Response of thin-film SQUIDs to applied fields and vortex fields:... John R. Clem

Response of thin-film SQUIDs to applied fields and vortex fields:... John R. Clem
Response of thin-film SQUIDs to applied fields and vortex fields: Linear SQUIDs
John R. Clem
Ames Laboratory - DOE and Department of Physics and Astronomy, Iowa State University, Ames Iowa 50011
Ernst Helmut Brandt
arXiv:cond-mat/0507524 v2 25 Sep 2005
Max-Planck-Institut für Metallforschung, D-70506 Stuttgart, Germany
(Dated: September 27, 2005)
In this paper we analyze the properties of a dc SQUID when the London penetration depth λ
is larger than the superconducting film thickness d. We present equations that govern the static
behavior for arbitrary values of Λ = λ2 /d relative to the linear dimensions of the SQUID. The
SQUID’s critical current Ic depends upon the effective flux Φ, the magnetic flux through a contour
surrounding the central hole plus a term proportional to the line integral of the current density
around this contour. While it is well known that the SQUID inductance depends upon Λ, we show
here that the focusing of magnetic flux from applied fields and vortex-generated fields into the central
hole of the SQUID also depends upon Λ. We apply this formalism to the simplest case of a linear
SQUID of width 2w, consisting of a coplanar pair of long superconducting strips of separation 2a,
connected by two small Josephson junctions to a superconducting current-input lead at one end and
by a superconducting lead at the other end. The central region of this SQUID shares many properties
with a superconducting coplanar stripline. We calculate magnetic-field and current-density profiles,
the inductance (including both geometric and kinetic inductances), magnetic moments, and the
effective area as a function of Λ/w and a/w.
PACS numbers: 74.78.-w
I.
INTRODUCTION
The research in this paper has been motivated by
several important recent developments in superconductivity: (a) the fabrication of thin-film SQUIDs (superconducting quantum interference devices) made of
high-Tc superconductors,1 (b) the study of noise generated by vortices in active and passive superconducting
devices,1,2,3,4,5,6,7,8,9,10 and (c) line-width reduction in
superconducting devices to eliminate noise due to vortices trapped during cooldown in the earth’s magnetic
field.11,12,13,14,15,16
The fact that the London penetration depth λ increases as T increases and diverges at Tc is an important
consideration for high-Tc SQUIDs operated at liquidnitrogen temperature. When λ is larger than the film
thickness d, the physical length that enters the equations
governing the spatial variation of currents and fields is
the Pearl length17 Λ = λ2 /d. Accordingly, the equations
governing the behavior of active or passive thin-film superconducting devices depend upon the ratio of Λ to the
linear dimensions of the device. In particular, the equations governing SQUIDs involve not just the magnetic
flux up through a contour within the SQUID, but the effective flux Φ, which is the sum of the magnetic flux and
a term proportional to the line integral of the current
density around the same contour. While the effective
flux Φ is similar to London’s fluxoid,18 which is quantized in multiples of the superconducting flux quantum
φ0 = h/2e, we show in the next section that Φ is not
quantized. We also give in Sec. II the basic equations,
valid for any value of Λ, that govern the behavior of a dc
SQUID.
A vortex trapped in the body of the SQUID during
cooldown through the superconducting transition temperature Tc in an ambient magnetic field generates a magnetic field and a screening current that together make a
sizable vortex-position-dependent contribution Φv to the
effective flux Φ. If such a vortex remains fixed in position and the temperature remains constant, this simply produces a harmless bias in Φ. On the other hand,
both vortex motion due to thermal agitation and temperature fluctuations generate corresponding fluctuations in
Φv and noise in the SQUID output. In Sec. II we show
that to calculate Φv , it is not necessary to calculate the
spatial dependence of the vortex-generated fields and currents. Instead, one may determine Φv with the help of
the sheet-current distribution of a circulating current in
the absence of the vortex.
In Sec. III we apply the basic equations of Sec. II
to calculate the properties of a model linear SQUID,
which has the basic topology of a SQUID but is greatly
stretched along one axis, such that the central portion
resembles a coplanar stripline. The advantage of using
such a model is that simple analytical results can be
derived that closely approximate the exact numerically
calculated quantities in the appropriate limits. In addition to calculating the field and current distributions
for several values of Λ, we calculate the total inductance,
geometric inductance, kinetic inductance, and magnetic
moment when the SQUID carries a circulating current.
We calculate the field and current distributions, magnetic
moment, and effective area Aeff = Φf /Ba when a perpendicular magnetic induction Ba is applied and the effective
flux Φf is focused into the SQUID. Finally, we calculate
the field and current distributions and the magnetic moment for the zero-fluxoid state when the junctions are
2
short-circuited and the sample remains in the state with
Φ = 0 when a perpendicular magnetic induction Ba is
applied.
In Sec. IV, we present a brief summary of our results.
II.
I
BASIC EQUATIONS
y
Our purpose in this section is to derive general equations that govern the behavior of a dc SQUID consisting
of thin superconducting films of thickness d less than the
weak-field London penetration depth λ, such that the
fields and currents are governed by the two-dimensional
screening length or Pearl length17 Λ = λ2 /d. We calculate both the current I = I1 + I2 through the SQUID [see
Fig. 1] and the circulating current19 Id = (I2 − I1 )/2 and
describe how to calculate the critical current Ic of the
SQUID for arbitrary values of the SQUID’s inductance
L. For this case, the contributions from line integrals of
the current density to the effective flux in the hole cannot
be neglected, and the kinetic inductance makes a significant contribution to L. When a perpendicular magnetic
induction Ba is applied, we calculate how much magnetic
flux is focused into the SQUID’s hole; this flux also can
be expressed in terms of the effective area20 of the hole.
We also show how to calculate how much magnetic flux
generated by a vortex in the main body of the SQUID is
focused into the hole.
Consider a dc SQUID in the xy plane, as sketched
in Fig. 1. We suppose that the SQUID is symmetric
about the y axis, which lies along the centerline. The
maximum Josephson critical current is I0 for each of the
Josephson junctions, shown as small black squares. The
currents up through the left and right sides of the SQUID
can be written as I1 = I/2 − Id and I2 = I/2 + Id .
When the magnitude of I, the total current through the
SQUID, is less than the critical current Ic , the equations
that determine I = I1 + I2 and the circulating current
Id = (I2 − I1 )/2 can be derived using a method similar
to that used in Ref. 21. We begin by writing the local
current density j in the superconductors (i.e., the main
body of the SQUID and the counterelectrode) as18
j = −(1/µ0 λ2 )[A + (φ0 /2π)∇γ],
(1)
where A is the vector potential, φ0 = h/2e is the superconducting flux quantum, and γ is the phase of the order
parameter. The quantity inside the brackets, which is
gauge-invariant, can be thought of as the superfluid velocity expressed in units of vector potential. From the
point of view of the Ginzburg-Landau theory, implicit in
the use of this London-equation approach is the assumption that the applied fields and currents are so low that
the magnitude of the order parameter is not significantly
reduced from its equilibrium value in the absence of fields
and currents.
To obtain the SQUID equations, we integrate the vector potential around a contour C that passes in a counterclockwise direction through both junctions, the main
I1
I2
x
C
b
a
I
FIG. 1: I enters the main body of the SQUID from the counterelectrode below through two Josephson junctions (small
black squares, labeled a and b) and divides into the currents
I1 = I/2 − Id and I2 = I/2 + Id as shown.
body of the SQUID, and the counterelectrode as shown in
Fig. 1 and write the result in two ways. Since B = ∇×A,
this integral yields, on the one hand, the magnetic flux
in the z direction
Z
Bz (x, y)dS,
(2)
S
where S is the area surrounded by the contour C and
Bz (x, y) is the z component of the net magnetic induction
in the plane of the SQUID produced by the sum of a
perpendicular applied field Ba and the self-field Bsz (x, y)
generated via the Biot-Savart law by the supercurrent
density j(x, y). On the other hand, for those portions of
the contour lying in the superconductors, we eliminate
the line integrals of A in favor of line integrals of j using
Eq. (1). We then express the line integrals of ∇γ in
terms of the values of γ at the junctions. Equating the
two expressions for the line integral of A, we find that
the effective flux Φ in the z direction through the SQUID
is given by
Φ = (φ0 /2π)(φ1 − φ2 ),
(3)
where
Φ=
Z
S
Bz (x, y)dS + µ0 λ2
Z
C
j · dl,
(4)
with the integration contour C now passing through
both superconductors but excluding the junction barriers. These equations are equivalent to Eq. (8.67) in
3
Ref. 21. The gauge-invariant phase differences across the
junctions b and a are, respectively,22
φ1 = γbc − γbs − (2π/φ0 )
Z
bs
A · dl,
Zbcas
φ2 = γac − γas − (2π/φ0 )
A · dl,
(5)
(6)
ac
where bc labels a point on the counterelectrode side of
the junction b and bs labels the point directly across
the insulator in the SQUID washer, and ac and as label corresponding points for junction a. According to
the Josephson equations,22 the junction supercurrents
are I1 = I0 sin φ1 and I2 = I0 sin φ2 . In the above
derivation we have assumed that the linear dimensions of
the Josephson junctions are much less than the Josephson penetration depth λJ ,22 and that the applied fields
are sufficiently small that the Josephson current densities and gauge-invariant phase differences are very nearly
constant across the junction areas.
The magnetic moment m = mẑ generated by the currents in the SQUID is23
Z
1
m=
r × jd3 r.
(7)
2
It can be shown with the help of the London fluxoid
quantization condition,18
Z
Z
2
Bz (x, y)dS + µ0 λ
j · dl = nφ0 ,
(8)
S′
C′
where n is an integer and C ′ is a closed contour that surrounds an area S ′ within the body of the SQUID, that
if there are no vortices present (i.e., when n = 0), the
expression for Φ in Eq. (4) is independent of the choice
of contour C. Any convenient path can be chosen for
C, provided only that the path remains in the superconducting material in the body of the SQUID and the
counterelectrode. On the other hand, when there are
vortices in the main body of the SQUID, the quantity Φ
increases by φ0 each time the contour C is moved from
a path inside the vortex axis to one enclosing the vortex
axis. Thus, without specifying the precise contour C, Φ is
determined only modulo φ0 . However, this is of no physical consequence, because the gauge-invariant phases φ1
and φ2 , which also enter Eq. (3), are also determined only
modulo 2π. The final equations determining the currents
I and Id are independent of the choice of contour C and
remain valid even when vortices are present in the main
body of the SQUID.
When the thickness d of the SQUID is much larger
than λ, the contours C and C ′ can be chosen to be at the
midpoint of the thickness, where j is exponentially small,
such that the line integrals of j can be neglected. The
resulting equations are then the familiar ones found in
many reference books, such as Refs. 21,24,25,26,27,28,29.
However, we are interested here in the case for which
d < λ, such that the fields and currents are governed by
the two-dimensional screening length or Pearl length17
Λ = λ2 /d. The term in Eq. (4) involving j then must
be carefully accounted for. For this case, j is very nearly
constant over the thickness and it is more convenient to
deal with the sheet-current density J(x, y) = jd, such
that Eqs. (4) and (8) take the form30
Z
Z
J · dl
(9)
Bz (x, y)dS + µ0 Λ
Φ=
C
S
and
Z
Bz (x, y)dS + µ0 Λ
Z
C′
S′
J · dl = nφ0 .
(10)
For the general case when the SQUID is subject to a
perpendicular applied magnetic induction Ba , carries a
current I unequally divided between the two arms, I1 =
I/2−Id and I2 = I/2+Id, where the circulating current19
is Id = (I2 − I1 )/2, and contains a vortex at the position
rv in the body of the SQUID, the effective flux Φ in the z
direction can be written as the sum of four independent
contributions:
Φ = Φ I + Φd + Φf + Φv ,
(11)
The first term on the right-hand side of Eq. (11) is that
which would be produced by equal currents I/2 in the y
direction on the left and right sides of the SQUID shown
in Fig. 1:
Z
Z
JI · dl,
(12)
BI (x, y)dS + µ0 Λ
ΦI =
S
C
where BI (x, y) is the z component of the self-field generated via the Biot-Savart law by the sheet-current density
JI (x, y), subject to the condition that the same current
I/2 flows through the two contacts a and b. For a symmetric SQUID, JI (x, y), the y component of JI (x, y), is
then an even function of x, and JI (x, y) and BI (x, y) are
odd functions of x. As a result, both terms on the righthand side of Eq. (12) vanish by symmetry, and ΦI = 0.
Since ∇ · JI = 0 except at the contacts a and b, we may
write JI = −(I/2)∇ × GI , where GI = ẑGI , such that
JI (x, y) = (I/2)ẑ×∇GI (x, y). The contours of the scalar
stream function GI (x, y) = const correspond to streamlines of JI (x, y), and we may choose GI = 0 for points
ri = (xi , yi ) all along the inner edges of the superconductors and GI = 1 for points ro = (xo , yo ) all along the
outer right edges and GI = −1 for points ro = (xo , yo )
all along the outer left edges.
The second term on the right-hand side of Eq. (11) is
due to the circulating current19 Id = (I2 − I1 )/2 in the
counterclockwise direction when unequal currents flow in
the two sides of the SQUID shown in Fig. 1:
Z
Z
Jd · dl,
(13)
Bd (x, y)dS + µ0 Λ
Φd =
S
C
where Bd (x, y) is the z component of the self-field generated via the Biot-Savart law by the circulating sheetcurrent density Jd (x, y) when a current Id flows through
4
contact a from the counterelectrode into the body of
the SQUID, passes around the central hole, and flows
through contact b back into the counterelectrode. The
magnetic moment md generated by the circulating current is proportional to Id , as can be seen from Eq. (7).
Since ∇ · Jd = 0 except at the contacts a and b, we may
write Jd = −Id ∇ × Gd , where Gd = ẑGd , such that
Jd (x, y) = Id ẑ × ∇Gd (x, y). The contours of the scalar
stream function Gd (x, y) = const correspond to streamlines of Jd (x, y), and we may choose Gd = 0 for points
ri = (xi , yi ) all along the inner edges of the superconductors and Gd = 1 for points ro = (xo , yo ) all along the
outer edges. Once a numerical result for Φd is found, the
result can be used to determine the inductance L of the
SQUID via L = Φd /Id , as was done for a circular ring in
Ref. 31. The resulting expression for L is the sum of the
geometric and kinetic inductances.
The third term on the right-hand side is a flux-focusing
term due to the applied field:
Z
Z
Jf · dl,
(14)
Bf (x, y)dS + µ0 Λ
Φf =
S
C
where Bf (x, y) is the z component of the net magnetic induction in the plane of the SQUID produced by the sum
of a perpendicular applied field Ba and the z component
of the self-field Bsf (x, y) generated via the Biot-Savart
law by the sheet-current density Jf (x, y) induced in response to Ba , subject to the condition that no current
flows through the junctions a and b. In other words, the
desired fields are those that would appear in response
to Ba if the junctions a and b were open-circuited. Since
∇·Jf = 0, we may write Jf = −∇×Gf , where Gf = ẑGf ,
such that Jf (x, y) = ẑ × ∇Gf (x, y). The contours of the
scalar stream function Gf (x, y) = const correspond to
streamlines of Jf (x, y), and we may chose Gf = 0 for all
points (x, y) along the inner and outer edges of the superconductor. Once a numerical result for Φf is found,
the result can be used to determine the effective area20
of the SQUID’s central hole, Aeff = Φf /Ba , as was done
for a circular ring in Ref. 31.
To prove that the effective area is also given by Aeff =
md /Id ,32 we Rconsider the electromagnetic energy cross
term Efd = (Bf · Bd /µ0 + µ0 λ2 jf · jd )d3 r, where the
integral extends over all space. Here, Bf (r) = Ba (r) +
Bsf (r) = ∇ × Af (r), where ja (r) = ∇ × Ba (r)/µ0 is
the current density in the distant coil that produces a
nearly uniform field Ba in the vicinity of the SQUID,
jf = ∇ × Bsf /µ0 is the induced current density in the
SQUID, and Bsf is the corresponding self-field under the
conditions of flux focusing, i.e., when jf = 0 through the
junctions. Also, Bd = ∇ × Ad is the dipole-like field
distribution generated by the circulating current Id with
density jd in the SQUID; at large distances from the
SQUID23 Ad = µ0 md × r/4πr3 . We evaluate Efd in two
ways, making use of the vector identities ∇ · (A × B) =
B · (∇ × A) − A · (∇ × B) and ∇ · (γj) = γ∇ · j + ∇γ · j,
and applying the divergence theorem, first with A = Ad ,
B = Bf , γ = γd , and j = jf , from which we obtain
Efd = Ba md , and then with A = Af , B = Bd , γ = γf ,
and j = jd , from which we obtain Efd = Φf Id with the
help of Eq. (3). Since Φf = Ba Aeff , the effective area
obeys Aeff = md /Id .
The fourth term on the right-hand side of Eq. (11) is
due to a vortex at position rv = x̂xv + ŷyv in the body
of the SQUID:
Z
Z
Jv · dl,
(15)
Bv (x, y)dS + µ0 Λ
Φv (rv ) =
C
S
where Bv (x, y) is the z component of the self-field generated by the vortex’s sheet-current density Jv (x, y) via
the Biot-Savart law when no current flows through the
junctions a and b. The desired fields are those that would
appear in response to the vortex if the junctions a and
b were open-circuited. Since ∇ · Jv = 0, it is possible
to express Jv (x, y) in terms of a scalar stream function,
as we did for Jf (x, y) and Jd (x, y). However, as shown
below, it is possible to use energy arguments to express
Φv (x, y) in terms of the stream function Gd (x, y).33
To obtain Φv (r) when a vortex is at the position
r = x̂x + ŷy, imagine disconnecting the counterelectrode
in Fig. 1 and attaching leads from a power supply to
the contacts a and b. The power supply provides a constant current Id in the counterclockwise direction, and
the sheet-current distribution through the body of the
SQUID is given by Jd (x, y) = Id ẑ × ∇Gd (x, y), as discussed above. We also imagine attaching leads from a
high-impedance voltmeter to the contacts a and b. If
the vortex moves, the effective flux Φv changes with
time, and the voltage read by the voltmeter will be34
Vab = dΦv /dt. The power delivered by the power supply
can be expressed in terms of the Lorentz force on the
vortex, Jd × ẑφ0 = Id φ0 ∇Gd ; i.e., the rate at which work
is done on the moving vortex is Id φ0 ∇Gd · dr/dt. Equating this to the power P = Id dΦv /dt = Id ∇Φv · dr/dt
delivered by the power supply to maintain constant current, we obtain the equation ∇Φv (r) = φ0 ∇Gd (r). Thus
Φv (r) = φ0 Gd (r)+const, where the constant can have
one of two possible values depending upon whether the
integration contour C is chosen to run inside or outside
the vortex axis at r = x̂x+ ŷy. Choosing C to run around
the outer boundary of the SQUID, we obtain
Φv (r) = φ0 Gd (r).
(16)
Since Gd (ro ) = 1 for points r = ro on the outer edges
of the SQUID and Gd (ri ) = 0 for points r = ri on the
inner edges (at the perimeter of the central hole or along
the edges of the slit), we have Φv (ro ) = φ0 and Φv (ri ) =
0. The derivation of Eq. (16 ) implicitly assumes that
the vortex-core radius is much smaller than the linear
dimensions of the SQUID.
We now return to the problem of how to find the currents I and Id in the SQUID, as well as the critical
current Ic . As discussed above, we have ΦI = 0 for a
symmetric SQUID. For simplicity, we assume first that
there are no vortices in the body of the SQUID, such that
5
y
1
0.8
0.6
Ic 2I0
-w
-a
a
w
x
0.4
0.2
0
0.1
0.2
0.3
Ff Φ0
0.4
0.5
FIG. 2: Ic /2I0 vs Φf /φ0 , calculated from Eqs. (17) and (18),
for πLI0 /φ0 = 0, 1, 2, 3, 4, and 5 (bottom to top).
Φv = 0 and Φ = Φf + Φd in Eq. (3), where Φd = LId .
From the sum and the difference of I1 and I2 we obtain
πΦ
πLId f
I = 2I0 cos
sin φ̄,
(17)
+
φ0
φ0
πΦ
πLId f
Id = −I0 sin
+
cos φ̄,
(18)
φ0
φ0
where φ̄ = (φ1 + φ2 )/2 is determined experimentally by
how much current is applied to the SQUID. When φ̄ = 0,
the current I is zero. As φ̄ increases, the magnitude of I
increases and reaches its maximum value Ic at a value
of φ̄ that must be determined by numerically solving
Eqs. (17) and (18). A simple solution is obtained for
arbitrary Φf only in the limit πLI0 /φ0 → 0, for which
I = Ic = 2I0 | cos(πΦf /φ0 )| and Id = 0 at the critical current. For values of πLI0 /φ0 of order unity, as is the case
for practical SQUIDs, one may obtain Ic for any value of
Φf by solving Eq. (18) self-consistently for Id for a series
of values of φ̄ and by substituting the results into Eq. (17)
to determine which value of φ̄ maximizes I. Equations
(17) and (18) have been solved numerically by de Bruyn
Ouboter and de Waele,24 (some of their results are also
shown by Orlando and Delin21 ), who showed that at Ic
Ic (Φf )
Id (Φf )
I1 (Φf )
I2 (Φf )
=
=
=
=
Ic (Φf + nφ0 ) = Ic (−Φf ),
Id (Φf + nφ0 ) = −Id (−Φf ),
I1 (Φf + nφ0 ) = I2 (−Φf ),
I2 (Φf + nφ0 ) = I1 (−Φf ),
(19)
(20)
(21)
(22)
where n is an integer. Hence all the physics is revealed
by displaying Ic (Φf ) over the interval 0 ≤ Φf ≤ φ0 /2, as
shown in Fig. 2.
When a vortex is present, Eqs. (17) and (18) still hold,
except that Φf in these equations is replaced by the sum
Φf + Φv . Thermally agitated motion of vortices in the
body of the SQUID can produce flux noise via the term
Φv (rv ) and the time dependence of the vortex position
FIG. 3: Sketch of central portion of the long SQUID considered in Sec. III.
rv . From Eq. (16) we see that the sensitivity of Ic to
vortex-position noise is proportional to the magnitude
of ∇Φv (r) = φ0 ∇Gd = Jd × ẑφ0 /Id . Thus Ic is most
sensitive to vortex-position noise when the vortices are
close to the inner or outer edges of the SQUID, where
the magnitude of Jd is largest. These equations provide
more accurate results for the vortex-position sensitivity
than the approximations given in Refs. 2 and 35.
So far, we have investigated how the general equations
governing the behavior of a dc SQUID are altered when
the contributions arising from line integrals of the current
density are included. As we have shown in Ref. 31, these
additional contributions are important when the Pearl
length Λ is an appreciable fraction of the linear dimensions of the SQUID. We have found that the basic SQUID
equations, Eqs. (17) and (18), are unaltered, except that
the magnetic flux (sometimes called Φext 21 ) generated in
the SQUID’s central hole by the externally applied field
in the absence of a vortex is replaced by the effective
flux Φf , given in Eq. (14). Similarly, we have shown that
the total inductance L of the SQUID has contributions
both from the magnetic induction (geometric inductance)
and the associated supercurrent (kinetic inductance). We
also have shown in principle how to calculate the effect of
the return flux from a vortex at position rv in the body
of the SQUID, and we have found that the effective flux
arising from the vortex is Φv (rv ), given in Eqs. (15) and
(16). To demonstrate that all the above quantities can be
calculated numerically for arbitrary values of Λ, we next
examine the behavior of a model SQUID as decribed in
Sec. III.
III.
LONG SQUID IN A PERPENDICULAR
MAGNETIC FIELD
Here we consider a long SQUID whose thickness d is
less than the London penetration depth λ and whose
topology is like that of Fig. 1 but which is stretched
6
to a large length l in the y direction, as sketched in
Fig. 3. SQUIDs of similar geometry have been investigated experimentally in Refs. 14,36,37,38,39. We treat
here the case for which the length l is much larger than
the width 2w of the body of the SQUID, and we focus
on the current and field distributions in and near the left
(−w < x < −a) and right (a < x < w) arms and near
the center of the SQUID, where to a good approximation
the current density jy is uniform across the thickness
and depends only upon x, and the magnetic induction
B = ∇ × A depends only upon x and z. In the equations that follow, we deal with the sheet-current density,
whose component in the y direction is Jy (x) = jy (x)d.
The self-field magnetic induction generated by Jy (x) is
BJ (x, z) = ∇ × AJ (x, z), where AJ (x, z), the y component of the vector potential obtained from Ampere’s law,
is
Z
µ0
C
AJ (x, z) =
Jy (x′ ) ln p
dx′ . (23)
2π
(x − x′ )2 + z 2
The integration here and in the following equations is
carried out only over the strips, and C is a constant with
dimensions of length remaining to be determined. In the
presence of a perpendicular applied field Ba = ẑBa =
∇× Af , the total vector potential is A = AJ + Af , where
Af = ŷBa x.
A.
Formal solutions
We now use the approach of Ref. 40 to calculate the inplane magnetic-induction and sheet-current distributions
appearing in Eqs. (11)-(14) in Sec. II. For all of these contributions we shall take into account the in-plane (z = 0)
self-field contribution AJ (x) = AJ (x, 0) to the y component of the vector potential, where
Z
C
µ0
dx′ .
(24)
Jy (x′ ) ln
AJ (x) =
2π
|x − x′ |
We first examine the equal-current case and consider
the contributions BI (x), the z component of BI (x), and
JI (x), the y component of JI (x), due to equal currents I/2 in the left and right sides of the SQUID.
Since JI (−x) = JI (x), the corresponding y component
of the vector potential is also a symmetric function of
x: AI (−x) = AI (x), where the subcripts I refer to
the equal-current case. There are no flux quanta between the strips (ΦI = 0), and the second term in
the brackets on the right-hand side of Eq. (1) vanishes;
(φ0 /2π)∇γ = 0. However, the constant C must be chosen such that JI (x) = −AI (x)/µ0 Λ in the superconductor. Combining this equation with Eq. (24), making
use ofR the symmetry JI (−x) = JI (x), and noting that
w
I = 2 a JI (x)dx, we obtain
Z wh
i
I
b
b2
1
′
′
′
ln =
ln 2
′ 2 + Λδ(x−x ) JI (x )dx
2π C
2π
|x
−x
|
a
(25)
for a < x < w. Here b can be chosen to be any convenient length, such as the length l or w, but not C. We
now define the inverse integral kernel K sy (x, x′ ) for the
symmetric-current case via
Z w
i
h 1
b2
′′
′
′′
ln ′′ 2
K sy (x, x′′ )
′ 2 + Λδ(x − x ) dx
2π
|x
−
x
|
a
= δ(x − x′ ). (26)
Applying this kernel to Eq. (25), we obtain
b Z w
I
JI (x) =
ln
K sy (x, x′ )dx′ .
2π
C
a
Rw
Since I = 2 a JI (x)dx, we find
b
.Z wZ w
ln
=π
K sy (x, x′ )dxdx′ ,
C
a a
(27)
(28)
such that
JI (x) =
I
2
Z
a
w
.Z wZ w
K sy (x, x′ )dx′
K sy (x′ , x′′ )dx′ dx′′ .
a
a
For a ≤ x ≤ w, the stream function is
Z
2 x
GI (x) =
JI (x′ )dx′ ,
I a
(29)
(30)
and for −w ≤ x ≤ −a, GI (−x) = −GI (x). The
corresponding z component of the magnetic induction
BI (x) can be obtained from the Biot-Savart law or,
since BI (x) = dAI (x)/dx, from Eq. (24). Note that
BI (−x) = −BI (x). Although the kernel K sy (x, x′ ) depends upon b, we find numerically that JI (x) and BI (x)
are independent of b.
We next examine the circulating-current case and consider the contributions Bd (x), the z component of Bd (x),
and Jd (x), the y component of Jd (x), due to a circulating current Id ; the current in the y direction on the right
side of the SQUID is I2 = Id and that on the left side
is I1 = −Id . The vector potential is still given by Eq.
(24), except that we add subscripts d. However, since
Jd (−x) = −Jd (x), the vector potential is now an antisymmetric function of x; Ad (−x) = −Ad (x). The circulating current is generated by the fluxoid Φd [see Eq. (13)]
associated with a nonvanishing gradient of the phase γ
around the loop. The second term inside the bracket on
the right-hand side of Eq. (1), (φ0 /2π)∇γ, is −ŷΦd /2l
for a < x < w and ŷΦd /2l for −w < x < −a. Thus,
Jd (x) = −[Ad (x) − Φd /2l]/µ0Λ for a < x < w. Combining this equation with that for Ad (x), noting that the
inductance of the SQUID is L = Φd /Id , and making use
of the symmetry Jd (−x) = −Jd (x), we obtain
Z wh
x + x′ i
LId
1
= µ0
ln +Λδ(x−x′ ) Jd (x′ )dx′ (31)
′
2l
2π
x−x
a
for a < x < w. We now define the inverse integral kernel
K as (x, x′ ) for the asymmetric-current case via
Z w
x′′ + x′ i
h 1
′′
′
+
Λδ(x
−
x
)
dx′′
ln ′′
K as (x, x′′ )
2π
x − x′
a
= δ(x − x′ ). (32)
7
Applying
this kernel to Eq. (31) and noting that Id =
Rw
J
(x)dx,
we obtain
d
a
Jd (x) = αId
Z
w
K as (x, x′ )dx′
(33)
a
and
L = 2αµ0 l,
(34)
where
α=1
.Z wZ
a
w
K as (x, x′ )dxdx′
(35)
a
is a dimensionless function of a, b, w, and Λ, which we
calculate numerically in the next section. For a ≤ x ≤ w,
the stream function is
Z
1 x
Gd (x) =
Jd (x′ )dx′ ,
(36)
Id a
and for −w ≤ x ≤ −a, Gd (−x) = Gd (x).
In this formulation, as in Ref. 31, L = Lm + Lk is
the total inductance. The geometricR inductance contriw
bution Lm = 2Em /Id2 , where Em = l a Jd (x)Ad (x)dx is
the stored magnetic energy, andRthe kinetic contribution
w
Lk = 2Ek /Id2 , where Ek = µ0 Λl a Jd2 (x)dx is the kinetic
energy of the supercurrent, can be calculated using Eq.
(33) from31,44,45
Z Z
µ0 l w w x + x′ Lm =
ln Jd (x)Jd (x′ )dxdx′ ,
πId2 a a
x − x′
(37)
Z w
2
2µ0 lΛ
2µ0 lΛ < Jd >
Lk =
,
(38)
J 2 (x)dx =
Id2 a d
(w − a) < Jd >2
where the brackets (<>) denote averages over the film
width. We can show that Lm + Lk = L with the help of
Eqs. (32) and (35).
The z component of the magnetic induction Bd (x) generated by Jd (x) can be obtained from the Biot-Savart
law, or, since Bd (x) = dAd (x)/dx, from Eq. (24). Note
that Bd (−x) = Bd (x).
When l ≫ w, the magnetic moment in the z direction
generated by the circulating current Id is (to lowest order
in w/l)
Z w
md = 2l
xJd (x)dx,
(39)
a
where the factor 2 accounts for the fact41 that the currents along the y direction and those along the x direction
at the ends (U-turn) give exactly the same contribution
to md , even in the limit l → ∞. To next higher order in
w/l one has to replace l in Eq. (39) by l − (w − a)q, where
(w − a)q/2 is the distance of the center of gravity of the
x-component of the currents near each end from this end.
For a single strip of width w−a, one has, e.g., q = 1/3 for
the Bean critical state (with rectangular current stream
lines) and q = 0.47 for ideal screening (Λ ≪ w).42
We next examine flux focusing. As discussed in Sec.
II, to calculate the effective area of the slot, we need to
calculate the fields produced in response to a perpendicular applied magnetic induction Ba , subject to the condition that no current flows through either junction. Since
this is equivalent to having both junctions open-circuited,
the problem reduces to finding the fields produced in the
vicinity of a pair of long superconducting strips connected
by a superconducting link at only one end, i.e., when the
slot of width 2a between the two strips is open at one
end. However, the desired fields may be regarded as the
superposition of the solutions of two separate problems
when the slot has closed ends: (a) the fields generated
in response to Ba , when Φ = 0 (the zero-fluxoid case)
and a clockwise screening current flows around the slot
[second term on the right-hand side of Eq. (41) below],
and (b) the fields generated in the absence of Ba , when
flux quanta in the amount of Φf are in the slot and a
counterclockwise screening current flows around the slot
[first term on the right-hand side of Eq. (41)]. The desired flux-focusing solution is obtained by setting the net
circulating current equal to zero.
The equations describing the fields in the flux-focusing
case are derived as follows. The z component of the
net magnetic induction Bf (x) is the sum of Ba and the
self-field Bsf (x) generated by Jf (x). The vector potential Ay (x) is the sum of Ba x, which describes the
applied magnetic induction, and the self-field contribution given by Eq. (24) but with subscripts f. Since
Jf (−x) = −Jf (x), the vector potential is again an antisymmetric function of x; Af (−x) = −Af (x). The fluxoid Φf [see Eq. (14)] contributes a nonvanishing gradient of the phase γ around the loop, such that the
second term inside the brackets on the right-hand side
of Eq. (1), (φ0 /2π)∇γ, is −ŷΦf /2l for a < x < w
and ŷΦf /2l for −w < x < −a. Equation (1) yields
Jf (x) = −[Ba x + Af (x) − Φf /2l]/µ0Λ for a < x < w.
Combining this equation with that for Af (x) [Eq. (24)],
making use of the symmetry Jf (−x) = −Jf (x), and introducing the effective area via Φf = Ba Aeff [see Sec. II],
we obtain
Ba (Aeff − 2lx)/2l =
Z wh
x + x′ i
1
′
+
Λδ(x
−
x
)
Jf (x′ )dx′
µ0
ln 2π
x − x′
a
(40)
for a < x < w. We again use the inverse integral kernel
K as (x, x′ ) for the asymmetric-current case [Eq. (32)] to
obtain
Z
Ba w Aeff
Jf (x) =
(41)
− x′ K as (x, x′ )dx′ .
µ0 a 2l
The effective area of the SQUID Aeff is found from the
condition
that the net current around the loop is zero
Rw
[ a Jf (x)dx = 0], which yields
Z wZ w
Aeff = 2αl
x′ K as (x, x′ )dxdx′ .
(42)
a
a
8
For a ≤ x ≤ w, the stream function is
Gf (x) =
Z
4
x
Jf (x′ )dx′ ,
Ba= 0, I > 0
I1 = I2 = I / 2
3
4G
a
J, B, 4G
2
and for −w ≤ x ≤ −a, Gf (−x) = Gf (x). The spatial
distribution of the resulting z component of the in-plane
magnetic induction is given by Bf (x) = Ba + Bsf (x),
where Bsf (x) can be obtained from the Biot-Savart law
or by substituting Jf (x) into Eq. (24) and making use of
Bsf (x) = dAf (x)/dx. Note that Bf (−x) = Bf (x). The
resulting magnetic moment mf in the z direction can be
calculated by replacing Jd by Jf in Eq. (39).
In the next section we also present numerical results
for field and current distributions in the zero-fluxoid case,
in which I = 0, Φv = 0, and the effective flux is zero:
Φ = Φd +Φf = 0. Such a case could be achieved by shortcircuiting the Josephson junctions in Fig. 1, cooling the
device in zero field such that initially Φ = 0, and then
applying a small perpendicular magnetic induction Ba .
A circulating current J(x), given by the second term on
the right-hand side of Eq. (41), would spontaneously arise
in order to keep Φ = 0, as in the Meissner state.
1
equal currents
(43)
J
0
1
J
1
0
B
0
1
Λ/w = 0
0.03
0.1
0.3
1
−1
−2
0
0.2
0.4
0.6
0
B
0.8
B
1
1.2
x/w
FIG. 4: Profiles of the sheet current JI (x), Eq. (49), stream
function GI (x), Eq. (50), and magnetic induction BI (x) for
the equal-current case (Ba = 0, I1 = I2 = I/2 > 0). Shown
are the examples a/w = 0.3 with Λ/w = 0 (solid lines with
dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3 (dotted
lines), and 1 (solid lines). Here B/µ0 and J are in units I2 /w
and G in units I2 .
where δij = 0 for i 6= j, δii = 1, and
B.
Qsy
ij =
Numerical solutions
In the previous section we have presented formal solutions for the sheet-current density Jy (x) in Eqs. (29),
(33), and (41), which are expressed as integrals involving
the geometry-dependent inverse kernels K sy (x, x′ ) and
K as (x, x′ ). As in Ref. 31 for thin rings, these integrals are
evaluated on a grid xi (i = 1, 2, . . . , N ) spanning only the
strip (but avoiding the edges, where the integrand may
have infinities), a < |xi | < w, such that for any function
Rw
PN
f (x) one has a f (x)dx = i=1 wi f (xi ). Here wi are the
weights, approximately equal to the local spacing of the
P
xi ; the weights obey N
i=1 wi = w − a. We have chosen
the grid such that the weights wi are narrower and the
grid points xi more closely spaced near the edges a and w,
where Jy (x) varies more rapidly. We have accomplished
this by choosing some appropriate continuous function
x(u) and an auxiliary discrete variable ui ∝ i − 21 , such
that wi = x′ (ui )(u2 − u1 ). We can choose x(u) such that
its derivative x′ (u) vanishes (or is reduced) at the strip
edges to give a denser grid there. By choosing an appropriate substitution function x(u) one can make the numerical error of this integration method arbitrarily small,
decreasing rapidly with any desired negative power of the
grid number N , e.g., N −2 or N −3 .
For the equal-current case, Eq. (25) becomes
Qsy
ii =
b2
1
, i 6= j ,
ln 2
2π |xi − x2j |
1
πb2
.
ln
2π xi wi
(45)
The optimum choice of the diagonal term Qsy
ii (i.e., with
|x2i − x2j | replaced by xi wi /π for i = j, which reduces
the numerical error from order N −1 to N −2 or higher
depending on the grid) is discussed in Eq. (3.12) of Ref.
41 for strips and in Eq. (18) of Ref. 31 for disks and
rings. The superscript (sy) is a reminder that this is
for a symmetric current distribution [JI (−x) = JI (x)].
sy
−1
Defining Kij
= (wj Qsy
, such that
ij + Λδij )
N
X
sy
Kik
(wj Qsy
kj + Λδkj ) = δij ,
(46)
k=1
and applying it to Eq. (44), we obtain
JI (xi ) =
Since I = 2
PN
i=1
ln
N
b X
I
K sy .
ln
2π
C j=1 ij
(47)
wi JI (xi ), we find
b
C
=π
N
N X
.X
sy
wi Kij
,
(48)
i=1 j=1
such that
b
I
ln =
2π C
N
X
j=1
N
(wj Qsy
ij + Λδij )JI (xj )
(44)
JI (xi ) =
N
N
I X sy . X X
sy
K
wk Kkl
.
2 j=1 ij
k=1 l=1
(49)
9
Λ/w = 0
4
Ba= 0, I = 0
− I1 = I2 > 0
3
circulating currents
J, B, 4G
1
0
2
Λ/w = 0.01
4G
1
1
J
B
0
0
1
0
Λ/w = 0
0.03
0.1
0.3
1
−1
Λ/w = 0.1
−2
0
0.2
0.4
0.6
1
0
B
0.8
B
1
1.2
x/w
FIG. 6: Profiles of Jd (x), Eq. (54), Gd (x), Eq. (57), and magnetic induction Bd (x) for the circulating-current case (Ba = 0,
−I1 = I2 > 0). Shown are the examples a/w = 0.3 with
Λ/w = 0 (solid lines with dots), 0.03 (dot-dashed lines), 0.1
(dashed lines), 0.3 (dotted lines), and 1 (solid lines). B/µ0
and J are in units I2 /w and G in units I2 .
Λ/w = 50
as
−1
Defining Kij
= (wj Qas
, such that
ij + Λδij )
FIG. 5: Magnetic field lines in the equal-current case for
a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and 50 (or ∞).
N
X
as
Kik
(wj Qas
kj + Λδkj ) = δij ,
(53)
k=1
It is remarkable that although the parameter b appears
in Eq. (45), the final result for JI (xi ) in Eq. (49) does
not depend upon b. The stream function GI (x) can be
evaluated as43
GI (xi ) =
i X
N
X
N X
N
.X
sy
wj Kjk
j=1 k=1
applying it to Eq. (51), and noting that Id
PN
i=1 wi Jd (xi ), we obtain
Jd (xi ) = αId
sy
wl Klm
.
LId
(wj Qas
= µ0
ij + Λδij )Jd (xj )
2l
j=1
(54)
and
Shown in Fig. 4 are plots of JI (x), GI (x), and the corresponding magnetic induction BI (x) vs x for a/w = 0.3
and various values of Λ/w = 0, 0.03, 0.1, 0.3, 1. The
curves for Λ = 0 exactly coincide with the analytic expressions of Appendix A. The magnetic field lines for this
case are depicted in Fig. 5.
For the circulating-current case, Eq. (31) becomes
N
X
as
Kij
j=1
(50)
l=1 m=1
N
X
=
L = 2αµ0 l,
(55)
N X
N
.X
(56)
where
α=1
as
wi Kij
.
i=1 j=1
The stream function Gd (x) can be evaluated as43
(51)
Gd (xi ) = α
i X
N
X
sy
wj Kjk
.
(57)
j=1 k=1
where
1
xi + xj
ln
, i 6= j ,
2π |xi − xj |
4πxi
1
,
ln
=
2π
wi
Qas
ij =
Qas
ii
(52)
The superscript (as) is a reminder that this is for an
asymmetric current distribution [Jd (−x) = −Jd (x)].
Shown in Fig. 6 are plots of Jd (x), Gd (x), and Bd (x)
vs x for a/w = 0.3 and various values of Λ/w. Note that
these curves look similar to those in Fig. 4, but they all
have opposite parity, as can be seen from the different
profiles B(x) near x = 0. The magnetic field lines for
this case are shown in Fig. 7.
As discussed in Sec. II, when a vortex is present in
the region a < |x| < w, the sensitivity of the SQUID’s
10
Λ/w = 0
3
(a)
Λ/w=1
0.3
2.5
0.1
L / ( µ0 l )
0.03
Λ/w = 0.01
2
1.5
Λ/w = 0
1
0.5
Λ/w = 0.1
0
0
FIG. 7: Magnetic field lines in the circulating-current case for
a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and 50 (or ∞).
Lm / ( µ0 l )
Λ/w = 50
2.5
Lk / ( µ0 l ),
3
1.5
0.2
0.4
(b)
a/w
0.6
0.8
Lk
1
L
k
Λ/w=1
0.3
0.1
2
L
m
1
0.5
1, 0.1, 0
Λ / w = 0.03
0
0
critical current Ic is proportional to the magnitude of
dΦv /dx = φ0 dGd /dx = φ0 Jd (x)/Id . From Fig. 6 we see
that when Λ ≪ w, this sensitivity is greatly enhanced
when the vortex is close to the edges a and w but that
when Λ ≥ w, the sensitivity is nearly independent of
position.
Shown as the solid curves in Fig. 8(a) are plots of the
inductance L vs a/w for various values of Λ/w = 0, 0.03,
0.1, 0.3, and 1. The solid curves in Fig. 9(a) show the
same L vs Λ/w (range 0.0045 to 2.2) for several values of
a/w = 0.01, 0.1, 0.4, 0.8, 0.95, and 0.99.
The geometric and kinetic contributions Lm and Lk
can be calculated separately from Eqs. (37) and (38)
Lm =
N N
2µ0 l X X
wi wj Qas
ij Jd (xi )Jd (xj ) ,
Id2 i=1 j=1
N
Lk =
(58)
N
2µ0 lΛ X X
wi Jd2 (xi ) ,
Id2 i=1 j=1
(59)
using Eqs. (54) and (56). We can show that Lm + Lk = L
using the property of inverse matrices that M · M −1 =
M −1 · M = I, where I is the identity matrix.
Shown as solid curves in Fig. 8(b) are Lm and Lk vs
a/w. For Λ = 0, when Lk = 0, L = Lm exactly coincides with Eq. (A10), which may be approximated by
Eq. (A11) for a/w < 0.7 [open circles in Fig. 8(a)] and
by Eq. (A12) for a/w > 0.7 [open squares in Fig. 8(a)].
The dotted curves in Fig. 8(b) for Lm and Lk are those
0.2
0.4
a/w
0.6
0.8
1
FIG. 8: (a) Solid curves show the inductance L = Lm + Lk
[Eq. (55)] vs a/w calculated for Λ/w = 0, 0.03, 0.1, 0.3, and
1. See the text for descriptions of analytic approximations
shown by the open circles, open squares, dots, and dashes.
(b) Solid curves show the geometric inductance Lm [Eq. (58)]
vs a/w for Λ/w = 0, 0.1, and 1 and the kinetic inductance
Lk [Eq. (59)] vs a/w for Λ/w = 0.03, 0.1, 0.3, and 1. See the
text for descriptions of analytic approximations shown by the
dotted and dashed curves.
of Eqs. (C3) and (C6) in the limit Λ/w → ∞, when the
circulating current density is uniform. The dotted curves
in Fig. 8(a) are obtained from L = Lm + Lk using the
approximations of Eqs. (C3) and (C6); they are an excellent approximation to L for Λ/w ≥ 0.03 except for small
values of a/w. Improved agreement for small values of
Λ/w and a/w is shown by the dashed curves in Fig. 8(a),
which show the approximation of Eq. (B7) for L, and in
Fig. 8(b), which show the approximation of Eq. (B12) for
Lk .
The solid curves in Fig. 9(b) show Lm and Lk vs Λ/w.
The geometric inductance Lm depends upon Λ but only
weakly, varying slowly between its Λ = 0 asymptote
[Eq. (A10), horizontal dot-dashed line] and its Λ = ∞
asymptote [Eq. (C3), horizontal dotted line]. For larger
values of of a/w, Lm is nearly independent of Λ. On
the other hand, the kinetic inductance Lm , is approxi-
11
1
1
10
(a)
0.9
a / w = 0.99
0.95
magnetic moment for the
circulating − current case
0.8
0.7
L / ( µ0 l )
0.8
0.4
0.6
0.1
m
0.01
10
0.5
Λ / w = 10
1
0.3
0.1
0.03
0.01
0.003
0
0.4
0
10
0
0.3
0.2
0.1
−2
−1
10
10
Lk
(b)
a/w=
0
0
0
10
Λ/w
Lk
Lm / ( µ0 l )
0.2
0.3
0.8
10
Lm
0.4
0.1
0.6
0.7
0.8
0.9
1
FIG. 10: The magnetic moment md for the circulating current
case (Ba = 0, −I1 = I2 = Id > 0) plotted versus a/w for
various values of Λ/w in units 2wlId . These curves coincide
with those in Fig. 13 below.
Lm
0.01
4
−1
10
B
−1
10
Λ/w
0
10
FIG. 9: (a) Solid curves show the inductance L = Lm + Lk
[Eq. (55)] vs Λ/w for a/w = 0.01, 0.1, 0.4, 0.8, 0.95, and
0.99. See the text for descriptions of analytic approximations
shown by the dotted and dashed curves. (b) Solid curves show
both the geometric inductance Lm [Eq. (58)] and the kinetic
inductance Lk [Eq. (59)] vs Λ/w for a/w = 0.01, 0.1, 0.4,
0.8, 0.95, and 0.99. See the text for descriptions of analytic
approximations shown by the dotted, dashed, and dot-dashed
curves.
mately proportional to Λ/w. The straight dotted lines
in Fig. 9(b), calculated from the large-Λ approximation
given in Eq. (C6), are a good approximation to Lm except for small values of Λ/w and a/w. The dotted curves
in Fig. 9(a) are obtained from L = Lm + Lk using the
approximations of Eqs. (C3) and (C6). Improved agreement for small values of Λ/w and a/w is shown by the
dashed curves in Fig. 9(a), which show the approximation of Eq. (B7) for L, and in Fig. 9(b), which show the
approximation of Eq. (B12) for Lk .
In Eq. (3) of Ref. 44, Yoshida et al. derived an approximate expression for the kinetic inductance when
Λ/w ≪ 1. We have found that their expression for Lk
is not an accurate approximation to our exact numerical results. To eliminate the logarithmic divergences
due to the inverse square-root dependence of the current density near the edges, Yoshida et al. followed an
J, B, −2G
−2
0.5
0.99
0
10
0.4
a/w
0.95
Lk / ( µ0 l ),
0.1
B >0
a
I = I =0
J
Λ/w = 0
3
0.03
0.1
0.3
2
1
1
B
B
2
flux focusing
0
0
1
1
0
1
−1
0
−2G
J
−2
0
0.2
0.4
0.6
x/w
0.8
1
1.2
FIG. 11: Profiles Jf (x), Eq. (62), Gf (x), Eq. (64), and magnetic induction Bf (x) for the flux-focusing case (Ba > 0,
I1 = I2 = 0). Shown are the examples a/w = 0.3 with
Λ/w = 0 (solid lines with dots), 0.03 (dot-dashed lines), 0.1
(dashed lines), 0.3 (dotted lines), and 1 (solid lines). B and
µ0 J are in units Ba , and G in units wBa /µ0 .
approach used by Meservey and Tedrow,45 and chose a
cutoff length of the order of d, the film thickness. When
d < λ, however, this approach cannot be correct, because the equations describing the fields and currents in
superconducting strips contain only the two-dimensional
screening length Λ = λ2 /d. The cutoff length therefore
must instead be chosen to be of the order of Λ, as we
have done in Appendix B.
The magnetic moment in the z direction generated by
12
1
Λ/w = 0
0.9
0.8
Aeff / 2wl
0.7
Λ/w = 0.01
Λ / w = 10
0.6
Λ / w >> 1
10
0.3
0.1
0.03
0.01
0.003
0.001
0.5
0.4
0.1
0.01
0
0.3
0.2
Λ/w=0
0
0.1
Λ/w = 0.1
0
0
0.1
0.2
0.3
0.4
a/w,
0.5
0.6
0.7
0.8
0.9
1
20*a / w + 0.3
FIG. 13: The effective area Aeff , Eq. (63), plotted versus the
gap half width a/w for several values of Λ/w = 0, 0.001, 0.003,
0.01, 0.03, 0.1, 0.3, and 10. The lower-right curves show the
same data shifted and stretched along a/w. The dots show
the exact result (A19) in the limit Λ/w → 0. For Λ/a ≥ 10
one has Aeff /2wl ≈ (1 + a/w)/2, Appendix C. These curves
coincide with Fig. 10 since Aeff = md /Id .
Λ/w = 0.5
0.6
FIG. 12: Magnetic field lines in the flux-focusing case for
a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and 0.5.
0.55
a / w = 0.2
0.5
0.45
md = 2
N
X
wi xi Jd (xi ).
Aeff / 2wl
the circulating current is, from Eq. (39),
(60)
i=1
0.1
0.4
0.03
0.35
0.3
0.01
As shown in Fig. 10, this magnetic moment vanishes very
slowly when the gap width and Λ go to zero, a/w → 0
and Λ/w → 0. This can be explained by the fact that for
Λ = 0 and a < x ≪ w one has Jd (x) ∝ 1/x, Eq. (A5).
The contribution of these small x to md , Eq. (60), stays
finite due to the factor x, but the total current Id to
which md is normalized, diverges when a/w → 0, thus
suppressing the plotted ratio md /Id . Interestingly, the
curves in Fig. 10 coincide with those in Fig. 13; see below.
Expressions for md in the limits Λ/w → 0 and Λ/w → ∞
are given in Eqs. (A13) and (C7).
For the flux-focusing case, Eq. (40) becomes
0.25
0.003
0.001
0.2
−5
10
−4
10
−3
Ba (Aeff /2l − xi ) = µ0
(wj Qas
ij + Λδij )Jf (xj ).
10
−1
Λ/w
10
0
10
1
10
FIG. 14: The effective area Aeff , Eq. (63), plotted versus Λ/w
(range 7 · 10−6 to 14) for several values of a/w = 0, 0.001,
0.003, 0.01, 0.03, 0.1, and 0.2. Same data as in Fig. 13.
where, since
PN
i=1
wi Jf (xi ) = 0, the effective area is
Aeff = 2αl
N
X
−2
10
N X
N
X
as
wi Kij
xj .
(63)
i=1 j=1
(61)
j=1
Applying Eq. (53), we obtain
The stream function Gf (x) can be evaluated as43
Gf (xi ) =
i
X
wj Jf (xj ).
(64)
j=1
Jf (xi ) =
N
N
Ba Aeff X as X as Kij xj ,
Kij −
µ0 2l j=1
j=1
(62)
Shown in Fig. 11 are plots of the flux-focusing Jf (x),
Gf (x), and Bf (x) vs x for a/w = 0.3 and Λ/w = 0, 0.03,
13
5
0.3
B (0)/B ≈ (w/a) / ln(4w/a)
f
a
3
0.3
10
1
0
1
0.2
a/w
a / w = 0.001
0.05
0.9
enhancement of B (x=0)
0.8
by flux focusing
f
0
Λ/w=0
0.001
0.003
0.7
0.5
0.01
0.4
0.03
Λ/w=0
0.1
0.3
2×
10
0.1
0.2
0.3
0.4
a / w,
0.5
0.6
0.7
0.8
−2
−1
10
10
Λ/w
0
1
10
10
1
5*a / w + 0.35
FIG. 15: The minimum of the magnetic induction in the fluxfocusing case, Bf (0) = Bf (x = 0), referred to the applied field
Ba and plotted versus the half gap width a/w. Top: The
ratio Bf (x = 0)/Ba , tending to unity for a/w → 1 and for
Λ/w ≫ 1, and diverging for a/w → 0 when Λ = 0. Bottom:
The same ratio multiplied by a/w to avoid this divergence
and fit all data into one plot. The lower right plot depicts
the small-gap data two times enlarged along the ordinate,
and shifted and five times stretched along the abscissa. The
circles show the approximation Bf (0)/Ba ≈ (w/a)/ ln(4w/a)
good for a/w ≤ 0.3.46
magnetic moment for the
flux − focusing case
Λ/w=0
0.003
0.01
0.03
0.1
0.3
1
10
2
0.9
−3
10
2.5
0.3
10
0.2
−4
10
FIG. 16: The minimum field Bf (0) = Bf (x = 0) for the fluxfocusing case as in Fig. 15 but plotted versus Λ/w (range
7 · 10−6 to 14) for several values of a/w = 0, 0.001, 0.003,
0.01, 0.03, 0.1, and 0.2.
0.6
0.1
−5
10
1.5
−m
f
0.01
0.003
0.3
a
[B (0) / B ] (a / w)
0.03
0.1
0.1
1
0
0
0.2
0.15
0.1
2
0.1
0.25
f
4
0.2
for Λ=0, a/w <
− 0.3 :
[Bf(0) / Ba] (a / w)
a
6
B (0) / B
0.35
Λ/w=0
0.003
0.01
0.03
0.01
0.3
1
10
7
1
0.1
0.5
0.3
1
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a/w
FIG. 17: The magnetic moment mf for the flux-focusing
case plotted versus a/w for various values of Λ/w in units
w2 lBa /µ0 .
0.1, 0.3, and 1. The first term in Eq. (62) equals the
circulating-current sheet-current density, Eq. (54), with
appropriate weight factor such that the total circulating
current vanishes, I1 = I2 = 0. The corresponding magnetic field lines are depicted in Fig. 12. Shown in Figs. 13
and 14 are plots of the effective area Aeff (a/w, Λ/w) versus a/w and Λ/w, respectively, in units of the maximum
possible area 2wl. In the limit Λ/w → 0, Aeff is given
by Eq. (A19), and when Λ/w → ∞, Aeff = l(w + a).
Note in Fig. 14 that Aeff increases with increasing Λ,
particularly for small gap widths 2a. Flux focusing is reflected by the fact that for small a/w → 0 the effective
area Aeff of the gap tends to a constant, except in the
limit Λ → 0, where it vanishes very slowly, Aeff /2wl ≈
(π/2)/ ln(4w/a) [Eq. (A19)]. When a/w → 0, the enhancement factor Aeff /2al → ∞ and thus diverges even
for Λ = 0. In the limit a/w ≪ 1, Aeff (0, Λ/w) tends to
a universal function [see Fig. 14]. Interestingly, Figs. 13
and 10 show identical curves; this is because the identity
Aeff = md /Id holds for all values of a/w and Λ/w, as
proved in general in Sec. II.
Figures 15 and 16 show the minimum of the magnetic
induction in the flux-focusing case, Bf (0) = Bf (x = 0)
[see Fig. 11], plotted versus a/w and Λ/w, respectively.
The ratio Bf (0)/Ba ≥ 1 tends to unity for a/w → 1
and for Λ/w ≫ 1, and it diverges for a/w → 0 when
Λ = 0. The curve for Λ = 0 exactly coincides with
the analytic expression Bf (0)/Ba = wE(k ′ )/aK(k ′ )
obtained from Eq. (A17). For a/w ≪ 1 this yields
Bf (0)/Ba ≈ (w/a)/ ln(4w/a), which is a good approximation for 0 < a/w ≤ 0.3.46 The magnetic moment mf
for the flux-focusing case, calculated from Eq. (60) but
14
5
4
−J
zero−fluxoid case
0.03
0
3
− J, B
Λ/w = 0
B >0
a
I1 = − I2 > 0
Λ/w = 1
0.3
2
0.1
0.03
0
1
B
B
0.1
0.3
−J
0
Λ/w = 0.01
−J
1
1
0
Λ/w = 0.1
B
0.2
0.4
0.6
x/w
0.8
1
1.2
FIG. 18: Profiles of the sheet-current density J(x) [second
term in Eq. (62)] and magnetic induction B(x) generated
when a perpendicular magnetic induction Ba is applied in
the zero-fluxoid case when Φ = 0, I1 = −I2 > 0, and I = 0.
Shown are the examples a/w = 0.3 with Λ/w = 0 (solid lines
with dots), 0.03 (dot-dashed lines), 0.1 (dashed lines), 0.3
(dotted lines), and 1 (solid lines). B and µ0 J are in units Ba .
with Jd (xi ) replaced by Jf (xi ), is shown in Fig. 17. Expressions for mf in the limits Λ/w → 0 and Λ/w → ∞
are given in Eqs. (A20) and (C9)
Shown in Fig. 18 are profiles for the zero-fluxoid case
with plots of J(x) and the corresponding B(x) generated by an applied magnetic induction Ba > 0 when
the junctions are short-circuited such that Φ = 0 and
I1 = −I2 > 0; for comparison see analogous profiles in
Sec. 2.5 of Ref. 47 for two parallel strips and in Sec.
IV of Ref. 48 and Sec. 4 of Ref. 31 for rings. That the
current density J(x) in the zero-fluxoid case is given by
the second term on the right-hand sides of Eqs. (41) and
(62), can be seen by setting Φf = Ba Aeff = 0 in Eqs.
(40), (41), (61), and (62). Depicted in Fig. 18 are the
examples a/w = 0.3 with Λ/w = 0, 0.03, 0.1, 0.3, and
1. Figure 19 shows the magnetic field lines for this case
and Fig. 20 the magnetic moment m. Expressions for J,
B, and m for the zero-fluxoid case in the limits Λ/w → 0
and Λ/w → ∞ are given in Appendixes A and C.
IV.
SUMMARY
In Sec. II of this paper we have presented general
equations governing the static behavior of a thin-film dc
SQUID for all values of the Pearl length Λ = λ2 /d, where
the London penetration depth λ is larger than d, the
film thickness. The SQUID’s critical current Ic depends
upon the effective flux Φ, which is the sum of the magnetic flux up through a contour surrounding the central
hole and a term proportional to the line integral of the
current density around this contour. For a symmetric
SQUID there are three important contributions to Φ: a
Λ/w = 0.5
FIG. 19: Magnetic field lines in the zero-fluxoid case Φ = 0,
Ba > 0, and I = 0 for a/w = 0.1 and Λ/w = 0, 0.01, 0.1, and
0.5.
3
2.5
magnetic moment for the
zero−fluxoid case B > 0
a
2
−m
−1
0
Λ/w=0
0.003
0.01
0.03
0.1
1.5
0.3
1
0.5
1
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a/w
FIG. 20: The magnetic moment m for the zero-fluxoid case
Φ = 0, Ba > 0, and I = 0 plotted versus a/w for various
values of Λ/w in units w2 lBa /µ0 . At a = Λ = 0 one has
−m = πw2 lBa /µ0 .
circulating-current term Φd , a vortex-field term Φv , and a
flux-focusing term Φf , all of which depend upon Λ. Since
Λ is a function of temperature, an important consequence
is that all of the contributions to Φ are temperaturedependent.
The circulating-current term Φd can be expressed in
15
terms of the SQUID inductance L and the circulating
current Id via Φd = LId . The SQUID inductance has
two contributions, L = Lm + Lk , where the first term
is the geometric inductance (associated with the energy
stored in the magnetic field) and the second is the kinetic
inductance (associated with the kinetic energy of the circulating supercurrent). Both contributions are functions
of Λ, since they both depend on the spatial distribution
of the current density. However, Lm depends only weakly
upon Λ, because for the same circulating current Id , the
energy stored in the magnetic field does not vary greatly
as Λ ranges from zero to infinity. On the other hand,
because the kinetic energy density is proportional to Λ,
Lk is also nearly proportional to Λ, with deviations from
linearity occurring only for small values of Λ/w.
The vortex-field term can be written as Φv = φ0 Gd ,
where Gd is a dimensionless stream function describing
the circulating sheet-current density Jd . Roughly speaking, when Λ is small, Ic is most strongly dependent upon
the vortex position when the vortex is close to the edges
of the film, but when Λ is large, Ic is equally sensitive
to the vortex position wherever the vortex is. Recent
experiments49 have used the relationship Φv = φ0 Gd to
determine the vortex-free sheet-current density Jd (x, y)
from vortex images obtained via low-temperature scanning electron microscopy.6,9 The experimental data obtained in magnetic fields up to 40 µT are in excellent
agreement with numerical calculations of Jd (x, y), confirming the validity of the above relationship, even in
the presence of many (up to 200) vortices in the SQUID
washer.
The flux-focusing term can be expressed as Φf =
Ba Aeff , where Ba is the applied magnetic induction and
Aeff is the effective area of the central hole of the SQUID.
Although Aeff is primarily determined by the dimensions
of the SQUID, it also depends upon the value of Λ.
To illustrate the Λ dependence of the above quantities,
in Sec. III of this paper we analyzed in detail the behavior of a long SQUID whose central region resembles a
coplanar stripline. We numerically calculated the profiles
of the sheet-current density, stream function, and magnetic induction in the equal-current, circulating-current,
flux-focusing, and zero-fluxoid cases for various representative values of Λ. We presented plots of the inductances
L, Lm , and Lk , the effective area Aeff , and the magnetic
moments for these cases. Useful analytic approximations
are provided for the Λ/w → 0 limit in Appendix A, for
small Λ/w and a/w in Appendix B, and for the Λ/w → ∞
limit in Appendix C.
We are in the process of applying the above theory to
square and circular SQUIDs, using the numerical method
of Ref. 50.
Acknowledgments
We thank D. Koelle for stimulating discussions. This
work was supported in part by Iowa State University
of Science and Technology under Contract No. W-7405ENG-82 with the U.S. Department of Energy and in part
by the German Israeli Research Grant Agreement (GIF)
No G-705-50.14/01.
APPENDIX A: THE LIMIT Λ/w = 0
In the ideal-screening limit Λ/w = 0, the y component
of the sheet-current density in the strips (a < |x| < w)
for the equal-current case is47
JI (x) =
|x|
I
,
2
2
π [(x − a )(w2 − x2 )]1/2
(A1)
and the z component of the magnetic induction in the
plane z = 0 of the strips is
µ0 I
x
, |x| > w, (A2)
2π [(x2 − a2 )(x2 − w2 )]1/2
= 0, a < |x| < w,
(A3)
x
µ0 I
=
, |x| < a, (A4)
2π [(a2 − x2 )(w2 − x2 )]1/2
BI (x) = −
and the
√ constant C in Eqs. (24), (25), (27), and (28) is
C = w2 − a2 /2.
For the circulating-current case, the y component of
the sheet-current density in the strips (a < |x| < w) in
the limit Λ/w = 0 is47
Jd (x) =
w2
2B0 x
,
µ0 |x| [(x2 − a2 )(w2 − x2 )]1/2
(A5)
and the z component of the magnetic induction in the
plane z = 0 of the strips is
w2
, |x| > w, (A6)
[(x2 − a2 )(x2 − w2 )]1/2
= 0, a < |x| < w,
(A7)
2
w
, |x| < a, (A8)
= B0 2
[(a − x2 )(w2 − x2 )]1/2
Bd (x) = −B0
where the parameter B0 , the magnetic flux Φd in the z
direction in the slot, the circulating current Id , and the
geometric inductance Lm are related by
Φd = Lm Id = 2B0 lwK(k)
(A9)
Lm = µ0 lK(k)/K(k ′),
(A10)
and
where K(k) is the complete elliptic integral of the first
kind √
of modulus k = a/w and complementary modulus
′
k = 1 − k 2 . The geometric inductance is well approximated for small a/w by
Lm = (πµ0 l/2)/ ln(4w/a),
(A11)
16
neglecting corrections proportional to a2 /w2 , and for
small (w − a)/w by
Lm = (µ0 l/π) ln[16/(1 − a2 /w2 )],
2
(A12)
2
neglecting corrections proportional to 1 − a /w . In the
limit that Λ = 0, the kinetic inductance vanishes (Lk =
0), and the inductance in Eq. (A10) becomes the total
inductance: L = Lm . The magnetic moment [see Eq.
(39)] can be obtained from Eqs. (13)-(16) of Ref. 47:
md = [πlw/K(k ′ )]Id .
(A13)
For the flux-focusing case, the y component of the
sheet-current density in the strips (a < |x| < w) in the
limit Λ/w = 0 is47
Jf (x) =
2Ba
x E(k ′ )w2 − 2K(k ′ )x2
,
µ0 K(k ′ ) |x| [(x2 − a2 )(w2 − x2 )]1/2
(A14)
and the z component of the magnetic induction in the
plane z = 0 of the strips is
Ba E(k ′ )w2 − 2K(k ′ )x2
,
K(k ′ ) [(x2 − a2 )(x2 − w2 )]1/2
|x| > w,
(A15)
= 0, a < |x| < w,
(A16)
Ba E(k ′ )w2 − 2K(k ′ )x2
=
,
K(k ′ ) [(a2 − x2 )(w2 − x2 )]1/2
|x| < a,
(A17)
Bf (x) = −
where E(k ′ ) is the complete elliptic integral
√ of the second kind of complementary modulus k ′ = 1 − k 2 and
modulus k = a/w. The magnetic flux in the z direction
in the slot is
Φf = πBa lw/K(k ′ ),
(A18)
and the effective area Aeff of the slot is
Aeff = Φf /Ba = πlw/K(k ′ ).
(A19)
Note that Aeff = md /Id . The magnetic moment generated by Jf (x) is
mf = −πl[w2 + a2 − 2w2 E(k ′ )/K(k ′ )]Ba /µ0 .
(A20)
For the zero-fluxoid case, the y component of the sheetcurrent density in the strips can be obtained from Sec.
2.5 of Ref. 47:
2Ba x x2 −[1−E(k)/K(k)]w2
J(x) = −
.
(A21)
µ0 |x| [(x2 − a2 )(w2 − x2 )]1/2
The corresponding z component of the magnetic induction is47
x2 −[1−E(k)/K(k)]w2
B(x) = Ba
, |x| > w, (A22)
[(x2 − a2 )(x2 − w2 )]1/2
= 0, a < |x| < w,
(A23)
[1−E(k)/K(k)]w2 −x2
, |x| < a. (A24)
= Ba
[(a2 − x2 )(w2 − x2 )]1/2
APPENDIX B: BEHAVIOR FOR SMALL Λ AND
SMALL a
In this section we present some expressions for L, Lk ,
and Lm that follow from approximating the circulatingcurrent distribution for small values of Λ and a.
When the slot is very narrow (a/w ≪ 1), we approximate the sheet-current density in the region a < x < w
generated by the fluxoid Φd via
w
Jd (x) = I0 p
,
(B1)
2
2
2
(x − a + δ )(w2 − x2 + δ 2 )
where I0 = 2Φd /πµ0 l and δ is a quantity of order Λ =
λ2 /d determined as follows. When l → ∞ and then a →
0 and w → ∞, an exact calculation yields for x > 0
Z ∞ −xt/Λ
e
dt
2Φd
.
(B2)
Jy (x) =
2
πµ0 lΛ 0
t +1
Rb
We find 0 Jy (x)dx = (2Φd /πµ0 l) ln(γb/2Λ) when b ≫
Λ, where γ = eC = 1.781..., and C = 0.577... is EuRb
ler’s constant. From Eq. (B1) we find 0 Jd (x)dx =
(2Φd /πµ0 l) ln(2b/δ) when a = 0 and δ ≪ b ≪ w.
Comparing these two integrals we obtain δ = (4/γ)Λ =
2.246Λ. Integrating Eq. (B1) from a to w to obtain Id ,
we find
w
Id = I0 √
[F (λa , q) − F (λw , q)],
(B3)
w2 + δ 2
where F (φ, k) is the elliptic integral of the first kind and
r
w 2 − a2 + δ 2
,
(B4)
λa = arcsin
w2 − a2 + 2δ 2
δ
,
(B5)
λw = arcsin √
2
w − a2 + 2δ 2
r
w2 − a2 + 2δ 2
q =
.
(B6)
w2 + δ 2
Expanding Eq. (B3) for a ≪ w and δ ≪ w, using
I0 = 2Φd /πµ0 l, and neglecting terms of order a2 /w2 and
δ 2 /w2 , we obtain
L = Φd /Id = (πµ0 l/2)/{ln[4w/(a + δ)] − δ/w}, (B7)
where δ = 2.246Λ. Note that Eq. (B7) reduces to Eq.
(A11) when Λ = 0.
From Eq. (B1) we obtain the approximation
Z w
w
(fa + fw ),
(B8)
Jd2 (x)dx = I02 2
(w − a2 + δ 2 )
a
where
fa
The magnetic moment generated by J(x) is
m = −πl[2w2 E(k)/K(k) − w2 + a2 ]Ba /µ0 .
(A25)
fw
√
2
2
w
−1 (w − a) δ − a
tan
= √
,
2
2
2
a(w − a) + δ
δ −a
a < δ,
(B9)
√
2
2
(w − a) a − δ
w
tanh−1
,
= √
2
2
a(w − a) + δ 2
a −δ
a > δ,
(B10)
√
2
2
(w − a) w + δ
w
. (B11)
tanh−1
= √
2
2
w(w − a) + δ 2
w +δ
17
Using Eqs. (B3) and (B8), we obtain from Eq. (38)
(w2 + δ 2 )
(fa + fw )
Λ
,
2
2
2
w (w − a + 2δ ) [F (λa , q) − F (λw , q)]2
(B12)
where δ = 2.246Λ. Although our intention in using the
ansatz of Eq. (B1) initially was to obtain an improved
approximation to Lk for small values of Λ and a, we
see from Figs. 8(b) and 9(b) that Eq. (B12) provides a
reasonably good approximation for all values of Λ and a.
neglecting corrections proportional to (1 − a/w)2 . From
Eq. (38) we obtain the kinetic inductance
Lk = 2µ0 l
Lk = 2µ0 lΛ/(w − a).
(C6)
When Λ ≫ w, the total inductance L is dominated by the
kinetic inductance (Lk ≫ Lm ), such that L ≈ Lk . Since
Jd is uniform, the magnetic moment is easily found from
Eq. (39) to be
md = l(w + a)Id .
(C7)
APPENDIX C: THE LIMIT Λ/w → ∞
In the weak-screening limit Λ/w → ∞, K sy (x, x′ ) =
Λ−1 δ(x − x′ ), the y component of the sheet-current density in the strips (a < |x| < w) in the equal-current case
is uniform, JI = I/2(w−a), the z component of the magnetic induction in the plane of the strips obtained from
the Biot-Savart law is
(x − w)(x + a) µ0 I
BI (x) =
ln (C1)
,
4π(w − a)
(x + w)(x − a)
and the constant C in Eqs. (24), (25), (27), and (28) is
C = w exp[−πΛ/(w − a)].
For the circulating-current case in the limit Λ/w →
∞, K as (x, x′ ) = Λ−1 δ(x − x′ ), α = Λ/(w − a), the y
component of the sheet-current density in the strips is
again uniform, Jd = Id /(w − a) for a < x < w, and the z
component of the magnetic induction in the plane of the
strips obtained from the Biot-Savart law is
Bd (x) =
x2 − w2 µ0 Id
ln 2
.
2π(w − a)
x − a2
(C2)
The geometric inductance is, from Eq. (37)
Lm =
h
4w2 µ0 l
2
w
ln
π(w − a)2
w 2 − a2
4a2 i
w + a
+ a2 ln
−2aw ln
, (C3)
w−a
w 2 − a2
which is independent of Λ. Equation (C3) is well approximated for small a/w by
Lm = (µ0 l/π)(1 + 2a/w) ln 4,
1
i
2
3 1
+ − (1 − a/w) ,
1 − a/w 2 2
x2 − w2 Ba h
(w + a) ln 2
4πΛ
x − a2
(x + w)(x − a) i
+ 2x ln − 4(w − a) . (C8)
(x − w)(x + a)
Bsf (x) =
The magnetic moment generated by Jf (x) in this limit is
mf = −[l(w − a)3 /6Λ](Ba /µ0 ).
(C9)
For the zero-fluxoid case in the limit Λ/w → ∞, the
applied field is only weakly screened, and the z component of the magnetic flux density is nearly equal to the
applied magnetic induction, B(x) ≈ Ba . The y component of the vector potential is approximately given by
A(x) = Ba x, and the y component of the induced sheetcurrent density, obtained from Eq. (1) with γ = 0, is
J(x) = −(Ba /µ0 Λ)x. To the next order of approximation, B(x) = Ba + Bs (x), where the self-field Bs is found
from the Biot-Savart law
Bs (x) =
(C4)
neglecting corrections proportional to a2 /w2 , and for
small (w − a)/w by
h
Lm = (µ0 l/π) ln
For the flux-focusing case in the limit Λ/w → ∞,
K as (x, x′ ) = Λ−1 δ(x − x′ ), the y component of the sheetcurrent density is Jf (x) = Ba (w + a − 2x)/2µ0 Λ for
a < x < w, the effective area is Aeff = l(w + a) = md /Id ,
and the z component of the magnetic induction in the
plane of the strips is Bf (x) = Ba + Bsf (x), where from
the Biot-Savart law
(x + w)(x − a) i
Ba h
x ln − 2(w − a) . (C10)
2πΛ
(x − w)(x + a)
The magnetic moment generated by J(x) in this limit is
m = −[2l(w3 − a3 )/3Λ](Ba /µ0 ).
(C11)
(C5)
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