PHYSICAL REVIEW B 68, 174514 共2003兲
Magnetic-field and current-density distributions in thin-film superconducting rings and disks
Ali A. Babaei Brojeny
Department of Physics, Isfahan University of Technology, Isfahan 84154, Iran
and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011-3160, USA
John R. Clem
Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011-3160, USA
共Received 23 June 2003; published 10 November 2003兲
We show how to calculate the magnetic-field and sheet-current distributions for a thin-film superconducting
annular ring 共inner radius a, outer radius b, and thickness dⰆa) when either the penetration depth obeys ␭
⬍d/2 or, if ␭⬎d/2, the two-dimensional screening length obeys ⌳⫽2␭ 2 /dⰆa for the following cases: 共a兲
magnetic flux ⌽ z (a) trapped in the hole in the absence of an applied magnetic field, 共b兲 zero magnetic flux in
the hole when the ring is subjected to an applied magnetic field H a , and 共c兲 focusing of magnetic flux into the
hole when a magnetic field H a is applied but no net current flows around the ring. We use a similar method to
calculate the magnetic-field and sheet-current distributions and magnetization loops for a thin, bulk-pinningfree superconducting disk 共radius b) containing a dome of magnetic flux of radius a when flux entry is
impeded by a geometrical barrier.
DOI: 10.1103/PhysRevB.68.174514
PACS number共s兲: 74.78.⫺w, 74.25.Ha, 74.25.Op
I. INTRODUCTION
Recently Babaei Brojeny et al.1 reported exact analytical
solutions for the magnetic-field and sheet-current-density
profiles for two current-carrying parallel coplanar thin-film
superconducting strips in a perpendicular magnetic field. Included were calculations for 共a兲 the inductance per unit
length when the two strips carry equal and opposite currents,
共b兲 the zero-flux-quantum state when no net magnetic flux
threads between the strips in a perpendicular applied field
H a , and 共c兲 the focusing of magnetic flux between the two
strips in a field H a when each strip carries no net current.
These problems are of relevance to the design of superconducting thin-film devices, especially superconducting quantum interference devices 共SQUIDs兲.
Of interest is the focusing of magnetic flux into the central
hole in washer-type2 SQUIDs and, in particular, the question
of how much flux ⌽ h goes into the hole when the SQUID is
in a perpendicular magnetic field H a ⫽B a / ␮ 0 and no net
current circulates around the hole. The flux-focusing problem
was examined by Ketchen et al.3 who expressed ⌽ h in terms
of an effective pickup area of the hole, A e f f ⫽⌽ h /B a , which
in general is larger than the actual area of the hole, A h , but
less than the area occupied by the washer, A w . Accounting
only for azimuthal currents, they considered a washer of circular geometry 共an annular ring兲 and derived a simple theoretical expression for the effective area, A e f f
⬇(8/␲ 2 )A h (A w /A h ) 1/2, the theoretical approximations used
being valid only for A h ⰆA w . Experiments on a series of
square washers with A w /A h up to 104 yielded results in excellent qualitative agreement with the prediction, but with
A e f f ⬇1.1A h (A w /A h ) 1/2.
Experiments by Dantsker et al.4 on SQUIDs made with
narrow superconducting lines separated by slots or holes 共for
trapping flux quanta during cooldown in the earth’s magnetic
field兲 have revealed that the presence of slots or holes increases the effective area over the value for a solid washer.
0163-1829/2003/68共17兲/174514共9兲/$20.00
This effect was confirmed experimentally by Jansman et al.5
who were able to account for the increased effective area by
treating the slotted washers as parallel circuits of pickup inductances.
In this paper we introduce an approach suitable for extension to calculations of the magnetic-field and sheet-currentdensity distributions in superconducting thin-film strips,
rings, and narrow lines. We consider the idealized case for
which the penetration depth ␭ obeys ␭⬍d/2 or, if ␭⬎d/2,
the two-dimensional screening length ⌳⫽2␭ 2 /d obeys ⌳
Ⰶa, such that the key boundary condition is that the normal
component of the magnetic induction is zero on the surface
of the superconductor. A complicating consequence is that
the sheet-current distribution in the superconductor has
inverse-square-root singularities at the edges. We show here
that for thin rings and disks, an approach taking into account
the inverse-square-root singularities from the beginning is a
simple and efficient alternative to mutual-inductance approaches such as those used in Refs. 5– 8.
The flux-focusing result of Ref. 3 was obtained by superposition. First, the induced current flowing in the clockwise
direction in an applied magnetic induction B a was calculated
assuming zero magnetic flux in the hole. This current was
approximated using the known result for a superconducting
disk with no central hole. Next, the induced current flowing
in the counterclockwise direction in the absence of an applied field was calculated assuming a given amount of magnetic flux ⌽ h in the hole. This current was approximated
using the known result for an infinite superconducting sheet
with a round hole in it. Finally, the relation between B a and
⌽ h was obtained by equating the magnitudes of the two circulating currents. In the present paper, we show how to calculate all properties without making the small-hole approximations used in Ref. 3. We show how to solve the fluxfocusing problem directly, as well as by superposition.
Another problem of interest is the calculation of the
magnetic-field and current-density distributions for the case
68 174514-1
©2003 The American Physical Society
PHYSICAL REVIEW B 68, 174514 共2003兲
ALI A. BABAEI BROJENY AND JOHN R. CLEM
of a bulk-pinning-free type-II superconducting disk of radius
b and thickness dⰆb in which the entry of magnetic flux is
impeded by a geometrical barrier.9,10 Analytic solutions for
the field and current distributions and the magnetization in
strips subject to a geometrical barrier have been studied for
the bulk-pinning-free case in Refs. 11 and 12 and for the case
of Bean-model bulk pinning (J c ⫽ const兲 in Ref. 13. Numerical results for the field and current distributions and the
magnetization in disks subject to both a geometrical barrier
and bulk pinning with a B-dependent J c have been presented
in Refs. 14 –18. In the following, we present an efficient
method for calculating the field and current distributions and
the magnetization in bulk-pinning-free disks subject to a
geometrical barrier.
Our paper is organized as follows. In Sec. II, we outline
our approach and set down the basic equations. In Sec. III,
we apply this approach to calculate the inductance of an
annular ring of arbitrary inner radius. In Sec. IV, we calculate
the current circulating around a ring remaining in the zeroflux-quantum state while subjected to a perpendicular magnetic field. In Sec. V, we consider the flux-focusing problem
and calculate the magnetic flux contained in the center of a
ring in an applied magnetic field when there is no net current
around the ring. In Sec. VI, we calculate the magnetization
loop for a bulk-pinning-free thin-film type-II superconducting disk subject to a geometrical barrier. We briefly discuss
our results in Sec. VII.
Another quantity of interest is the magnetic flux up through a
circle of radius ␳ in the plane z⫽0, 19,20
⌽ z 共 ␳ 兲 ⫽ ␮ 0 H a ␲␳ 2 ⫹
␮0
2
冕
a
G A 共 ␳ , ␳ ⬘ 兲 ⫽ 共 ␳ ⫹ ␳ ⬘ 兲关共 2⫺k 2 兲 K 共 k 兲 ⫺2E 共 k 兲兴
1
2␲
冕
b
a
G 共 ␳ , ␳ ⬘ 兲 K ␾共 ␳ ⬘ 兲 d ␳ ⬘,
K̃ ␾ 共 u 兲 ⫽
4g 共 u 兲
␲ u 冑共 u ⫺ã 2 兲共 1⫺u 2 兲
2
where H a is the applied field, K ␾ ( ␳ ) is the sheet-current
density in the counterclockwise direction,
G 共 ␳ , ␳ ⬘ 兲 ⫽K 共 k 兲 / 共 ␳ ⫹ ␳ ⬘ 兲 ⫺E 共 k 兲 / 共 ␳ ⫺ ␳ ⬘ 兲 ,
共2兲
k⫽2 共 ␳␳ ⬘ 兲 1/2/ 共 ␳ ⫹ ␳ ⬘ 兲 ,
共3兲
冕
b
a
K ␾共 ␳ 兲 d ␳ ,
h m共 u 兲 ⫽
2
m z⫽ ␲
冕
共4兲
冕␳
a
K ␾共 ␳ 兲 d ␳ .
共5兲
174514-2
2
␾ m共 u 兲 ⫽
␲
␲ /2
0
gm
冉 冊
G 共 u, v 兲
␲ /2
4
i m⫽
␲
and the magnetic moment along the z direction is
2
兺
m⫽1
u⫺ã
共8兲
m⫺1
共9兲
.
1⫺ã
冉 冊
冕 冉 冊
冕 冉 冊
冕
冉 冊
␲2
f m⫽
b
,
Although we are not certain that such a choice gives an exact
solution in general, it reduces to known exact solutions in
various limits 关 a→0, b→⬁, or (b⫺a)Ⰶb], all of which
have inverse-square-root singularities at the sample edges.
To determine the N coefficients, N⫺1 equations are obtained
by setting H z ( ␳ n )⫽0, where ␳ n ⫽a⫹n(b⫺a)/N and n
⫽1,2, . . . ,N⫺1. The Nth equation depends on the case under consideration; for case 共a兲 we use Eq. 共4兲 for given I, for
case 共b兲 we use Eq. 共6兲 and set ⌽ z (a)⫽0, for case 共c兲 we
use Eq. 共4兲 and set I⫽0, and for case 共d兲 we use Eq. 共9兲 and
set g(ã)⫽0.
For numerical evaluation of the integrals in Eqs. 共1兲, 共4兲–
共6兲, it is convenient to change variables using the substitution
v ⫽ ␳ ⬘ /b⫽ 冑ã 2 ⫹(1⫺ã 2 )sin2␾ and to define the functions
and K and E are complete elliptic integrals of the first and
second kind with modulus k. An important boundary condition we will use in this paper is that H z ( ␳ )⫽0 for a⬍ ␳
⬍b. The total current in the counterclockwise direction is
I⫽
共7兲
where u⫽ ␳ /b and ã⫽a/b and g(u) is a polynomial containing N terms,
g共 u 兲⫽
共1兲
共6兲
and k is given in Eq. 共3兲.
In the following sections we present solutions of the
above equations and determine the corresponding sheetcurrent density K ␾ ( ␳ ) for four cases: 共a兲 self-inductance L
⫽⌽ z (a)/I when H a ⫽0, 共b兲 the zero-flux-quantum state
关 ⌽ z (a)⫽0 兴 in an applied field H a , 共c兲 flux focusing in an
applied field 关calculation of ⌽ z (a) when I⫽0], and 共d兲
geometrical-barrier effects in a thin disk of radius b containing a Lorentz-force-free magnetic-flux dome of radius a. In
each case, we assume a spatial dependence of the reduced
sheet-current density of the form
II. BASIC EQUATIONS
H z 共 ␳ 兲 ⫽H a ⫹
G A共 ␳ , ␳ ⬘ 兲 K ␾共 ␳ ⬘ 兲 d ␳ ⬘,
where
N
We consider a thin-film superconducting annular ring in
the plane z⫽0, centered on the z axis, with inner and outer
radii a and b and thickness dⰆa. We assume that either ␭
⬍d/2 or ⌳Ⰶa if ␭⬎d/2, as discussed in the Introduction.
By the Biot-Savart law, the z component of the magnetic
field in the plane z⫽0 is19,20
b
␲ /2
0
␲ /2
0
v ⫺2 d ␾ ,
共10兲
m⫺1
v ⫺2 d ␾ ,
v ⫺ã
共11兲
m⫺1
1⫺ã
G A 共 u, v 兲
m⫺1
1⫺ã
1⫺ã
0
4
␲
v ⫺ã
v ⫺ã
v ⫺ã
1⫺ã
d␾,
共12兲
m⫺1
v ⫺2 d ␾ ,
共13兲
PHYSICAL REVIEW B 68, 174514 共2003兲
MAGNETIC-FIELD AND CURRENT-DENSITY . . .
and
␣ nm ⫽h m 共 u n 兲 ,
共14兲
where u n ⫽ ␳ n /b⫽ã⫹n(1⫺ã)/N, and n⫽1,2, . . . ,N⫺1.
For a⬍ ␳ ⬍b (ã⬍u⬍1), Eqs. 共10兲 and 共13兲 are principalvalue integrals, evaluated by splitting the ␾ integral into two
parts, one from 0 to ⌽(u⫺ ⑀ ) and the other from ⌽(u⫹ ⑀ ) to
␲ /2, where
⌽ 共 u 兲 ⫽sin⫺1
冑
u 2 ⫺ã 2
共15兲
1⫺ã 2
and ⑀ is an infinitesimal. For the results presented here we
have used ⑀ ⫽10⫺7 .
III. INDUCTANCE OF AN ANNULAR RING
To calculate the inductance, we set H a ⫽0 in Eq. 共1兲 and
define K I ␾ ⫽(I I /b)K̃ I ␾ , where the subscript I henceforth labels all quantities that are specific to calculations of the inductance. To evaluate the coefficients g Im in
N
兺
g I共 u 兲 ⫽
m⫽1
g Im
冉 冊
u⫺ã
m⫺1
1⫺ã
,
共16兲
we use the N equations
N
兺
m⫽1
␣ Inm g Im ⫽ ␤ In ,
共17兲
n⫽1,2, . . . ,N, where ␣ Inm ⫽ ␣ nm and ␤ In ⫽0 for n⬍N, and
␣ INm ⫽i m and ␤ IN ⫽1 for n⫽N. These equations are obtained from Eqs. 共1兲, 共8兲–共10兲, and 共14兲 and H z ( ␳ n )⫽0 for
n⬍N, and from Eqs. 共4兲, 共8兲, 共9兲, and 共11兲 for n⫽N.
Numerical results for H̃ Iz ⫽bH Iz /I I , K̃ I ␾ , and g I vs u
⫽ ␳ /b for a⫽b/2 (ã⫽0.5) are shown in Fig. 1. In this calculation, as well as in all others in Secs. III–VI, the magnitude of the reduced magnetic field for a⬍ ␳ ⬍b was less than
10⫺5 except for ␳ very close to a or b, where the numerical
results for the principal-value integrals in Eq. 共10兲 became
less accurate. Results for g Im vs ã are shown in Fig. 2. The
inductance is calculated from
L⫽⌽ Iz /I I ⫽ ␮ 0 b⌽̃ Iz 共 ã 兲 ,
FIG. 1. Reduced magnetic field H̃ Iz ⫽bH Iz /I I , reduced sheetcurrent density K̃ I ␾ , and polynomial g I 共multiplied by 3兲 vs u
⫽ ␳ /b obtained while calculating the inductance for a superconducting ring with ã⫽a/b⫽0.5.
⫹ã)/2 and w⫽(b⫺a)⫽b(1⫺ã).兴 Equation 共20兲 can be obtained from21 L⫽ ␮ 0 R 关 ln(8R/r)⫺2兴, the inductance of a superconducting ring of radius R and wire radius rⰆR, by
replacing r by w/4. 22 The empirical formula
L 2 ⫽ ␮ 0 b 关 ã⫺0.197ã 2 ⫺0.031ã 6 ⫹ 共 1⫹ã 兲 tanh⫺1 ã 兴 ,
共21兲
where ã⫽a/b, fits our numerical results for L within 0.06%,
and a plot of it is indistinguishable from the solid curve in
Fig. 3.
The magnetic moment associated with the circulating current can be calculated from
共18兲
where
N
⌽̃ Iz 共 u 兲 ⫽
兺
m⫽1
␾ m 共 u 兲 g Im ,
共19兲
and is shown in Fig. 3 as a function of ã⫽a/b. Dashed lines
in Fig. 3 show expressions valid in the limits of small and
large ã: For ãⰆ1, the inductance approaches L 0 ⫽2 ␮ 0 a 关or
⌽̃ Iz (ã)⫽2ã], as obtained by Ketchen et al.,3 and for ã
→1, the inductance approaches
L 1 ⫽ ␮ 0 R 关 ln共 8R/w 兲 ⫺ 共 2⫺ln 4 兲兴 ,
8
共20兲
as obtained by Brandt for a superconducting annulus of
mean radius R and width wⰆR. 关Here R⫽(a⫹b)/2⫽b(1
FIG. 2. Coefficients g Im in the polynomial of Eq. 共16兲 vs ã
⫽a/b obtained while calculating the inductance of a superconducting ring.
174514-3
PHYSICAL REVIEW B 68, 174514 共2003兲
ALI A. BABAEI BROJENY AND JOHN R. CLEM
FIG. 3. Reduced inductance of a superconducting ring, L/ ␮ 0 b
vs ã⫽a/b, calculated from Eqs. 共18兲 and 共19兲. Dashed curves show
approximations valid in the limits ã→0 and ã→1.
N
m Iz ⫽I I ␲ b
2
兺
m⫽1
共22兲
f m g Im .
Consider an annular ring that has been cooled into the
superconducting state in the absence of a magnetic field,
such that no magnetic flux is trapped anywhere in the ring.
When a perpendicular magnetic field H a is applied, a circulating current is induced, but the ring remains in the Meissner state, and the magnetic flux up through the hole remains
zero 共there are no flux quanta in the hole兲. The induced
sheet-current density is K Z ␾ ⫽H a K̃ Z ␾ , where the subscript Z
henceforth labels all quantities that are specific to calculations for the zero-flux-quantum state. To evaluate the coefficients g Zm in
g Z共 u 兲 ⫽
兺
m⫽1
g Zm
冉 冊
u⫺ã
N
Ĩ Z ⫽
IV. ZERO-FLUX-QUANTUM STATE
N
FIG. 4. Reduced magnetic field H̃ Zz ⫽H Zz /H a , reduced sheetcurrent density K̃ Z ␾ , and polynomial g Z 共multiplied by 3兲 vs u
⫽ ␳ /b for the zero-flux-quantum state with ã⫽a/b⫽0.5.
兺
m⫽1
i m g Zm ,
共25兲
and is shown in Fig. 6 as a function of ã⫽a/b. Dashed lines
in Fig. 6 show expressions valid in the limits of small and
large ã: For ãⰆ1, the induced current approaches I Z
⫽⫺4H a b/ ␲ 共or Ĩ Z ⫽⫺4/␲ ), as obtained by Ketchen et al.,3
and for ã→1, the induced current approaches I Z
⫽⫺ ␲ R 2 B a /L 1 .
V. FLUX FOCUSING
We now solve for the current and field distribution when a
superconducting annular ring is placed in a perpendicular
m⫺1
1⫺ã
,
共23兲
we use the N equations
N
兺
m⫽1
␣ Znm g Zm ⫽ ␤ Zn ,
共24兲
n⫽1,2, . . . ,N, where ␣ Znm ⫽ ␣ nm and ␤ Zn ⫽⫺1 for n⬍N,
and ␣ ZNm ⫽ ␾ m (ã) and ␤ ZN ⫽⫺ ␲ ã 2 for n⫽N. These equations are obtained from Eqs. 共1兲, 共8兲–共10兲, and 共14兲 and
H z ( ␳ n )⫽0 for n⬍N, and from Eqs. 共6兲, 共8兲, 共9兲, and 共13兲 for
n⫽N.
Numerical results for H̃ Zz ⫽H Zz /H a ,K̃ Z ␾ , and g Z vs u
⫽ ␳ /b for a⫽b/2 (ã⫽0.5) are shown in Fig. 4. Results for
g Zm vs ã are shown in Fig. 5. The magnitude 兩 I Z 兩 of the
induced current is obtained from I Z ⫽H a bĨ Z , where
FIG. 5. Coefficients g Zm in the polynomial of Eq. 共23兲 vs ã
⫽a/b for the zero-flux-quantum state.
174514-4
PHYSICAL REVIEW B 68, 174514 共2003兲
MAGNETIC-FIELD AND CURRENT-DENSITY . . .
FIG. 6. Magnitude of the reduced current, ⫺I Z /bH a , vs ã
⫽a/b for the zero-flux-quantum state calculated from Eq. 共25兲.
Dashed curves show approximations valid in the limits ã→0 and
ã→1.
magnetic field H a subject to the condition that there is no net
current circulating around the ring. We wish to determine
how much magnetic flux is focused into the hole in the
middle of the ring. The sheet-current density in this case is
K F ␾ ⫽H a K̃ F ␾ , where the subscript F henceforth labels all
quantities that are specific to calculations of flux focusing. To
evaluate the coefficients g Fm in
N
g F共 u 兲 ⫽
兺
m⫽1
g Fm
冉 冊
u⫺ã
m⫺1
1⫺ã
,
共26兲
we use the N equations
FIG. 7. Reduced magnetic field H̃ Fz ⫽H Fz /H a , reduced sheetcurrent density K̃ F ␾ , and polynomial g F 共multiplied by 3兲 vs u
⫽ ␳ /b for flux focusing with ã⫽a/b⫽0.5.
Aef f
⌽ Fz 共 a 兲
1
⫽
⫽1⫹
2
Ah
␮ oH a␲ a
␲ ã 2
N
兺
m⫽1
␾ m 共 ã 兲 g Fm ,
共29兲
which is shown in Fig. 9 as a function of 1/ã⫽b/a. Dashed
lines in Fig. 9 show expressions valid in the limits of small
and large ã: For ãⰆ1, A e f f /A h approaches (8/␲ 2 )(b/a) 共or
8/␲ 2 ã), as obtained by Ketchen et al.,3 and for ã→1,
A e f f /A h approaches (R/a) 2 „or 关 (1⫹ã)/2ã 兴 2 …, where R
⫽(a⫹b)/2 is the mean radius of the ring.
The flux-focusing problem also can be solved from a linear superposition of the fields calculated in Secs. III and IV.
From K F ␾ ⫽K I ␾ ⫹K Z ␾ and the condition I F ⫽I I ⫹I Z ⫽0 we
obtain g F (u)⫽⫺Ĩ Z g I (u)⫹g Z (u), g Fm ⫽⫺Ĩ Z g Im ⫹g Zm , and
the result
N
兺
m⫽1
␣ Fnm g Fm ⫽ ␤ Fn ,
共27兲
n⫽1,2, . . . ,N, where ␣ Fnm ⫽ ␣ nm and ␤ Fn ⫽⫺1 for n⬍N,
and ␣ FNm ⫽i m and ␤ FN ⫽0 for n⫽N. These equations are
obtained from Eqs. 共1兲, 共8兲–共10兲, and 共14兲 and H z ( ␳ n )⫽0
for n⬍N, and from Eqs. 共4兲, 共8兲, 共9兲, 共11兲, and I⫽0 for n
⫽N.
Numerical results for H̃ Fz ⫽H Fz /H a ,K̃ F ␾ , and g F vs u
⫽ ␳ /b for a⫽b/2 (ã⫽0.5) are shown in Fig. 7. Results for
g Fm vs ã are shown in Fig. 8. The magnetic flux focused into
the hole is ⌽ Fz (a)⫽ ␮ o H a b 2 ⌽̃ Fz (ã), where
N
⌽̃ Fz 共 ã 兲 ⫽ ␲ ã ⫹
2
兺
m⫽1
␾ m 共 ã 兲 g Fm .
共28兲
The effective area of the hole 共which corresponds to the
effective pickup area of a SQUID made of a circular
washer兲, defined via ⌽ Fz (a)⫽ ␮ 0 H a A e f f , is always larger
than the actual area of the hole, A h ⫽ ␲ a 2 . We find
FIG. 8. Coefficients g Fm in the polynomial of Eq. 共26兲 vs ã
⫽a/b for flux focusing.
174514-5
PHYSICAL REVIEW B 68, 174514 共2003兲
ALI A. BABAEI BROJENY AND JOHN R. CLEM
FIG. 9. Reduced effective area A e f f /A h vs 1/ã⫽b/a for flux
focusing calculated from Eq. 共29兲 or 共30兲. Dashed curves show
approximations valid in the limits ã→0 and ã→1.
Aef f
Ĩ Z ⌽̃ Iz 共 ã 兲
⫽⫺
,
Ah
␲ ã 2
共30兲
which gives numerically the same values as Eq. 共29兲.
VI. GEOMETRICAL BARRIER
We next present an efficient method for calculating the
magnetic-field and current-density distributions and the magnetization of a bulk-pinning-free type-II superconducting
disk subject to a geometrical barrier, which impedes the entry of vortices into the disk. We consider a disk 共radius b and
thickness dⰆb) in the plane z⫽0, centered on the z axis,
initially in the Meissner state. We assume that the London
penetration depth obeys ␭⬍d/2 or, if ␭⬎d/2, that the twodimensional screening length ⌳⫽2␭ 2 /d obeys ⌳Ⰶb. When
a perpendicular magnetic field H a is applied, a sheet-current
density19
K ␾ 共 ␳ 兲 ⫽⫺
4H a
␲
␳
冑b
2
共31兲
⫺␳2
is induced. The resulting magnetic field in the plane z⫽0,
determined from Eq. 共1兲, is H z ( ␳ )⫽0 for ␳ ⬍b and23
再 冋
H z 共 ␳ 兲 ⫽H a 1⫹
2
␲
1
冑共 ␳ /b 兲
2
⫺1
⫺sin⫺1
冉 冊册冎
b
␳
共32兲
for ␳ ⬎b.
A geometrical barrier prevents vortices from entering the
film until the magnetic field at the edge 共accounting for demagnetizing effects兲 reaches the value H s . We expect that
H s ⫽H c1 , the lower critical field, if there is no BeanLivingston barrier, or H s ⬇H c , the bulk thermodynamic
critical field, if the edge is without defects and thermal acti-
vation is negligible. An equivalent criterion is that the magnetic flux begins to penetrate when the magnitude of the
sheet-current density at the edge reaches the value K s
⫽2H s . 24 To estimate H z or K ␾ at the edge of the film, we
note that the approximations that led to Eqs. 共31兲 and 共32兲
break down and that the inverse-square-root divergences in
these equations are cut off when ␳ is within ␦ of the edge,
where ␦ is the larger of d/2 or ⌳. Accordingly, we approximate H z at the edge of the film by replacing ␳ in the squareroot denominator of Eq. 共32兲 by b⫹ ␦ and using ␦ Ⰶb, such
that H z (edge兲 ⬇(H a / ␲ ) 冑2b/ ␦ . Similarly, we approximate
K ␾ at the edge of the film by replacing ␳ in the square-root
denominator of Eq. 共31兲 by b⫺ ␦ and using ␦ Ⰶb, such that
K ␾ (edge)⬇⫺(2H a / ␲ ) 冑2b/ ␦ . Whichever criterion is used
关 H z 共edge兲 ⫽H s or 兩 K ␾ (edge) 兩 ⫽K s ⫽2H s ], we estimate that
the geometrical barrier is overcome when the applied field is
equal to H 0 ⫽ ␲ H s 冑␦ /2b. 共In this paper we have chosen ˜␦
⫽ ␦ /b⫽0.01, such that H 0 ⫽0.222H s . See Fig. 13.兲
When H a ⬎H 0 such that H z (edge)⬎H s , vortices nucleate at the edge of the disk and move rapidly towards the
center of the disk under the influence of the Lorentz force per
unit length, f⫽JH ⫻ ␾0 , where JH ⫽“⫻Hre v , ␾0 is a vector
of magnitude ␾ 0 ⫽h/2e along the vortex axis, and Hre v is the
thermodynamic magnetic field in equilibrium with the
magnetic-flux density B inside the superconductor. As more
vortices enter, the return field outside the disk generated by
the vortices inside the disk gradually reduces the value of the
field at the edge to H s , thereby halting further vortex nucleation. If bulk pinning is negligible, the case considered in
this paper, the vortices adjust their positions such that the
magnetic-flux density 共averaged over the intervortex distance兲 in the plane of the disk B z ( ␳ ) has its maximum value
at the center, decreases monotonically to zero at ␳ ⫽a, and
remains zero for a⬍ ␳ ⬍b. The corresponding sheet-current
density K H ␾ ⫽J H ␾ d is zero for ␳ ⭐a, such that the Lorentz
force on any vortex vanishes and no further motion occurs.
Screening supercurrents still flow, however, in the vortexfree region a⬍ ␳ ⬍b.
To good approximation when dⰆb, the resulting
magnetic-field and supercurrent distributions are the same as
those generated by a thin superconducting annular ring (a
⬍ ␳ ⬍b) in a perpendicular applied field H a , when the solutions are subject to the constraint that the sheet-current density K ␾ is zero at ␳ ⫽a. The Biot-Savart law 关Eq. 共1兲 and its
extension to 兩 z 兩 ⬎0] guarantees that the current density JB
⫽“⫻B/ ␮ 0 is zero everywhere except within the ring a
⬍ ␳ ⬍b; thus K B ␾ ⫽J B ␾ d is zero for ␳ ⭐a. Because JH and
JB in thin films are dominated by the curvature of Hre v and
B/ ␮ 0 , rather than by the gradients “H re v and “B/ ␮ 0 , 11,25 it
can be shown that the difference between K H ␾ and K B ␾ is of
the order of (d/b)H a , decreases for B⬎2B c1 as Hre v approaches B/ ␮ 0 , and is negligible for the thin films considered in this paper (d/b⫽0.01). Nevertheless, our simplified
approach would be incapable of calculating details in the
structure that has been observed in the magnetic flux-density
distribution at the vortex-lattice melting transition.26 To treat
such a problem would require a more refined approach such
as that in Refs. 14 and 15, which calculates the local JB
174514-6
PHYSICAL REVIEW B 68, 174514 共2003兲
MAGNETIC-FIELD AND CURRENT-DENSITY . . .
FIG. 10. Reduced magnetic field H̃ Gz ⫽H Gz /H a , reduced sheetcurrent density K̃ G ␾ , and polynomial g G vs u⫽ ␳ /b for a pin-free
disk of radius b with a geometrical barrier and a flux dome of
reduced radius ã⫽a/b⫽0.5.
currents flowing at the vortex solid-liquid interface and distinguishes between Hre v and B/ ␮ 0 .
The magnetic-field and supercurrent distributions for the
case of a thin pin-free disk subject to a geometrical barrier
therefore can be calculated efficiently by using an approach
similar to that used in Secs. II–V. When a Lorentz-force-free
dome of magnetic flux occupies the region ␳ ⬍a, the sheetcurrent density in the region a⬍ ␳ ⬍b is K G ␾ ⫽H a K̃ G ␾ ,
where the subscript G henceforth labels all quantities that are
specific to the geometrical-barrier problem. To evaluate the
coefficients g Gm in
N
g G共 u 兲 ⫽
兺
m⫽1
g Gm
冉 冊
u⫺ã
FIG. 11. Coefficients g Gm in the polynomial of Eq. 共33兲 vs ã for
a pin-free disk of radius b with a geometrical barrier and a flux
dome of reduced radius ã⫽a/b. We require g G1 ⫽0.
⌽ Gz 共 a 兲 ⫽ ␮ o H a b 2 ⌽̃ Gz 共 ã 兲 ,
共35兲
where
N
⌽̃ Gz 共 ã 兲 ⫽ ␲ ã ⫹
2
兺
m⫽1
␾ m 共 ã 兲 g Gm ,
共36兲
and the average magnetic-flux density in the disk is B a v
⫽⌽ Gz (a)/ ␲ b 2 . Figure 12 shows how B a v /B a and
H Gz (0)/H a , where H Gz (0) is the magnetic field at the center of the disk, depend upon ã.
m⫺1
1⫺ã
,
共33兲
we use the N equations
N
兺
m⫽1
␣ Fnm g Gm ⫽ ␤ Gn ,
共34兲
n⫽1,2, . . . ,N, where ␣ Gnm ⫽ ␣ nm and ␤ Gn ⫽⫺1 for n
⬍N, and ␣ GNm ⫽ ␦ 1m and ␤ GN ⫽0 for n⫽N. These equations are obtained from Eqs. 共1兲, 共8兲–共10兲, and 共14兲 and
H z ( ␳ n )⫽0 for n⬍N, and from Eqs. 共8兲, 共9兲, and K̃ G ␾ (ã)
⫽0 for n⫽N.
Numerical results for H̃ Gz ⫽H Gz /H a ,K̃ G ␾ , and g G vs u
⫽ ␳ /b for a⫽b/2 (ã⫽0.5) are shown in Fig. 10. In these
calculations, we have made no distinction between Hre v and
B/ ␮ 0 , which corresponds to assuming that B⬇ ␮ 0 H. However, in cases for which B differs significantly from ␮ 0 H,
our plots of H Gz /H a 共such as in Fig. 10兲 would correspond
most closely to plots of the reduced flux density B Gz /B a .
Results for g Gm vs ã are shown in Fig. 11. The magnetic flux
contained within ␳ ⬍a can be obtained from
FIG. 12. Reduced average flux density B a v /B a 共solid兲 and reduced flux density at the center H Gz (0)/H a 共dashed兲 of a pin-free
disk of radius b with a geometrical barrier and a flux dome of
reduced radius ã⫽a/b.
174514-7
PHYSICAL REVIEW B 68, 174514 共2003兲
ALI A. BABAEI BROJENY AND JOHN R. CLEM
FIG. 13. Calculated hysteresis in the reduced magnetization
M Gz / ␹ 0 H s vs reduced applied field H a /H s for a pin-free disk with
a geometrical barrier 共solid兲. The dashed curve shows a reversible
minor hysteresis ‘‘loop’’ occurring when the applied field is reduced
after the applied field H a has reached H 1 along the field-increasing
magnetization curve. As H a decreases, the flux dome expands, but
the flux contained within the dome remains constant.
We next calculate the average magnetization, i.e., the
magnetic moment divided by the volume of the disk, M Gz
⫽m Gz / ␲ b 2 d, where m Gz is calculated from Eq. 共5兲. The
initial magnetization of the disk in the Meissner state (0
⭐H a ⭐H 0 , see Fig. 13兲, calculated from Eq. 共31兲, is27 M Gz
⫽⫺␹0Ha , where ␹ 0 ⫽8b/3␲ d; i.e., the external magnetic
susceptibility28 in this case is ␹ ⫽⫺ ␹ 0 . Whenever there is a
dome of magnetic flux within the region ␳ ⬍a, the average
magnetization, obtained from Eqs. 共5兲, 共8兲, 共12兲, and 共33兲,
may be calculated from
f 2 ⬇1,
H a ⬇H 0 冑2/ ␲ 冑1⫺ã,
and
M Gz↑ / ␹ 0 H s
⬇⫺(3 ␲ 2 /8)˜␦ H s /H a , where ˜␦ ⫽ ␦ /bⰆ1.
For H 0 ⬍H a ⬍H irr along the field-decreasing magnetization curve at the critical exit condition, we assume that
the radius a of the vortex-filled region has reached within
␦ of the radius b of the disk; i.e., ã⫽1⫺˜␦ . Using Eq. 共37兲
with g Gm ⬇⫺ ␲˜␦ ␦ m2 and f 2 ⬇1, we obtain M Gz↓ / ␹ 0 H s
⬇⫺(3 ␲ 2 /8)˜␦ H a /H s . See Fig. 13.
The field-increasing and field-decreasing magnetization
curves in Fig. 13 meet at H a ⫽H irr , the irreversibility field.
The criteria we used for the critical entry and exit conditions
lead to the result that H irr ⬇H s , where the magnetization is
given by M Gzirr / ␹ 0 H s ⬇⫺(3 ␲ 2 /8)˜␦ . However, the above
expressions for H irr , M Gzirr , and M Gz↓ are the least reliable
results of our paper, because all these quantities are very
sensitive to the precise conditions for entry and exit at the
edge of the disk, including such details as the shape of the
edge.9–12,14,15 The magnetic moment responsible for the
magnetization M Gzirr and M Gz↓ is produced by currents that
flow only within a very narrow band around the disk’s edge,
where a theory more accurate than ours is needed.
The minor hysteresis loop, shown as the dashed curve in
Fig. 13, can be calculated as follows. We start at a point on
the field-increasing magnetization curve where the flux dome
has radius a 1 . The magnetic flux contained within the dome
⌽ Gz (a 1 ), the magnetization M Gz1 , and the corresponding
applied field H 1 are obtained from Eqs. 共35兲, 共37兲, and 共38兲,
where ã, f m , and g Gm are all evaluated at ã⫽ã 1 ⫽a 1 /b. As
the applied field H a is reduced from its starting value H 1 , the
radius a of the flux dome expands, but the magnetic flux
within the dome remains constant. For each value of ã
⬎ã 1 , we recalculate f m , g Gm , and ⌽̃ Gz (ã). We then use Eq.
共35兲 to obtain the corresponding value of the applied field,
H a ⫽H 1 ⌽̃ Gz 共 ã 1 兲 /⌽̃ Gz 共 ã 兲 ,
and Eq. 共37兲 to obtain the corresponding value of the magnetization.
N
M Gz ⫽
3␲
␹ H
f g ,
8 0 a m⫽1 m Gm
兺
共37兲
where f m and g Gm depend implicitly upon ã.
For H 0 ⬍H a ⬍H irr along the field-increasing magnetization curve at the critical entry condition 共see Fig. 13兲, H a and
ã are related via
H a ⫽⫺H 0 冑1⫺ã 2
冒兺
N
m⫽1
g Gm .
共38兲
This equation follows from the condition that 兩 K G ␾ (edge) 兩
⫽2H s , where K G ␾ 共edge兲 is obtained by evaluating Eq. 共8兲
at u⫽1 but replacing 冑1⫺u 2 in the denominator by 冑2 ␦ /b,
as in the evaluation of H 0 . When ã⫽0, we see by comparing Eqs. 共8兲 and 共31兲 that g G (u)⫽⫺u 3 , such that g Gm
⫽⫺ ␦ m4 共see also Fig. 11兲, f 4 ⫽8/3␲ , and M Gz↑ ⫽⫺ ␹ 0 H 0 at
H a ⫽H 0 . In the limit as ã→1, K̃ G ␾ ⬇⫺2 冑u⫺a/ 冑1⫺u,
such that g G (u)⬇⫺ ␲ (u⫺ã), g Gm ⬇⫺ ␲ (1⫺ã) ␦ m2 ,
共39兲
VII. DISCUSSION
In this paper, we have presented an efficient method for
the calculation of magnetic-field and current-density profiles
for thin-film rings in the Meissner state and for bulk-pinningfree disks subject to a geometrical barrier. In each case, the
sheet-current density was expressed in the form of Eq. 共8兲,
where the quantity g(u) in the numerator is a polynomial of
degree N⫺1.
For all the calculations presented in the figures, for which
we assumed N⫽5, we found that the magnitude of g 5 was
less than 0.0012 in each case 共see Figs. 2, 5, 8, and 11兲 and
that its contribution to g(u) was less than 1.1%. Using N
⫽6 yields values of g 6 whose magnitudes are much smaller
than those of g 5 , and the values of the calculated physical
quantities are altered only in the sixth decimal place.
Moreover, we offer the conjecture that the problems we
solved numerically in Secs. III–VI might be solved analytically with functions g I ,g Z ,g F , and g G that are third-order
polynomials in u; i.e., the sums in Eqs. 共9兲, 共16兲, 共23兲, 共26兲,
174514-8
PHYSICAL REVIEW B 68, 174514 共2003兲
MAGNETIC-FIELD AND CURRENT-DENSITY . . .
and 共33兲 might simply terminate with N⫽4. As evidence in
support of this conjecture, we note that our calculations for
ã⫽a/b⫽0.1 and 0.5 with N⫽4, 5, 6, and 7 yielded values
of L/ ␮ 0 b 关Eqs. 共18兲 and 共19兲兴 that differed only in the fifth
decimal place. Similarly, values of I Z /H a b 关Eq. 共25兲兴,
A e f f /A h 关Eq. 共29兲兴, and M Gz↑ / ␹ 0 H s 关Eqs. 共37兲 and 共38兲兴
calculated for ã⫽a/b⫽0.1 and 0.5 with N⫽4, 5, 6, and 7
differed at most only in the fourth significant figure. It is
possible that the values we obtained for g 5 ,g 6 , and g 7 in
Secs. III–VI were nonzero only because of small numerical
errors introduced because we performed the integrals in Eqs.
共10兲–共13兲 numerically rather than analytically.
Although in this paper we have considered only bulkpinning-free thin-film rings and disks, it should be possible
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ACKNOWLEDGMENTS
We thank J. Clarke and V. G. Kogan for stimulating discussions. This manuscript has been authored in part by Iowa
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