APPLIED PHYSICS LETTERS
VOLUME 72, NUMBER 13
30 MARCH 1998
Magnetization curves and hysteretic losses in superconducting films
with edge barrier
I. L. Maksimova) and A. A. Elistratov
Nizhny Novgorod University, Nizhny Novgorod, 603600 Russia
~Received 7 January 1998; accepted for publication 30 January 1998!
The influence of both bulk and edge pinning on the response of a thin-film superconductor to an
oscillating magnetic field is considered. The magnetic-flux-defreezing field and the flux-exit field
are defined. The hysteresis and magnetization curves of a sample are constructed for the entire cycle
of the magnetic field. From this, we obtain the dependence of the hysteresis losses on the field
amplitude. © 1998 American Institute of Physics. @S0003-6951~98!01413-2#
To obtain an unambiguous solution, Eq. ~1! should be
supplemented by conditions reflecting the magnetic history.
For the flux-entry case these conditions are8,9
The hysteresis behavior of low-dimensional type-II superconductors is determined by the joint influence of both
bulk and edge pinning of the Pearl–Abrikosov vortices. It is
known that hysteretic losses take place in hard superconductors for any small amplitude of an alternating magnetic field.1
The surface barrier ~in bulk samples!2 or the edge barrier ~in
low-dimensional superconductors!3,4 causes a significant delay of the flux entry and exit, which results in a threshold
character of the dissipation in type-II superconductors. The
combined influence of both irreversibility mechanisms on the
electromagnetic response of macroscopic superconductors
~with size greatly exceeding the London depth l! was considered in detail by Clem2 within the framework of the local
critical-state model, which applies to longitudinal geometry;
for the discussion of nonlocal effects see Ref. 5. However,
for thin-film superconductors in perpendicular magnetic
field, this important problem was not yet investigated. Some
progress in describing the critical-state structure in lowdimensional superconductors has been reported in cases
when the edge barrier can be neglected.6,7 Meanwhile, the
presence of the edge barrier supplemented by a long-range
intervortex repulsion makes the solution of the problem
rather nontrivial.
We consider a superconducting strip of thickness d(0
<z<d) and of width 2W( u Y u <W), placed in a perpendicular magnetic field H5(0,0,H). When the field exceeds the
first-vortex-entry field H 1 ~which is of the order of 1 kOe in
a sample with smooth edges!,3 then flux penetrates from the
edges and concentrates inside the film. The equilibrium distribution of the magnetic-flux density B(y) and of the sheet
current i(y) is governed by the one-dimensional version of
the Maxwell–London equation, which for sufficiently wide
films reads,
2
E
1
21
i~ t !dt
5H2B ~ y ! ,
t 2y
u i ~ y ! u Þi p ,
u y u ¹ @ Q 1 ,Q 2 # ,
n ~ y ! Þ0,
i ~ y ! 5i p sign y,
~2a!
u y u P @ Q 1 ,Q 2 # ,
~2b!
where sign y denotes the sign of y. Condition ~2a! reflects
the absence of fluxoids inside the region occupied by the
shielding currents i(y).i p 5 j p d, since pinning centers are
ineffective to trap the vortices. According to Eq. ~2b! vortices are concentrated in the region Q 1 < u y u <Q 2 with the
density n(y) necessary to sustain the depinning current i p .
The solution of Eq. ~1! during the increase of field can
be found by means of the Cauchy integral inversion method9
taking into account the conditions ~2!
n~ y !5
1
F0
A
~ y 2 2Q 21 !~ Q 22 2y 2 !
y 2 ~ 12y 2 !
@ H22i p C ~ y 2 !# , ~3!
where
C~ y 2!5
E A
2
Q2
t ~ 12t !
dt
~ t2Q 21 !~ Q 22 2t ! ~ t2y 2 !
2
Q1
.
The distribution of the sheet current, obtained by the direct
inversion of Eq. ~1! has the following form:
i ~ y ! 56
1
2p
A
~ Q 21 2y 2 !~ Q 22 2y 2 !
y 2 ~ 12y 2 !
@ H22i p C ~ y 2 !# .
~4!
Sign ‘‘2’’ corresponds to the region u y u <Q 1 , and ‘‘1’’
corresponds to the region Q 2 < u y u . The size of the vortexfilled region Q 1 < u y u <Q 2 is to be determined from the current distribution specifics. Particularly, the compatibility
condition H52i p C(0), reflecting the symmetry of the problem, eliminates the unphysical current-density singularity at
y50. Another condition follows from the current-density
saturation near the edges: i( u y u '12L/W)'i E , where L
52l 2 /d!W is the effective screening length and i E is the
flux-entry sheet current.8
Thus, the presence of a high edge barrier results in a
nontrivial flux distribution, namely, two strips Q 1 < u y u
<Q 2 , oriented symmetrically along the X axis @see curve 1
in Fig. 1~a!#. Such a state is essentially controlled by edge
~1!
where y5Y /W is a dimensionless coordinate, i(y)' jd, j is
the bulk current density, and the magnetic-flux density per
unit length ~the magnetic induction! B5F 0 n(y) is determined by the fluxoid density n(y).
a!
Electronic mail: ilmaks@phys.unn.runnet.ru
0003-6951/98/72(13)/1650/3/$15.00
n ~ y ! 50,
1650
© 1998 American Institute of Physics
Appl. Phys. Lett., Vol. 72, No. 13, 30 March 1998
I. L. Maksimov and A. A. Elistratov
1651
FIG. 1. Fluxoid distribution inside the film (y.0); H 1 /i p 510. ~a!: ~1! H 1 ,H<H 0 , vortices entry (H increases!; ~2! H df,H<H 0 , frozen–flux regime (H
decreases!; ~3! H ex,H<H df , conserved-flux stage. ~b!: ~4! H * ,H<H ex , vortices exit ( u 2 5 u 2max,1); ~5! H (2)
en ,H<H * , vortices exit ( u 2 , u 2max); ~6!
2H 0 ,H<H (2)
en , antivortices entry.
pinning10 and differs qualitatively from the conventional
Bean critical state.6,7 Quite nontrivial magnetization features
are expected in such a system. Below we will concentrate on
the irreversible characteristics of thin films in an ac field.
During decrease of the external magnetic field from its
maximum value H 0 , the vortices are ‘‘frozen’’ at their initial
localization positions within the region Q 10< u y u <Q 20 @see
curve 2 in Fig. 1~a!#; the index ‘‘0’’ corresponds to the distribution of vortices and currents at H5H 0 . The ‘‘frozen’’
flux will remain until the resulting density of the sheet current i t in the region adjacent to the vortex localization edge
y'Q 20 will reach the magnitude 2i p . The distribution of
the current in this regime satisfies the superposition principle
i t ~ y,H ! 5i t ~ y,H 0 ! 1
H2H 0
i m~ y ! .
H
~5!
Here, i m (y)5H y/ @ 2 p ((12y 2 ) 1/2# is the Meissner component of the total-sheet–current-density i t that satisfies Eqs.
~1!–~2!;8,9 the expression for it in the weak-pinning limit
(H 1 /i p @1) is reported in Ref. 11.
The magnetic-flux-defreezing field H df is found with the
help of Eq. ~5!, by equating i t (Q 20 ,H df)52i p . This is a
necessary condition at which the vortices located in the region u y u 'Q 20 move towards the edge of a film. Taking into
account the obvious equality i t (Q 20 ,H 0 )5i p one finds:
H df5H 0 24 p i p
A
12Q 220
.
Q 220
~6!
We should emphasize that the obtained result is valid for an
arbitrary bulk pinning intensity.
Let us describe the field-decrease stage taking into account the flux-defreezing effect. With a goal to obtain the
new distribution of flux and current density, we should
modify conditions ~2a! and ~2b! correspondingly:
n ~ y ! 50,
i ~ y ! Þi p sign y,
u y u P @ 0,Q 10# ø @ Q 2 ,1# ,
~7a!
n ~ y ! 5n 0 ~ y ! ,
i ~ y ! Þi p sign y,
n ~ y ! Þn 0 ~ y ! ,
i ~ y ! 52i p sign y,
u y u P @ Q 10 ,Q 0 # ,
~7b!
u y u P @ Q 0 ,Q 2 # .
~7c!
The exact distributions n(y) and i(y) can be found by solving Eqs. ~1! and ~7! with the help of the Cauchy integral
inversion method.9 It is not too instructive to give here a
quite cumbersome closed-form analytical solution in the case
of arbitrary pinning strength. Therefore, in what follows we
prefer to present only final results of the calculation.
In accordance with the exact solution during further drop
of a field @H<H df the area of ‘‘frozen’’ flux concentration
(Q 10< u y u <Q 0 ) is narrowed down, while the area of ‘‘defrozen’’ flux (Q 0 < u y u <Q 2 ) expands both inward (Q 0
<Q 20) and outward ~Q 2 .Q 20 ; see curve 3 on the Fig.
1~a!#. Inside the region Q 0 < u y u <Q 2 , one obviously has
i t (y)52i p , which corresponds to the critical state during
the ‘‘vortices exit.’’ The described vortex behavior is governed by the flux conservation condition: F in
t (H 0 )5F t (H),
which taking account of the ‘‘area rule’’ @see Fig. 1~a!# reduces to the equation
E
Q 20
Q0
n 0 ~ y ! dy5
E
Q2
Q0
n ~ y ! dy.
~8!
Detailed analysis shows that this condition can be maintained only in a certain interval of fields H ex,H,H df . At
H,H ex the total flux trapped in the film starts to reduce
appreciably with the decrease of H, so H ex can be defined as
the field of vortex exit out of the sample. Numerical solution
of Eq. ~8! shows that at the flux-exit field, the vortex distribution edge (y5Q 2max) may not coincide with the film edge
(Q 2max,1) when the bulk pinning strength is sufficiently
high. This is in contrast with the vortex-exit conditions in the
mixed state.3,4 The set of fluxoid–density profiles for H
<H ex is shown in Fig. 1~b! ~see curves 4 and 5!.
For H<H ex , the flux-defreezing boundary Q 0 decreases
monotonically; Q 2 stays constant in some interval of fields
(H * ,H<H ex) but further reduces towards the sample center ~at H<H * ! until vortices of opposite sign ~antivortices!
start to enter the sample at H<H (2)
en @see distribution 6 in
Fig. 1~b!#. Calculations show that H (2)
en (H 0 ) dependence is
rather weak; this reflects the negligible effect of the almost
totally extinguished remnant flux at the field H5H (2)
en .
As a result of the annihilation of vortices and antivortices, the magnetization curve 2M (H) exhibits a sharp bend
near H (2)
en ~see Fig. 2!. Correspondingly, a sharp decrease of
the trapped flux takes place during the decrease of H in the
same region, see Fig. 3. It is obvious that in the field interval
H ex,H<H 0 , the magnetization changes have reversible
character; hence, dissipation losses are practically absent ~if
we neglect viscous losses occurring at fields H ex,H<H df!.
Thus, during field decrease the dominant contribution to the
energy dissipation caused by magnetic hysteresis and vortex
annihilation in the film takes place in the field range 2H 0
1652
Appl. Phys. Lett., Vol. 72, No. 13, 30 March 1998
FIG. 2. Magnetization curves; H 1 /i p 55. ~1! H 0 515i p . ~2! H 0 58i p . Arrows indicate the characteristic fields for the case 1.
<H<Hex . Similar analysis of the situation during the field
increase from H52H 0 to H51H 0 yields the dominant
contribution of the field interval H (2)
ex <H<H 0 to the hysteresis loss power, where H (2)
is
the
field of antivortex exit.
ex
The behavior of the magnetization curve 2M (H) and of
hysteresis curve F(H) under cyclic field variation in the
interval 2H 0 <H<H 0 is presented in Figs. 2 and 3 for finite
bulk pinning H 1 /i p 55. Note the existence of linear portions
in 2M (H) at H df,H,H 0 @see Eq. ~3!# and at 2H 0 <H
<H (2)
ex . These linear parts are explained by the dominating
contribution of the Meissner currents to the sample magnetization in the above field range. In the case of the extremely
large oscillation amplitude H 0 /i p 515, this linear behavior is
replaced by a pronounced plateau in the field range H (2)
ex
<H<0 that emerges in accordance with the Faraday law
due to the redistribution of the trapped flux inside the film. It
is seen from Fig. 2 that the magnetization curve 2M (H) in
the high-amplitude case has the shape of a curved carpet.
Such a profile essentially differs from the ‘‘pillow-shaped’’
magnetization curve typical for thin films without edge bar-
I. L. Maksimov and A. A. Elistratov
rier ~the Bean case!6,7,12 or for bulk samples.1
The behavior of the hysteresis curve F(H) ~see Fig. 3!
in the upper half-plane (F.0), resembling the vertical tail
of a plane ~‘‘plane-tail’’ feature!, differs qualitatively from
the corresponding curve in bulk superconductors,2 similar to
a whale-tail profile ~‘‘whale-tail’’ feature!. We wish to emphasize that this unusual behavior of the hysteresis curve in
films is mainly due to the presence of edge pinning.3 Indeed,
in the close vicinity of the flux-exit field H ex( u H ex
2H u !H ex) one has dF b /dH'0 ~and d 2 F b /dH 2 ,0! for
bulk samples,2 while for films with an edge barrier one has
dF f /dH52 m 0 W.0.3,4
The energy Q(H 0 )51/(4p )rHdF, dissipated per unit
length of the film during one cycle of quasistationary variation of the field, exhibits a threshold behavior with respect to
the field magnitude H 0 . This reflects the edge-pinning specifics, since the losses on magnetic-field reversal arise only if
H 0 .H 1 . 2,3 For H 0 2H 1 !H 1 the dependence Q(H 0 ) is
practically linear, approaching the asymptotical behavior
Q(H 0 );(H 0 2H 1 ) r with r'0.7 at H 0 @3H 1 . Comparison
with the results of Ref. 6, revealing nonthreshold behavior
Q(H 0 );H q0 ~where q54 for H 0 !i p , and q'1 for H 0 @i p !
in films without an edge barrier, demonstrates clearly the
edge-barrier effect upon the dissipative characteristics of
low-dimensional superconductors.
In conclusion, we propose the generalized critical-state
model including both bulk and edge pinning in lowdimensional superconductors. This model allows us to evaluate quantitatively the magnetic and dissipative characteristics
of thin-film superconductors.
The authors are grateful to Dr. G. M. Maksimova for
useful discussions of the model used, and to Professor J.
Clem for permanent interest in the work and for correspondence. The authors are obliged to Dr. E. H. Brandt for reading the manuscript, for valuable comments, and for numerous suggestions to improve its contents. The work is
supported by the Science Ministry of Russia ~Project No.
98-012! and the Russian Foundation for Basic Research
~Grant No. 97-02-17437!. Partial support of the International
Center for Advanced Studies ~INCAS! through Grant No.
97-2-10 is gratefully acknowledged.
1
FIG. 3. Hysteresis curves; H 1 /i p 55. ~1! H 0 515i p . ~2! H 0 58i p . ~3! H 0
56i p . Arrows indicate the characteristic fields for the case 1.
A. V. Gurevich, R. G. Mints, and A. L. Rakhmanov, Stability of Composite Superconductors ~Nauka, Moscow, 1987!, p. 28.
2
J. R. Clem, J. Appl. Phys. 50, 3518 ~1979!.
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