JOURNAL OF APPLIED PHYSICS
VOLUME 90, NUMBER 7
1 OCTOBER 2001
Magnetic response of hard superconductors subjected to parallel rotating
magnetic fields
A. F. Carballo-Sánchez, F. Pérez-Rodrı́guez,a) and A. Pérez-González
Instituto de Fı́sica, Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, Puebla 72570, México
共Received 7 June 2001; accepted for publication 13 July 2001兲
The manifestation of flux-line cutting in the magnetic behavior of a type-II superconductor, either
共i兲 subjected to a rotating magnetic field, or 共ii兲 undergoing slow oscillations in a static magnetic
field, is investigated theoretically. We have applied both the generalized double critical-state model
and the two-velocity hydrodynamic one to interpret available experimental results for oscillating
disks of Nb. The hydrodynamic model generates only closed hysteresis loops, after the first full
oscillation, in accordance with the experimental hysteresis loops observed at a relatively small
amplitude of oscillation, ␪ max⫽45°. However, at larger amplitudes of oscillation, several measured
loops are evidently open. This behavior as well as their asymmetric form could be reproduced only
by the generalized double critical-state model. The limits of applicability of both models are
discussed. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1400094兴
I. INTRODUCTION
value. Within the framework of the generalized double
critical-state model 共GDCSM兲 the electric fields, which result from time-varying B profiles, obey Faraday’s law and
continuity across boundaries; this is not generally satisfied
by the original DCSM.12 A clear evidence of the improvement of the GDCSM in describing the magnetic response of
type-II superconductors, as is compared with the DCSM, is
the situation where the direction of the applied magnetic field
of fixed intensity oscillates with large amplitude. In this case
the GDCSM yields open magnetization curves in agreement
with experimental hysteresis curves,6,12 whereas the original
DCSM predicts closed hysteresis loops.6 GDCSM has been
widely exploited for finding the magnetic response of hard
superconductors in the presence of flux-line cutting effects
共see, for example, Refs. 15–21兲.
The magnetic behavior of type-II superconductors is so
complex that it cannot be explained completely by the
GDCSM. So, for example, this phenomenological model
fails to account for the intricate evolution and quasisymmetric suppression of the magnetic moment of hard superconductors under the action of a dc-bias magnetic field H z and a
sweeping field H y perpendicular to it ( 兩 H y 兩 ⱗH z ). 22–30 The
disagreement between the GDCSM and experiment is particularly significant at small values of the oscillating component H y of the applied magnetic field, i.e., when the tilt of
the total field is small ( 兩 H y 兩 ⰆH z ). The observed quasisymmetric suppression of the magnetic moment could be explained by employing another approach, namely the twovelocity hydrodynamic model 共TVHM兲.30–33 The success of
the TVHM is due to the generation of so-called collapse
zones, where the magnitude B(x) of the magnetic induction
B is homogeneous, as well as zones where its behavior is
similar to that predicted by the GDCSM.
Recently,34 both the TVHM and GDCSM were applied
to interpret magnetization curves of superconducting plates,
cooled in a fixed magnetic field H z parallel to their plane
and, later, subjected to cycles of a transverse magnetic field
H y with a large amplitude H y,max⬎Hz . There, it was shown
During more than two decades, many research groups
have tried to explain the complicated magnetic behavior of
irreversible type-II superconductors under the action of an
applied magnetic field which is made to vary in strength as
well as in direction. Despite the effort of these scientific
groups, the magnetic response of hard superconductors still
remains a very intriguing topic. At present, it is well established that the magnetic behavior of these materials is controlled by two fundamental processes: flux pinning and fluxline cutting. Flux pinning is due to the presence of
inhomogeneities in the material and gives rise to the critical
state1 in which the current density J⬜ perpendicular to the
magnetic induction B attains its maximum value Jc⬜ . The
other fundamental process is connected with the cross joining of adjacent nonparallel vortices at their intersection.2,3 In
the regions, where flux-line cutting occurs, the current density has a component parallel to the magnetic induction 共B兲
and the direction ␣ of B varies spatially. LeBlanc and coworkers in a set of works,4 –7 developed a model by postulating that both the gradients in the density of the magnetic
flux 共dB/dx for the planar geometry兲 and in the angular orientation of the flux lines (d ␣ /dx) exist in critical states. The
double critical-state model 共DCSM兲 allows us to reproduce
the majority of their extensive data for different situations
and geometries. In particular, the DCSM explains satisfactorily the hysteresis losses of hard superconductors, which undergo slow oscillations in a static magnetic field6 or are subjected to a time varying magnetic field orthogonal to a static
bias field.7
The double critical-state model was improved by Clem
and Pérez-González8 –14 by allowing the existence of metastable zones where the magnitude of one or both current
densities, perpendicular (J⬜ ) and parallel (J 储 ) to the magnetic induction B, are smaller than the corresponding critical
a兲
Electronic mail: fperez@acuario.ifuap.buap.mx
0021-8979/2001/90(7)/3455/7/$18.00
3455
© 2001 American Institute of Physics
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3456
J. Appl. Phys., Vol. 90, No. 7, 1 October 2001
Carballo-Sánchez, Pérez-Rodrı́guez, and Pérez-González
that the collapse zones, predicted by the TVHM, disappear
when the magnitude of H y is large enough ( 兩 H y 兩 ⲏH z ) and,
consequently, their effect on magnetization curves M z (H y ) is
not appreciable. In this case, the hysteresis curves of
M z (H y ), generated by the hydrodynamic model and the generalized double critical-state approach, have a similar ‘‘butterfly’’ behavior and qualitatively reproduce experimental
data.
The aim of the present work is to study the magnetic
response of hard superconductors under the action of a rotating magnetic field. Here we will apply both competing models: the generalized double critical-state approach 共Sec. II兲
and the two-velocity hydrodynamic one 共Sec. III兲. In particular, it is of great interest to verify whether the hydrodynamic
model can also explain the magnetic behavior of hard superconductors in a magnetic field of fixed strength, which oscillates with a large amplitude and, hence, flux-line cutting effects are well developed. We will analyze magnetization
hysteresis loops for different amplitudes of oscillation and
compare our results with related experiments6 共Sec. IV兲. The
limits of applicability of both the TVHM and the GDCSM
are discussed.
II. GENERALIZED DOUBLE CRITICAL-STATE MODEL
Consider an irreversible type-II superconducting slab,
whose surfaces are located at x⫽0 and x⫽d. The slab is
subjected to an external field parallel to the y-z plane
Ha ⫽H a ␣ˆ s ,
␣ˆ s ⫽ 共 ŷ sin ␣ s ⫹ẑ cos ␣ s 兲 .
共1兲
Here H a and ␣ s are the magnitude and the tilt angle of Ha
with respect to the z axis. It is assumed that the magnetic
induction B⫽ ␮ 0 H inside the sample, and any surface barriers against flux entry of exit, are neglected. For the planar
geometry, the magnetic induction depends only upon the coordinate x and time t. Thus
B共 x,t 兲 ⫽B 共 x,t 兲 ␣ˆ 共 x,t 兲 ,
共2兲
where B(x,t) is the magnitude of B(x,t) and
␣ˆ 共 x,t 兲 ⫽ŷ sin ␣ 共 x,t 兲 ⫹ẑ cos ␣ 共 x,t 兲
共3兲
is its direction.
After writing the current density J and the electric field
E in terms of their components parallel and perpendicular to
the local field B 共2兲, we get
J⫽J 储 ␣ˆ ⫹J⬜ 共 ␣ˆ ⫻x̂ 兲 ,
共4兲
E⫽E 储 ␣ˆ ⫹E⬜ 共 ␣ˆ ⫻x̂ 兲 .
共5兲
Using Eqs. 共4兲, 共5兲, Ampere’s law (J⫽ⵜ⫻H), and Faraday’s law (ⵜ⫻E⫽⫺ ⳵ B/ ⳵ t), one obtains
⳵B
⫽⫺ ␮ 0 J⬜ ,
⳵x
共6兲
⳵␣
⫽ ␮ 0J 储 ,
⳵x
共7兲
B
⳵ E⬜
⳵␣
⳵B
⫹E 储
⫽⫺
,
⳵x
⳵x
⳵t
共8兲
E⬜
⳵␣ ⳵E储
⳵␣
⫺
⫽⫺B
.
⳵x
⳵x
⳵t
共9兲
The generalized double critical state model12 employs
Eqs. 共6兲–共9兲 and makes use of the assumption that the electric field E 共5兲 behaves as
E⬜ ⫽
and
E 储⫽
再
再
␳⬜ 关 兩 J⬜ 兩 ⫺J c⬜ 兴 sign共 J⬜ 兲 ,
兩 J⬜ 兩 ⬎J c⬜ ,
0,
0⭐ 兩 J⬜ 兩 ⭐J c⬜
␳ 储 关 兩 J 储 兩 ⫺J c 储 兴 sign共 J 储 兲 ,
兩 J 储 兩 ⬎J c 储 ,
0,
0⭐ 兩 J 储 兩 ⭐J c 储 .
.
共10兲
共11兲
Here J c⬜ and J c 储 are the critical current densities, across and
along B, respectively. The parameter J c⬜ (J c 储 ) determines
the threshold for depinning 共flux-line cutting兲 of 共in兲 the vortex array.8 The quantities ␳⬜ and ␳ 储 are effective flux-flow
and flux-line-cutting resistivities of the material. The system
of Eqs. 共6兲–共9兲, 共10兲, and 共11兲 is solved numerically for slow
variations of the external field Ha 共1兲, i.e., for small components of the induced electric field: 兩 E⬜ 兩 Ⰶ ␳⬜ J c⬜ and 兩 E 储 兩
Ⰶ ␳ 储 J c 储 . The resulting spatial distributions of B and ␣ are
then nearly relaxed and essentially independent of ␳⬜ and
␳储 .
III. TWO-VELOCITY HYDRODYNAMIC MODEL
This theoretical model for describing the magnetic response of hard superconductors was proposed in Refs. 31
and 33. Within this macroscopic approach, the vortex system
inside a hard superconducting slab (0⬍x⬍d), which is subjected to an external field Ha 共1兲, is characterized by the
mean vortex density n(x,t) and two velocities, namely the
average translational V(x,t) and relative U(x,t) velocities.
The introduction of the velocity U is a fundamental requirement for taking into account flux-line cutting effects.31 Because of the use of V and U, all the vortices are separated
into two groups, A and B, with different velocities
V A ⫽V⫹
U
,
2
V B ⫽V⫺
U
2
共12兲
and distinct tilt angles ␣ A (x,t) and ␣ B (x,t) with respect to
the z axis, respectively. For simplicity, the vortex densities,
n A (x,t) and n B (x,t), in both groups are assumed to be equal
to n(x,t)/2 (n A ⫽n B ⫽n/2), where the total density n(x,t)
satisfies the continuity equation
冋
册
⳵n
⳵
共 V A ⫹V B 兲
⫽⫺
n
.
⳵t
⳵x
2
共13兲
In what follows, it is convenient to characterize the system
by the average angle ␣ (x,t)⫽ 关 ␣ A (x,t)⫹ ␣ B (x,t) 兴 /2 and the
difference ⌬ ␣ (x,t)⫽ ␣ A (x,t)⫺ ␣ B (x,t) instead of the mean
vortex angles ␣ A and ␣ B for the groups A and B. So, the
equation for the angle transport can be written as33
⳵n␣
1 ⳵ 关 n ␣ 共 V A ⫹V B 兲兴 1 ⳵ 关 n⌬ ␣ 共 V A ⫺V B 兲兴
⫽⫺
⫺
.
⳵t
2
⳵x
4
⳵x
共14兲
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J. Appl. Phys., Vol. 90, No. 7, 1 October 2001
Carballo-Sánchez, Pérez-Rodrı́guez, and Pérez-González
Transport Eqs. 共13兲 and 共14兲 are complemented by electrodynamic equations, which are derived from the condition
of the force balance for each of the intersecting vortex lattices A and B.33 One of the electrodynamic equations is given
by
⳵B
␮ 0 J c⬜
⫽⫺
关 F 共 V A 兲 ⫹F 共 V B 兲兴 .
⳵x
2
共15兲
Here B is the modulus of the magnetic induction B(x,t) 共2兲
and is proportional to the vortex density
B 共 x,t 兲 ⫽⌽ 0 n 共 x,t 兲 ,
共16兲
where ⌽ 0 is the magnetic flux quantum. The quantity J c⬜ in
Eq. 共15兲 denotes the critical current density perpendicular to
B, and the function
F 共 ␷ 兲 ⫽sign* 共 ␷ 兲
共17兲
coincides with sign共␷兲 everywhere except at ␷⫽0, where
sign共0兲 lies within the interval 共⫺1,1兲.
The other electrodynamic equation has the form33
⌬␣B
⳵␣
⫹p
⳵x
冑
n
关 B⫺ ␮ 0 H a cos共 ␣ ⫺ ␣ s 兲兴 ⌬ ␣ 2
8
⫻sign共 V A ⫺V B 兲
⫽⫺ ␮ 0 J c⬜ 关 F 共 V A 兲 ⫺F 共 V B 兲 ⫹p sign共 V A ⫺V B 兲兴 ,
共18兲
where the quantities H a and ␣ s are the magnitude and the tilt
angle, respectively, of the external applied field Ha 共1兲 parallel to the surfaces of the superconducting slab. The parameter p in Eq. 共18兲 denotes the averaged probability of fluxline cutting at the vortice intersection.
The system of nonlinear Eqs. 共13兲–共15兲 and 共18兲 relates
five functions to be found: V A , V B , B⫽n⌽ 0 , ␣, and ⌬␣.
Therefore, an additional equation is needed to close such a
system. In Refs. 31 and 33, it was assumed that the difference ⌬␣ is proportional to the derivative of the average angle
⌬ ␣ ⫽⫺l sign共 V A ⫺V B 兲
⳵␣
,
⳵x
共19兲
where the phenomenological parameter l symbolizes the
mean free path of vortices between two successive intersection 共flux-line cutting兲 events. Labeling, without loss of generality, the vortex sublattices A and B at each point x so that
⌬ ␣ ⫽ ␣ A ⫺ ␣ B ⬍0 关i.e., sign(V A ⫺V B )⫽sign( ⳵ ␣ / ⳵ x)兴, and
substituting Eq. 共19兲 into Eq. 共18兲, we get
再
B⫺
l
l*
冎冉 冊 冉 冊
冉 冊册
关 B⫺ ␮ 0 H a cos共 ␣ ⫺ ␣ s 兲兴 l
⫽ ␮ 0 J c⬜
冋
⳵␣
⳵x
2
⳵␣
F 共 V A 兲 ⫺F 共 V B 兲 ⫹p sign
⳵x
sign
,
⳵␣
⳵x
2 冑2
p 冑n
.
Here, however, the nonlinear system of Eqs. 共13兲–共15兲 and
共20兲 will be solved numerically for arbitrary values of the
ratio l/l * . With this aim, it is convenient to replace Eq. 共17兲
by a relation between ␷ and F as
␷⫽
再
␯ 关 兩 F 兩 ⫺1 兴 sign共 F 兲 ,
兩 F 兩 ⬎1,
0,
0⭐ 兩 F 兩 ⭐1,
共22兲
where ␯ is an auxiliary parameter. In the case when the external applied field Ha varies so slowly that the velocities V A
and V B are much smaller than the parameter ␯ 共V A Ⰶ ␯ , V B
Ⰶ ␯ 兲, the calculated profiles of B(x) and ␣ (x) are quasirelaxed and independent of the chosen value for ␯ .
IV. COMPARISON WITH EXPERIMENT
In this section we will apply both the double criticalstate model and the two-velocity hydrodynamic approach for
explaining some experimental results of Ref. 6, where the
magnetic behavior of superconducting disks of Nb oscillating slowly over various angular displacements in static magnetic fields Ha directed parallel to the disk plane 共in the z
direction兲 and perpendicular to the axis of rotation was investigated. Panels 共a兲 in Figs. 1– 4 show the quantities 具 B y 典
and ⫺ ␮ 0 具 M z 典 ⫽ ␮ 0 H a ⫺ 具 B z 典 versus the angle ␪ of rotation,
which were obtained in Ref. 6 by continously measuring the
average spatial components of the magnetic induction, 具 B y 典
and 具 B z 典 , for a disk of thickness d⫽0.25 mm. The measurements started in the nonmagnetic initial state which is
reached after cooling the superconducting disk in the fields
␮ 0 Ha ⫽0.261 T 共Figs. 1 and 2兲 and 0.149 T 共Figs. 3 and 4兲
through the transition temperature T c . The oscillation amplitudes ␪ max are 45° and 120°, respectively.
In order to apply the GDCSM 共Sec. II兲 and the TVHM
共Sec. III兲 to the experiment 共Ref. 6兲 we can fix the sample
and rotate the external magnetic field Ha ⫽H a ␣ˆ s 共1兲 through
an angle ␣ s ⫽⫺ ␪ instead of holding the magnetic field fixed
and rotating the sample. Indeed, the experimental quantities
具 B y 典 and ⫺ ␮ 0 具 M z 典 correspond, respectively, to the theoretical quantities
具 B ⬘y 典 ⫽
1
d
冕
d
0
dxB ⬘y 共 x 兲 ,
⫺ 具 ␮ 0 M z⬘ 典 ⫽ ␮ 0 H a ⫺
1
d
共23兲
冕
d
dxB z⬘ 共 x 兲 ,
共24兲
B ⬘y ⫽ ␣ˆ s ⫻x̂•B⫽B 共 x 兲 sin关 ␣ 共 x 兲 ⫺ ␣ s 兴 ,
共25兲
B z⬘ ⫽ ␣ˆ s •B⫽B 共 x 兲 cos关 ␣ 共 x 兲 ⫺ ␣ s 兴 .
共26兲
0
where
共20兲
where
l *⫽
3457
共21兲
As was estimated in Refs. 31 and 33, the parameter l is
of the order of l * (l⬃l * ). If we assume l⫽l * in Eq. 共20兲,
we get the system of equations employed in Refs. 30 and 34.
In the GDCSM 共Sec. II兲, the spatial distributions B(x)
and ␣ (x), and the average values 具 B ⬘y 典 共23兲 and ⫺ 具 ␮ 0 M z⬘ 典
共24兲 were calculated by using B dependencies for the critical
current densities, perpendicular and parallel to B, given
by6,12
冉
J c⬜ 共 B 兲 ⫽J c⬜ 共 0 兲 1⫺
冊
B
,
B c2
共27兲
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3458
J. Appl. Phys., Vol. 90, No. 7, 1 October 2001
Carballo-Sánchez, Pérez-Rodrı́guez, and Pérez-González
FIG. 1. Locus of experimental values of 具 B y 典 vs ␪ for nonmagnetic inilial
state 共Ref. 6兲 and the corresponding 具 B ⬘y 典 ( ␪ ) calculated with the generalized
double critical-state model 共b兲 and the two-velocity hydrodynamic one 共c兲.
Here ␮ 0 H a ⫽261 mT, ␪ max⫽45°.
J c 储 共 B 兲 ⫽k c 储 共 0 兲
冉
B
B
1⫺
␮0
B c2
冊
共28兲
with J c⬜ (0)⫽1.6⫻109 A/m2 , k c 储 (0)⫽5.2⫻104 rad/m, and
B c2 ⫽0.35 T. The results for 具 B ⬘y 典 and ⫺ 具 ␮ 0 M z⬘ 典 are presented in panels 共b兲 of Figs. 1– 4.
In the hydrodynamic model 共Sec. III兲, we used the same
critical current for depinning of flux lines as in the GDCSM
关Eq. 共27兲兴, and after several trial calculations we chose a
probability of flux-line cutting p⫽0.01 and a mean free path
of vortices l⫽0.4l * 关see Eqs. 共19兲–共21兲兴. Shown in panels
共c兲 of Figs. 1– 4 are the curves for 具 B ⬘y 典 ( ␪ ) and ⫺ 具 ␮ 0 M z⬘ 典
⫻( ␪ ) corresponding to the TVHM.
FIG. 2. Locus of experimental values of ⫺ ␮ 0 具 M z 典 vs ␪ for nonmagnetic
initial state 共Ref. 6兲 and the corresponding ⫺ ␮ 0 具 M z⬘ 典 ( ␪ ) calculated with the
generalized double critical-state model 共b兲 and the two-velocity hydrodynamic one 共c兲. Here ␮ 0 H a ⫽261 mT, ␪ max⫽45°.
As is seen in Figs. 1 and 2 for the case of an amplitude
of oscillation ␪ max⫽45°, the experimental hysteresis loops
关subfigures 共a兲兴 are closed. The curves 共b兲 predicted by the
GDCSM are very similar to the experimental ones. Nevertheless, one of the loops, namely the loop for ⫺ 具 ␮ 0 M z⬘ 典 ,
exhibits an appreciable lack of closure after the first full oscillation between the limits ␪ ⫽⫾ ␪ max 关see Fig. 2共b兲兴. We
have verified that this theoretical loop, however, closes after
various subsequent oscillations. On the other hand, the
TVHM generates absolutely closed hysteresis loops 关subfigures 共c兲兴 after the first complete oscillation of amplitude
␪ max .
Another interesting feature of the experimental curves
⫺ 具 ␮ 0 M z 典 ( ␪ ) 关Fig. 2共a兲兴 is the intersection, at ␪ ⫽ ␪ i
⬃⫺5°, of the lines, traced by decreasing the angle ␪ to
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J. Appl. Phys., Vol. 90, No. 7, 1 October 2001
FIG. 3. The same as in Fig. 1, except here ␮ 0 H a ⫽149 mT, ␪ max⫽120°.
⫺45° and then increasing it to 45°. This feature at ␪ ⫽ ␪ i
⬍0 agrees with the predictions of the generalized double
critical-state model 关Fig. 2共b兲兴. By contrast, within the twovelocity hydrodynamic model this kind of intersection occurs
at the zero angle of rotation, ␪ i ⫽0° 关Fig. 2共c兲兴, and therefore
the hysteresis loops for ⫺ 具 ␮ 0 M z⬘ 典 are symmetric under reflection with respect to the vertical line ␪⫽0.
For the oscillation amplitude ␪ max⫽120°, the experimental loops 关panels 共a兲 in Figs. 3 and 4兴 for 具 By 典 and
⫺ 具 ␮ 0 M z 典 are clearly open after the first full cycle between
⫾ ␪ max . Note that the maximum value of 具 B y 典 observed in
the initial rotation to ␪⫽⫹120° is larger than the maximum
reached in the next rotation to ⫹120°. Besides, in the
interval ⫺ ␪ max⬍␪⬍0 the intersection of the curves
⫺ 具 ␮ 0 M z 典 ( ␪ ) takes place at an angle of rotation of relatively
large magnitude 共at ␪ ⫽ ␪ i ⬃⫺40°兲. All these features are
satisfactorily reproduced by the generalized double criticalstate model 关see subfigures 共b兲兴. It is very interesting that the
Carballo-Sánchez, Pérez-Rodrı́guez, and Pérez-González
3459
FIG. 4. The same as in Fig. 2, except here ␮ 0 H a ⫽149 mT, ␪ max⫽120°.
TVHM, unlike the experiment and the GDCSM, generates a
closed hysteresis loop for 具 B ⬘y 典 关Fig. 3共c兲兴, which envelops
the curve traced during the initial rotation from 0° to 120°. In
addition, the closed loops for ⫺ 具 ␮ 0 M z⬘ 典 ( ␪ ), described by
the TVHM, conserve its reflection symmetry with respect to
the line ␪⫽0.
To explain the behavior of 具 B ⬘y 典 ( ␪ ) and ⫺ 具 ␮ 0 M z⬘ 典 ( ␪ )
predicted by the two models, the profiles B(x) and ␣ (x) at
different rotation angles ␪ with ␮ 0 H a ⫽0.149 T and ␪ max
⫽120° are exhibited in Figs. 5 and 6. Here, curves 0 show
the initial homogeneous distributions of B(x) and ␣ (x),
which are identical in both model calculations. Curves 1–5
correspond, respectively, to the consecutive rotation angles
␪ ⫽⫹ ␪ max (⫽⫹120°), 0°, ⫺ ␪ max , 0° and ⫹ ␪ max .
In the GDCSM, after the first rotation from 0° to ⫹120°,
two V-shaped minima in the B(x) pofile 关curve 1 of Fig.
5共a兲兴 appear at x⫽x ␷ and x⫽d⫺x ␷ . The gradient dB/dx in
the regions adjacent to the V-shaped minima 共0⬍x⬍x 0 and
d⫺x 0 ⬍x⬍d兲 takes the critical values ⫾ ␮ 0 J c⬜ (B). On the
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3460
J. Appl. Phys., Vol. 90, No. 7, 1 October 2001
Carballo-Sánchez, Pérez-Rodrı́guez, and Pérez-González
FIG. 6. The evolution of ␣ (x) distribution as ␪ ( ␣ s ⫽⫺ ␪ ) is varied. The
curve labeled 0 corresponds to the initial nonmagnetic 共after field cooling兲
distribution at ␪⫽0. The curves 1–5 correspond, respectively, to the values
⫹ ␪ max(⫽120°), 0, ⫺ ␪ max , 0, ⫹ ␪ max of the angle ␪. The evolution is calculated within the two-velocity hydrodynamic model with the parameters as
in Figs. 3共c兲 and 4共c兲.
FIG. 5. The evolution of the spatial distributions of 共a兲 B and 共b兲 ␣ as ␪
( ␣ s ⫽⫺ ␪ ) is varied. The curves labeled 0 correspond to the initial nonmagnetic 共after field cooling兲 distributions at ␪⫽0. The curves 1–5 correspond,
respectively, to the values ⫹ ␪ max(⫽120°), 0, ⫺ ␪ max , 0, ⫹ ␪ max of the angle
␪. The evolution is calculated within the generalized double critical-state
model with the parameters as in Figs. 3共b兲 and 4共b兲.
other hand, the magnitude of the angle ␣ 关curve 1 of Fig.
5共b兲兴 decreases from its maximum value at the sample surfaces ( ␣ s ⫽⫺ ␪ max⫽⫺120°) to the zero value at the points
x⫽x c and x⫽d⫺x c . Hence, inside the sample we identify
共i兲 zones, 0⬍x⬍x c and d⫺x c ⬍x⬍d, in which both flux
transport and flux-line cutting are occurring; 共ii兲 zones, x c
⬍x⬍x 0 , where only flux transport occurs, and 共iii兲 a central
zone between x⫽x 0 and x⫽d⫺x 0 , in which neither flux
transport nor flux-line cutting is occurring. Note that as ␪
oscillates between the values ⫾ ␪ max⫽⫾120° the V-shaped
minima of B(x) 关Fig. 5共a兲兴 move away from sample surface
x⫽0(x⫽d) until x ␷ (d⫺x ␷ ) reaches x c (d⫺x c ), the boundary for the zone in which the profile ␣ (x) 关Fig. 5共b兲兴 is
altered and flux-line cutting occurs. This motion of the
V-shaped minima makes the profile B(x) be different at the
beginning 共curve 1兲 and the end 共curve 5兲 of the first full
cycle with amplitude ␪ max . Therefore, the loops for
具 B ⬘y 典 ( ␪ ) and ⫺ 具 ␮ 0 M z⬘ 典 ( ␪ ), predicted by the generalized
double critical-state model 关Figs. 3共b兲 and 4共b兲兴, are open
after the first full oscillation as the experimental curves are
关Figs. 3共a兲 and 4共a兲兴. The hysteresis loops will close after
various cycles, when x ␷ reaches x c . This result agrees with
the experiment,6 where it is commented that three or more
oscillations are required before closing takes place.
The spatial distribution for the angle ␣ of the magnetic
induction, calculated within the hydrodynamic model, behaves as is shown in Fig. 6. This behavior of ␣ (x), as ␪ is
varied, turns out to be periodic as is also predicted by the
generalized double critical-state approach 关compare Figs.
5共b兲 and 6兴. However, unlike the GDCSM, the magnitude B
of the magnetic induction is, surprisingly, unaffected by the
sample rotation within the hydrodynamic model. So, the profile B(x) at any angle ␪ is the same as the initial homogeneous one 关B(x)⫽ ␮ 0 H a , see curve 0 in Fig. 5共a兲兴. Therefore, the TVHM generates only closed loops for both
具 B ⬘y 典 ( ␪ ) and ⫺ 具 ␮ 0 M z⬘ 典 ( ␪ ) 关Figs. 3共c兲 and 4共c兲兴 after the
first cycle of amplitude ␪ max .
The spatial distributions of B(x) and ␣ (x) calculated
with both models for ␮ 0 H a ⫽0.261 T and ␪ max⫽45° are
qualitatively similar to the corresponding profiles in Figs. 5
and 6. Hence, the hysteresis loops 具 B ⬘y 典 ( ␪ ) 共Fig. 1兲 and
⫺ 具 ␮ 0 M z⬘ 典 ( ␪ ) 共Fig. 2兲, generated by the GDCSM 共TVHM兲,
are open 共closed兲 after the first complete cycle between
⫾45°.
It should be noted that among all oscillation amplitudes
considered in the experiment,6 only for the smallest amplitude ␪ max⫽45° both 具 B y 典 ( ␪ ) and ⫺ 具 ␮ 0 M z 典 ( ␪ ) turned out to
be closed after the first full oscillation 关Figs. 1共a兲 and 2共a兲兴 as
is described by the hydrodynamic model. In other
cases, which correspond to larger oscillation amplitudes,
at least one of the measured loops, either 具 B y 典 ( ␪ ) or
⫺ 具 ␮ 0 M z 典 ( ␪ ), is clearly open. The later observations indicate that the GDCSM describes better than the TVHM the
behavior of hard superconductors, undergoing oscillations of
sufficiently large amplitude in static magnetic fields. Indeed,
only the generalized double critical-state model can generate
open hysteresis loops after the first full oscillation. Besides,
in some cases the lack of closure of the loops cannot be very
noticeable 关see, for example, Figs. 1共b兲 and 4共b兲兴. This fact
may explain the measured ‘‘closed’’ hysteresis loops since
the detection of its failure to close could require a higher
accuracy of the experiment.
V. CONCLUSION
We have investigated theoretically the behavior of
type-II superconductors either subjected to a rotating magnetic field or undergoing slow oscillations in a magnetic field
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J. Appl. Phys., Vol. 90, No. 7, 1 October 2001
of fixed magnitude. This investigation was carried out by
applying two competing models, namely the generalized
double critical-state 共GDCSM兲 model and the two-velocity
hydrodynamic 共TVHM兲 approach, which describe the magnetic response of hard superconductors by taking into account not only flux pinning, but also flux-line cutting effects.
We have compared the predictions of both approaches with
available experimental results6 for disks of Nb, which are
cooled through the critical temperature T c in a magnetic field
Ha parallel to their plane and, later, oscillate slowly over
various angular displacements in the same field. Measured
closed hysteresis curves for both the average magnetization
along the external field ( 具 M z 典 ) and the average magnetic
induction in the perpendicular direction ( 具 B y 典 ) versus the
angle of rotation ␪ were satisfactorily reproduced by the hydrodynamic model for a relatively small amplitude of oscillation, ␪ max⫽45°. At larger amplitudes of oscillation, at least
one of the measured hysteresis curves, either 具 M z 典 ( ␪ ) or
具 B y 典 ( ␪ ), is clearly open after the first full cycle of ␪ between
⫾ ␪ max . This evident lack of closure of the hysteresis loops
as well as their asymmetric form at large amplitudes of oscillation were reproduced only by the generalized double
critical-state model. According to this model, the complicated behavior of the hysteresis loops is connected with the
appearance of zones where the magnitude of magnetic induction 共B兲 is consumed. These zones are close to the sample
boundaries and reach a maximal size after various oscillations. The consumption of B, which is a direct consequence
of flux-line cutting,8 is very noticeable for large amplitudes
of oscillation.
The results of the present work and those of Refs. 30 and
34 allow us to define certain limits of applicability for the
GDCSM and TVHM. The hydrodynamic model turns out to
be rather good for describing the magnetic behavior of hard
superconductors subjected to cycles of a field H y perpendicular to a fixed field H z and with a small amplitude
H y,maxⰆHz30 共hence where the tilt angle ␪ of the applied field
is small兲. Besides, its predictions for the case of type-II superconductors under the action of a rotating magnetic field
with relatively small oscillation amplitude ( ␪ maxⱗ45°) agree
qualitatively with the experiment. On the other hand, the
generalized double critical-state model can reproduce qualitatively and quantitatively experimental magnetization
curves when the tilt angle ␪ of the applied field is varied over
a large range and, hence, flux-line cutting effects are well
developed.34 As was shown above, unlike the TVHM, the
GDCSM accounts for the evolution of magnetization hysteresis loops for type-II superconductors undergoing oscillations of very large amplitude ( ␪ maxⲏ100°) in a static magnetic field.
Carballo-Sánchez, Pérez-Rodrı́guez, and Pérez-González
3461
ACKNOWLEDGMENTS
This work was partially supported by the Consejo Nacional de Ciencia y Tecnologı́a 共CONACYT兲 under Grant
No. 32123-E. A. F. C. -S. acknowledges support from
CONACYT.
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