PHYSICAL REVIEW B, VOLUME 63, 144501
Josephson junction in a thin film
V. G. Kogan, V. V. Dobrovitski, and J. R. Clem
Ames Laboratory - DOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011
Yasunori Mawatari
Frontier Technology Division, Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba, Ibaraki 305-8568, Japan
R. G. Mints
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
共Received 15 November 2000; published 5 March 2001兲
The phase difference ␸ (y) for a vortex at a line Josephson junction in a thin film attenuates at large
distances as a power law, unlike the case of a bulk junction where it approaches exponentially the constant
values at infinities. The field of a Josephson vortex is a superposition of fields of standard Pearl vortices
distributed along the junction with the line density ␸ ⬘ (y)/2␲ . We study the integral equation for ␸ (y) and
show that the phase is sensitive to the ratio l/⌳, where l⫽␭ 2J /␭ L , ⌳⫽2␭ L2 /d, ␭ L , and ␭ J are the London and
Josephson penetration depths, and d is the film thickness. For lⰆ⌳, the vortex ‘‘core’’ of the size l is nearly
temperature independent, while the phase ‘‘tail’’ scales as 冑l⌳/y 2 ⫽␭ J 冑2␭ L /d/y 2 ; i.e., it diverges as T
→T c . For lⰇ⌳, both the core and the tail have nearly the same characteristic length 冑l⌳.
DOI: 10.1103/PhysRevB.63.144501
PACS number共s兲: 74.50.⫹r, 74.76.⫺w
I. INTRODUCTION
l⫽
Recent interest in Josephson junctions in superconducting
films has been driven by experiments probing the properties
of grain boundaries and, in particular, the order parameter
symmetry.1–4 In these experiments, the junction plane was
normal to the film faces 共unlike traditional thin-film largearea Josephson junctions in which the junction plane is parallel to the faces of two films deposited on top of each other兲.
The junctions in fact are lines separating two thin-film banks
touching only along the edges. The Josephson vortices at
such boundaries are quite different from those at familiar
bulk junctions, because the stray magnetic field of a vortex
results in an integral equation governing the phase
distribution;5 i.e., the problem becomes nonlocal 共as opposed
to the well studied local sine-Gordon equation for junctions
between bulk superconductors兲. The theory of thin-film junctions is just emerging; there have been no attempts made to
connect the phase difference at the junction line with the
measurable field outside the film. In fact, the data obtained
on films are commonly analyzed with the help of bulk formulas; see, e.g., Refs. 1 and 3. One of the motivations for the
present work was to fill in this missing link.
In the following section we describe the approach we employ for thin films and demonstrate it by solving the wellknown problem of the Pearl vortex.6 Besides the transparency and some advantages in providing analytic results for
the fields in real space, the method can be readily applied to
the problem of a thin-film junction. This is done in the next
section, where we rederive the integral equation of Mints and
Snapiro5 for the phase difference ␸ at the junction line, and
establish the relation between the phase and the measurable
outside magnetic field. The theory contains two characteristic lengths: one is related to physical properties of the
junction,
0163-1829/2001/63共14兲/144501共9兲/$20.00
c␾0
⫽
2
16␲ 2 j c ␭ L
␭ 2J
␭L
共1兲
,
where ␭ J is the Josephson length of a junction made of bulk
banks of the same material and with the same critical current
density j c , and ␭ L is the London penetration depth of the
banks. The other length is that of Pearl which describes the
film:
⌳⫽
2␭ L2
d
共2兲
,
with d being the film thickness.
In the next section we study the distribution of the phase
difference ␸ (y) along the junction. We show that at large
distances the phase approaches the limiting values of 0 or
2 ␲ obeying a power law:
␸ 共 y→⬁ 兲 ⬇2 ␲ ⫺
␸ 共 y→⫺⬁ 兲 ⬇
2l⌳
y2
2l⌳
y2
.
,
共3兲
共4兲
This constitutes a major difference from the phase distribution in bulk junctions, where ␸ (y) approaches exponentially
the limiting values at infinities. We argue that this behavior
is prescribed by the stray field outside the film. As is seen
from Eqs. 共3兲 and 共4兲, the characteristic length scale for the
large-distance phase variation is 冑l⌳.
We then consider the asymptotic behavior of the phase in
two limiting cases. We show that for lⰆ⌳, the central part of
the Josephson vortex 共the core兲 is of a size l which is nearly
temperature independent. Since the phase tail has a scale
冑l⌳(T), the vortex structure changes with T. Unlike Joseph-
63 144501-1
©2001 The American Physical Society
KOGAN, DOBROVITSKI, CLEM, MAWATARI, AND MINTS
son vortices in bulk junctions, in thin films the phase distribution is not a universal function of coordinates with a
unique temperature dependent length scale. Instead this distribution is described by two lengths, the ratio of which is T
dependent.
In applications, the length l may reach a micron size,
while the Pearl length ⌳ might not exceed ␭ L by much.
Hence, the limit lⰇ⌳ is also of interest. We show that in this
case the scale of the phase variation in the core is of the same
order as in the phase tail; i.e., it is 冑l⌳. This is done with the
help of a variational technique.
In Section IV we provide examples of ␸ (y) obtained by
solving numerically the integral equation in accordance with
our asymptotic and variational estimates.
II. THIN FILMS
As was stressed by Pearl,6 a large contribution to the energy of a vortex in a thin film comes from the stray fields. In
fact, the problem of a vortex in a thin film is reduced to that
of the field distribution in free space subject to certain
boundary conditions at the film surface.7 Since curl h
⫽div h⫽0 outside, one can introduce a scalar potential for
the outside field:
h⫽ⵜ ␺ ,
ⵜ 2 ␺ ⫽0.
PHYSICAL REVIEW B 63 144501
Here the superscripts ⫾ stand for the upper and lower faces
of the film, z⫽⫾d/2.
If the environments of the upper and lower half-spaces are
⫹
identical, we have h ⫺
x,y ⫽⫺h x,y and
2␲
g ⫽⫺h ⫹
y ,
c x
␺ 共 r,z 兲 ⫽
冕
共 2␲ 兲2
␺ 共 k兲 e ik•r⫺kz ,
h z ⫺⌳
h⫹
4 ␲ ␭ L2
c
curl j⫽ ␾ 0 ẑ ␦ 共 r兲 ,
2␲⌳
curlz g⫽ ␾ 0 ␦ 共 r兲 ,
c
共8兲
where g(r) is the sheet current density.
Since all derivatives ⳵ / ⳵ z are large relative to the tangential ⳵ / ⳵ r, the Maxwell equation curl h⫽4 ␲ j/c is reduced to
conditions relating the sheet current to discontinuities of the
tangential field:
4␲
⫹
g ⫽h ⫺
y ⫺h y ,
c x
4␲
⫺
g ⫽h ⫹
x ⫺h x .
c y
共9兲
共12兲
the superscript P is added for convenience of reference to the
Pearl vortex. The distribution of the potential ␺ everywhere
and, in particular, at the film surface follow readily:
冕
␾
冕
␲
␺ P 共 r,z⫽0 兲 ⫽⫺ ␾ 0
⫽⫺
⫽
0
2
d 2k
e ik•r
4 ␲ 2 k 共 1⫹k⌳ 兲
J 0 共 kr 兲
1⫹k⌳
⬁ dk
0
冋 冉 冊 冉 冊册
␾0
r
r
Y0
⫺H0
4⌳
⌳
⌳
,
共13兲
where we have used Ref. 11, 6.562.2. Here, H0 ,Y 0 are
Struve and second-kind Bessel functions; their difference is
well studied; see Ref. 8. The field distribution in real space at
the film surface is given in Appendix A.
III. THIN-FILM JUNCTION
Let a thin film have a line junction along the y axis.
The London equation everywhere on the film except the
junction reads:
where ẑ is the unit vector along the vortex axis. Averaging
over the thickness d, we obtain
h z⫹
共11兲
␾0
;
k 共 1⫹k⌳ 兲
␺ P 共 k兲 ⫽⫺
共6兲
共7兲
⳵hz
⫽ ␾ 0 ␦ 共 r兲 .
⳵z
Applying the 2D Fourier transform and recalling that h z (k)
⫽⫺k ␺ (k) for the upper half-space, we obtain:
共5兲
with k⫽(k x ,k y ), r⫽(x,y), and k⫽ 兩 k兩 . Here ␺ (k) is the
two-dimensional 共2D兲 Fourier transform of ␺ (r,z⫽0). In the
lower half-space we have to replace z by ⫺z in Eq. 共6兲.
Let the film thickness d be small relative to the bulk penetration depth of the film material ␭ L ; for simplicity, the
latter is assumed isotropic. For a vortex at r⫽0, the London
equations for the film interior read:
共10兲
In this case we can consider only the upper half-space and
omit the subscript ⫹ by the field components.
We substitute Eq. 共10兲 into Eq. 共8兲 and use div h⫽0:
Consider a thin film situated at z⫽0. The general form of
the potential which vanishes at z→⫹⬁ of the empty upper
half-space is
d 2k
2␲
g ⫽h ⫹
x .
c y
h z⫹
2␲⌳
curlz g⫽0,
c
x⫽0.
共14兲
At the junction line x⫽0, the current g y is discontinuous.
One can write for the whole x,y plane:
h z⫹
2␲⌳
curlz g⫽ f 共 y 兲 ␦ 共 x 兲 ,
c
共15兲
where the function f (y) is still to be determined. To this end,
integrate Eq. 共15兲 over the area within the contour following
the junction banks along x⫽⫾0 and crossing the junction at
y 1 and y 2 ; see Fig. 1. The magnetic flux through this contour
is zero, and we obtain:
144501-2
JOSEPHSON JUNCTION IN A THIN FILM
PHYSICAL REVIEW B 63 144501
␾ 0 ˜␸ ⬘ 共 k y 兲
.
2 ␲ k 共 1⫹k⌳ 兲
␺ 共 k兲 ⫽⫺
FIG. 1. The junction is shown by a thick solid line. The dashed
line shows the contour used to obtain Eq. 共16兲.
2␲⌳
c
冕
y2
y1
冕
关 g y 共 ⫹0,y 兲 ⫺g y 共 ⫺0,y 兲兴 dy⫽
y2
y1
This gives the outside field distribution in terms of the yet
unknown phase difference ␸ .
One can write Eq. 共25兲 as ␺ (k)⫽ ˜␸ ⬘ (k y ) ␺ P (k)/2 ␲ ,
where ␺ P (k) for the Pearl vortex is given in Eq. 共12兲. This
suggests that convolution argument might be useful in relating the field of the junction to that of Pearl vortices. To this
end, we take
␺ 共 r,z 兲 ⫽⫺
f 共 y 兲 dy
共16兲
冕
␾ 0 ˜␸ ⬘ 共 k y 兲 e ⫺kz ik•r
e ,
共 2 ␲ 兲 2 2 ␲ k 共 1⫹k⌳ 兲
d 2k
˜␸ ⬘ 共 k y 兲 ⫽
共17兲
⫺
We now use the London relation
g⫽⫺
c␾0
4␲ ⌳
2
冉
ⵜ␪⫹
冊
2␲
A ,
␾0
共18兲
2␲
␾0
冕
⫹0
⫺0
to obtain
d ␸ 4 ␲ 2⌳
⫽
关 g y 共 ⫹0,y 兲 ⫺g y 共 ⫺0,y 兲兴 .
dy
c␾0
共20兲
Equations 共17兲 and 共20兲 now yield:
f 共 y 兲⫽
␾0
␸ ⬘共 y 兲 .
2␲
共21兲
Thus, we have instead of Eq. 共14兲:
h z⫹
␾0
d␸
2␲⌳
curlz g⫽
.
␦共 x 兲
c
2␲
dy
共22兲
⳵hz ␾0
d␸
␦共 x 兲
⫽
,
⳵z 2␲
dy
⫺⬁
ds ␸ ⬘ 共 s 兲 e ⫺ik y s ,
共27兲
d 2 r⬘␺ P 共 r⬘ ,z 兲 e ⫺ik•r⬘ ,
共28兲
冕
冕
⬁
⫺⬁
ds
␸ ⬘共 s 兲 P
␺ 共 x,y⫺s,z 兲 .
2␲
j c d sin␸ 共 y 兲 ⫽g x 共 0,y 兲 ⫽⫺
⫽⫺
共23兲
⫽
c ⫹
h 共 0,y 兲
2␲ y
ic
2␲
c␾0
4␲2
冕
冕
d 2k
4␲2
k y ␺ 共 k兲 e ik y y
˜ ⬙共 k y 兲
d 2k ␸
e ik y y ,
2 k 共 1⫹k⌳ 兲
4␲
共30兲
The 2D Fourier transform now yields
⫺ 共 k⫹k 2 ⌳ 兲 ␺ 共 k兲 ⫽
␾0
˜␸ ⬘ 共 k y 兲 ,
2␲
共29兲
Thus, the field of the Josephson junction is a superposition of
fields of Pearl vortices distributed along the junction with the
line density ␸ ⬘ (y)/2␲ .
This remarkable conclusion could have been made on the
basis of comparison of Eqs. 共8兲 and 共15兲 which suggests that
the Pearl solution for h z at the film surface is the Green’s
function for h z (r,0) for arbitrary sources, in our case
␸ ⬘ (y) ␦ (x)/2␲ . The result 共29兲 is more general, since it pertains to all components of the field everywhere outside the
film, the surface included. It is worth noting that for bulk
junctions, a similar result has been obtained by Gurevich: the
field in the junction is a superposition of fields of Abrikosov
vortices distributed along the junction with the line density
␸ ⬘ (y)/2␲ .9
To obtain an equation for ␸ (y), we write:
This equation serves as the boundary condition for the
Laplace problem of the outside field.
As in the Pearl problem, we first rewrite Eq. 共22兲 replacing the sheet currents with tangential fields according to Eq.
共10兲 and using div h⫽0:
h z ⫺⌳
⬁
␾ 0 e ⫺kz
⫽⫺
k 共 1⫹k⌳ 兲
␺ 共 x,y,z 兲 ⫽
dx A x 共 x,y 兲 共19兲
冕
and, after integration over k, obtain
and the definition of the gauge invariant phase difference
␸ 共 y 兲 ⫽ ␪ 共 ⫺0,y 兲 ⫺ ␪ 共 ⫹0,y 兲 ⫺
共26兲
substitute here
for any y 1 and y 2 . This gives:
2␲⌳
关 g y 共 ⫹0,y 兲 ⫺g y 共 ⫺0,y 兲兴 ⫽ f 共 y 兲 .
c
共25兲
共24兲
where Eq. 共25兲 has been used. We now substitute the inverse
transforms
where ˜␸ ⬘ (k y ) is the Fourier transform of d ␸ /dy. Thus we
have:
144501-3
˜␸ ⬙ 共 k y 兲 ⫽
冕
⬁
⫺⬁
dy ␸ ⬙ 共 y 兲 e ⫺ik y y ,
KOGAN, DOBROVITSKI, CLEM, MAWATARI, AND MINTS
冕
4⌳
⫽
k 共 1⫹k⌳ 兲
冋 冉 冊 冉 冊册
d 2 r e ⫺ik•r H0
r
r
⫺Y 0
⌳
⌳
共31兲
PHYSICAL REVIEW B 63 144501
which means that there the vortex flux ␾ 0 is distributed uniformly over the solid angle 2 ␲ . We then obtain for 兩 y 兩 →⬁:
into Eq. 共30兲 and integrate over k 共which is equivalent to
utilizing the convolution theorem兲. We obtain:
sin ␸ 共 y 兲 ⫽
l⫽
冕
l
2
⬁
⫺⬁
c␾0
16␲ 2 j c ␭ L2
ds ␸ ⬙ 共 s 兲 Q
冉 冊
兩 y⫺s 兩
,
⌳
, Q 共 w 兲 ⫽H0 共 w 兲 ⫺Y 0 共 w 兲 .
共32兲
This integral equation for the phase has been obtained by
Mints and Snapiro using a different technique.5
Although both H0 (w) and Y 0 (w) oscillate, the kernel
Q(w) decreases monotonically. This is seen from the integral representation:8
冕
2
Q共 w 兲⫽
␲
e ⫺wt
⬁
dt
0
冑1⫹t 2
冉
.
冊
2
w
ln ⫹ ␥ ,
␲
2
Q 共 w 兲 ⬃2/␲ w.
冕
⫺⬁
冏
␾ 0 ˜␸ ⬘ 共 k y 兲
2 ␲ 共 1⫹k⌳ 兲
2 w⫹1
,
ln
␲
w
d v ␸ ⬙ 共 v 兲 Q 共 兩 u⫺ v 兩 兲 ,
␮ ⫽l/2⌳,
共35兲
sin ␸ 共 u 兲 ⫽ ␮
⫽␾0 ,
共41兲
k→0
˜␸ ⬙ 共 k y 兲 ⫽0.
共42兲
共37兲
⬁
0
d v Q 共 v 兲关 ␸ ⬙ 共 v ⫹u 兲 ⫺ ␸ ⬙ 共 v ⫺u 兲兴 ,
h z 共 0,y 兲 ⫽
共43兲
␾0
␸ ⬘共 y 兲
4␲␭L
共44兲
共the junction thickness is assumed small relative to ␭ L ). In
thin-film junctions this relation does not hold. Instead, we
have, combining Eqs. 共17兲, 共21兲, and 共10兲,
h x 共 ⫹0,y,0兲 ⫺h x 共 ⫺0,y,0兲 ⫽
␾0
␸ ⬘共 y 兲 .
2␲⌳
共45兲
By symmetry h x (⫹0,y,0)⫽⫺h x (⫺0,y,0); therefore,
共38兲
共39兲
冕
where it was assumed that ␸ ⬘ ( v ) is an even and ␸ ⬙ ( v ) is an
odd function of v . This form shows that ␸ (0)⫽ ␲ .
In bulk junctions the vortex field h z at the junction plane
is related to the gradient of the phase difference:
共36兲
which shows that only the ratio of these lengths is relevant.
Hereafter, we use the notation y,s for coordinates in common units, whereas the variables u⫽y/⌳, v ⫽s/⌳ will be
kept dimensionless. When needed we will use also various
rescaled variables denoting them as ␰ , ␩ .
The solution ␸ (y) should satisfy certain conditions at y
→⫾⬁. Since the Josephson current g x (0,y)⬀sin ␸ should
vanish at infinities, we can choose ␸ (⫹⬁)⫽2 ␲ and
␸ (⫺⬁)⫽0. At large distances
␾0
2␲
g 共 0,y 兲 ⫽⫺h y 共 0,y 兲 ⬇⫺
sign y,
c x
2␲y2
˜␸ ⬘ 共 k y 兲 ⫽2 ␲ ,
共34兲
which gives correct leading-order asymptotics at w→0 and
w→⬁.
Thus, the equation for the phase contains two independent
lengths l and ⌳. If ⌳ is chosen as a unit length, the equation
acquires the form
sin ␸ 共 u 兲 ⫽ ␮
⫺k ␺ 共 k兲 兩 k→0 ⫽
By splitting the integration domain in Eq. 共37兲 into
v ⬍u and v ⬎u, we rearrange it to the form:
With better than 9% accuracy, the kernel can be approximated by a simple function:
⬁
共40兲
The relations given in Eqs. 共3兲 and 共4兲 immediately follow. It
is worth recalling that in the bulk junctions the phase ␸
⫽4 tan⫺1 (e y/␭ J ) approaches the limiting values at infinities
as e ⫺ 兩 y 兩 /␭ J .
Another relation follows from fluxoid quantization, which
states that the total flux crossing the film is ␾ 0 . Since the
total flux is 兰 d 2 rh z (r)⫽h z (k⫽0) we have
共33兲
where ␥ is the Euler constant. For large w, only small values
of t contribute to the integral 共33兲, and we obtain:8
Q共 w 兲⬇
gx
2l⌳
⬇⫺ 2 sign y.
j cd
y
which implies that as k y →0,
At small arguments, H0 ⬀w 2 while Y 0 (w) diverges:
Q 共 w→0 兲 ⬇⫺
sin ␸ ⫽
h x 共 ⫹0,y,0兲 ⫽
␾0
␸ ⬘共 y 兲 .
4␲⌳
共46兲
In principle, all fields and currents can be calculated with
the help of Eq. 共25兲. Since due to the flux quantization
⬁
˜␸ ⬘ (k y ⫽0)⫽2 ␲ , Eq. 共42兲, the integrals 兰 ⫺⬁
dy can be
evaluated without actual knowledge of ␸ (y). In particular, it
is easily shown that
冕
⬁
⫺⬁
␺ 共 x,y,z 兲 dy⫽
冕
⬁
⫺⬁
␺ P 共 x,y,z 兲 dy,
共47兲
⬁
dy of the
which implies that similar relations hold for 兰 ⫺⬁
field components.
144501-4
JOSEPHSON JUNCTION IN A THIN FILM
PHYSICAL REVIEW B 63 144501
A. Asymptotic solution for lÕ⌳™1
The length l has a remarkable property of being weakly
temperature dependent.10 Indeed, as T approaches T c , the
product j c ␭ L2 is constant because j c ⬀⌬ 2 ⬀(T c ⫺T), while
␭ L2 ⬀1/(T c ⫺T). On the other hand, the Pearl length ⌳⬀␭ L2
⬀1/(T c ⫺T) and diverges at T c . Therefore, as T→T c , the
ratio l/⌳→0. Also, this ratio might be small for sufficiently
thin films for any T. Since the exact solution of the integral
Eq. 共37兲 is not available, the search of an approximate result
in the limit ␮ →0 is well justified.
It is worth noting that similar to the standard 共bulk兲 Josephson junction, the phase ␸ varies rapidly only near the
vortex center at y⫽0 共within the ‘‘Josephson core’’兲 and the
change is slow outside this domain. We will see that the
domain of rapid change is of size ⬃ ␮ Ⰶ1, whereas in the
rest of the junction the phase varies as a power law. This
suggests employing an asymptotic procedure utilizing two
different length scales.12
Within this method, one looks for the solution of the form
␸⫽
兺
n⫽0
␸ n⫽
兺
n⫽0
␮ n 共 c n ⫹t n 兲 ,
␮ ⫽l/2⌳Ⰶ1,
共48兲
where the functions c n (u) and t n (u) approximate the behavior of ␸ within the core and in the tail, respectively, and u
⫽y/⌳. In particular, this implies imposing the correct
boundary conditions at u⫽0 only upon functions c(u),
whereas the conditions at ⬁’s should be obeyed only by
contributions t(u). Still, neither should diverge in the domain of the other 共thus providing a uniform asymptotic convergence of the so constructed approximation兲. Besides, all
␸ n should have the correct symmetry ( ␸ n⬘ must be an even
and ␸ n⬙ an odd function of u).
We expect the core to occupy a domain of the size ␮ 共or
l in common units兲, an assumption to be confirmed. To find
an equation for c 0 (u), we introduce ‘‘stretched’’ variables
u
␰⫽ ,
␮
␩⫽
v
␮
c 0 ( ␰ ) at large distances, e.g., c 0 ( ␰ →⫺⬁)⬃1/␰ , disagrees
with requirements 共3兲 and 共4兲. Therefore, we proceed to the
next approximation:
␸ ⫽ ␸ 0 ⫹ ␸ 1 ⫽c 0 ⫹ ␮ 共 t 1 ⫹c 1 兲
and substitute this in Eq. 共37兲:
sin c 0 ⫹ ␸ 1 cos c 0 ⫽ ␮
冕
⬁
⫺⬁
d␩ Q共 ␮兩␰⫺␩兩 兲
d 2c 0
.
d␩2
共50兲
For ␮ →0, we can use the asymptotic form 共34兲 of the kernel
for small arguments. Since c ⬙0 ( ␩ ) is odd in ␩ , the constant
terms in the kernel 共34兲 yield zero after integration, and we
obtain
sin c 0 共 ␰ 兲 ⫽⫺
2
␲
冕
⬁
⫺⬁
d ␩ ln兩 ␰ ⫺ ␩ 兩
d 2c 0
.
d␩2
共51兲
This equation has an exact solution13,9,5
c 0 共 ␰ 兲 ⫽2 tan⫺1 共 ␰ /2兲 ⫹ ␲ .
共52兲
⬁
⫺⬁
冕
dv
d 2c 0
Q 共 兩 u⫺ v 兩 兲
dv2
⬁
⫺⬁
dv
d 2␸ 1
Q 共 兩 u⫺ v 兩 兲 . 共54兲
dv2
We now use Eq. 共51兲 to rewrite this as:
t 1 cos c 0 ⫺ ␮
⫽
冕
冕
⬁
⫺⬁
dv
⬁
⫺⬁
dv
d 2t 1
Q 共 兩 u⫺ v 兩 兲
dv2
冋
册
2
d 2c 0
ln兩 u⫺ v 兩 ,
2 Q 共 兩 u⫺ v 兩 兲 ⫹
dv
␲
共55兲
where we took into account that by design c 1 Ⰶt 1 at large
distances. In the limit ␮ →0, the integral at the left-hand side
contains an extra factor ␮ and, therefore, can be disregarded,
whereas
4␮
dc 0
⫽ 2
du 4 ␮ ⫹u 2
冏
⫽2 ␲ ␦ 共 u 兲 .
共56兲
␮ →0
Hence, for ␰ Ⰷ ␮ , where cos c0⫽(␰2⫺4)/(␰2⫹4)⬇1, we
obtain:
t 1 ⫽2 ␲
dQ 共 兩 u 兩 兲
4
⫹
sign u.
d兩u兩
兩u兩
共57兲
Thus, at this stage of the expansion we have:
␸ ⫽ ␸ 0 ⫹ ␮ t 1 ⫽2 tan⫺1
One can set t 0 (u)⬅0 and obtain:
sin c 0 共 ␰ 兲 ⫽
冕
⫹␮
共49兲
.
共53兲
u
dQ 共 兩 u 兩 兲 4 ␮
⫹
.
⫹ ␲ ⫹2 ␲ ␮
2␮
du
u
共58兲
The last term here compensates both the ‘‘wrong’’ behavior
of 2 tan⫺1 (u/2␮ ) at large distances and the divergence of
Q ⬘ ( 兩 u 兩 ) at u⫽0. One can see that magnetostatics requirements 共3兲 and 共4兲 are now satisfied.
One should note, however, that while having the correct
behavior at large distances, ␸ 0 ⫹ ␮ t 1 acquires a finite discontinuity at the origin:
共 ␸ 0 ⫹ ␮ t 1 兲 u⫽⫾0 ⫽ ␲ ⫾4 ␮ .
共59兲
This mismatch is proportional to the small parameter ␮ and,
in principle, could be cured by the core contribution c 1 . We
omit this difficult calculation, because our major goal of establishing the characteristic lengths of the phase variation is
already achieved. Near the vortex center we have:
Note that by construction, this formula approximates the actual solution in the core; although the boundary conditions at
infinities and at zero are satisfied, the asymptotic behavior of
144501-5
␸ 共 y→0 兲 ⫽ ␲ ⫹
2y
,
l
共60兲
KOGAN, DOBROVITSKI, CLEM, MAWATARI, AND MINTS
whereas at large distances we have confirmed asymptotics
共3兲 and 共4兲. In other words, l is the characteristic length
within the core, whereas at large distances the scale is 冑l⌳.
B. The case lÕ⌳š1
The length l⬀1/j c depends on the junction quality and is
often large. It may exceed considerably the Pearl length ⌳;
this is the case in many experimental situations.3 It is of
interest to consider Eq. 共37兲 in the limit ␮ →⬁.
Since sin ␸⬍1 at the LHS of Eq. 共37兲, the equation can be
satisfied only if ␸ ⬙ →0, i.e., if ␸ is nearly linear in u in a
broad domain adjacent to u⫽0. In physical terms, this means
that the vortex core is likely to be large. Out of the core, ␸ is
close to the limiting values of 0 and 2 ␲ at infinities. To
‘‘shrink’’ the core domain we introduce new variables:
␰ ⫽u/ 冑␮ ,
␩ ⫽ v / 冑␮ .
sin ␸ 共 ␰ 兲 ⫽ 冑␮
冕
⫺⬁
d␩
␸ 共 u 兲 ⫽2 tan⫺1 共 u/L 0 兲 ⫹ ␲ ,
which satisfies the boundary conditions at infinities and varies as ␲ ⫹2u/L 0 near the origin. After the calculation outlined in Appendix C, we obtain a relation between L 0 and the
parameter ␮ :
␲ L 0 共 1⫹4L 20 兲
.
2␮⫽
␲ ⫹4L 0 ln 2L 0
sin ␸ 0 共 ␰ 兲 ⬇
⬇
冕
d ␩ ␸ ⬙0 共 ␩ 兲
⫺⬁
兩␰⫺␩兩
1
兩␰兩
冕
⬁
2 ␮ ⫽L 0 ,
⫺⬁
d␩2
Q 共 冑␮ 兩 ␰ ⫺ ␩ 兩 兲 .
共62兲
d ␩ ␸ 0⬙ 共 ␩ 兲 1⫹
冊
␩
2␲
⫹ . . . ⬇⫺
;
␰
兩␰兩␰
共63兲
we have integrated by parts ␩ ␸ 0⬙ and used ␸ 0⬙ (⫺ ␩ )
⫽⫺ ␸ ⬙0 ( ␩ ). This result coincides with requirements 共3兲 and
共4兲.
The problem of the core structure can be addressed as
follows. The functional, minimization of which leads to Eq.
共37兲 for the phase, reads:
W兵␸其⫽
冕
⬁
⫺⬁
⫹
␮
2
du 共 1⫺cos ␸ 兲
冕
⬁
⫺⬁
du ␸ ⬘ 共 u 兲
冕
⬁
⫺⬁
␮ Ⰶ1,
␲ L 20
,
2␮⫽
ln 2L 0
共67兲
␮ Ⰷ1.
共68兲
IV. NUMERICAL RESULTS
d 2␸
冉
共66兲
It is seen that for ␮ Ⰶ1, L 0 must be small, too. Likewise,
large ␮ require L 0 Ⰷ1. The limiting cases are:
We have solved numerically the integral Eq. 共32兲 by an
iterative method. Starting from a certain trial function ␸ 0 (y),
we obtain the phase difference after i⫹1 iterations as
␸ i⫹1 共 y 兲 ⫽ ␸ i 共 y 兲 ⫹AD 兵 ␸ i 共 y 兲 其 ,
In the limit 兩 ␰ 兩 →⬁, we can replace the kernel Q(z) with the
large argument asymptotics 共35兲. As a result, the parameter
␮ drops off 共this is precisely why the scaling factor has been
chosen as 1/冑␮ ):
⬁
共65兲
共61兲
Equation 共37兲 now takes the form:
⬁
PHYSICAL REVIEW B 63 144501
d v ␸ ⬘ 共 v 兲 Q 共 兩 u⫺ v 兩 兲 .
共64兲
It is shown in Appendix B that 共within a constant factor兲 W
is in fact the total energy consisting of the Josephson, kinetic, and magnetic contributions. We can now choose a set
of trial functions ␸ 0 (u) containing a variational parameter
which we call L 0 ; the functions should be linear in u at short
distances. Substituting these functions in Eq. 共64兲 we find
W(L 0 ) minimization of which gives the best value for L 0 for
a given set.
As an example of this procedure we choose
D 兵 ␸ 共 y 兲 其 ⬅⫺sin ␸ 共 y 兲 ⫹
l
2
冕
⫹⬁
⫺⬁
ds ␸ ⬙ 共 s 兲 Q
共69兲
冉 冊
兩 y⫺s 兩
,
⌳
共70兲
where A is a constant. Equation 共37兲 is equivalent to
D 兵 ␸ (y) 其 ⫽0. If the constant A is small enough to stabilize
the iterative procedure, the 兩 D 兵 ␸ i (y) 其 兩 becomes smaller for
larger i. The solution 兩 D 兵 ␸ i (y) 其 兩 ⬍ ⑀ with an arbitrary accuracy 0⬍ ⑀ Ⰶ1 is obtained by iterating the procedure until
兩 D 兵 ␸ i (y) 其 兩 becomes less than ⑀ .
The open circles of Fig. 2 show the results of numerical
solution of Eq. 共37兲 for the phase difference ␸ (y ⬘ ) with y ⬘
⫽y/ 冑l⌳ and l/⌳⫽0.01. The solid curve is calculated according to the approximation 共52兲. It is worth observing that
the approximation is not only good for small y; it is still
fairly good for 冑l⌳⬍y⬍4 冑l⌳ 共see the inset兲 and deteriorates slowly at large distances. As we will see below, the
large distance behavior has little effect on integrated quantities such as the total vortex energy, mainly because the Josephson currents at these distances are exceedingly small.
Figure 3 shows numerical solutions for l/⌳
⫽0.01,0.1,1,10, and 100. One sees that the slope ␸ ⬘ (y) at
y⫽0 is suppressed; i.e., the vortex ‘‘core’’ expands with
increasing ratio l/⌳.
To illustrate the core expansion with increasing l/⌳ we
plot in Fig. 4 the slopes d ␸ /dy ⬘ at the origin obtained from
the numerical 共‘‘exact’’兲 solutions 共open circles兲 along with
the slopes calculated using Eq. 共66兲 obtained using the variational procedure described above 共solid curve兲. We see that
the trial functions 共65兲 reproduce well ␸ ⬘ (0) for small ␮
⫽l/2⌳, as they should because these functions are close to
the actual ␸ (y) for this case. It is worth noting, however,
that even for large ␮ the ansatz 共65兲 provides a reasonable
estimate for the slope ␸ ⬘ (0).
144501-6
JOSEPHSON JUNCTION IN A THIN FILM
PHYSICAL REVIEW B 63 144501
FIG. 2. The phase difference ␸ versus y ⬘ ⫽y/ 冑l⌳ for l/⌳
⫽0.01 or ␮ ⫽0.005. For this and the following plots, 冑l⌳ is chosen
as unit of length for convenience of comparison. The open circles
are obtained by solving numerically the integral Eq. 共37兲. The solid
curve is the approximation 共52兲, which reads in terms of y ⬘ as ␸ 0
⫽2 tan⫺1 (y ⬘ / 冑2 ␮ )⫹ ␲ . The inset shows the phase for y ⬘ ⬎1 or for
y⬎ 冑l⌳. The dashed curve is the standard bulk soliton ␸ b
⫽4 tan⫺1 关 exp(y ⬘ / 冑2 ␮ ) 兴 which has the same slope at the origin,
but approaches the value of 2 ␲ exponentially rapidly as y ⬘ →⬁.
As is shown in Appendix B, the total energy of a Josephson vortex in a thin film reads:
␾ 20
␾ 20
W共 ␮ 兲
W⫽
.
E⫽
16␲ 3 l
8 ␲ 2⌳ 4 ␲ ␮
FIG. 4. The slopes ␸ ⬘ (y ⬘ ⫽0) for the set of ␮ ⫽l/2⌳ shown in
Fig. 3 obtained by solving numerically Eq. 共37兲 共open circles兲. The
solid line is obtained with the help of the variational result 共66兲.
we found L 0 which minimizes the energy. Then we evaluated E numerically using the exact kernel Q. Figure 5 shows
that these two approximations yield nearly the same energies; the relative difference between them is plotted in the
inset and shows that the thin film ansatz yields lower energies for ␮ Ⰶ1, whereas the bulk ansatz is better for large ␮ .
V. SUMMARY
共71兲
Here, the prefactor ␾ 20 /8␲ 2 ⌳ is a natural energy scale because the self-energy of a Pearl vortex is given by this prefactor 关multiplied by ln(⌳/␰) with ␰ being the coherence
length兴. We have calculated E numerically using the trial
function 共65兲 with a too slow 1/y asymptotics; we tried also
the bulk soliton 4 tan⫺1 关 exp(u/L0)兴, which decays at large
distances faster than the needed 1/y 2 . For each trial function,
FIG. 3. Numerical solutions ␸ (y ⬘ ), y ⬘ ⫽y/ 冑l⌳⫽u 冑⌳/l, of Eq.
共37兲 for l/⌳⫽0.01 共open circles兲, 0.1, 1, 10, and 100 共crosses兲. The
inset (l/⌳⫽1) shows that the approximation 共52兲, the solid curve,
which is good for l/⌳⫽0.01 fails for l/⌳⫽1.
To summarize, we reiterate the following points:
The field associated with a Josephson vortex is a superposition of fields of Pearl vortices distributed along the junction with the line density ␸ ⬘ (y)/2␲ .
The Josephson vortex in thin films extends to much larger
distances than in the bulk due to the (L ⬁ /y) 2 decay of the
FIG. 5. The energy E of Eq. 共71兲 in units of ␾ 20 /8␲ 2 ⌳ as a
function of ␮ ⫽l/2⌳ evaluated for the film ansatz ␸
⫽2 tan⫺1 (u/L 0 )⫹ ␲ with L 0 chosen to minimize the energy functional 共64兲. The inset shows the relative difference D⫽(E
⫺E b )/E b between E and the energy E b calculated in the same way
with the bulk ansatz ␸ b ⫽4 tan⫺1 exp(u/L0).
144501-7
KOGAN, DOBROVITSKI, CLEM, MAWATARI, AND MINTS
phase difference at large distances. This power law is imposed by the magnetostatics of the stray fields outside and as
such is the same for any film thickness 共as long as dⰆ␭ J and
the z dependence of the phase can be disregarded兲. The characteristic length L ⬁ for the phase attenuation at large distances is
L ⬁ ⬇ 冑l⌳⬃␭ J
冑
␭L
.
d
PHYSICAL REVIEW B 63 144501
L 0 ⬇L ⬁ ⬇ 冑l⌳.
h z 共 r,0 兲 ⫽
ACKNOWLEDGMENTS
Ames Laboratory is operated for U.S. DOE by the Iowa
State University under Contract No. W-7405-Eng-82. The
work of V.G.K. and R.G.M. were supported in part by Grant
No. 96-00048 from the United States-Israel Binational Science Foundation 共BSF兲, Jerusalem, Israel. R.G.M. acknowledges hospitality and support of the International Institute of
Theoretical and Applied Physics of the Iowa State University. Y.M. is pleased to acknowledge the support of Science
and Technology Agency in Japan for his visit to Iowa State
University.
APPENDIX A
h r 共 r,0兲 ⫽
再
冋 冉 冊 冉 冊 册冎
We have for rⰆ⌳:
.
␾ 0⌳
2␲ r3
冋 冉 冊册
1⫺O
⌳2
r2
共A3兲
.
冋 冉冊 冉冊 册
⳵␺
␾0
r
r
2
⫽
H
⫺Y
⫺
. 共A4兲
1
1
⳵r 4␲⌳2
⌳
⌳
␲
The asymptotics are readily obtained:
h r 共 r,0兲 ⫽
冋
冉 冊册
␾0
⌳2
r2
r
r
⫹O
1⫺ ⫺
ln
␲ ⌳r
⌳ 2⌳ 2 2⌳
r2
for rⰆ⌳, and
h r 共 r,0兲 ⫽
␾0
2␲r2
冋
1⫺
3⌳ 2
r2
⫹O
冉 冊册
⌳4
r4
共A5兲
共A6兲
for rⰇ⌳. As expected, the behavior at large distances corresponds to the Coulomb field of a point ‘‘charge’’ creating
the flux ␾ 0 in a solid angle 2 ␲ . Note also the 1/r divergence
of h r at r→0.
APPENDIX B
The Josephson coupling energy is
E J⫽
␾ 20
16␲ l
3
冕
⬁
dy
共 1⫺cos ␸ 兲 .
⫺⬁ ⌳
共B1兲
The magnetic field energy in the upper half-space is expressed in terms of the potential ␺ :
冕
dV
h
⫽
8␲
2
⫽
冕
dV
ⵜ• 共 h␺ 兲
⫽⫺
8␲
1
8␲
冕
d 2k
4␲2
冕
d 2r
h 共 r,0兲 ␺ 共 r,0兲
8␲ z
k ␺ 共 k兲 ␺ 共 ⫺k兲 .
共B2兲
The total field energy E F is twice this amount. The kinetic
energy E K of the supercurrents is the integral over the film
volume of the quantity 2 ␲ ␭ L2 j 2 /c 2 ⫽ ␲ ⌳g 2 /c 2 d. Since according to Eqs. 共10兲 g 2 ⫽c 2 (h 2x ⫹h 2y )/4␲ 2 (h x,y are taken at
z⫽0), we find readily:
The component h z at the film surface is
␾0 1
␲
r
r
h z 共 r,0 兲 ⫽
⫺
H0
⫺Y 0
2 ␲ ⌳ r 2⌳
⌳
⌳
共A2兲
The tangential component at the film surface is
共73兲
The results obtained in this work for the thin-film limit,
dⰆ␭ L , should hold also for thicker films as long as one can
disregard the z dependence of the phase. Without going into
formal details of the difficult problem of a junction in a slab
of finite thickness, we may guess that the z dependence of ␸
is weak when dⰆ␭ J , since ␭ J is the shortest length at which
the phase can vary (␭ J is assumed to exceed ␭ L ). This makes
our results applicable to experimental situations as those of
Ref. 3 where junctions in YBCO films with ␭ L ⬇0.15 ␮m
and ␭ J ⬃1 ␮m have been studied.
.
It is interesting to note that the field in this domain diverges
faster than in the bulk vortex for rⰆ␭ L , where h z ⬀ln(r/␭L).
For rⰇ⌳, one can use the asymptotic expansion 共12.1.30兲 of
Ref. 8:
共72兲
Hence, for moderately thin films (d⬃␭ L ), the length L ⬁ is of
the order of the bulk Josephson length.
The characteristic length L 0 at small distances 共the core
size兲 is l for lⰆ⌳ as is seen from Eq. 共52兲. This is the case
in very thin films and for any film thickness close enough to
T c . Thus, for lⰆ⌳, the Josephson vortex is characterized by
two lengths, L ⬁ ⫽ 冑l⌳ at large distances and L 0 ⫽l at short
ones. These two lengths have different T dependencies, and
therefore the vortex structure changes with temperature.
Hence, the situation in films is distinctly different from that
in bulk junctions, where the structure is universal for all T
and is characterized by a single length ␭ J (T).
For lⰇ⌳, the characteristic length at all distances is of the
same order:
冉 冊 冉 冊册
冋
␾0
r
r
r
1⫹ ln
⫹O
2␲⌳ r
⌳
⌳
⌳
h z 共 r,0 兲 ⫽
E K⫽
共A1兲
⌳
4␲
冕
d 2k
4␲
2
k 2 ␺ 共 k兲 ␺ 共 ⫺k兲 .
共B3兲
It is now straightforward to show with the help of Eqs. 共25兲,
共27兲, and 共31兲 that the total energy
144501-8
JOSEPHSON JUNCTION IN A THIN FILM
E⫽E J ⫹E F ⫹E K ⫽
␾ 20
16␲ 3 l
PHYSICAL REVIEW B 63 144501
共B4兲
W,
Here, all integrals are from ⫺⬁ to ⬁ and we have changed
variables: s⫽u⫹ v , t⫽u⫺ v . After integration over s one
obtains:
where W is given in Eq. 共64兲.
APPENDIX C
W 2 ⫽8 ␲ ␮ L 0
The first integral in Eq. 共64兲 is easily evaluated for the
ansatz 共65兲:
W 1 ⫽2 ␲ L 0 .
共C1兲
One can estimate the contribution W 2 of the double integral
as follows. Since for the ansatz 共65兲 ␸ ⬘ (u)⫽2L 0 /(u 2 ⫹L 20 ),
we have
W 2 ⫽2 ␮ L 20
⫽16␮ L 20
冕冕
⬁ dt
0
Q共 t 兲
4L 20 ⫹t 2
.
共C3兲
Further analytical progress is difficult to make, and we resort
to the approximation 共36兲:
W 2 ⫽8 ␮
du d v Q 共 兩 u⫺ v 兩 兲
冕
⬁
0
dw
1⫹w
2wL 0 ⫹1
.
2wL 0
ln
2
共C4兲
共 L 20 ⫹u 2 兲共 L 20 ⫹ v 2 兲
冕冕
ds dt Q 共 兩 t 兩 兲
关 4L 20 ⫹ 共 s⫹t 兲 2 兴关 4L 20 ⫹ 共 s⫺t 兲 2 兴
.
共C2兲
1
冕
C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Janes, A. Gupta, T.
Shaw, J.Z. Sun, and M.B. Ketchen, Phys. Rev. Lett. 73, 593
共1994兲.
2
J.H. Miller, Jr., Q.Y. Ying, Z.G. Zou, N.Q. Fan, J.H. Xu, M.F.
Davis, and J.C. Wolfe, Phys. Rev. Lett. 74, 2347 共1995兲.
3
J.R. Kirtley, C.C. Tsuei, M. Rupp, J.Z. Sun, Lock See Yu-Jahnes,
A. Gupta, M.B. Ketchen, K.A. Moler, and M. Bhushan, Phys.
Rev. Lett. 76, 1336 共1996兲.
4
J. Mannhart, H. Hilgenkamp, B. Mayer, Ch. Gerber, J.R. Kirtley,
K.A. Moler, and M. Sigrist, Phys. Rev. Lett. 77, 2782 共1996兲.
5
R.G. Mints and I.B. Snapiro, Phys. Rev. B 49, 6188 共1994兲; R.G.
Mints and I.B. Snapiro, ibid. 51, 3054 共1995兲.
6
J. Pearl, Appl. Phys. Lett. 5, 65 共1964兲.
7
V.G. Kogan, Phys. Rev. B 49, 15 874 共1994兲.
8
Handbook of Mathematical Functions, edited by M. Abramowitz
and A. Stegun 共U.S. GPO, Washington, D.C., 1965兲.
We now minimize W⫽W 1 ⫹W 2 with respect to L 0 and
obtain the relation 共66兲 between L 0 and ␮ . Numerical comparison shows that ␮ (L 0 ) so obtained differs by less than 3%
from the result of using the exact kernel Q in Eq. 共C3兲.
A. Gurevich, Phys. Rev. B 46, 3187 共1992兲.
For grain boundaries in YBCO, l is T independent for large misalignment angles. For small angles, the boundaries behave as
superconductor-normal-superconductor
junctions,
unlike
superconductor-insulator-superconductor structures which are
discussed in the text. On the other hand, j c of small angle
boundaries might be large, which reduces l and makes the case
l/⌳Ⰶ1 possible not only near T c .
11
I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series, and
Products 共Academic Press, New York, 1980兲.
12
N.N. Bogoliubov, and Y.A. Mitropolsky, Asymptotic Methods in
the Theory of Non-linear Oscillations 共Hindustan Publishing
Corp., Delhi, 1961兲.
13
A. Seeger, Teorie der Gitterfehlstellen, in Handbuch der Physik
共Springer, Berlin, 1955兲, Vol. 7, Part 1.
9
10
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