 2 ln 211 ! LW r c
When H,H c1 f no vortices nucleate in a film, therefore H c1 f
is the lowest limit for the penetration field.
When the first vortex is pinned at the edge the vortical
current joins with the shielding one. The edge current rises,
therefore, the probability of nucleation of successive vortices
increases. The critical state forms at the edge. In the region
x<W2w the critical current is described by Eq. ~6!. In the
belt x>W2w the current depends on unknown parameters
of superconductor in the defect regions. As a contribution of
the edge current to the saturated magnetization is as small as
2w/W!1 for a wide strip, it is quite reasonable to neglect
the distribution of the edge current and to estimate it by some
average value. We restrict the edge current by I c at x5W
2w and calculate in the Appendix the ratio of the barrierrelated to the pinning-related magnetization:
F
S
m s 1m rs 2L
w
2
.
F 01
F 12
m s 2m rs i p W
L
p
DG
.
~12!
Here functions F are taken at w/L. As follows from Fig. 5,
where the ratio is presented for different w/L, the barrier
slightly affects the magnetization if vortices arise at the distance ;L from the edge. When edges are not specially prepared by lithography, the magnetization is well described by
the conventional critical-state model, since w@L for such
films.
V. DISCUSSION
Because of the presence of the shielding current flowing
throughout the film the barrier discussed above differs from
the Bean-Livingston barrier in bulk superconductors.10–12
Vortices in films of the layered superconductor consist of
pancakes which are able to surmount the edge barrier one
after another. Therefore, the obtained results are hardly applied to films of layered superconductors.
Let us estimate some barrier-related parameters. There is
the most interest in HTSC thin-film superconducting devices
operated at liquid-nitrogen temperature. So we take a strip of
a YBa2 Cu3 O72 d film with thickness d50.15 m m and width
2W5100 m m as an example.
For HTSC films with the c axis perpendicular to the film
the restriction on a film thickness d!2l is rewritten as d
!2l/ Ae . Due to strong anisotropy ~the anisotropy parameter e .0.04 for YBa2 Cu3 O72 d ) HTSC films of thickness
d;l are quite thin. High-quality epitaxial YBa2 Cu3 O72 d
films consist of islandlike grains 0.25 m m in diameter.19
Thus the film thickness and the diameter of grains are of the
same order.
At T577.4 K for l.0.22 m m ~Ref. 20! and a typical
value of the pinning-related critical current density19,21 J c
.23106 A/cm2 , one calculates L.0.65 m m and i p .3.
Taking w/L50.25 which corresponds to w;d, we estimate
the penetration field H p 5H c1 f .0.1 Oe ~Ref. 22! and obtain
that the barrier-related saturated magnetization is only 1% of
1512
A. V. KUZNETSOV, D. V. EREMENKO, AND V. N. TROFIMOV
the pinning-related one. Thus the barrier slightly affects the
high-field magnetization of YBa2 Cu3 O72 d films. Note, that
the barrier-related magnetization can be observed in films
with lower J c . For example, we calculated (m s 1m rs )/(m s
2m rs ).0.17 at T515 K for the Nd1.85Ce0.15CuO42 d film
strip with W/L5100.23
To estimate the barrier influence on low-field magnetization, ac losses, and noises, one should numerically solve the
critical-state equations2 with the critical current given by Eq.
~6!. When a flux front position is located near the edges, it is
evident that the barrier holds in flux penetration. Indeed, the
authors of Ref. 13 observed the barrier-related decrease of
magnetic hysteresis in superconducting quantum interference
device magnetometers caused by penetration of vortices near
the edges. One expects also that the barrier influence is significant in high-frequency fields when pinning is suppressed.
In this case the barrier-related frequency is apparently restricted by a time of the film-vortex nucleation which is of
the order of 1029 s. 15
VI. CONCLUSION
In this paper we considered the Bean-Livingston barrier in
a thin superconducting film. The present analysis is a part of
the general problem of flux penetration through an edge barrier in a thin flat superconductor. The cases of type-I and
type-II superconductors have been analyzed earlier ~see Refs.
5,24, and references therein!.
In conclusion we point out the main results obtained in
the present work.
~1! The Gibbs energy of a vortex situated near the edge of
a thin superconducting film strip has been calculated and the
Bean-Livingston barrier has been described.
~2! The critical-state model has been extended to account
for the edge barrier and the magnetization has been calculated.
~3! An entry of the first vortex into a film has been analyzed and the penetration field has been calculated.
Integrating by parts the first integral we accounted that
«(W)50. Using F 0 ( z ).(2/p ) @ ln(2/z )2 g # , z 5r c /2L!1
for the first term and * u0 F 0 d z .(2/p ) @ ln(2u)1g#, u5W/L
@1 for the second term16 one writes
m s 1m rs .2
S
~A1!
and
2
m s 1m rs 5
f 0 Wd
52
E
W
0
2
« 8 xdx52
f 0 Wd
F S D E
f0
rc
L
F0
2
2 m 0 Ld
2L
W
E
W
0
W/L
0
~A2!
D
~A3!
When the critical current is restricted at the edge region
W2w<x by the constant value I c 5« 8 / f 0 6I p , where « 8 is
taken at W2w, one writes for the barrier-related magnetization
m s 1m rs 5
2
f 0 Wd
52
2
FE
W2w
0
F
x« 8 ~ x ! dx1« 8
E
E
W
f0
~ W2w ! F 0 2L
2 m 0 LWd
S
w ~ 2W2w ! 2
2F 1
2L
p
DG
xdx
W2w
W/L
w/L
G
F 0~ z ! d z
.
Here functions F are taken at w/L. Using the integral
representation16 of F 0 one writes
E
b
a
F 0d z 5
2
p
E
5
2
p
E
`
e 2t dt
0
E Az z
d
b
2
a
1t 2
b1 Ab 2 1t 2
e 2t ln
dt,
a1 Aa 2 1t 2
`
0
and
APPENDIX
m s 2m rs 52WI p /d,
D
m s 1m rs
4L
L
.
ln 10.81 .
m s 2m rs p i p W r c
m s 1m rs 52
In the case of the ideal edge barrier the critical current
I c 5« 8 / f 0 6I p flows throughout the strip. From Eqs. ~4! and
~9! one calculates for I p 5const
S
f0
4L
L 2W
ln
2 ln
.
p m 0 Ld e g r c W e g L
As L/r c ;2l 2 /d j &2 k , we estimate for k 5l/ j >50 and
W/L>50 that the last term at the right-hand side is less than
1.5% of the first one. Neglecting it and dividing Eq. ~A2! by
Eq. ~A1! we write for the ideal barrier
ACKNOWLEDGMENTS
The authors are indebted to Dr. A. A. Ivanov and K. V.
Klementev for helpful discussions. We are grateful to Professor A. P. Menushenkov for his support of the present investigation.
PRB 59
F5
FE
L 2
W p
2
2
S
S
D G
W/L1 A~ W/L ! 2 1t 2
e 2t ln
0
w2
2L
`
F
f0
w
2
F 01
F 12
2F ,
2 m 0 Ld
L
p
F 12
w/L1 A~ w/L ! 1t
2
p
2
DG
2
dt1
w
F
L 0
~A4!
.
We estimate that F can be neglected with an accuracy of 4%
or better for w/L>0.1 and W/L>50. Dividing Eq. ~A4! by
Eq. ~A1! we finally write
«dx
G
F 0~ z ! d z .
F
S
m s 1m rs 2L
w
2
.
F 01
F 12
m s 2m rs i p W
L
p
DG
.
~A5!
PRB 59
ONSET OF FLUX PENETRATION INTO A THIN . . .
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1
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1513
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