PHYSICAL REVIEW B
VOLUME 55, NUMBER 18
1 MAY 1997-II
ac response of thin superconductors in the flux-creep regime
A. Gurevich
Applied Superconductivity Center, University of Wisconsin, Madison, Wisconsin 53706
E. H. Brandt
Max Planck Institute für Metallforschung, D-70506 Stuttgart, Germany
~Received 25 October 1996!
We calculate both analytically and numerically the ac susceptibility x ( v ) and the nonlinear electromagnetic
response of thin superconductor strips and disks of constant thickness in a perpendicular time-dependent
magnetic field B a (t)5B 0 cosvt, taking account of the strong nonlinearity of the voltage-current characteristics
below the irreversibility line. We consider integral equations of nonlinear nonlocal flux diffusion for a wide
class of thermally activated creep models. It is shown that thin superconductors, despite being fully in the
critical state, exhibit a universal Meissner-like electromagnetic response in the dissipative flux-creep regime.
The expression for the linear ac susceptibility during flux creep appears to be similar to the susceptibility of
Ohmic conductors, but with the relaxation time constant replaced by the time t elapsed after flux creep has
started. This result is independent of any material parameter or temperature or dc field. For v t@1, we obtain
x ( v )'211pln(qivt)/(ivt), where p and q are constants. Above a critical ac amplitude B 0 5B l , the local
response of the electric field becomes nonlinear, and there are two distinctive nonlinear regimes at B 0 .B l ,
where B l ;s(d/a) 1/2B p , B p is a characteristic field of full flux penetration, s(T,B)5 u dlnj/dlntu is the dimensionless flux-creep rate and d and a are the sample thickness and width, respectively. For B l ,B 0 ,B h ( v ) the
response of the electric field is strongly nonlinear but nonhysteretic, since the ac field B a (t) does not cause a
periodic inversion of the critical state. As a result, the magnetic moment exhibits a Meissner-like nondissipative response, in stark contrast to the Bean model. For B 0 .B h ( v ) the ac field causes hysteresis dissipation due
to a periodic remagnetization of the critical state that gives rise to the hysteretic magnetic response of the Bean
model at B 0 @B h . Here B h ( v ) weakly depends on v and is of order (d/a) 1/2B p for a very wide frequency
range, well below the irreversibility field, where s(T,B)!1. Magnetization and ac losses at B 0 @B h are
calculated accounting for the nonlinearity of E(J) at J,J c and a crossover between flux flow and flux creep
at J.J c . All these regimes were confirmed by our computer simulations of nonlinear flux diffusion in strips
and disks. @S0163-1829~97!10917-1#
I. INTRODUCTION
Measurements of the linear ac magnetic susceptibility
x ( v ) of superconductors can provide important information
about mechanisms of vortex dynamics and pinning.1,2 The
experimental data are usually analyzed within the framework
of two opposite theoretical approaches which represent two
limiting regimes of magnetic-flux dynamics in superconductors. The first one is based on the critical state model,3,4
which completely neglects dissipative processes in the subcritical state j, j c by implying a stepwise dependence
j5 j c E/E of the current density j on the electric field E. Here
j c (T,B) is a critical current density which depends on the
temperature T and magnetic field B, but is assumed to be
independent of E. The second approach describes the linear
flux dynamics above the irreversibility line, where the
current-voltage characteristic is Ohmic.1,5–13 These two models correspond to the opposite regimes of very strong and
very weak pinning, respectively. The ac response in the
crossover region about the irreversibility field B i (T) was
considered in Ref. 14.
In high-temperature superconductors the critical state
model remains approximately valid below the irreversibility
field, B,B i (T), although thermally activated flux creep can
considerably affect the macroscopic electrodynamics of these
0163-1829/97/55~18!/12706~13!/$10.00
55
materials.1,2 In the flux-creep regime the E( j) characteristics
becomes strongly nonlinear and can generally be described
by
E ~ j ! 5E c exp„2U ~ j ! /T….
~1!
Here the flux-creep activation barrier U( j) is a nonlinear
function of j such that U( j c )50, and E c is the electric-field
criterion which defines the apparent j c by E( j c )5E c . The
differential resistivity r (E)5 ] E( j)/ ] j equals
r~ E !5
U U
E ]U
E
' .
T ]j
j1
~2!
For any power law U( j), for example, vortex-glass or collective creep models in which U( j)5U 0 ( j c / j) m ~see, e.g.,
Ref. 1!, the resistivity r (E) turns out to be linear in E with
an accuracy to some slowly varying function j 1 (lnE) that
comes from the specific form of U( j) in Eq. ~1!. In a very
wide region of E, this weak dependence of j 1 on E can be
neglected, thus j 1 can be expressed via the observed fluxcreep rate j 1 (T,B)5 u d j/dlntu, except for exponentially small
fields E,E c exp(21/s). Here s(T,B)5 u dlnj/dlntu is the dimensionless flux-creep rate, which is much smaller than
unity well below B i (T). For B!B i , the approximately linear
dependence r }E 121/n , holds for a power law E5( j/ j c ) n E c
12 706
© 1997 The American Physical Society
55
ac RESPONSE OF THIN SUPERCONDUCTORS IN THE . . .
FIG. 1. The functions f s ( h ) and f d ( h ) defined by Eq. ~8! for a
strip and a disk.
as well, since n@1. In this case15–18 j}t 21/(n21) at large
creep times, thus s51/(n21)!1. These E( j) characteristics
comprise all known models of thermally activated flux
dynamics,1,2 for which the linear dependence r (E)}E turns
out to be universal. Therefore, the Maxwell equations with
r 5E/ j 1 provide a model-independent description of the
evolution of E(r,t). This simplifying fact, for example, allows one to calculate flux-creep time constants directly from
the observed macroscopic parameters without need of any
microscopic assumptions or models.15,16
The nonzero resistance r (E) at j, j c gives rise to a time
decay of j(t) and E(t). For instance, if the applied dc magnetic field B a (t) is held constant at times t.0, flux creep
induces an electric field
E 0 ~ h ,t ! 5 a f ~ h ! / ~ t1 t ! ,
~3!
where f ( h ) is a universal function, the prefactor
a 5ad j 1 m 0 /2p depends on the specimen, and
t 5a m 0 j 1 /Ḃ a is a time constant which determines a transient
regime after the linearly increasing field B a (t)5tḂ a was
stopped at t50.15–17 For strips of width 2a and disks of
radius a and of thickness d!a in a transverse field, the
functions f s ( h ) and f d ( h ) are plotted in Fig. 1 versus the
reduced spatial variable h 5x/a and h 5r/a, respectively.
The relaxation of current density or magnetization can be
obtained by substituting Eq. ~3! into specific E( j) characteristics.
An applied ac field B a (t) induces an electric-field perturbation d E(r,t) on a time-dependent background electric field
E 0 (t). The time evolution of E 0 (t) is accompanied by a
corresponding decrease of the differential resistivity r }1/t,
which in turn determines the dynamics of d E. In this case
not only the calculation, but also the notion of a magnetic
susceptibility x ( v ) become nontrivial, although the essential
physics can be clarified by the following qualitative arguments. In Ohmic conductors, the complex susceptibility
x ( v t 0 ) is a function of a dimensionless variable v t 0 , where
t 0 5ad m 0 /2pr is the Ohmic relaxation time constant.2,18
Now if we substitute Eqs. ~2! and ~3! for t@ t and f ;1 in
the t 0 , we obtain that the relaxation time ‘‘constant’’ for
12 707
superconductors is independent of any material properties
and should just be replaced by the time t elapsed after flux
creep has started. This result confirmed by exact
calculations,19,20 implies that the dissipative imaginary part
of x ( v t) decreases as t increases and vanishes at v t→`. In
this limit the real part x ( v t)→21 corresponds to a
Meissner-like response in the dissipative flux-creep regime
due to an ideal skin effect in the limit v t→`, since r }1/t.
This example shows that x of the flux-creep state not only
depends on v , but also changes with time.
Because of the universality of the dynamics of E(r,t), the
problem of finding the proper linear-response function in the
flux-creep regime can be formulated quite generally in terms
of a nonlinear diffusion of the electric field described by the
Maxwell equations, which take account of the strong nonlinearity of E( j) due to flux pinning and thermally activated
flux motion. Depending on the ac frequency v and amplitude
B 0 , this dynamic approach gives a variety of different regimes, from the linear Meissner response at small B 0 to a
hysteretic magnetic behavior of the Bean model. This also
enables us to take into account the important effect of the
sample geometry, which can qualitatively change the character of flux diffusion from a local flux diffusion in a slab in a
parallel magnetic field to a nonlocal flux diffusion in thin
superconductors in perpendicular fields.2,9,16,17 A nonlinear
quasistatic response well below the irreversibility field can
be calculated in the framework of the Bean model for thin
superconductors in a perpendicular field.21–34 For this geometry, the linear ac response, above the irreversibility field was
considered in Refs. 18, 30, 31.
However, the theoretical understanding of ac experiments
performed in the flux-creep regime still remains incomplete,
since the dynamic magnetic response changes with time and
can be both linear and nonlinear, depending on the ac amplitude, B 0 . For small ac signals this problem has been addressed before, in particular, the linear response was calculated both for parallel19 and perpendicular20 geometries, and
the generation of the third harmonics was studied in Refs.
35–37. The parallel field orientation was considered in Refs.
39–43. In this paper, which extends our previous short
communication,20 we consider both linear and nonlinear ac
response for the most common experimental situation of thin
flat superconductors ~strips and disks! in a perpendicular
magnetic ac field. We solve analytically and numerically
the integrodifferential equation of nonlocal flux
diffusion,2,17,18,38 and obtain the ac response in the flux-creep
regime for both small and large ac signals. The paper is
organized as follows.
In Sec. II we give a general linear-response theory for
d E(r,t) in superconductor strips and disks in a strong perpendicular dc magnetic field B dc superimposed with a small
ac component B a (t). It is shown that, unlike linear Ohmic
conductors, the amplitude of the induced ac electric field
d E(t) decreases with time as 1/t, even when the amplitude
of the external ac magnetic field remains constant.
In Sec. III we calculate the ac component of the magnetic
moment d M (t) and find that both the amplitude of d M and
the linear magnetic susceptibility x ( v ) are nearly independent of time t despite the strong decreases of E 0 and d E
during flux creep. Moreover, this x ( v ) for a superconductor
in the completely penetrated critical state, is close to the
12 708
A. GUREVICH AND E. H. BRANDT
susceptibility x 521 of a superconductor in the Meissner
state, which is independent of both v and material parameters. The frequency dependence of x ( v )5 x 8 2i x 9 is similar to that of Ohmic strips and disks, but with the Ohmic
relaxation time replaced by t.
In Sec. IV we consider the transition to the nonlinear
magnetic response for larger ac signals B 0 .B l . We found
two qualitatively different regimes which occur at intermediate and large B 0 . The first one takes place at
B l ,B 0 ,B h ( v ) and corresponds to a highly nonlinear response of d E(r,t), but a Meissner-like nondissipative response in M (t) for which the ac field does not cause a
change of sign of E(x,t) during the ac cycle. Only above the
critical ac amplitude B 0 .B h ( v ), a periodic remagnetization
of the critical state occurs which gives rise to the common
hysteretic behavior of the Bean model.3,4
In Sec. V we present results of computer simulations of
flux creep in strips and disks in the presence of an applied ac
field. We have observed a variety of different dynamic regimes, including the Meissner response at small and intermediate B 0 and a transition to the hysteretic Bean-like response
at higher ac amplitudes. We traced essentially different onsets of nonlinearity in d E and M , in agreement with our
analytical results.
II. LINEAR ac RESPONSE
In this paper we consider flat superconductors in a uniform perpendicular dc magnetic field B dc superimposed with
an ac component B a (t). It is assumed that the superconductor is fully in the critical state, which implies that B dc is
well above the characteristic field of complete flux penetration B p 5 m 0 j c d/ p ~for thin strips!, where j c is the critical
current density and d is the thickness of the sample.2 In this
case one can neglect the effect of the geometrical barrier44–48
and describe the superconductor as a nonmagnetic nonlinear
conductor, i.e., we put B5 m 0 H ~equivalent to a lower critical field H c1 50) and specify a current-voltage law
E5E( j). For instance, in our computer simulations we
chose a conventional power law E( j)5E c ( j/ j c ) n , which implies a logarithmic activation energy U( j)5U 0 ln(jc /j) in Eq.
~1! with n5U 0 /T. For n51 the Ohmic response in the regimes of flux flow or thermally assisted flux flow6 is reproduced, and for n→`, Bean’s critical state model results.3
For this power law E( j), one gets ] E/ ] j5nE/ j}E 121/n / j 1
with j 1 5 j c /(n21), since j' j c during flux creep.
We use the Maxwell equations in an integral form that
describes nonlinear nonlocal diffusion of the electric field
E(r,t) inside thin flat superconductors,9,18
m 0a
E ~ h ,t ! 52 g h aḂ a ~ t ! 1
2p
E
1
0
Q 0 ~ h ,u ! J̇ ~ u,t ! du. ~4!
55
Q 0 ~ h ,u ! 5
SD E
~5!
122sin2 w
dw
~ 12k 2 sin2 w ! 1/2
~6!
h 2u
,
h 1u
u
h
1/2
p /2
k
0
for strips and disks, respectively, with k 2 54 h u/( h 1u) 2 .
Substituting J̇5Ė ] J/ ] E5Ė/ r in Eq. ~4! and using the
approximation r 5E/ j 1 , we obtain the equation which will
be used throughout this paper:2,17,18
E ~ h ,t ! 52 g aḂ a ~ t ! h 1 a
E
1 Ė ~ u,t !
0
E ~ u,t !
Q 0 ~ h ,u ! du,
~7!
with a 5a m 0 J 1 /2p , J 1 5 j 1 d. For Ḃ a 50, Eq. ~7! has a universal solution of the form of Eq. ~3! with17
f ~ h ! 52
E
1
0
Q 0 ~ h ,u ! du.
~8!
For a strip, f ( h )5 f s ( h ) can be calculated explicitly,
f s ~ h ! 5 ~ 11 h ! ln~ 11 h ! 2 ~ 12 h ! ln~ 12 h ! 22 h lnu h u .
~9!
The function f d ( h ) for a disk is given in Ref. 30 and depicted in Fig. 1.
Now we consider the linear ac response d E( h ,t) on the
time-dependent background electric field E 0 ( h ,t). We look
for solutions of Eq. ~7! in the form
E ~ h ,t ! 5E 0 ~ h ,t ! 1 d E ~ h ,t !
~10!
with d E!E 0 . Inserting Eq. ~10! into Eq. ~7!, we obtain the
following linear integral equation for d E with timedependent coefficients:
d E ~ h , u ! 52 p ~ t ! h 1
E
1
0
@~ t 1t ! d Ė ~ u,t !
1 d E ~ u,t !# Q ~ h ,u ! du,
~11!
where Q( h ,u)5Q 0 ( h ,u)/ f (u), and p(t)5 g aḂ a (t).
To solve the nonuniform integral equation ~11!, we write
d E( h ,t) in the form
d E ~ u,t ! 5g ~ u ! ( c n ~ t ! w n ~ u ! .
n
~12!
Here w n (u) and L n are the normalized orthogonal eigenfunctions and eigenvalues of the following linear integral
equation,
w n ~ h ! 52L n
Ew
1
Here J( h ,t)5 j( h ,t)d5 j(E)d is the sheet current, g 51 and
1/2 for strips of width 2a and disks of radius a, respectively,
h 5x/a5r/a is the dimensionless coordinate across the strip
or disk, and Ḃ a (t)5dB a /dt is the ramp rate of the applied
uniform magnetic field B a (t). The kernel Q 0 ( h ,u) depends
on the sample geometry and equals30
U U
Q 0 ~ h ,u ! 5ln
0
E
1
0
Q̃ ~ h ,u ! w n ~ u ! du,
n ~ u ! w m ~ u ! du5 d mn ,
~13!
~14!
where d mn is the Kronecker symbol and Q̃( h ,u)
5Q̃(u, h )5Q( h ,u)g(u)/g( h ) is a symmetrized kernel with
g(u)5 f s (u) 1/2 for a strip and g(u)5 @ f d (u)/u # 1/2 for a disk.
It can be shown that L 1 51, and the first eigenfunction is
ac RESPONSE OF THIN SUPERCONDUCTORS IN THE . . .
55
w 1 (u)5 f s (u) 1/2 for a strip and w 1 (u)5 @ u f d (u) # 1/2 for a
disk. Physically, the w n (u) describe the dissipative modes in
the flux-creep regime, d E n ( h ,t)5 w n ( h ) @ t /(t1 t ) # L n 11 .
Equation ~13! may be evaluated numerically by introducing a grid of N510 . . . 200 points u i , e.g., u i 53x 2i 22x 3i ,
x i 5(i21/2)/N, i51, 2, . . . N.18 This choice of nonequidistant points considerably increases the numerical accuracy
since the derivatives of the eigenfunctions have logarithmic
singularities at u50 and u51. The eigenvalue problem now
amounts to the inversion of the n3n matrix
Q̃ i j 5Q̃(u i ,u j )w j where w j 56x i (12x i ) is the weight function entering the integral due to the substitution. In this way
we obtain the eigenvalues L 1 . . . L 5 51,1.9029, 2.6673,
3.3589, 4.0005 ~1, 1.8156, 2.4885, 3.0932, 3.6561! for the
strip ~disk!, and L n '0.6n for n@1. The eigenfunctions
w n (u) vary rapidly near u50 and have logarithmically diverging slopes at u50 and u51.
The coefficients c n (t) in Eq. ~12! can be found by multiplying Eq. ~11! by w n ( h ) and integrating over h with the use
of Eqs. ~13! and ~14!. This yields
~ t1 t ! ċ n 1 ~ 11L n ! c n 52d n L n p ~ t ! ,
d n5
Ew
n~ h ! h
dh.
g~ h !
1
0
~15!
~16!
Here
the
first
‘‘oscillator
strengths’’
dn
are
d 1 . . . d 5 50.4247, 0.1291, 0.0651, 0.0409, 0.0288, ~0.4476,
0.1308. 0.0661, 0.0425, 0.0310! for the strip ~disk!, and
d n '0.36/n 3/2 for n@1.
Substituting the solution of Eq. ~15! in Eq. ~12!, we obtain
a general solution of the linear-response problem,
d E ~ h ,t ! 52g ~ h ! (
n
3ga
E
t
0
Ḃ a ~ t 8 !~ t 8 1 t ! L n dt 8 .
~17!
Equation ~17! describes time evolution of the induced ac
electric field d E( h ,t) in the flux-creep regime, provided the
external ac field B a (t) was switched on any time after
t50. In particular, for a stepwise increase of B a (t) from 0 to
B s at t5t i , we get Ḃ a (t)5B s d (t2t i ). In this case Eq. ~17!
gives the following long-time asymptotics of d E( h ,t) for
t@ t ,
d E ~ h ,t ! 52 g aB s
~ t i1 t !d 1 f ~ h !
.
t2
FIG. 2. The complex function e( h , v t)5e 8 1ie 9 , Eq. ~21!.
Plotted are e 8 ~solid lines! and e 9 ~dashed lines! versus the spatial
coordinate h 5x/a5r/a at v t53.1, 10, and ` for the disk, and
e 8 at v t5` for the strip ~dotted line!.
c n 52i v jad n L n B 0 e i v t
F
~18!
This formula describes a power-law decay of induced perturbations of d E( h ,t). The corresponding current perturbation
d J(t)5J 1 d E/E 0 }1/t turns out to be independent of the spatial coordinates.
Let us consider in more detail the important case of a
periodic ac signal B a (t)5B 0 exp(ivt) during the steady-state
flux creep (t@ t ) when transient processes can be neglected.
For v t@1, an asymptotic solution of Eq. ~15! can be written
in the form
G
Ln
1
2
1 ... .
L n 1i v t ~ L n 1i v t ! 3
~19!
Substituting Eq. ~19! into Eq. ~12!, we obtain
d E ~ h ,t ! 52
aB 0 exp~ i v t !
e~ h,vt !,
t
e~ h !5gg~ h !
d nL nw n~ h !
~ t 1t ! L n 11
12 709
(n
d nL nw n~ h !
,
11L n /i v t
~20!
~21!
where we retained only the first term in the brackets in Eq.
~19!, thus neglecting higher-order terms in 1/v t!1.
Equations ~19! and ~20! have a number of distinctive features as compared to those for linear Ohmic conductors.
First, although the amplitude of the external signal remains
constant, the amplitude of the induced d E( h ,t) decays with
time as 1/t, since the differential resistivity r (E)5E 0 / j 1
decreases with time as 1/t. Second, the induced d E(t) turns
out to be proportional to B a (t), but not to Ḃ a (t) like in
Ohmic conductors, and the amplitude of d E(t) is independent of v for v t→`. This gives rise to a characteristic
‘‘Meissner’’ ac magnetic response of a superconductor in the
dissipative flux-creep regime.
The universal function e( h ) of Eq. ~20! is shown in Fig.
2 for strips (e s ) and disks (e d ). These profiles of the ac
electric field are related to the profiles of the unperturbed
E 0 } f ( h ), Eq. ~3!, by
e ~ h ,` ! 5 ~ h g 8 / p !~ 12 h 2 ! 21/2 f ~ h ! ,
~22!
where g 8 51 for a strip and g 8 52/p for a disk. From Eq.
~22! it follows that in the ratio d E/E 0 5 d J/J 1 the factor
f ( h )/t cancels, yielding
d J ~ h ,t ! 5
h
2 g 8B a~ t !
,
m 0 d ~ 12 h 2 ! 1/2
~23!
A. GUREVICH AND E. H. BRANDT
12 710
55
which is just the current density which ideally screens the
applied ac field from the interior of the sample.18,30 This
Meissner ac component superimposes to the relaxing background j 0 ( h ,t) which is obtained by inverting the relation
E( j 0 )5E 0 ( h ,t). As seen from Fig. 2, the perturbation
d E( h ,t) at finite v t is rounded and finite at the edges, gaining a dissipative part, since e( h , v t) becomes complex.
Similar profiles of j( h , v ) were obtained in Refs. 2,18 for
the Ohmic strip and disk, and for the rectangle and square in
Refs. 30,31,34.
III. ac MAGNETIC MOMENT
AND LINEAR SUSCEPTIBILITY
Using the above results, we can now calculate the ac component d M (t) of the total magnetic moment
M5(1/2) * @ j„E…3r# d 3 r. Using d j5 j 1 d E/E 0 , we get
d M (t) for a long strip of length L and of width 2a:
d M ~ t ! 52La 2 J 1
E
d E ~ h ,t !
h
dh,
E 0 ~ h ,t !
0
1
~24!
where E 0 ( h ,t) and d E( h ,t) are given by Eqs. ~3! and ~21!,
respectively. For a disk of radius a, the prefactor 2L in Eq.
~24! should be replaced by p a and the weight h by h 2 . For
instance, the induced magnetic moment d M caused by a
stepwise change of B a (t) at t5t i , is obtained from Eqs. ~3!,
~18!, and ~24! in the form
d M ~ t ! 52
2 p d 1 LaB s ~ t i 1 t !
m0
t
~25!
for a strip, and with L replaced by p a/3 for a disk. For
t i @ t , the value d M (t) in Eq. ~25! becomes independent of
superconducting parameters.
To obtain d M for a periodic ac signal, we can use Eq.
~21!, which yields for a strip
d M ~ t ! 52 b s p La 2 m 21
0 B a ~ t ! 1O ~ 1/v t ! .
~26!
Likewise, we obtain for a disk of radius a:
d M ~ t ! 52 b d p a 3 m 21
0 B a ~ t ! 1O ~ 1/v t ! .
~27!
Both parameters b s and b d are given by the sum
b 54 ( d 2n L n .
n
~28!
Evaluating this numerically, we obtain for the strip b s 51
and for the disk b d 532/(3 p 2 )51.08 to eight significant digits on a grid of N5200 points. These values of b coincide
with those obtained for the ideal Meissner state in strips and
disks.18,30
As seen from Eqs. ~26! and ~27!, the small ac component
d M (t) is independent both of the ac frequency v and of the
superconducting parameters. Moreover, unlike d E(t), the
amplitude of d M (t) remains independent of t. In this case a
superconductor is fully in the critical state and exhibits a
Meissner-type magnetic response to small ac magnetic fields
in the dissipative flux-creep regime.
Now we calculate the complex susceptibility,
x ( v )52 d M ( v t)/ d M ( v t→`), where M ( v t→`) is given
FIG. 3. The complex susceptibility x ( v t)5 x 8 2i x 9 Eq. ~30!
for v t>1.
by Eqs. ~26! and ~27!, so that x 521 corresponds to an ideal
diamagnet. Substituting Eqs. ~19! and ~20! into Eq. ~24!, we
obtain
x ~ v t ! 52
4
b
d 2L n
(n 11Ln n /i v t ,
~29!
where b s 51 and b d 532/(3 p 2 )51.08 as above.
The susceptibility ~29! for a nonlinear conductor looks
similar to that of Refs. 30, 31 for the x ( v ) of a linear conductor with t5 t 0 , where t 0 is the Ohmic time constant.
There are, however, essential differences:
~i! The eigenvalues L n and oscillator strengths d n in these
two expresions are obtained from integral equations with different kernels Q 0 ( h ,u) and Q̃( h ,u), respectively. This difference results from the spatial variation of the flux-creep
resistivity r } f ( h ).
~ii! The relaxation time t 0 5ad m 0 /2pr of Ref. 31 can be
complex and frequency dependent, but it does not depend on
space and time. By contrast, t 0 in superconductors is replaced by the time t elapsed after flux creep has started.
~iii! Whereas the susceptibility in Ref. 31 applies to all
frequencies 0< v <`, our present results were derived for
v @1/t, since x ( v t) makes sense only if at least a few ac
cycles are completed during the creep time t.
Evaluating the sum ~29! numerically, we find the following expression accurate with a relative error of ,231022
for v t>10, and ,331024 for v t>100,
x ~ v t ! 5211 p
ln~ qi v t !
.
ivt
~30!
Here p s 50.2804, q s 519.85 for strips, and p d 50.2728,
q d 528.74 for disks. The real and imaginary parts of
x 5 x 8 2i x 9 Eq. ~30! shown in Fig. 3 are given by
x 8 ~ v t ! 5211 p
p
,
2vt
~31!
55
ac RESPONSE OF THIN SUPERCONDUCTORS IN THE . . .
x 9 ~ v ! 5p
ln~ q v t !
.
vt
~32!
This linear ac susceptibility x (i v t) depends only on the
creep time t and the sample shape, but is independent of any
material parameter and of T and B. Such a universality was
also pointed out for long specimens in a parallel field for
which flux diffusion is local.19 Therefore, for the longitudinal
geometry the general asymptotic of x ( v t) at v t@1 is
x ~ v t ! 5211 ~ 12i !
const
Av t
,
~33!
where the constant depends only on the shape of the specimen cross section.
A. Nonlinear nonhysteretic response
We calculate the nonlinear ac response by looking for the
solution of Eq. ~7! for E( h ,t) in the form
E ~ h ,t ! 5 a
The linear magnetic response implies that the induced
electric field d E( h ,t) remains much smaller than the background field E 0 , so that the total field E( h ,t) undergoes only
small temporal oscillations. At larger ac amplitudes we enter
the nonlinear regime for which the induced d E(t) becomes
comparable with E 0 . In this case there are two qualitatively
different nonlinear ac responses characteristic of intermediate B l ,B 0 ,B h and large B 0 .B h ac amplitudes, respectively. For B l ,B 0 ,B h , the value d E( h ,t) becomes of order E 0 , but the signs of E(t) and J(t) do not change during
the ac cycle. By contrast, for B 0 .B h the ac signal causes
periodic remagnetization ~inversion! of the critical state,
similar to that of the Bean model. In the following, we consider these nonlinear regimes separately.
The characteristic ac amplitude B g above which d E becomes of order E 0 , can be estimated by comparison of Eqs.
~21! and ~23! with Eq. ~3!, which yields
B g . m 0 j 1 d/ p .
~34!
The critical amplitude B g is of order sB p , where
B p 5 m 0 j c d/ p is the characteristic field of full flux penetration into thin superconductors,27–29 and s(T,B)5dlnM/dlnt
is the dimensionless flux-creep rate. For example, taking
d51 m m, j c 533106 A/cm 2 , and s50.1, we get B g ;10 Oe.
The value B g corresponds to the onset of nonlinearity in
the entire sample, so the condition B 0 ! m 0 j 1 d provides the
applicability of the linear-response theory for d E and d M or
x . However, as shown below, this global estimate does not
provide the first onset of nonlinearity. The reason is that the
nonlinearity in d E( h ,t) manifests itself first at the sample
edges, thus the local condition d E(1,t),E 0 gives a weaker
criterion B 0 ,B l .(d/a) 1/2B g , which depends on the sample
aspect ratio. At the same time, the strongly nonlinear response in the electric field E( h ,t) at B 0 @B l does not automatically imply a nonlinear response in the observed magnetic moment M (t). As shown below, the approximately
linear Meissner magnetic response x '21 can in fact occur
in a much larger range of amplitudes B 0 <B h
; m 0 j c d 3/2/ p a 1/2 if s(T,B)!1.
f ~ h ! 2h ~ t ! m ~ h ! 1 c ~ h ,t !
e
.
tS ~ h !
~35!
Here h(t)52 p g B a (t)/ m 0 J 1 , and
S~ h !5
E
v
2p
2p/v
e 2h ~ t ! m ~ h ! dt.
~36!
0
Notice that for c 50 and v t@1, the time averaging of
E( h ,t) gives the background electric field E 0 ( h ,t). The
value m( h ) in Eq. ~35! is chosen to satisfy the integral equation
E
1
0
IV. NONLINEAR MAGNETIC RESPONSE
12 711
Q 0 ~ h ,u ! m ~ u ! du52 h ,
~37!
which coincides with the equation for the Meissner screening
sheet current J( h ) at H a ,H c1 . The solution of Eq. ~37! is30
J ~ h ! /H a 5m ~ h ! 5 g 8 h / ~ p A12 h 2 ! ,
~38!
with g 8 51 for a strip and g 8 52/p for a disk. Substituting
Eq. ~35! into Eq. ~7! we obtain the following equation for
c ( h ,t):
E
1
0
Q 0 ~ h ,u ! ċ ~ h ,u ! du5
F
G
f ~ h ! e 2hm1 c
21 .
t
S
~39!
For c 50, the time average of the expression in the brackets
in Eq. ~39! is zero, so for v t@1, the value c ( h ,t) is small
and the nonlinear Eq. ~39! can be solved iteratively in c . As
a first approximation one can therefore put c 50 in the righthand side of Eq. ~39!. The resulting equation solved in the
Appendix, has the following asymptotic solution for a strip
in the harmonic ac field h(t)5h 0 cosvt at v t@1:
c ~ h ,t ! 5
`
4h
p
2
3
A12 h
E
(
2 n51
1 f ~u!
0
sinn v t
nvt
A12u 2 ]
h 2u
2
2
I n @ h 0 m ~ u !#
du, ~40!
] u I 0 @ h 0 m ~ u !#
where I n (x) are modified Bessel functions and
h 0 52 p B 0 /J 1 m 0 is a dimensionless ac amplitude.
As seen from Eq. ~40!, c ( h ,t) decays with time, vanishing at v t→`. Therefore, for v t@1, we can put c 50 in Eq.
~35! which is thus an asymptotic solution of Eq. ~3! describing the nonlinear ac response.
For harmonic ac signal h(t)5h 0 cosvt, Eq. ~35! becomes
E ~ h ,t ! 5 a
f ~ h ! e 2h 0 m ~ h ! cosv t
.
t I 0 @ h 0 m ~ h !#
~41!
To clarify the meaning of c and to trace the onset of nonlinearity, we first consider the linear regime h 0 !1 for which
Eq. ~41! can be expanded in h 0 . Then Eq. ~35! reduces to Eq.
~10!, where
d E ~ h ,t ! 5 @ 2h 0 m ~ h ! cosv t1 c ~ h ,t !# E 0 ~ h ,t ! . ~42!
12 712
A. GUREVICH AND E. H. BRANDT
The first term in Eq. ~42! corresponds to the nondissipative
screening current Eq. ~23! for v t→`. Then we notice that
for h 0 !1, only the term with n51 in Eq. ~40! contributes to
c ( h ,t), since I n (z)}z n at z!1. As a result, c ( h ,t) for a
strip takes the form
c ~ h ,t ! 5
4hh0
sinv t
3A
2 vt
p 12 h
E
1
0
f ~ u ! du
. ~43!
~ h 2 2u 2 !~ 12u 2 !
Hence c ( h ,t) corresponds to the dissipative 1/v t correction
to the Meissner response, which has a p /2 phase shift as
compared to h(t), in agreement with Eq. ~21!.
As follows from Eqs. ~35! and ~43!, the nonlinearity first
manifests itself at the sample edges, where the Meissner
screening current m( h ) Eq. ~38! diverges. So, formally, the
ac response of a thin superconductor is always nonlinear in
the limit d→0. However, when the nonzero sample thickness d or finite v t is accounted for, both m( h ) and c ( h )
attain finite maxima near the sample edges ~see Fig. 2!. Here
m(1)'(a/2p 2 d) 1/2, and the logarithmically divergent integral ~43! is cut off by replacing the upper limit by
12max@ d/a,( v t) 21/2# . With this cutoff, the condition of
linearity at the sample edges, h 0 m(1)!1, yields the criterion
B 0 ,B l . Here the critical amplitude,
B l . m 0 j 1 d 3/2/ p a 1/2
~44!
is by the factor (d/a) 1/2 smaller than B g given by Eq. ~34!,
which corresponds to the onset of nonlinearity in the whole
sample.
Now we consider the nonlinear regime at h 0 .1, for
which Eq. ~41! predicts a strongly anharmonic response in
E( h ,t). For instance, if cosvt.0, the electric field E( h ,t)
becomes exponentially small, whereas for cosvt,0, the
magnitude of E( h ,t),
E ~ h ,t ! ' @ 2 p m ~ h ! h 0 # 1/2E 0 ~ h ,t !
~45!
increases with h 0 and exhibits a 1/t decay characteristic of
the background field E 0 ( h ,t)5 a f ( h )/t. Interestingly,
E( h ,t) given by Eq. ~35! does not change sign for any largeamplitude h(t), thus exhibiting a qualitatively different behavior as compared to the Bean model which predicts a partial or complete periodic remagnetization of the critical state
for any ac amplitude. This point will be addressed in the next
section, where it will be shown that the transition to the
hysteretic Bean response occurs for B 0 .B h when Eq. ~35!
becomes invalid. Nevertheless, there is a pretty wide region
of ac amplitudes, B l ,B 0 ,B h in which Eq. ~35! gives an
asymptotically exact description of a nonhysteretic but
strongly nonlinear ac response in E( h ,t). In the rest of this
subsection we calculate macroscopic characteristics of this
state.
Substituting Eq. ~35! into the current-voltage characteristic J(E)5J c 1J 1 ln(E/Ec), and integrating over h , we obtain
the magnetic moment M (t),
M ~ t ! 5M̃ c 2M 1 ln~ t/t 0 ! 2 z B a ~ t ! ,
M̃ c 5M c 2M 1
E
1
0
~ n 11 ! h n lnI 0 ~ mh 0 ! d h .
~46!
~47!
55
Here M c is the critical state magnetic moment,
M 1 5dM /dlnt is the flux-creep rate, with n 51,
M c 5a 2 LJ c , M 1 5a 2 LJ 1 for a strip of length L and width
2a, and n 52, M c 5 p a 3 J c /3, M 1 5 p a 3 J 1 /3 for a disk of
radius a. The values z 5 p La 2 / m 0 for a strip and
z 532a 3 /(3 p m 0 ) for a disk correspond to the ideal Meissner
response @see Eqs. ~26! and ~27!#, and
t 0 5C m 0 aJ 1 /E c ,
~48!
where J 1 5 j 1 d, and C5C s 50.25 for a strip and
C5C d 50.178 for a disk.17
As seen from Eq. ~46!, the ac component of M (t) is linear
in B a (t) and coincides with that of the ideal Meissner state.
The rest of M (t) exhibits the usual logarithmic time decay
similar to that in the absense of ac field, but with a renormalized M̃ c , which decreases quadratically with B 0 at
h 0 !1 and linearly for h 0 @1. The latter effect gives rise to a
parallel downshift of the M (t) curves with B 0 ~see below!.
Thus, a superconductor being in a dissipative flux-creep regime exhibits a universal Meissner ac response not only for
B 0 ,B l , but also in a highly nonlinear region B l ,B 0 ,B h .
In this case the ac part of M (t) appears to be linear in
B a (t) and decoupled from the logarithmic relaxation of the
background critical state, with the ac field not affecting the
creep rate d ^ M & /dlnt of the averaged ~over some ac periods!
magnetic moment ^ M & . The critical amplitude B h corresponds to the onset of a hysteretic Bean response due to a
periodic remagnetization of the critical state which is considered in the next section.
For the Meissner response, the dissipated power ^ P &
}1/v t averaged over one ac cycle vanishes for v t@1, since
the ac field does not cause the remagnetization of the critical
state. To calculate
^ P & 5t 21
ac
E E
t ac
0
dt
a
2a
J ~ x,t ! E ~ x,t ! dx
for a strip, we first average the product EJ over the ac period
t ac52 p / v , where E( h ,t) is given by Eq. ~41!, and
t
J5J c 1J 1 ln(E/Ec). Then ^ P & 5(1/t ac) * 0acP(t)dt can be written in the form ^ P & 5 P 0 1 ^ d P & , where
P 0 52aL
E
1
0
E 0 ~ h ,t ! J @ E 0 ~ h ,t !# d h
~49!
is the power dissipated in the absence of the ac field. Since
E 0 }1/t decays with time, P 0 }1/t becomes negligible in the
limit v t@1. The dissipated power d P(t) caused by the ac
field is given by
d P ~ t ! 52aL
E
1
0
d h $ @ E ~ h ,t ! 2E 0 ~ h ,t !# J ~ E 0 ! 2J 1 E ~ h ,t !
3@ h 0 mcosv t1lnI 0 ~ h 0 m !# 2 c % .
~50!
To obtain the power ^ d P & averaged over the ac cycle, we
note that for v t@1, the slowly varying factor 1/t in
E 0 ( h ,t) can be regarded as constant. Since ^ c E & 50 and
^ (E2E 0 )J(E 0 ) & 50, we arrive at
55
ac RESPONSE OF THIN SUPERCONDUCTORS IN THE . . .
^ d P & 52aLJ 1
E
1
0
d h E 0 ( h ,t)
G
F
h 0m ~ h ! I 1~ h 0m ~ h !
I 0~ h 0m ~ h !
2lnI 0 „h 0 m( h )… .
~51!
The ac correction ^ d P & is positive and exhibits the same
1/t decay as E 0 (t) and P 0 (t).
Therefore, thin superconductors in the flux-creep state do
not exhibit usual hysteretic ac losses if B 0 ,B h . For this
reason the ac field does not affect the flux-creep rate of the
averaged magnetic moment ^ M & which follows from Eq.
~46!. This behavior is in stark contrast to the well-known
result of the Bean model at small B 0 , for which the dissipated power is P}B 30 for long slabs and cylinders in parallel
field and strips with elliptical cross section in perpendicular
field,4 or P}B 40 for thin strips of constant thickness in perpendicular field,21,32,34 is nonzero for any ac amplitude
B 0 ,B p .
B. Nonlinear hysteretic response
To consider the transition from the above Meissner response to the hysteretic remagnetization of the critical state,
we trace how Eq. ~35! breaks down at larger ac amplitudes
B 0 .B h . The reason is that the r (E)}E used to obtain Eq.
~35! actually corresponds to exponential E –J characteristics,
E(J)5E c exp@(J2Jc)/J1#, for which E does not go to zero at
J50. This fact does not affect the above results as long as
the electric field E( h ,t) remains larger than the threshold
field E min;Ecexp(2Jc /J1), below which the r (E) exhibits a
slightly slower than linear decrease with E, for example, r
}E 121/n and r }Eln212m(E/Ec) for the power-law and
vortex-glass E(J) curves, respectively. However, for large
B 0 , the electric field E( h ,t) given by Eq. ~35! becomes
exponentially small at the sample edges, thus Eq. ~35! is
invalid for B 0 .B h which corresponds to E(1,t)
,E min . A qualitative estimate for B h can be obtained
from Eq. ~35! for mh 0 @1, which yields at the edges
E(1).E 0 exp@22h0m(1)#/A2 p m(1)h 0 , where m(1).(a/
d) 1/2/ p accounts for the finite sample thickness, and
h 0 52 p B 0 /J 1 . Hence, the edge field E(1) becomes smaller
than E min for B 0 .B h , where
B h .B p
A
d 1
a ln~ t/t 0 !
~52!
with t 0 given by Eq. ~48!. The value B h slowly decreases
with time, however for s5J 1 /J c !1, B h is larger by the
factor 1/s than the nonlinearity onset B l for times
t,t 0 exp(1/s) which can be much larger than the time window accessible in magnetization or flux-creep measurements.
In this case the ac field does not cause a remagnetization of
the critical state if B 0 ,B h .(d/a) 1/2B p , while a superconductor exhibits the Meissner-like response considered in the
previous sections. Thus, there is a wide region of ac amplitudes B l ,B 0 ,B h in which the response in E becomes
strongly nonlinear, but the electric field E(x,t) ~and thus also
the current! in the whole sample does not change its sign
during the ac cycle.
12 713
For B 0 .B h , the ac field periodically changes the sign of
the magnetization currents flowing in the lateral regions
l, u h u ,a, with the nonlinear ac ‘‘penetration depth’’
l a 5a2l increasing with B 0 . For s!1, the length l(B 0 ) can
be calculated from the Bean model, which gives the following current distribution in a strip27–29 on the decreasing
branch of B a (t) for h .0,
J ~ h ! 52J c ,
S
h.h0 ,
12 h 20
4
J ~ h ! 5J c 2J c tan21 h 2
p
h 02 h 2
D
~53!
1/2
,
h,h0 ,
~54!
where h 0 51/cosh(Ba/2B p ). From J(l)50, we find the point
x5l at which J( h ) changes sign,
l5a/cosh1/2~ B 0 /B p ! .
~55!
Notice that the width 2l(B 0 ) is larger than the width of the
flux-free region 2a/cosh(Bdc /B p ), which is negligibly small
in the case of a strong applied dc field B dc @B p considered in
this paper. The penetration field is B p 5 m 0 J c / p for strips27
and B p 5 m 0 J c /2 for disks.25 As seen from Eq. ~55!, the width
of the hysteretic lateral regions l a 5a2l'aB 20 /4B 2p grows
quadratically with B 0 at B 0 !B p . For B 0 @B p , the width of
the central nonhysteretic region 2l.2a A2exp(2B0/2B p ) decreases exponentially with B 0 . In this model with d→0, the
value l a is nonzero for any B 0 . For finite d, Eq. ~55! makes
sense only if l a .d, whereas for smaller B 0 for which Eq.
~55! gives l a ,d, the ac field in fact does not cause a periodic
remagnetization of the critical state even at the strip edges.
The critical amplitude B h ;(d/a) 1/2B p above which one has
l a (B 0 ).d, is of the order of the value given by Eq. ~52!,
except for a slowly changing logarithmic factor which accounts for flux creep neglected in the Bean model.
The time evolution of E(x,t) calculated from Eq. ~4! and
shown in Fig. 4 for the power law E5E c (J/J c ) n and different positions x across the strip, shows that the behavior of
E(t) does qualitatively change as x is increased. In the central region u x u ,l, the electric field E(t) oscillates but does
not change its sign. This regime corresponds to the abovedescribed Meissner response, for which the dissipation P
}1/t decreases with t. By contrast, in the lateral regions, the
electric field E(t) periodically changes its sign, giving rise to
a hysteretic remagnetization of the critical state and a constant dissipated power ^ P & . The case shown in Fig. 4 corresponds to the incomplete remagnetization of the critical state
caused by the ac field.
C. Large-amplitude ac signal
We consider the effect of the finite creep rate s(T,B) on
the magnetic moment M and dissipated power P for a largeamplitude ac signal B 0 @B p which causes a periodic remagnetization of the entire sample @ l(B 0 )!a # . In this case one
can neglect the screening of E by the magnetization currents,
so E52a h Ḃ a . We focus here on two characteristic limiting
cases for which the amplitude of the induced ac electric field
can be either much smaller (aḂ 0 !E c ) or larger (aḂ 0 .E c )
than the crossover field E c between flux-flow and flux-creep
regimes. These two regimes are determined by different parts
of the E(J) characteristics which will be considered separately.
A. GUREVICH AND E. H. BRANDT
12 714
55
FIG. 5. The dimensionless function q(b) of Eq. ~60!, for different creep rates s50.01, 0.05, 0.1, and 0.2.
FIG. 4. The time dependence of the electric field E( h ,t) at four
positions h 5x/a50.02, 0.2, 0.76, and 0.91 ~from top to bottom! in
a strip for creep exponent n511. The history of the applied field
B a (t) was as follows: After a period of constant ramp rate Ḃ a that
caused saturation of E5xḂ a and of J5E 1/11'J c , the ramping was
stopped at time t50; at t52, a small ac magnetic field
B 0 sin@v(22t)# with amplitude B 0 50.2 and frequency v /2p 51
was added to the dc field. Note the strong increase of the amplitude
of E from the center to the edge and the periodic change of sign
occurring in the outer region u x u .l, Eq. ~55!.
The case aḂ 0 !E c corresponds to the subcritical region
J,J c for which the E(J) curve can be well approximated by
the power-law dependence E5E c (J/J c ) n with n@1 well below the irreversibility field, B dc!B irr . For a strip, we obtain
M 52a 2 LJ c * 10 (E/E c ) 1/n h d h , and ^ P & 52 ^ M Ḃ a & , where
S D
2nJ c La 2 B 0 v a
M ~ t !5
Ec
~ 112n !
^ P&5
1/n
sin v t,
~56!
1/n
S D
nLaJ c E c G 2 ~ 111/2n ! a v B 0
p ~ 112n ! G ~ 211/n !
Ec
Now we consider the opposite case of large ac signals
aḂ 0 .E c for which M and P are determined by the crossover region between the flux-flow (J.J c ) and flux-creep
(J,J c ) parts of the E(J) curve. A theoretical description of
this crossover region is unknown, so we use here a simple
interpolation formula
E ~ J ! 5E c ln@ 11e ~ J2J c ! /J 1 # ,
~58!
which gives the exponential E2J curve in the subcritical
flux-creep region J,J c 2J 1 and describes the linear fluxflow regime E5(J2J c ) r f for J.J c 1J 1 . Here r f is the
flux-flow resistivity, and E c 5J 1 r f is a characteristic crossover electric field between the flux-flow and flux-creep
regimes.15 For this E(J), we can write M and P for a strip in
the form
M 5a 2 LJ c q ~ b ! ,
P5a 2 LJ c u Ḃ a u q ~ b ! ,
~59!
where b5a u Ḃ a u /E c is the dimensionless ramp rate, and the
function q(b) is given by
q ~ b ! 511sln~ e b 21 ! 2bs
E
1
0
h 2d h
12e 2b h
.
~60!
The asymptotics of q(b) are
1/n11
,
~57!
where G(x) is the gamma function. For n@1, Eq. ~57! gives
the quasilinear dependence of the dissipated power ^ P & on
B 0 and a stepwise magnetization curve M }sin1/n ( v t) close
to that of the ideal Bean model M (B 0 )}sgn(B 0 ) for
B 0 @B p . 3,4 The weak dependence of M on v results from
the finite exponent n. Although weak on the ‘‘human’’ time
scale t,104 2105 s at n@1, this frequency dependence nevertheless results in the vanishing of M in the static limit
v →0, in which a superconductor with flux creep always
exhibits reversible magnetic behavior.
q ~ b ! 511
2sb
,
3
q ~ b ! 511slnb2s/2,
b@1,
b!1.
~61!
~62!
Here q51 corresponds to the Bean model, while the corrections proportional to s5J 1 /J c 51/(n21) describe the contributions of flux creep (b!1) and flux flow (b@1). These
terms make M dependent on the ramp rate and result in a
deviation from the Bean linear dependence of P on Ḃ a . The
function q(b) for different creep rates s is shown in Fig. 5.
55
ac RESPONSE OF THIN SUPERCONDUCTORS IN THE . . .
12 715
FIG. 7. Same as Fig. 6, but for larger amplitude B 0 50.01 and at
times t55.225 and t55.725. Plotted is M and 20Pt.
FIG. 6. Top: The magnetic moment M and dissipation P versus
time t for the same strip with n511 and same B a (t) as in Fig. 4, for
small ac amplitude B 0 50.002. Plotted is M and 33Pt. Bottom: The
spatial profiles of the oscillating sheet current J and electric field
E5J 11 at two times t55.125 ~solid curves! and t55.625 ~dashed
curves!, approximately corresponding to minimum and maximum
E at the edges.
V. COMPUTATION OF PERTURBED CREEP
To check our analytical results for the linear and nonlinear ac response, we have performed computations of flux
creep in a uniform magnetic dc field B dc with an ac field
B a (t) superimposed. We solve numerically the integral
equation ~4!, which applies to thin strips and disks, with the
power law E(J)5E c (J/J c ) n inserted as described in Refs.
18, 30, 33 and 45. Extensions to films of square and rectangular shape34 and to superconductors of finite thickness18,38
were given recently. Using in our computations the reduced
units a5E c 5J c 51, we are left with only two control parameters B 0 and n. The units are thus a for length, J c 5 j c d
for magnetic field and sheet current, aḂ a for electric field,
and J c /Ḃ a for time. Since only the combination E c /J nc enters, choosing a different voltage criterion E c would only
redefine J c and the time scale.
For zero ac field, our simulations confirm the universal
creep law ~3! as described in Ref. 17. When a small ac field
B a (t)5B 0 cos(vt1w) is switched on, then after a transient
time t '1, the ac components of the induced electric field
d E( h ,t), current density d J( h ,t), and magnetic moment
d M (t) approach the universal profiles considered above. For
example, the computed electric field E depicted in Fig. 6 is
the superposition of the unperturbed E 0 } f ( h )/t ~Fig. 1!, and
the oscillating perturbation d E( h ,t) ~Fig. 2!, calculated for
the strip and disk from the eigenvalue problem ~13!. Various
onset times and phase shifts of B a (t) yielded essentially the
same ac response after a transient time of less than one cycle.
In the figures we used B a (t)52B 0 sin„v (t22)… switched on
at time t52 with v 52 p .
The time evolution of the distributions of J( h ,t) and
E( h ,t) for different ac amplitudes B 0 are shown in Figs. 4
and 6–8. The superposition of the decaying background electric field E 0 ( h ,t) and the induced ac field makes the oscillations of the total E( h ,t) asymmetric in time. For small ac
amplitudes B 0 ,B h the perturbations d E( h ,t) are largest
near the edges, where the current density during the lower
half cycle can be suppressed almost to zero, while during the
upper half cycle the ~positive! perturbation adds only a few
percent to J'J c . For B l ,B 0 ,B h , the ac response of the
electric field is strongly nonlinear, but E( h ,t) does not
change its sign during the ac cycle. In this case, which corresponds to the above Meissner regime, the ac component of
M is linear in B 0 , and the dissipation ^ P & averaged over the
ac cycle decreases as 1/t, see Figs. 6–9. Thus, flux creep
suppresses the hysteretic ac losses for B 0 ,B h . Figure 10
shows the magnetic response to a usual ac field ~top! and to
a variable-frequency field B a (t)5B 0 sin(2plnt) over four
time decades ~bottom!. As seen from Fig. 10, the small ac
signal gives rise to a parallel downshift of the M (t) curves,
but it does not affect the relaxation rate d ^ M & /dlnt of the
averaged magnetic moment. Thus our computations are in
full agreement with the above description of the nonlinear
nonhysteretic response. At long creep times the nonlinearity
in the ac response slowly increases since the background
M 0 (t) decreases and eventually can become smaller than the
perturbation d M .
12 716
A. GUREVICH AND E. H. BRANDT
FIG. 8. Same as Fig. 6 but for large amplitude B 0 50.5. Top:
Plotted is M and P. Bottom: Profiles of J and E5J 11 at ten times
t55.1751i/10, i51, 2, . . . 10 during one cycle. At such large ac
amplitudes the asymmetrically oscillating sheet current J( h ,t)
yields a nearly symmetrically oscillating electric field E.
Above a critical amplitude B 0 , the ac field causes a periodic change of the signs of E( h ,t) and J( h ,t) at the sample
edges, the width of this hysteretic region increasing with
B 0 . This case corresponds to a crossover from the Meissner
response to a hysteretic remagnetization of the critical state.
For large B 0 ~Fig. 8! we observe a hysteretic ac response for
55
FIG. 10. Relaxation of the magnetic moment M of a strip in the
presence of an ac field B a (t). Top: B a (t)52B 0 sin(2pt) switched
on at t52 with amplitude B 0 50, 0.005, 0.01, 0.02, and 0.04. Bottom: Relaxation of M over a larger time window for
B a (t)5B 0 sin(2plnt) switched on at t51 with B 0 50, 0.02, and
0.04. Note that for B 0 ,B h the averaged slope d ^ M & /dlnt is independent of B 0 .
which the profiles of E( h ,t) are nearly straight lines in the
remagnetized lateral regions, and the J( h ,t) profiles are
close to those of the Bean model.28,29 Figure 8 clearly shows
the incomplete penetration of the large-amplitude ac field,
for which the dissipated power P(t), unlike in the
dissipation-free Meissner regime, remains finite and constant
during the ac cycle, see also Fig. 9. For large ac amplitudes
B 0 @B l and n@1, the critical state model gives a good quantitative description of the ac response,27–29,32 and the fluxcreep effects can be neglected.
VI. CONCLUDING REMARKS
FIG. 9. The average dissipation ^ P & 5(1/t) * t0 P(t)dt versus
time t for ac amplitudes B 0 50.03, 0.1, 0.2, 0.25, 0.3 with the same
B a (t) as in Figs. 4 and 6–8.
As follows from the results presented above, the strong
nonlinearity of the V-I characteristic in the subcritical flux
creep regime, J,J c , gives rise to a variety of different ac
responses of thin flat superconductors in a perpendicular
magnetic field. At small ac amplitudes, B 0 ,B l
;s m 0 j c d 3/2/ p a 1/2, the linear response depends not only on
frequency but also on time and corresponds to almost perfect
diamagnetism at v t@1. For intermediate ac amplitudes,
B l ,B 0 ,B h (t); m 0 j c d 3/2/ @ p a 1/2ln(t/t0)#, Eq. ~52!, a superconductor exhibits the same Meissner-like response for the
magnetic moment M as a function of the ac field B a (t),
although the response for the electric field E(x,t) becomes
strongly nonlinear in B a (t). This dynamic regime occurs
only for a nonzero frequency v , since B h ( v )→0 for
v →0. For even larger amplitudes, B 0 .B h , the ac field
causes a periodic remagnetization ~inversion! of the critical
ac RESPONSE OF THIN SUPERCONDUCTORS IN THE . . .
55
state, and the superconductor exhibits the usual hysteretic
response of the Bean model. All these regimes were also
observed in our computer simulations of the nonlinear flux
diffusion described by Eq. ~4! with E}J n inserted.
The effect of the flux-creep background is that flat thin
superconductors in perpendicular magnetic fields do not exhibit hysteretic ac losses for small amplitudes B 0 ,B h . This
finding is in contrast to the Bean model, which does not take
into account the nonlinearity of E(J) at J,J c and thus
yields ac losses at any amplitude.
`
e
zcosu
5I 0 ~ z ! 12
(
n51
I n ~ z ! cosn u ,
~A2!
which enables us to integrate Eq. ~A1! over t and obtain
c ~ h ,t ! 5
ACKNOWLEDGMENTS
`
4h
(
p 2 A12 h 2 n51
3
The work of A.G. was supported by the U.S. DOE, Department of High Energy Physics, and by the NSF IRG Program ~DRM-9214707!.
12 717
E
1 f ~u!
0
@ ci~ n v t ! 2ci~ n v t 0 !#
A12u 2 ]
I n @ h 0 m ~ u !#
du, ~A3!
] u I 0 @ h 0 m ~ u !#
u 2h2
2
where ci(z) is the integral cosine function,49
APPENDIX
The linearized integral equation ~39! for c ( h ,t) can be
inverted30 to give
c ~ h ,t ! 5
2h
p 2 A12
3
E
E
h
2
1 f ~u!
0
E
z cost21
0
t
dt.
~A4!
t dt 8
ti
t8
A12u 2 du ]
u 22 h 2
e 2h ~ t 8 ! m ~ u !
du.
]u
S~ u !
~A1!
Now we assume h(t)5h 0 cosvt and use the identity
1
ci~ z ! 5 g 1lnz1
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