PHYSICAL REVIEW B, VOLUME 64, 024505
Theory of type-II superconductors with finite London penetration depth
Ernst Helmut Brandt
Max-Planck-Institut für Metallforschung, D-70506 Stuttgart, Germany
共Received 14 February 2001; published 13 June 2001兲
Previous continuum theory of type-II superconductors of various shapes with and without vortex pinning in
an applied magnetic field and with transport current is generalized to account for a finite London penetration
depth ␭. This extension is particularly important at low inductions B, where the transition to the Meissner state
is now described correctly, and for films with thickness comparable to or smaller than ␭. The finite width of the
surface layer with screening currents and the correct dc and ac responses in various geometries follow naturally
from an equation of motion for the current density in which the integral kernel now accounts for finite ␭. New
geometries considered here are thick and thin strips with applied current, and ‘‘washers,’’ i.e. thin-film squares
with a slot and central hole, as used for superconducting quantum interference devices.
DOI: 10.1103/PhysRevB.64.024505
PACS number共s兲: 74.60.Ec, 74.60.Ge
I. INTRODUCTION
The statics and dynamics of the Abrikosov vortex lines1
and of the two-dimensional 共2D兲 pancake vortices in layered
superconductors2 recently was formulated within continuum
approximation in terms of an equation of motion of the current density j inside the superconductor.3 The resulting integral equation for the space- and time-dependent j(x,y,z,t)
contains the assumed constitutive law for the electric field Ev
caused by the motion of vortices. In general, the law E
⫽Ev (j,B) depends on the current density j and on the local
induction B and describes, e.g., free flux flow, pinning, and
thermally activated depinning of vortices.
Besides this, the equation of motion for j depends on the
geometry of the given problem. Various geometries have
been considered: Thin long strips,4 thin circular disks and
rings,5 thin rectangular platelets or films,6 thick strips,7 and
thick circular disks or short cylinders,8 always in a magnetic
field applied perpendicular to the plane of the superconductor. Further numerical solutions for the statics of thin-film
superconductors of various shapes within the Bean model are
given by Prigozhin.9 In all these geometries, the current density j or the sheet current in films J(x,y,t)⫽(J x ,J y )
⫽ 兰 j(x,y,z,t)dz has only one component 共flowing along the
strip or circulating in the disk兲 or may be derived from a
scalar potential g(x,y,t) according to J x ⫽ ⳵ g/ ⳵ y, J y
⫽⫺ ⳵ g/ ⳵ x since “•J⫽0. The equation of motion describes
thus a scalar ( j or g) that depends on only one or two spatial
coordinates.
A more general continuum formulation including also
transport currents applied by contacts, applying in principle
to arbitrary 3D geometry, and accounting also for a finite
lower critical field H c1 and for a general reversible thermodynamic field H(B), is presented in Ref. 10, where it is
applied to the problem of the geometric barrier.11 The elegance of the formulation4–8,10 in terms of an equation of
motion for the current density is that it explicitly accounts
for the applied magnetic field or transport current, which in
other formulations enter via a separate boundary condition
for a differential equation. Our integral equation automatically accounts for the inhomogeneous magnetic field outside
the superconductor, without any need to calculate this infi0163-1829/2001/64共2兲/024505共15兲/$20.00
nitely extended field explicitly during the time integration
since all spatial integrations extend only over the finite specimen cross section. The applied field and applied current enter via the screening currents that flow on the specimen surface, the information about the geometry is contained in a
precalculated integral kernel, and the properties of the superconductor enter via one single constitutive law E⫽E v ( j,B)
if B⫽ ␮ 0 H may be assumed, or via two laws if some nontrivial law B(H)⫽ ␮ 0 H is used.
So far, this continuum description4–8,10 considered the
limit where both the vortex spacing a and the London penetration depth ␭ are assumed to be much smaller than all
other relevant lengths. There are, however, two situations
where a finite London depth ␭ should be considered.
共i兲 In bulk superconductors, at small inductions B the
Meissner state should follow in the limit B→0. So far, with
the above assumption E⫽E v ( j,B) one has E⫽0 for B→0 if
the free-flux-flow law or similar laws E v ⬀B are used. In real
type-II superconductors, however, one may have E⫽0 in a
surface layer of thickness ␭ if j varies with time, since one
has E⫽ ␮ 0 ␭ 2 ⳵ j/ ⳵ t in the Meissner-London limit 共this London equation means that the electric field accelerates the
Cooper pairs兲.
共ii兲 In thin films with thickness d, both cases d⬎␭ and
d⬍␭ should be described correctly by a continuum theory.
In particular, for dⰆ␭ the effective penetration depth of thin
films2,12 ⌳⫽␭ 2 /d may become macroscopically large and
comparable with the film width.
In the present paper our previous continuum electrodynamics of type-II superconductors4–8,10 is generalized to allow for an arbitrary London penetration depth ␭. The resulting integral equations for j in various geometries correctly
describe all limiting cases and are easily solved numerically.
With finite ␭ the stability and speed of the numerics is even
improved since this ␭ provides a well-defined inner cutoff
length, whereas with ␭⫽0 the inner cutoff depends on the
chosen grid spacing.
In the following sections, equations of motion for the current density in type-II superconductors with finite London
depth ␭ are derived for various geometries. For illustration,
some selected numerical results are presented for each geom-
64 024505-1
©2001 The American Physical Society
ERNST HELMUT BRANDT
PHYSICAL REVIEW B 64 024505
etry, namely, current and magnetic-field profiles, magnetization curves, ac susceptibilities, and the current stream lines in
thin films.
II. VOLTAGE-CURRENT LAW
In type-II superconductors within continuum approximation the local electric field E is composed of two parts, originating from the vortex motion and from the Meissner surface
currents,
E⫽Ev ⫹EM .
共1兲
The first term Ev (j,B) in general is a nonlinear function
of the current density j due to pinning and thermal or quantum depinning 共creep兲. It may be anisotropic 共a兲 if the Hall
effect of vortex motion is accounted for, or 共b兲 if the superconductor is anisotropic, or 共c兲 if j is not perpendicular to B.
This latter case is still little understood but should occur in
most really three-dimensional geometries, e.g., in a superconductor cube with applied magnetic field; one possible
model description in this case uses two critical current densities j c⬜ and j c 储 for the current components perpendicular
and parallel to B. Though the parallel current density j 储 does
not exert a Lorentz force on the vortices it nevertheless can
trigger a helical instability13,14 and vortex cutting,15 which
leads to a finite j c 储 . Furthermore, as shown by Gurevich16
some combinations of nonlinearity and anisotropy in
voltage-current models may lead to unstable current distributions. In the following geometry examples, for transparency I
shall consider isotropic Ev since the extension to anisotropic
Ev is straightforward. Isotropic voltage-current laws have the
form
Ev 共 j,B 兲 ⫽ ␳ v 共 j,B 兲 j,
共2兲
where ␳ v ( j,B) is the resistivity caused by vortex motion, and
j⫽ 兩 j兩 , B⫽ 兩 B兩 , with j⬜B. In the free-flux-flow limit 共zero
pinning or high current density兲 one has ␳ v ⫽ ␳ ff
⬇(B/B c2 ) ␳ n where B c2 is the upper critical field and ␳ n is
the normal resistivity just above B c2 . For thermally activated
depinning many experiments and theoretical models yield a
logarithmic activation energy U( j)⫽U 0 ln(jc /j), which gives
E v ⫽E c exp(⫺U/kT)⫽Ec(j/jc)n with creep exponent n
⫽U 0 /kTⰇ1 and 共arbitrary兲 ‘‘threshold criterion’’ E c
⫽E( j c ). With Eq. 共2兲 this means ␳ v ⫽ ␳ c ( j/ j c ) ␴ where ␴
⫽n⫺1Ⰷ1. A simple model that combines this creep model
and the free-flux-flow limit reads10
␳n
共 j/ j c 兲 ␴
␳ v 共 j,B 兲 ⫽
B
.
B c2 1⫹ 共 j/ j c 兲 ␴
共3兲
The second term in Eq. 共1兲 is caused by the current flowing in the Meissner state and is given by the dynamic London equation,
EM ⫽ ␮ 0 ␭ 2
⳵j
.
⳵t
共4兲
This term describes the acceleration of the massive charge
carriers 共Cooper pairs兲 by an electric field, ⳵ j/ ⳵ t⬀E. Thus, a
FIG. 1. Geometry of long superconductor strips with rectangular
cross section 2a⫻2b in a perpendicular applied magnetic field H a
共left兲 or with applied electric current I a ⫽I⫽ 兰 j(x,y)dx dy 共right兲.
finite London depth ␭ in principle may be introduced into
our continuum description4–8,10 by using the full voltagecurrent law,
E⫽Ev 共 j,B兲 ⫹ ␮ 0 ␭ 2
⳵j
.
⳵t
共5兲
This replacement works well in the case of linear response, where it yields a frequency-( ␻ -兲dependent complex
resistivity
␳ 共 ␻ 兲 ⫽ ␳ ff⫹i ␻ ␮ 0 ␭ 2
共6兲
in the absence of vortex pinning. In this case the linear integral equation for j reduces to an eigenvalue equation from
which the complex ac susceptibility of the superconductor
for a given geometry follows as an explicit sum.17
When E v ( j,B) is nonlinear, the integral equation for
j(r,t) has to be solved by numerical integration over time. It
turns out that this time integration becomes unstable when
the modified voltage-current law 共5兲 is inserted. However,
when the term containing ␭ 2 is incorporated into the integral
kernel as shown in Sec. III, then the numerics remains stable
and even become more stable and faster than it was with ␭
⫽0.
III. THICK STRIPS WITH FINITE ␭
To fix ideas we first derive the modified integral equation
for the geometry of an infinite strip or bar with rectangular
cross section filling the volume ⫺a⭐x⭐a, ⫺b⭐y⭐b,
⫺L/2⭐z⭐L/2, LⰇa,b, see Fig. 1. In the general 3D geometry, with r⫽(x,y,z), the vector potential A j (r) caused by
2
the current density j(r)⫽⫺ ␮ ⫺1
0 ⵜ Aj in the gauge “•A⫽0
is given by
Aj共 r兲 ⫽ ␮ 0
冕
d 3r ⬘
j共 r⬘ 兲
4 ␲ 兩 r⫺r⬘ 兩
,
共7兲
with the integral taken over the volume where the current
flows. Indeed, using ⵜ 2 兩 r⫺r⬘ 兩 ⫺1 ⫽⫺4 ␲ ␦ 3 (r⫺r⬘ ) ( ␦ 3 is the
3D delta function兲 one verifies that Eq. 共7兲 yields ⵜ 2 Aj(r)
⫽⫺ ␮ 0 j(r). For a long strip or bar, both j⫽ j(x,y)ẑ and A
024505-2
THEORY OF TYPE-II SUPERCONDUCTORS WITH . . .
PHYSICAL REVIEW B 64 024505
⫽A(x,y)ẑ are directed along z. Integrating Eq. 共7兲 over z
and writing from now on r⫽(x,y) we obtain
A j 共 r兲 ⫽ ␮ 0
冕
d 2 r ⬘ j 共 r⬘ 兲 Q bar共 r,r⬘ 兲
1
L/2
1
L
asinh
⬇
ln
.
2␲
兩 r⫺r⬘ 兩 2 ␲ 兩 r⫺r⬘ 兩
冕 ⬘冕
a
b
dx
0
0
Ȧ⫽⫺E v 共 j,B 兲 ⫺ ␮ 0 ␭ 2 ⳵ j/ ⳵ t,
共14兲
A j 共 r,t 兲 ⫽⫺
共11兲
with x ⫾ ⫽x⫾x ⬘ , y ⫾ ⫽y⫾y ⬘ . For strips with transport current I a along z one has j(x,y)⫽ j(⫺x,y)⫽ j(x,⫺y)⫽ j
(⫺x,⫺y), thus
1
L8
ln 2
2
2
2
2
2
2
2 .
4␲ 共x⫺
⫹y ⫺
⫹y ⫹
⫹y ⫺
⫹y ⫹
兲共 x ⫺
兲共 x ⫹
兲共 x ⫹
兲
共12兲
Note that the strip length L has dropped out in Eq. 共11兲 but
not in Eq. 共12兲. Therefore, some electrodynamic properties
of long strips with applied current 共e.g., their self induction兲
depend logarithmically on the strip length L, while for strips
in a magnetic field usually the limit L→⬁ may be taken.
Strips with oblique applied field Ha ⫽(H ax ,H ay ), or with
both applied H a and I a , have a lower symmetry and the
integration 共8兲 then has to be taken over the half or full cross
section of the strip rather than over a quarter.
To obtain an explicit dynamic equation for the current
density j(x,y,t) one has to incorporate the applied vector
potential Aa , which for the strip may be chosen along z,
Aa ⫽A a (x,y,t)ẑ. In general, A a (x,y,t) may have two parts
originating from an applied perpendicular magnetic field or
induction Ba (t)⫽ ␮ 0 Ha (t)⫽(B ax ,B ay ) and from an applied
electric field Ea ⫽E a (t)ẑ which drives the transport current
Ia ,
␮0
冕
E v 共 j,B 兲 dt⫺ ␮ 0 ␭ 2 j 共 r,t 兲 ⫺A a 共 r,t 兲 . 共16兲
d 2 r ⬘ j 共 r⬘ ,t 兲关 Q bar共 r,r⬘ 兲 ⫹␭ 2 ␦ 2 共 r⫺r⬘ 兲兴
⫽⫺
冕
E v 共 r,t 兲 dt⫺A a 共 r,t 兲 ,
K 共 r,r⬘ 兲 ⫽ 关 Q bar共 r,r⬘ 兲 ⫹␭ 2 ␦ 2 共 r⫺r⬘ 兲兴 ⫺1 ,
共18兲
we arrive at the equation of motion for j(r,t),
⳵ j 共 r,t 兲
⫽⫺ ␮ ⫺1
0
⳵t
冕
d 2 r ⬘ K 共 r,r⬘ 兲关 E v 共 j,B 兲 ⫹Ȧ a 共 r⬘ ,t 兲兴
共19兲
with Ȧ a (r⬘ ,t)⫽⫺x ⬘ Ḃ ay (t)⫹y ⬘ Ḃ ax (t)⫺E a (t) and with Q bar
from Eqs. 共9兲, 共11兲, or 共12兲. In Eq. 共19兲 only the electric field
caused by the vortex motion E v enters explicitly. The London length ␭ enters the inverse kernel K(r,r⬘ ) defined by
冕
d2r⬙K 共 r,r⬙ 兲关 Q bar共 r⬙ ,r⬘ 兲 ⫹␭ 2 ␦ 2 共 r⬙ ⫺r⬘ 兲兴 ⫽ ␦ 2 共 r⫺r⬘ 兲 .
共20兲
The kernel K(r,r⬘ ) may be computed by a matrix inversion as follows. First, a spatial grid ri ⫽(x i ,y i ) is chosen
with appropriate weights w i such that the integrals over the
strip cross section are well approximated by a sum,
冕
d 2 r f 共 r兲 ⬇
兺i
f 共 ri 兲 w i .
共21兲
Then we express the definition 共20兲 by such a sum,
2
兺i K li共 Q bar
i j w i ⫹␭ ␦ i j 兲 ⫽ ␦ l j ,
共13兲
共17兲
where ␦ 2 (r)⫽ ␦ (x) ␦ (y) is the 2D delta function. Taking the
time derivative of Eq. 共17兲 and introducing the inverse integral kernel18,19
A a 共 x,y,t 兲 ⫽A Ba ⫹A Ea ,
E a ⫽⫺Ȧ Ea ,
冕
Inserting this in Eq. 共14兲 and shifting the term ␮ 0 ␭ 2 j(r,t) to
the left under the integral we obtain
共10兲
2
2
2
2
⫹y ⫺
⫹y ⫹
兲共 x ⫹
兲
1 共x⫹
sym
H
Q bar ⫽Q bar⫽ ln 2
2
2
2
4 ␲ 共 x ⫺ ⫹y ⫺
⫹y ⫹
兲共 x ⫺
兲
共15兲
thus the right-hand side of Eq. 共14兲 may be written as
sym
dy ⬘ j 共 x ⬘ ,y ⬘ 兲 Q bar
共 x,y;x ⬘ ,y ⬘ 兲 .
Ba ⫽ⵜ⫻ 共 A Ba ẑ兲 ,
d 2 r ⬘ j 共 r⬘ ,t 兲 Q bar共 r,r⬘ 兲 ⫽A j 共 r,t 兲 ⫽A⫺A a .
共9兲
For strips with a magnetic field H a applied along y, one has
j(x,y)⫽⫺ j(⫺x,y)⫽ j(x,⫺y)⫽⫺ j(⫺x,⫺y), thus
sym
I
⫽Q bar
⫽
Q bar
冕
For the strip the induction law Ḃ⫽⫺“⫻E yields E⫽⫺Ȧ.
Inserting E from Eq. 共5兲 we have for strips
The integral 共8兲 is over the rectangular strip cross section,
⫺a⭐x⭐a and ⫺b⭐y⭐b, but actually Eq. 共8兲 applies to
strips with cross sections of any shape. If the current distribution is symmetric the integration 共8兲 may be restricted to
one quarter of the rectangular cross section, e.g.,
A j 共 x,y 兲 ⫽ ␮ 0
␮0
共8兲
with the 2D integral kernel
Q bar共 r,r⬘ 兲 ⫽
A Ea ⫽⫺ 兰 t0 E a (t ⬘ )dt ⬘ . Writing the total vector potential as A
⫽A j ⫹A a we obtain from Eq. 共8兲,
共22兲
where Q i j ⫽Q(ri ,r j ) and ␦ i j equals 1 if i⫽ j and 0 otherwise. Solving Eq. 共22兲 for the matrix K i j one finds
where the dot denotes the time derivative. For example, with
B a 储 y one has A Ba ⫽⫺xB a , and with E a (t⫽0)⫽0 one has
024505-3
2
⫺1
K i j ⫽ 共 Q bar
.
i j w i ⫹␭ ␦ i j 兲
共23兲
ERNST HELMUT BRANDT
PHYSICAL REVIEW B 64 024505
The accuracy of this method is considerably increased by
choosing a nonequidistant grid with narrow spacing near the
specimen surface and by taking appropriate diagonal terms
Q bar
ii as described in the appendix of Ref. 8共a兲. From Eq. 共23兲
one sees that finite ␭ 2 increases the 共positive兲 diagonal terms
of the matrix to be inverted; this makes the matrix inversion
more stable.
For numerics we need the equation of motion 共19兲 for
j(r,t) in discrete form,
⳵ ji
⫽⫺ ␮ ⫺1
0
⳵t
兺j K i j 关 E v共 j j ,B j 兲 ⫹Ȧ a j 兴 ,
共24兲
where the vectors j i (t)⫽ j(ri ,t), B i (t)⫽B(ri ,t), and
Ȧ ai (t)⫽Ȧ a (ri ,t)⫽⫺x i Ḃ ay (t)⫹y i Ḃ ax (t)⫺E a (t) are functions of the time t. The matrix K i j , Eq. 共23兲, is independent
of time and has to be computed only once for a given geometry and given ␭. Equation 共24兲 is easily integrated over time
t starting with j i (t⫽0)⫽0 and then switching on the applied
fields B a and/or E a . The resulting magnetic moment per unit
length m(t)⫽(m x ,m y ) and total transport current I a (t)
共along z) are then obtained as integrals over the strip cross
section,
m共 t 兲 ⫽
⬇
I a共 t 兲 ⫽
冕 冕
a
⫺a
dx
b
⫺b
dy 共 ⫺x̂y⫹ŷx 兲 j 共 x,y,t 兲
兺i 共 ⫺x̂y i ⫹ŷx i 兲 j i共 t 兲 w i ,
冕 冕
a
⫺a
dx
b
⫺b
dy j 共 x,y,t 兲 ⬇
兺i
共25兲
j i共 t 兲 w i .
共26兲
Note that the contribution to m(t) of the U-turn of the currents at the strip ends 共integrals over the x and y components
of j, amounting to exactly 21 m) is already considered in Eq.
共25兲. Note further that, though in experiments usually the
applied current I a (t) is imposed, the theory considers a spatially constant electric field E a (t) along z, which drives the
current I a (t) that results from the calculation.
Figures 2–6 show the current density j(x,y) and the magnetic field lines of B 共i.e., the contour lines of A) for strips
with aspect ratio b/a⫽0.4 in various cases, cf. Fig. 1. In Fig.
2 the strip is in the Meissner state, i.e., no vortices have
penetrated. Shown is the screening current density for finite
London depth ␭⫽0.025a for a strip in applied field 共top兲 and
for a strip with applied current 共bottom兲. Similar figures for
strips with quadratic cross section are depicted in Ref. 19.
Note the sharp peak of j(x,y) in the four corners, which has
finite height. This high local current density favors the nucleation of vortex-quarter loops from the corners of the strip
when the applied magnetic field or current exceed a certain
threshold.
Figures 3 and 4 show current density and field lines for
strips in an increasing applied field H a for two values of the
London depth ␭/a⫽0.025 and 0.1. Since the assumed
voltage-current law is very steep, E v ⬀ j n with n⫽101, a flat
saturation of j(x,y) occurs at the critical value j c , like in the
Bean model. The field of full penetration of this strip is7
FIG. 2. A strip with aspect ratio b/a⫽0.4, cf. Fig. 1, in the
Meissner state with London penetration depth ␭⫽0.025a. Shown is
the current density in a quarter of the cross section and the
magnetic-field lines 共inset兲. Top: In perpendicular applied magnetic
field H a . Bottom: With applied current.
H p ⫽( j c b/ ␲ ) 关 (2a/b)arctan(b/a)⫹ln(1⫹a2/b2)兴⫽0.4945a j c
for b/a⫽0.4 and H p ⫽2.52(2b j c ) for b/a⫽0.001.
Figures 5 and 6 show the same strips but with increasing
applied current I a and with no field applied. In this case,
when the critical current I c ⫽4ab j c of the strip is reached,
vortex rings penetrate continuously and annihilate in the center of the strip. With further increasing I a ⬎I c , the dissipation and voltage drop increase steeply. More results for strips
with both applied field and current will be published elsewhere.
Figures 7 and 8 show magnetization loops m(H a ) of
strips in perpendicular applied field H a (t)⫽H 0 sin ␻t at two
amplitudes H 0 , for five values of the London depth ␭, and
for two creep exponents n entering in E v ( j)⫽E c ( j/ j c ) n . Figure 7 is for a thick strip (b/a⫽0.4) and Fig. 8 for a thin strip
(b/a⫽0.001). With our dimensionless units a⫽ j c ⫽E c
⫽ ␮ 0 ⫽1, the used circular frequency ␻ ⫽1 corresponds to
␻ ⫽E c /( ␮ 0 j c a 2 ) in physical units. Note that with increasing
␭ the hysteresis loop becomes more narrow and finally collapses to a line, i.e., the magnetic response becomes revers-
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THEORY OF TYPE-II SUPERCONDUCTORS WITH . . .
PHYSICAL REVIEW B 64 024505
FIG. 3. Current density j(x,y) 共left兲 and magnetic field lines 共right兲 of a superconductor strip
with aspect ratio b/a⫽0.4 in increasing magnetic
field H a ⫽0.05, 0.15, and 0.35 in units of a j c ( j c
⫽ critical current density兲. The dashed lines are
contours of the current density at j/ j c ⫽⫾0.75,
⫾0.45, and ⫾0.15. The superconductor is characterized by a pinning-caused voltage-current law
E⬀( j/ j c ) 101 and by a small London penetration
depth ␭/a⫽0.025. The slight oscillation of j in
the lower left plot is an artifact caused by a too
large time step in the time integration.
ible. This is so since with increasing ␭ the screening current
density decreases and can no longer depin vortices, except
when H a is large or n is small.
␮0
冕
a
⫺a
dx ⬘ J 共 x ⬘ ,t 兲
⫽⫺
IV. THIN STRIPS
This section considers the limit of thin strips, bⰆa, with
B a applied perpendicular 共along y) and I a applied parallel
共along z) to the strip. In this limit only the current density
integrated over the film thickness matters, called sheet current and directed along z 共like j),
J 共 x,t 兲 ⫽
冕
b
⫺b
j 共 x,y,t 兲 dy.
Integrating Eq. 共17兲 over y we obtain 共see also Ref. 20兲
共27兲
冕
冋
L
1
ln
⫹⌳ ␦ 共 x⫺x ⬘ 兲
2 ␲ 兩 x⫺x ⬘ 兩
E v 共 x,t 兲 dt⫺A a 共 x,t 兲 ,
册
共28兲
where ⌳⫽␭ 2 /d is the effective penetration depth of thin
films with thickness d⫽2b⬍␭. Here I have used the approximate constancy of the current density along y, j(x,y)
⬇J(x)/d. Initially it was not clear to me if thin-film expressions of the type 共28兲 describe also the dynamics of superconductor strips and not only the statics, which was successfully considered, e.g., in the static Bean model calculations
of thin disks,21 thin strips,22–25 and ellipses.26 But recently
we have proven27 that not only j(r,t) but also the electric
field E(r,t) is practically constant over the film thickness;
deviations from this constancy occur only near the 共penetrating or exiting兲 flux front but are restricted to a transverse
length scale of order d. This result was obtained in the limit
024505-5
ERNST HELMUT BRANDT
PHYSICAL REVIEW B 64 024505
FIG. 4. As Fig. 3 but for larger London depth
␭/a⫽0.1.
␭→0, but it applies all the more for finite ␭. Thus, the dynamics of the sheet current J⬇ jd is well described by Eq.
共28兲.
Inverting Eq. 共28兲 and taking the time derivative we obtain the equation of motion for the sheet current in thin
strips,
␮ 0 J̇ 共 x,t 兲 ⫽⫺
冕
a
⫺a
dx ⬘ K 共 x,x ⬘ 兲关 E v 共 x ⬘ ,t 兲 ⫹Ȧ a 共 x ⬘ ,t 兲兴
with the inverse integral kernel
冋
1
L
ln
⫹⌳ ␦ 共 x⫺x ⬘ 兲
K 共 x,x ⬘ 兲 ⫽
2 ␲ 兩 x⫺x ⬘ 兩
width, 0⭐x⭐a. For B a ⫽0, I a ⫽0 one has J(x)⫽⫺J(⫺x)
and a symmetric kernel 关cf. Eq. 共11兲兴
冋
1
x⫹x ⬘
ln
⫹⌳ ␦ 共 x⫺x ⬘ 兲
K B 共 x,x ⬘ 兲 ⫽
2 ␲ 兩 x⫺x ⬘ 兩
册
共30兲
and with E v (x ⬘ ,t)⫽E v ( j,B) depending on j⫽J(x ⬘ ,t)/d and
B⫽ 兩 B(x ⬘ ,t) 兩 , and with Ȧ a (x ⬘ ,t)⫽⫺x ⬘ Ḃ a (t)⫺E a (t). For
strips with either applied field B a or applied current I a , the
integration in Eq. 共29兲 may be restricted to half the strip
共31兲
.
For B a ⫽0, I a ⫽0 one has J(x)⫽J(⫺x) and a symmetric
kernel 关cf. Eq. 共12兲兴
冋
L2
1
ln
⫹⌳ ␦ 共 x⫺x ⬘ 兲
K I 共 x,x ⬘ 兲 ⫽
2 ␲ 兩 x⫺x ⬘ 兩 共 x⫹x ⬘ 兲
共29兲
⫺1
册
⫺1
册
⫺1
.
共32兲
Note that the strip length L has dropped out in the kernel 共31兲
but not in the kernel 共32兲. The complex resistivity of thin
strips thus in general depends on the logarithm of the strip
length.
Figures 9 and 10 show the profiles of the sheet current
J(x) and perpendicular induction B y (x) of thin strips with
increasing applied magnetic field 共Fig. 9兲 or current 共Fig. 10兲
024505-6
THEORY OF TYPE-II SUPERCONDUCTORS WITH . . .
PHYSICAL REVIEW B 64 024505
FIG. 5. Current density j(x,y) 共left兲 and magnetic field lines 共right兲 of a superconductor strip
with aspect ratio b/a⫽0.4 with applied current
I a ⫽0.22, 0.66, and 0.94 in units of the critical
current I c ⫽4ab j c . The dashed lines are contours
of the current density at j/ j c ⫽0.75, 0.45, and
0.15. As in Figs. 3 and 4 a voltage-current law
E⬀( j/ j c ) 101 was used and a small London penetration depth ␭/a⫽0.025 was used.
for various values of the effective penetration depth ⌳
⫽␭ 2 /d; d⫽2b is the strip thickness and 2a the strip width,
here b/a⫽0.001. Note that for larger ⌳ the profiles J(x)
become smoother, but the bend of J(x) at the point where J
starts to deviate from the critical sheet current J c ⫽d j c remains sharp. For ⌳/a⬎0.2, the sections of J(x) where 兩 J 兩
⬍J c are almost straight lines in Fig. 9 and almost parabolas
in Fig. 10.
Figure 11 shows J(x) and B y (x) in thin strips that are
exposed to a high magnetic field H a ⰇH p before a current I a
is applied, ranging from 0 to the critical current I c
⫽4ab j c . First, when I a ⫽0, one has J(x)⬇J c sgn(x), and
finally, when I a ⬇I c , one has J(x)⬇J c 共equal signs would
apply in the Bean limit n→⬁). At intermediate currents I a
⬍I c , in the half strip 0⬍x⬍a one has J⬇J c ⫽const, and in
the other half ⫺a⬍x⬍0 the profile J(x) is symmetric about
the point x⫽⫺a/2. For ⌳⫽0, nⰇ1, the known Bean profiles for thin strips with transport current apply to this half
strip.22–25
V. AXIAL SYMMETRY
The above method for incorporating finite ␭ can also be
applied to axially symmetric problems, i.e., to superconductors with an axis of rotational symmetry in a magnetic field
applied parallel to this axis 共along y). A simple example is
disks or short cylinders with radius a and thickness d⫽2b,
but also other shapes like rings, toruses, cones, ellipsoids,
etc., are easily described by introducing a y-dependent radius
a(y). In all these cases the current density and vector potential have only one component directed along the azimuthal
unit vector ␸ˆ , j⫽ j(r,y) ␸ˆ and A⫽A(r,y) ␸ˆ , with r⫽(x 2
⫹z 2 ) 1/2. Integrating the 3D Eq. 共7兲 over the angle ␸
⫽arctan(z/x) we obtain the vector potential A j caused by the
current density j circulating in an axially symmetric conductor,
024505-7
A j 共 r,y 兲 ⫽ ␮ 0
冕
b
⫺b
dy ⬘
冕
a(y ⬘ )
0
dr ⬘ j 共 r ⬘ ,y ⬘ 兲 Q ax共 r,y;r ⬘ ,y ⬘ 兲
共33兲
ERNST HELMUT BRANDT
PHYSICAL REVIEW B 64 024505
FIG. 6. As Fig. 5 but for larger London depth
␭/a⫽0.1.
with the kernel8
Q ax共 r,y;r ⬘ ,y ⬘ 兲 ⫽ f 共 r,r ⬘ ,y⫺y ⬘ 兲 ,
f 共 r,r ⬘ , ␩ 兲 ⫽
冕
⫺r ⬘ cos ␸
␲ d␸
0
2 ␲ 共 ␩ 2 ⫹r 2 ⫹r ⬘ 2 ⫺2rr ⬘ cos ␸ 兲 1/2
共34兲
.
共35兲
When the conductor has a symmetry plane at y⫽0, then the
integration over y ⬘ in Eq. 共33兲 may be restricted to 0⭐y
⭐b if the kernel Q ax is replaced by a symmetric kernel,
Q cyl共 r,y;r ⬘ ,y ⬘ 兲 ⫽ f 共 r,r ⬘ ,y⫺y ⬘ 兲 ⫹ f 共 r,r ⬘ ,y⫹y ⬘ 兲 . 共36兲
If in addition the radius is constant 共like in disks and cylinders兲, Eq. 共33兲 simplifies to
A j 共 r,y 兲 ⫽ ␮ 0
冕 ⬘冕
b
a
dy
0
0
Comparing Eqs. 共33兲 and 共37兲 with Eqs. 共8兲 and 共10兲 for the
strip or bar, one easily verifies that the analog to Eq. 共17兲 for
axial symmetry is identical to Eq. 共17兲 but with Q bar(r,r⬘ )
replaced by Q ax(r,r⬘ ), Eq. 共34兲, or Q cyl(r,r⬘ ), Eq. 共36兲,
where now r⫽(r,y), r⬘ ⫽(r ⬘ ,y ⬘ ). In axial symmetric problems no transport current and thus no electric field is applied
but only a magnetic field or induction Ba
⫽“ 关 A a (r,y,t) ␸ˆ 兴 ⫽(B ar ,B ay ). In general this applied field
may be inhomogeneous, e.g., when levitation forces are to be
computed. If Ba is homogeneous one has A a (r,y,t)⫽
⫺(r/2)B a (t). With this notation, for the general axially symmetric geometry the equation of motion for the circulating
current density is identical to Eq. 共19兲 but with a different
inverse kernel
K 共 r,r⬘ 兲 ⫽ 关 Q ax共 r,r⬘ 兲 ⫹␭ 2 ␦ 2 共 r⫺r⬘ 兲兴 ⫺1 ,
dr ⬘ j 共 r ⬘ ,y ⬘ 兲 Q cyl共 r,y;r ⬘ ,y ⬘ 兲 .
共37兲
共38兲
with Q ax from Eq. 共34兲. If the superconductor has a symmetry plane at y⫽0 and the applied field is homogeneous 共or
024505-8
THEORY OF TYPE-II SUPERCONDUCTORS WITH . . .
PHYSICAL REVIEW B 64 024505
FIG. 7. Magnetization curves m(H a ) of a superconducting thick strip with aspect ratio b/a
⫽0.4 in a perpendicular field H a (t)⫽H 0 sin ␻t at
two amplitudes H 0 /H p ⫽2 共left兲 and H 0 /H p ⫽1
共right兲 with H p ⫽0.4945a j c , the field of full penetration. Fields in units of a j c and magnetic moment m in units of a 3 j c 共per unit length兲. Shown
are virgin curves and hysteresis loops for two
creep exponents n⫽101 共top兲 and n⫽7 共bottom兲
and for London depths ␭/a⫽0.02, 0.1, 0.2, 0.4,
and 0.6.
symmetric about the plane y⫽0) one may replace Q ax in Eq.
共38兲 by Q cyl from Eq. 共36兲,
K 共 r,r⬘ 兲 ⫽ 关 Q cyl共 r,r⬘ 兲 ⫹␭ 2 ␦ 2 共 r⫺r⬘ 兲兴 ⫺1 .
共39兲
These inverse kernels K(r,r⬘ ) may be computed by the same
method discussed below Eq. 共20兲 if the correct integration
area 共or spatial grid兲 is used as defined in Eqs. 共33兲 and 共37兲.
For the best choice of the diagonal terms Q ii see the appendix of Ref. 8共a兲.
The axial magnetic moment of a disk or any other rotationally symmetric conductor is
m共 t 兲⫽␲
冕 冕
b
⫺b
a(y)
dy
0
dr r 2 j 共 r,y 兲 .
共40兲
If the applied field is periodic, H a (t)⫽H 0 sin ␻t, the 共in general nonlinear兲 complex susceptibilities of the disk may be
defined as ␹ ␯ ⫽ ␹ ⬘␯ ⫺i ␹ ⬙␯ , ␯ ⫽1,2,3 . . . ,8
␹ ␯共 H 0 , ␻ 兲 ⫽
i
␲H0
冕
2␲
0
m 共 t 兲 e ⫺i ␯␻ t d 共 ␻ t 兲 .
共41兲
Here ␹ 1 is the fundamental susceptibility and the ␹ ␯ with ␯
⬎1 correspond to higher harmonics, which are absent for
linear response. The ␹ ␯ usually are normalized such that for
H 0 →0 or ␻ →⬁ the ideal diamagnetic susceptibility
␹ 1 (0,␻ )⫽⫺1 results. This is achieved by dividing all ␹ ␯ ,
Eq. 共41兲, by the absolute value of the initial slope
关 ⳵ m(H a )/ ⳵ H a 兴 H a ⫽0 .
As an example, Fig. 12 shows the real and imaginary
parts of the nonlinear fundamental susceptibility ␹ 1 (H 0 , ␻ )
⫽ ␹ ⫽ ␹ ⬙ ⫺i ␹ ⬘ of a thick disk with b/a⫽0.5 plotted versus
the ac amplitude H 0 for constant frequency ␻
⫽E c /( ␮ 0 j c a 2 ), creep exponent n⫽11, and for various London depths ␭/a⫽0.025, . . . ,1. The same data are depicted
in Fig. 13 as a polar plot, ␹ ⬙ versus ⫺ ␹ ⬘ . In both presentations ␹ (H 0 ) sensitively depends on the parameters n and ␭/a
and on the geometry. More data ␹ ␯ (H 0 , ␻ ) are available
from the author.
VI. THIN PLATES AND FILMS
The geometry of thin plates or planar films with arbitrary
shape in a perpendicular magnetic field H a 储 z differs from the
geometries of Sec. III–V, in that, now the current density is
no longer a scalar but has two components, j(x,y,t)
⫽( j x , j y ). However, because of the strict relation div j⫽0,
this planar j may be derived from a scalar potential 共or magnetization, stream function兲 g(x,y,t). Since we are interested
in the thin-film limit we consider the sheet current J(x,y,t)
⫽(J x ,J y )⫽ 兰 j(x,y,z,t)dz, J x ⫽ ⳵ g/ ⳵ y, J y ⫽⫺ ⳵ g/ ⳵ x. The
stream lines of J(x,y) coincide with the contour lines
g(x,y)⫽const. Thus, one may choose g(x,y)⫽0 on the
edge of the film since the current flows along the edge. In
this section I generalize the theory of Ref. 6 to finite London
depth ␭.
The magnetic field H⫽H z (x,y)ẑ in the film plane z⫽0 is
024505-9
ERNST HELMUT BRANDT
PHYSICAL REVIEW B 64 024505
FIG. 8. Magnetization curves m(H a ) of a superconducting thin strip with aspect ratio b/a
⫽0.001 in a perpendicular field H a (t)
⫽H 0 sin ␻t at two amplitudes H 0 ⫽3 共left兲 and
H 0 ⫽1 共right兲 in units of the critical sheet current
J c ⫽d j c (d⫽2b). Magnetic moment m in units
of a 2 J c 共per unit length兲, H p ⫽2.52J c . Virgin
curves and hysteresis loops for two creep exponents n⫽101 共top兲 and n⫽11 共bottom兲 and five
effective penetration depths ⌳⫽␭ 2 /d: ⌳/a
⫽0.02, 0.1, 0.2, 0.4, and 0.8.
related to the local magnetization g(x,y) by an integral over
the film area S,
H z 共 r兲 ⫽H a ⫹
冕
S
d 2 r ⬘ Q film共 r,r⬘ 兲 g 共 r⬘ 兲
共42兲
with r⫽(x,y) and r⬘ ⫽(x ⬘ ,y ⬘ ). The integral kernel
Q film共 r,r⬘ 兲 ⫽ lim
2z 2 ⫺ ␳ 2
z→0 4 ␲ 共 z
2
⫹ ␳ 2 兲 5/2
,
共43兲
with ␳ 2 ⫽(x⫺x ⬘ ) 2 ⫹(y⫺y ⬘ ) 2 , gives the magnetic field generated by a magnetic dipole of unit strength positioned in the
plane z⫽0 at (x ⬘ ,y ⬘ ). Explicit expressions for films with
rectangular shape are given by Eqs. 共42兲–共46兲 of Ref. 6 in
the form of Fourier series. To incorporate a finite London
depth ␭ we write the voltage-current law of the film in the
form of Eq. 共5兲 but now as a function of the sheet current
J(x,y)⫽j(x,y)d(x,y). We consider first a uniform isotropic
film with constant thickness d. The local electric field is then
E共 J,B 兲 ⫽ ␳ s 共 J,B 兲 J共 r,t 兲 ⫹ ␮ 0 ⌳J̇共 r,t 兲 ,
Ḃ⫽⫺“⫻E, which in the film plane z⫽0 means Ḃ z
⫽ ⳵ E x / ⳵ y⫺ ⳵ E y / ⳵ x. With Eq. 共44兲 and J x ⫽ ⳵ g/ ⳵ y, J y ⫽
⫺ ⳵ g/ ⳵ x, we obtain
Ḃ z 共 r,t 兲 ⫽“ 关 ␳ s “g 共 r,t 兲兴 ⫹ ␮ 0 ⌳“ 2 ġ 共 r,t 兲 .
Taking the time derivative of Eq. 共42兲, inserting Ḃ z ⫽ ␮ 0 Ḣ z
from Eq. 共45兲, and combining the two terms containing ġ,
one arrives at
冕
S
d 2 r ⬘ 关 Q film共 r,r⬘ 兲 ⫺ ␦ 2 共 r⫺r⬘ 兲 ⌳“ 2 兴 ġ 共 r⬘ ,t 兲
⫽ f 共 r,t 兲 ⫺Ḣ a 共 r,t 兲 ,
共46兲
with
f 共 r,t 兲 ⫽ ␮ ⫺1
0 “ 关 ␳ s 共 r,t 兲 “g 共 r,t 兲兴 .
共47兲
Inverting this we obtain the equation of motion for g(x,y,t),
ġ 共 r,t 兲 ⫽
共44兲
where ␳ s ⫽ ␳ /d is the sheet resistivity and ⌳⫽␭ 2 /d is the
effective magnetic penetration depth of the film. This electric
field is related to the induction B⫽ ␮ 0 H by the induction law
共45兲
冕
S
d 2 r ⬘ K 共 r,r⬘ 兲关 f 共 r⬘ ,t 兲 ⫺Ḣ a 共 r⬘ ,t 兲兴
共48兲
with f (r,t) from Eq. 共47兲 and with the inverse kernel
024505-10
K 共 r,r⬘ 兲 ⫽ 关 Q film共 r,r⬘ 兲 ⫺ ␦ 2 共 r⫺r⬘ 兲 ⌳ⵜ 2 兴 ⫺1 .
共49兲
THEORY OF TYPE-II SUPERCONDUCTORS WITH . . .
PHYSICAL REVIEW B 64 024505
FIG. 9. Profiles of sheet current J(x) 共dashed
lines兲 and perpendicular induction B y (x) 共solid
lines兲 of a thin strip in increasing perpendicular
magnetic field H a ⫽0.10, 0.22, 0.35, 0.49, 0.65,
0.82, 1.0, 1.23, 1.46, and 1.72 共curves from bottom to top兲. J and H a are in units of J c
⫽d j c (d⫽2bⰆa), B is in units of ␮ 0 J c . For
six effective penetration depths ⌳⫽␭ 2 /d: ⌳/a
⫽0, 0.05, 0.1, 0.2, 0.4, and 1. Creep exponent
n⫽51.
This kernel K(ri ,r j ) may be evaluated on a grid ri as a
Fourier series with discrete k vectors K. If the Fourier coefficients of Q film(ri ,r j ) are Q KK⬘ , given by Eq. 共46兲 of Ref. 6
for a rectangular platelet, then the Fourier coefficients of
K(ri ,r j ) are the inverse matrix
关 Q KK⬘ ⫹⌳K 2 ␦ KK⬘ 兴 ⫺1 .
E y ⫽ ␳ sy J y ⫹ ␮ 0 ⌳ y J̇ y ,
⌳ x 共 r兲 ⫽
B z 共 r,t 兲 ⫽
␳ xx 共 J,B,r兲
,
d 共 r兲
d 共 r兲
冉
冕
S
␳ y y 共 J,B,r兲
d 共 r兲
␭ 2y 共 r兲
d 共 r兲
冉
冊
共53兲
.
冊
⳵
⳵g
⳵ ġ
␳ sx ⫹ ␮ 0 ⌳ x
,
⳵y
⳵y
⳵y
共54兲
共55兲
with
Q̃ 共 r,r⬘ 兲 ⫽Q film共 r,r⬘ 兲 ⫺ ␦ 2 共 r⫺r⬘ 兲
共51兲
共52兲
⌳ y 共 r兲 ⫽
d 2 r ⬘ Q̃ 共 r,r⬘ 兲 ġ 共 r⬘ ,t 兲 ⫽ f 共 r,t 兲 ⫺Ḣ a 共 r,t 兲 ,
f 共 r,t 兲 ⫽ ␮ ⫺1
0
␳ sy 共 r兲 ⫽
,
⳵
⳵g
⳵ ġ
␳ sy ⫹ ␮ 0 ⌳ y
⳵x
⳵x
⳵x
⫹
with the two sheet resistivities
␳ sx 共 r兲 ⫽
␭ 2x 共 r兲
The generalized Eqs. 共45兲 and 共46兲 are then
共50兲
Figure 14 shows the current stream lines in a thin rectangle with side ratio b/a⫽0.7 for two effective penetration
depths ⌳/a⫽0 and ⌳/a⫽0.1 and three applied fields
H a /J c ⫽0.001, 0.5, and 1.5 共ideal screening, partial penetration, full penetration兲. Note that finite ⌳ rounds the sharp
corners of the rectangular stream lines and slightly delays
vortex penetration.
If the superconductor film is nonuniform 关e.g., has spatially varying thickness d(x,y)] and/or anisotropic 关e.g., has
two 共in general nonlinear兲 resistivities ␳ xx and ␳ y y and/or
two London depths ␭ x and ␭ y ], then a rather general currentvoltage law E(J,B,r)⫽(E x ,E y ) is
E x ⫽ ␳ sx J x ⫹ ␮ 0 ⌳ x J̇ x ,
and two effective penetration depths
冉
冊
⳵
⳵
⳵
⳵
⌳y ⫹ ⌳x
,
⳵x ⳵x ⳵y ⳵y
共56兲
冋 冉 冊 冉 冊册
⳵
⳵g
⳵
⳵g
␳ sy
⫹
␳ sx
⳵x
⳵x
⳵y
⳵y
.
共57兲
The equation of motion for g(x,y,t) now is still given by Eq.
共48兲 but with f (r,t) from Eq. 共57兲 and with the inverse ker-
024505-11
ERNST HELMUT BRANDT
PHYSICAL REVIEW B 64 024505
FIG. 10. As Fig. 9 but for strips with increasing applied current I a /I c ⫽0, 0.2, 0.4, 0.6, 0.8,
0.9, 0.965, and 1. I c ⫽4ab j c is the critical current
of the strip.
nel K(r,r⬘ )⫽Q̃(r,r⬘ ) ⫺1 , Eq. 共56兲. The kernels Q̃ and K may
be evaluated by Fourier transformation. Some figures of current stream lines and field profiles for thin rectangles with
anisotropic pinning and ⌳⫽0 are depicted in Ref. 28.
⫺1
can be obtained
When ⌳⫽0, the inverse kernel K⫽Q film
29
by iteration, see also the appendix of Ref. 27,
冋
1
␦ 共 r⫺r⬘ 兲 ⫹
K 共 r,r⬘ 兲 ⫽
C 共 r兲
冕
K 共 r⬙ ,r ⬘ 兲 ⫺K 共 r,r⬘ 兲
4 ␲ 兩 r⫺r⬙ 兩 3
S
册
h i⫽
d r⬙ ,
2
共58兲
starting with K(r,r⬘ )⫽0. Here C(r) is defined as an integral
over the infinite area S̄ outside the film or as a contour integral along the film edge,
4 ␲ C 共 r兲 ⫽
冕
S̄
d 2r ⬘
R3
⫽
冕
2␲
0
d␾
R共 ␾ 兲
1
4␲
关共 a⫺ px 兲 ⫺2 ⫹ 共 b⫺qy 兲 ⫺2 兴
兺
p,q
兺j Q i j w j g j ,
共59兲
共60兲
g i⫽
兺j K i j h j .
共61兲
The inverse kernel K i j ⫽(Q i j w j ) ⫺1 cannot be obtained directly by discretizing Q film(r,r⬘ ), Eq. 共43兲, but it may be
obtained by iterating Eq. 共58兲. This iteration can be solved in
one step by inverting a matrix,
冋 冉
K i j⫽ ␦ i j C i⫹
with R⫽ 兩 r⫺r⬘ 兩 . For rectangular films ( 兩 x 兩 ⭐a, 兩 y 兩 ⭐b) one
has explicitly
C 共 x,y 兲 ⫽
For numerics one needs the discretized versions of Eq.
共42兲 and its inverse. Introducing a 2D grid ri ⫽(x i ,y i ) with
weights w i 关 兺 i w i ⫽S, cf. Eq. 共21兲兴 and writing Q i j
⫽Q(ri ,r j ), K i j ⫽K(ri ,r j ), h i ⫽H z (ri )⫺H a , and g i ⫽g(ri )
one has
w l q il
兺
l⫽i
冊
⫺ 共 1⫺ ␦ i j 兲 w j q i j
册
⫺1
,
共62兲
where C i ⫽C(ri ) and q i j ⫽1/(4 ␲ 兩 ri ⫺r j 兩 3 ). Note that the
terms in Eq. 共62兲 must not be rearranged since q ii ⫽⬁.
Using this matrix K i j the current stream lines in ideally
screening flat films of arbitrary shape or of films with
trapped vortices are easily calculated. If the applied field is
constant, h i ⫽⫺H a , the stream lines are the contour lines of
the trace of matrix K i j ,
with p,q⫽⫾1. Expression 共60兲 may also be used for rectangular films with holes or slots if the integral in Eq. 共58兲 is
restricted to the area of the real film.
024505-12
g i ⫽⫺H a
兺j K i j .
共63兲
THEORY OF TYPE-II SUPERCONDUCTORS WITH . . .
PHYSICAL REVIEW B 64 024505
FIG. 12. The nonlinear complex ac susceptibility ␹ (H 0 , ␻ )
⫽ ␹ ⬘ ⫺i ␹ ⬙ of a type-II superconducting thick disk with radius a and
half thickness b⫽0.5a plotted versus the amplitude H 0 of the ac
magnetic field in units of the Bean field of full penetration H p
⫽0.722a j c . The curves are for different London depths ␭/a
⫽0.025, 0.05, 0.1, 0.15, 0.2, 0.35, 0.5, 0.7, and 1. E v ⬀ j n , n⫽11,
␻ ⫽E c /( ␮ 0 j c a 2 ).
FIG. 11. As Fig. 9 but for a thin strip in large magnetic field
H a ⰇH p to which an increasing current is applied, I a /I c ⫽0, 0.2,
0.4, 0.6, 0.8, 0.9, 0.97, and 1 共from bottom to top兲 with I c
⫽4ab j c . ⌳/a⫽0, 0.05, and 0.2. n⫽51.
If a vortex with flux ⌽ 0 is trapped at a grid point r j in the
film, the magnetic field is H z (r)⫽⌽ 0 ␦ (r⫺r j )
⬇(⌽ 0 /w j ) ␦ i j at r⫽ri , since we assume here ⌳⫽0. Thus,
g i is just one row of the matrix,
g i ⫽ 共 ⌽ 0 /w j 兲 K i j .
Figure 15 shows the so called washer geometry that is
used for SQUID’s 共Superconducting Quantum Interference
Devices兲,30 namely, a superconducting thin square or rectangle 共or circular disk兲 with a slot and central hole. See Refs.
31–34 for theories of such washers with finite effective penetration depth ⌳. After nucleation at one border of the slot, a
vortex without pinning will move towards a maximum of
g(r). For the square washer in Fig. 15 共top left兲 there is one
flat maximum of g(r) to the left of the central hole. For the
rectangular washer 共bottom兲 there are three such maxima, to
the left of the center and above and below the slot, which are
separated by two saddle points above and below the central
hole. A vortex may thus enter near the middle of the slot,
jump to the nearby maximum of g, and then jump over the
共64兲
In general, the stream function g i is the linear superposition
of the contributions of the applied field and trapped vortices,
and possibly of currents applied by contacts.
The following relations are useful: The difference g i ⫺g j
equals the total current crossing any line connecting the
points ri and r j ; the function ⫾g(r) is the potential in which
a probe vortex 共or flux bundle兲 of appropriate sign moves;
the Lorentz force on a vortex is perpendicular to the contour
lines of g(r); the matrix K i j is proportional to the interaction
energy between two vortices at ri and r j . All these statements apply to any film shape and to any ⌳ if the general
kernel K, Eq. 共49兲, is used. For ⌳⫽0, the kernel K is given
by Eqs. 共58兲 or 共62兲.
Thus, a vortex will nucleate at the position on the film
edge where the stream lines are densest, and then move to an
extremum of g(r) if it is not pinned by material inhomogeneities.
FIG. 13. The nonlinear susceptibility of Fig. 12 plotted as ␹ ⬙
versus ⫺ ␹ ⬘ .
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ERNST HELMUT BRANDT
PHYSICAL REVIEW B 64 024505
FIG. 14. The current stream lines in a thin superconductor rectangle (b/a⫽0.7) with ⌳/a⫽0 共left兲 and ⌳/a⫽0.1 共right兲 at applied perpendicular fields 共from top to bottom兲 H a /J c ⫽0.001, 0.5,
and 1.5.
saddle point to the wide maximum at the left. The fielddriven thermally activated motion of vortices generates lowfrequency noise in the SQUID, which may be suppressed by
introducing pinning centers, e.g., tiny holes.35
In the examples of Fig. 15, ⌳⫽0 共or ⌳Ⰶa) was assumed. In this case the vortices interact with the screening
currents of the applied field and with other vortices only via
the stray field outside the film. In general, at distances r
Ⰶ⌳ the point vortices in thin films interact via a logarithmic
potential similar to the interaction of line vortices in the bulk.
At distances rⰇ⌳ the interaction is mediated by the stray
field, is of long-range, and depends on the shape of the film.
This long range part of the vortex interaction 共or flux-bundle
interaction兲 is given by the integral kernel K, Eqs. 共58兲 and
共62兲. For infinitely extended films the ⌳-dependent vortex
interaction and the magnetic field of a vortex in thin films
(dⰆ␭) were derived by Pearl,12 see also Refs. 2 and 36. For
infinite films of arbitrary thickness the vortex interaction and
magnetic field are obtained in Refs. 37 and 38.
VII. SUMMARY
In this paper finite London penetration depth ␭ is introduced into known continuum methods that compute the electromagnetic response of type-II superconductors in various
geometries. In addition, solution methods for some new geometries are presented, namely, thick and thin strips with
transport current and with both applied current and applied
magnetic field, and thin films of arbitrary shape, e.g., the
washer geometry used for SQUID’s. The inverse integral
FIG. 15. Thin superconducting square (a⫽b) and rectangle
(a/b⫽3) with slit and hole 共washer兲 in the Meissner state with ⌳
⫽0. Shown are the current stream lines for constant applied field
共top left, bottom兲 and for zero applied field but with one or two
vortices trapped at various positions.
kernel K(r,r⬘ ) or the matrix K i j defined in Eqs. 共58兲 and
共62兲 have the simple physical interpretation of the interaction
energy between two point vortices in a film of given shape.
The stream function g(r) of Eqs. 共63兲 and 共64兲 is just the
potential in which a probe vortex 共or flux bundle兲 in the film
moves and which is caused by the applied perpendicular
magnetic field 共and/or applied current兲 and by the other vortices. In the depicted case of small ⌳⫽␭ 2 /d, this interaction
is mediated only via the magnetic stray field outside the thin
film, which crucially depends on the shape of the film. For
finite films with arbitrary ⌳, the vortex interaction is implicitly given by the inverse kernel K(r,r⬘ ), Eq. 共49兲. Explicit
⌳-dependent expressions for concrete film shapes will be
given elsewhere.
ACKNOWLEDGMENTS
The author wishes to acknowledge the hospitality of the
Institute for Superconducting and Electronic Materials, University of Wollongong, Australia, where part of this work
was performed, and financial support from the Australian
Research Council, IREX Program.
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PHYSICAL REVIEW B 64 024505
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