Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field
The magnetic response of irreversible type-II superconductor slabs subjected to in-plane rotating magnetic field is investigated by applying the circular, elliptic, extended-elliptic, and rectangular flux-line-cutting criticalstate models. Specifically, the models have been applied to explain experi...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Цитувати: | Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field / R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, F. Pérez-Rodríguez // Физика низких температур. — 2011. — Т. 37, № 11. — С. 1190–1200. — Бібліогр.: 34 назв. — англ. |
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Cortés-Maldonado, R. Espinosa-Rosales, J.E. Carballo-Sánchez, A.F. Pérez-Rodríguez, F. 2017-05-31T09:02:17Z 2017-05-31T09:02:17Z 2011 Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field / R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, F. Pérez-Rodríguez // Физика низких температур. — 2011. — Т. 37, № 11. — С. 1190–1200. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 74.25.Ha, 74.25.Op, 74.25.Wx https://nasplib.isofts.kiev.ua/handle/123456789/118794 The magnetic response of irreversible type-II superconductor slabs subjected to in-plane rotating magnetic field is investigated by applying the circular, elliptic, extended-elliptic, and rectangular flux-line-cutting criticalstate models. Specifically, the models have been applied to explain experiments on a PbBi rotating disk in a fixed magnetic field Ha, parallel to the flat surfaces. Here, we have exploited the equivalency of the experimental situation with that of a fixed disk under the action of a parallel magnetic field, rotating in the opposite sense. The effect of both the magnitude Ha of the applied magnetic field and its angle of rotation αs upon the magnetization of the superconductor sample is analyzed. When Ha is smaller than the penetration field HP, the magnetization components, parallel and perpendicular to Ha, oscillate with increasing the rotation angle. On the other hand, if the magnitude of the applied field, Ha, is larger than HP, both magnetization components become constant functions of α s at large rotation angles. The evolution of the magnetic induction profiles inside the superconductor is also studied. This work was partially supported by Consejo Nacional de Ciencia y Tecnología (CONACYT, Mexico). en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Современные проблемы низкотемпературной физики конденсированного состояния Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field Article published earlier |
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| title |
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field |
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Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field Cortés-Maldonado, R. Espinosa-Rosales, J.E. Carballo-Sánchez, A.F. Pérez-Rodríguez, F. Современные проблемы низкотемпературной физики конденсированного состояния |
| title_short |
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field |
| title_full |
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field |
| title_fullStr |
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field |
| title_full_unstemmed |
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field |
| title_sort |
flux-cutting and flux-transport effects in type-ii superconductor slabs in a parallel rotating magnetic field |
| author |
Cortés-Maldonado, R. Espinosa-Rosales, J.E. Carballo-Sánchez, A.F. Pérez-Rodríguez, F. |
| author_facet |
Cortés-Maldonado, R. Espinosa-Rosales, J.E. Carballo-Sánchez, A.F. Pérez-Rodríguez, F. |
| topic |
Современные проблемы низкотемпературной физики конденсированного состояния |
| topic_facet |
Современные проблемы низкотемпературной физики конденсированного состояния |
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2011 |
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English |
| container_title |
Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The magnetic response of irreversible type-II superconductor slabs subjected to in-plane rotating magnetic field is investigated by applying the circular, elliptic, extended-elliptic, and rectangular flux-line-cutting criticalstate models. Specifically, the models have been applied to explain experiments on a PbBi rotating disk in a fixed magnetic field Ha, parallel to the flat surfaces. Here, we have exploited the equivalency of the experimental situation with that of a fixed disk under the action of a parallel magnetic field, rotating in the opposite sense. The effect of both the magnitude Ha of the applied magnetic field and its angle of rotation αs upon the magnetization of the superconductor sample is analyzed. When Ha is smaller than the penetration field HP, the magnetization components, parallel and perpendicular to Ha, oscillate with increasing the rotation angle. On the other hand, if the magnitude of the applied field, Ha, is larger than HP, both magnetization components become constant functions of α s at large rotation angles. The evolution of the magnetic induction profiles inside the superconductor is also studied.
|
| issn |
0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/118794 |
| citation_txt |
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field / R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, F. Pérez-Rodríguez // Физика низких температур. — 2011. — Т. 37, № 11. — С. 1190–1200. — Бібліогр.: 34 назв. — англ. |
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2025-11-27T03:08:44Z |
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2025-11-27T03:08:44Z |
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| fulltext |
© R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, and F. Pérez-Rodríguez, 2011
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11, p. 1190–1200
Flux-cutting and flux-transport effects in type-II
superconductor slabs in a parallel rotating magnetic field
R. Cortés-Maldonado1, J.E. Espinosa-Rosales2, A.F. Carballo-Sánchez3,
and F. Pérez-Rodríguez1,
1Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apdo. Post. J-48, Puebla, Pue. 72570, Mexico
2Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla
Apdo. Post. 1152, Puebla, Pue., 72000, Mexico
3Universidad del Istmo, Campus Tehuantepec, Tehuantepec, Oax., 70760, Mexico
E-mail: fperez@ifuap.buap.mx
Received April 19, 2011
The magnetic response of irreversible type-II superconductor slabs subjected to in-plane rotating magnetic
field is investigated by applying the circular, elliptic, extended-elliptic, and rectangular flux-line-cutting critical-
state models. Specifically, the models have been applied to explain experiments on a PbBi rotating disk in a
fixed magnetic field Ha, parallel to the flat surfaces. Here, we have exploited the equivalency of the experimen-
tal situation with that of a fixed disk under the action of a parallel magnetic field, rotating in the opposite sense.
The effect of both the magnitude Ha of the applied magnetic field and its angle of rotation αs upon the magneti-
zation of the superconductor sample is analyzed. When Ha is smaller than the penetration field HP, the magneti-
zation components, parallel and perpendicular to Ha, oscillate with increasing the rotation angle. On the other
hand, if the magnitude of the applied field, Ha, is larger than HP, both magnetization components become con-
stant functions of αs at large rotation angles. The evolution of the magnetic induction profiles inside the super-
conductor is also studied.
PACS: 74.25.Ha Magnetic properties including vortex structures and related phenomena;
74.25.Op Mixed states, critical fields, and surface sheaths;
74.25.Wx Vortex pinning (includes mechanisms and flux creep).
Keywords: flux cutting, flux transport, vortex pinning, critical state, hard superconductor.
1. Introduction
The discovery of the phenomenon known as quasisym-
metrical collapse of magnetization [1], which is observed
in superconductors subjected to crossed magnetic fields
and well interpreted within the simple Bean’s critical-state
model [2,3], has been a turning point in the understanding
of the magnetic behavior of hard (irreversible type-II) su-
perconductors. Until then, the generalized double critical-
state model (GDCSM) [4–8], which is based on fundamen-
tal physical concepts such as flux transport and flux-line-
cutting [9,10], was successfully employed to explain a va-
riety of experiments where flux cutting occurs [11–16]. An
important feature of the GDCSM is the assumption that
flux cutting and flux depinning do not affect each other.
Besides, the GDCSM is inherently anisotropic because the
thresholds for these two effects are given by two indepen-
dent parameters, namely the critical current densities paral-
lel cJ and perpendicular cJ ⊥ to the local magnetic induc-
tion .B However, since the GDCSM cannot reproduce the
features of magnetic moment collapse [17,18], whereas
isotropic Bean’s model does it, the main assumption of the
GDCSM has been questioned, motivating the development
of new critical-state models in the past few years.
In Ref. 19, the so-called elliptic flux-line-cutting criti-
cal-state model was proposed. This model introduces the
anisotropy, induced by flux-line-cutting effects, by using a
procedure similar to that for structurally anisotropic super-
conductors [20,21], i.e. the magnitude of the critical cur-
rent density cJ , being the only parameter used within the
isotropic Bean’s model, is substituted by a symmetrical
tensor ( )c ikJ with principal values cJ and ,cJ ⊥ corres-
ponding to the directions along and across the local mag-
netic induction .B In good agreement with the experiment
on Yba2Cu3O7–δ samples [1,17], the elliptic critical-state
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11 1191
model predicts the quasisymmetrical suppression of the
average magnetization < >,zM for paramagnetic and di-
amagnetic initial states, by sweeping a transverse field yH
of magnitude much smaller than dc-bias magnetic field
zH [19,22]. When the magnitudes of the crossed fields
yH and zH are comparable, the value of the magnetiza-
tion < >zM after many cycles of the transverse field yH
turns out to be positive for both diamagnetic and paramag-
netic initial states if > .c cJ J ⊥ To our knowledge, the ob-
servation of such a paramagnetism of hard superconductors
was first reported in Refs. 23, 24. The elliptic model also
describes the behavior of < > ( )y yM H and < > ( )z yM H
in crossed fields yH and zH [19,25], which was observed
in the experiments on a VTi ribbon with nonmagnetic initial
state [14,26]. Here, the good agreement with the experiment
was achieved by using a relatively large anisotropy para-
meter / = 6.c cJ J ⊥ It should be noticed that the Bean’s
critical state model predicts neither the phenomenon of the
paramagnetism of hard superconductors nor the behavior of
the components of the average magnetization found in
Refs. 14, 26. Furthermore, as it is shown in [19,27], the el-
liptic critical-state model successfully describes the magnet-
ic response of superconducting disks undergoing oscillations
in a magnetic field of fixed magnitude for nonmagnetic,
paramagnetic, and diamagnetic initial states [11].
Despite the great success of the elliptic model [19], it
turns out that there exist phenomena, associated with flux
cutting, which are not completely described within such a
model. So, in a very recent work [28], the elliptic critical-
state model and other four theoretical approaches for de-
scribing the critical state of type-II superconductors
(GDCSM, extended GDCSM [29,30], extended elliptic
critical-state model [28,31], and an elliptic critical-state
model based on the variational principle [32]) were tested.
There, the angular dependencies of the critical current den-
sity cJ and the electric field E (for J just above )cJ
were measured, using an epitaxially grown YBCO thin
film, and compared with the predictions of the five theo-
ries. The measurements of angular dependence of the criti-
cal-current density cJ demonstrated a behavior rather sim-
ilar to that assumed by the elliptic critical-state models.
Besides, the smooth angular dependence of the ratio of the
transverse to the longitudinal components of the electric
field /y zE E for J just above ,cJ predicted by the three
elliptic models, was verified in the experiment [28]. How-
ever, the original critical-state model [19] leads to small
values of the ratio /y zE E in comparison with the experi-
mental data and the results obtained from the other two
elliptic models. On the basis of this detailed comparison
between experiment and the five theories, it was concluded
in Ref. 28 that the experiment favors only one of the mod-
els, namely the extended elliptic critical-state model.
The aim of the present work is to investigate the beha-
vior of a hard superconductor in a parallel rotating magnet-
ic field (or equivalently, the response of a rotating super-
conductor in a fixed magnetic field) and to compare the
predictions of four critical-state models with experiment.
Concretely, we shall consider the Bean's critical-state
model [2,3], the original elliptic critical-state model
[19,22], the recently-proposed extended elliptic model
[28,31], as well as the GDCSM [4–8], whose main charac-
teristics and assumptions will be revisited in Sec. 2. We
shall numerically solve Maxwell equations with the ma-
terial equation postulated by each of the considered criti-
cal-state models to calculate magnetization curves for a
superconductor disk rotating in a fixed magnetic field as in
the experiment [33] (Sec. 3). Here, we shall analyze the
effect of the magnitude aH of the applied magnetic field
upon the dependencies of the magnetization components,
parallel and perpendicular to ,aH on the rotation angle of
the superconductor disk. The evolution of magnetic induc-
tion profiles will also be studied to explain the magnetic
response of the rotating hard-superconductor sample.
2. Theoretical formalism
Let us consider a superconducting slab of thickness ,d
which occupies the space 0 < <x d and is subjected to a
magnetic field aH parallel to its surfaces:
ˆ ˆ ˆ= = [ sin( ) cos( )],a a s a s sH H α + αH a y z (1)
where sα is the angle of the applied magnetic field aH
with respect to the z-axis. Hence, the magnetic induction
( , )x tB inside the superconducting slab can be expressed as
ˆ ˆ= ( , )[ sin( ( , )) cos( ( , ))],B x t x t x tα + αB y z (2)
where B and α are respectively the magnitude and the
tilt angle of the magnetic induction. It is convenient to
write the electric field ( , )x tE and the electrical current
density ( , )x tJ in terms of their components parallel and
perpendicular to the local magnetic induction ( , ):x tB
ˆˆ( , ) = ( , ) ( , ) ( , ) ( , ),x t E x t x t E x t x t⊥+E a b (3)
ˆˆ( , ) = ( , ) ( , ) ( , ) ( , ),x t J x t x t J x t x t⊥+J a b (4)
where ˆ ˆˆ( , ) = ( , ).x t x t×b x a Inside the superconductor sam-
ple, we shall assume that the magnetic induction and the
magnetic field satisfy the relation 0( , ) = ( , ),x t x tμB H
which is good enough for applied magnetic fields much
larger than the first critical field 1( ).a cH H Moreover,
any surface barrier against the flux entry (or exit) will be
neglected. According to the planar geometry of the prob-
lem, we can rewrite Ampere and Lorentz laws,
0( , ) = ( , ),x t x t∇× μB J (5)
( , ) = ,x t
t
∂
∇× −
∂
BE (6)
as follow
R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, and F. Pérez-Rodríguez
1192 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11
0= ,B J
x ⊥
∂
−μ
∂
(7)
0= ,B J
x
∂α
−μ
∂
(8)
= ,
E BE
x x t
⊥∂ ∂α ∂
+ −
∂ ∂ ∂
(9)
= .
E
E B
x x t⊥
∂∂α ∂α
− −
∂ ∂ ∂
(10)
To solve the resulting system of differential equations for
,E B and ,J one should add the material equation. Be-
low, we shall use the material equations corresponding to
the circular, elliptic, extended-elliptic, and rectangular
flux-line-cutting critical-state models.
2.1. Circular model
The first model for describing the magnetic behavior of
superconductors in multicomponent situations was proposed
by Bean [2,3]. According to it, the critical current density J
points always along the local electric field .E Hence,
= .cJ
E
EJ (11)
The magnitude of the critical current density = cJ J is the
unique phenomenological parameter used and may depend
on the magnitude of the magnetic induction B . In the pla-
nar geometry (see Eqs. (1)–(4)), the assumption = cJ J
corresponds to a circle in the J J⊥ − plane.
In numerically solving the system of Eqs. (7)–(10) for
the electromagnetic fields, it is necessary to rewrite
Eq. (11) as
= ( ) ,E J
J
JE (12)
0, ( )
( ) =
( ( )), ( )
c
c c
J J B
E J
J J B J J B
≤⎧
⎨ρ − ≥⎩
(13)
where ρ is an effective resistivity. It should be mentioned
that for slow variations of the surface boundary conditions,
producing a small magnitude of the induced electric field
( ),cE Jρ the magnetic induction profiles are practically
relaxed and independent of the parameter ρ [34].
2.2. Elliptic model
The elliptic flux-line cutting critical-state model
[19,22,25] postulates:
= ( ) ,k
i c ik
E
J J
E
(14)
where
,( ) = ( ) , , = , .c ik c i ijJ J B i kδ ⊥ (15)
Here ikδ is the Kronecker delta symbol. Within the elliptic
critical-state model (14), the magnitude of the critical cur-
rent density cJ draws an ellipse on the J J⊥ − plane.
This model makes use of two phenomenological parame-
ters, namely the extreme values cJ ⊥ and cJ for the ra-
dius of the ellipse drawn by the magnitude of the critical
current density. In the numerical calculations for solving
the system of Eqs. (7)–(10), the relation (14) is rewritten in
the form
1= ( )( ) ,i c ik kE E J J J− (16)
0, ( , )
( ) = ,
( ( , )), ( , )
c
c c
J J B
E J
J J B J J B
≤ φ⎧
⎨ρ − φ ≥ φ⎩
(17)
where 1( )c ikJ − is the inverse of the matrix ( )c ikJ in (14).
The magnitude of the critical current density, ( , ),cJ B φ is
given by the expression
1/2
2 2
2 2
( ) ( )cos sin( , ) = .
( ) ( )
c
c c
J B
J B J B
−
⊥
⎡ ⎤φ φ⎢ ⎥φ +
⎢ ⎥⎣ ⎦
(18)
Here, φ denotes the angle of the critical current density J
with respect to the direction of the flux density .B If
= ,c cJ J⊥ the elliptic critical-state model (16) goes over
into the Bean’s (circular) critical-state model (12). Besides,
the calculations of electromagnetic fields with J close
to cJ are also independent of the auxiliary parameter ρ
in Eq. (17).
2.3. Extended elliptic model
The elliptic critical-state model, described in previous
subsection, has recently been extended in Refs. 28, 31 by
introducing the general relations
= ,E J⊥ ⊥ ⊥ρ (19)
= ,E Jρ (20)
where ⊥ρ and ρ are nonlinear effective resistivities, hav-
ing a ratio = /r ⊥ρ ρ independent of J just above cJ as
it was experimentally found [28]. A model for the effective
resistivities is given by [31]
0, 0 | |
= ,
(| | ) sign( ), | |
cd
d cd cd
J J
E
J J J J J
⊥
⊥
⊥ ⊥ ⊥
≤ ≤⎧
⎨ρ − ≥⎩
(21)
0, 0 | |
= .
(| | ) sign( ), | |
cc
c cc cc
J J
E
J J J J J
≤ ≤⎧⎪
⎨ρ − ≥⎪⎩
(22)
Here, the subscripts “d” and “c” respectively refer to de-
pinning and cutting. Besides, = ( , ) | sin( ) |cd cJ J B φ φ and
= ( , ) | cos( ) |,cc cJ J B φ φ where ( , )cJ B φ is defined ac-
cording to the elliptic critical-state model as in Eq. (18). If
| | / 1,c cJ J J− the extended elliptic critical-state model
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11 1193
reduces to the original one (Eqs. (16) and (17)) by replac-
ing dρ and cρ in Eqs. (21) and (22) with /c cJ J ⊥ρ and
/c cJ Jρ , correspondingly. Hence, in the case of the orig-
inal elliptic model, the ratio = /r ⊥ρ ρ at > cJ J is equal
to /c cJ J⊥ . On the other hand, the extended elliptic criti-
cal-state model is capable to modify the relation between
the components of the electric field E and the current den-
sity J with the aid of the additional parameter .r
2.4. Rectangular model
The generalized double critical-state model [4–8] uses
two phenomenological parameters, namely the critical val-
ues, cJ and cJ ⊥ , of the electrical current density along
and perpendicular to the local magnetic induction. Within
this model, each component of the electrical current densi-
ty is determined by its own electric field as
= sign( ),cJ J E⊥ ⊥ ⊥ (23)
= sign( ).cJ J E (24)
Evidently, the magnitude of the critical current density trac-
es a rectangle in the J J⊥ − plane. The parameter cJ ⊥
determines the threshold for depinning of vortices, whereas
cJ indicates the onset of flux-line cutting in the vortex ar-
ray. In calculating the electromagnetic fields within the
GDCSM, the material equation (23) is written in the form
0, 0 | |
= ,
(| | ) sign( ), | |
c
c c
J J
E
J J J J J
⊥ ⊥
⊥
⊥ ⊥ ⊥ ⊥ ⊥ ⊥
≤ ≤⎧
⎨ρ − ≥⎩
(25)
0, 0 | |
= .
(| | )sign( ), | |
c
c c
J J
E
J J J J J
≤ ≤⎧⎪
⎨ρ − ≥⎪⎩
(26)
The quantities ⊥ρ and ρ are effective flux-flow and flux-
line-cutting resistivities of the material. However, unlike
the above-commented critical-state models, the GDCSM
allows the existence of zones in the J J⊥ − plane where
either flux cutting or flux transport exclusively occur. The
latter is possible due to the assumption of the GDCSM that
the threshold for flux depinning, cJ ⊥ (flux cutting, cJ ) is
independent of the component J ( )J⊥ (compare
Eqs. (25) and (26) with Eqs. (21) and (22) where cdJ and
ccJ depend on the angle = arctan( / )J J⊥φ ).
3. Numerical results and comparison with experiment
In the present section we will apply the flux-line-cutting
critical-state models, commented above, to explain expe-
rimental magnetization curves [33] of a PbBi supercon-
ducting disk, rotating in the presence of an external mag-
netic field ,aH which is oriented parallel to the disk plane
(along the z-axis) and perpendicular to the axis of rotation.
3.1. Experimental results
Figure 1,a exhibits a standard magnetization curve,
which was measured in Ref. 33, for a PbBi disk of thick-
ness = 0.8d mm. The hysteresis in Fig. 1,a clearly corres-
ponds to the magnetization curve of a type-II irreversible
superconductor since its return crosses over and remains in
the paramagnetic region as a result of the strong flux pin-
ning. In the experiment, the isotropy of the PbBi disk was
also verified by comparing standard magnetization curves
with aH directed along different diameters of the disk.
Panels (a)–(c) in Fig. 2 show graphs of the magnetiza-
tion components, 0< > = < > /y yM B μ and < > =zM−
0= < > / ,a zH B− μ versus the angle θ of rotation, meas-
ured in the work [33] for the PbBi disk, rotating in the
magnetic field .aH The measurements started in the non-
magnetic initial state which is reached after cooling the
superconductor at the fields / =a PH H 0.5 (panel a), 1.0
(panel b), and 2.0 (panel c), where PH ( 0 = 0.1015PHμ T
[33]) is the penetration field. The initial state is supposed
to be nonmagnetic because no Meissner effect (flux expul-
sion) was observed after field cooling, within the accuracy
( < > 1MΔ ≤ Gauss) of the experiment.
Fig. 1. Standard magnetization curves (a) for a PbBi disk, taken
from Ref. 33. Theoretical magnetization curves (b) obtained with
a critical current density ( )cJ B⊥ as in Eq. (31).
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14
–0.06
–0.03
0
0.03
0.06
0.09
Experiment
a
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14
–0.06
–0.03
0
0.03
0.06
0.09
b
Model
�0Hz, T
�
0
<
>
,
T
M
z
�0Hz, T
�
0
<
>
,
T
M
z
R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, and F. Pérez-Rodríguez
1194 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11
As it is seen in Fig. 2, for the smallest value of aH
(= 0.5 ,PH panel a), both magnetization components have
a nonmonotonic behavior as functions of .θ Such a beha-
vior of magnetization has also been observed in Ref. 11
during the initial rotation of a Nb disk undergoing slow
oscillations in a parallel field. The dependence of the mag-
netization on θ radically changes at larger values of .aH
So (see Fig. 2,b), at =a PH H the functions < > ( )yM θ
and < > ( )zM− θ initially grow with θ and later (at
> 150 )θ they practically become constants with close
values (< > < >).y zM M≈ − Also note that < >yM has
a maximum at 75 .θ ≈ For aH larger than the penetration
field PH (panel c), the function < > ( )zM− θ takes val-
ues smaller than those for < > ( ).yM θ Both of them are
almost constant functions, except at small rotation angles
because of their fast initial growth. Thus, the maximum of
< >yM is shifted to a smaller value of θ ( 40 ).≈
3.2. Theoretical predictions
The models described in the previous section can be
applied to explain the experimental results (Fig. 2) if we
fix the sample and rotate the external magnetic field aH
(1) by an angle =sα −θ instead of fixing the magnetic
field and rotating the superconducting sample. Then, the
experimental values < >yM and < >zM− should re-
spectively correspond to the quantities:
0 0
1< >= ( ),
d
y yM dxB x
d
′
μ ∫ (27)
0 0
1< >= ( ),
d
z a zM H dxB x
d
′− −
μ ∫ (28)
where
ˆ ˆ= = ( )sin[ ( ) ],y s sB B x x′ × ⋅ α −αa x B (29)
ˆ= = ( )cos[ ( ) ].z s sB B x x′ ⋅ α −αa B (30)
The calculations of magnetization components < >yM
and < >zM− with the critical-state models, discussed in
Sec. 2, require the employment of the parameters ( )cJ B⊥
and ( ),cJ B depending on the magnetic induction. The
former, ( ),cJ B⊥ is determined from the experimental
curves of magnetization versus the applied field, varying
along one direction only as in Fig. 1 (In this case, flux cut-
ting does not occur and, consequently, the depinning ef-
fects are completely responsible for the magnetic response
of the superconductor.) The standard magnetization curves
are well reproduced by any one of the critical-state models
(see above) with
0
(0)
( ) = ,
(1 / )
c
c n
P
J
J B
B H
⊥
⊥
⊥+ μ (31)
7(0) = 47.11 10cJ ⊥ ⋅ A/m2, and = 2n⊥ (compare panels
(a) and (b) of Fig. 1). Other parameters of the critical state
models are found by adjusting theoretical magnetization
curves to the experimental ones (Fig. 2).
3.2.1. Circular model. Within the Bean,s circular criti-
cal-state model (11), there is only one phenomenological
parameter, i.e. ( ) = ( ) = ( ).c c cJ B J B J B⊥ Then, ( )cJ B
has the form (31) with the same values for the parameters
(0),cJ ⊥ and .n⊥
Figure 3 shows our numerical results for < >yM and
< >,zM− obtained with the Bean critical-state model. At
first glance, it seems that the circular model qualitatively
reproduces the experimental magnetization curves (Fig. 2).
However, there are important differences between its pre-
dictions and the experiment. Thus, for example, the “oscil-
lations” of the magnetization components (Fig. 3,a) have
small amplitudes compared with the experimental ones.
Fig. 2. PbBi rotational curves measured in Ref. 33.
0 45 90 135 180 225 270 315 360
–0.2
0
0.2
0.4
0.6
0.8
Experiment
, deg�
a
0 45 90 135 180 225 270 315 360
0
0.2
0.4
0.6
0.8
Experiment
b
0 45 90 135 180 225 270 315 360
0
0.2
0.4
0.6
0.8
Experiment
c
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 0.5
< >My
– < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 1.0
< >My
– < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 2.0
< >My
– < >Mz
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11 1195
Besides, at = 0.8a PH H the functions < > ( )yM θ and
< > ( )zM− θ approximate each other but at relatively
large rotation angles > 300 .θ Finally, when the applied
field has an amplitude larger than pH (see panel c), the
magnetization components are rather small in magnitude
and their initial growth, before the saturation, occurs in a
very small interval of θ (< 20 ).
3.2.2. Elliptic model. The calculations of magnetization
components < >yM and < >zM− within the elliptic
flux-line-cutting critical-state model (14) are shown in
Fig. 4. Here, we used the same ( )cJ B⊥ as in Eq. (31) and
( )cJ B of the form
0
(0)
( ) =
(1 / )
c
c n
P
J
J B
B H+ μ
(32)
with (0) = 1.5 (0)c cJ J ⊥ and = 1n . This choice provides
a good agreement between experimental (Fig. 2) and theo-
retical (Fig. 4) curves. Thanks to the use of a second para-
meter ( cJ ), the elliptic model is able to generate the “os-
cillations” of the magnetization components (Fig. 4,a) with
amplitude close to that observed in the experiment (panel
(a) in Fig. 2). Notice that < >yM and < >zM− ap-
proach each other at > 150θ with 0 = 1.05 PH H in good
concordance with the measurements (see Fig. 2,b, corres-
ponding to 0 = ).PH H In addition, when 0 = 2.0 pH H
(panel (c) in Fig. 4), the difference between < >yM and
< >zM− at > 45θ is as large as in the experiment
(Fig. 2,c).
Fig. 3. Curves of the average magnetization components versus
the rotation angle, calculated with Bean’s critical-state model.
0 45 90 135 180 225 270 315 360
–0.2
0
0.2
0.4
0.6
0.8 a
Circular model
0 45 90 135 180 225 270 315 360
0
0.2
0.4
0.6
0.8 b
Circular model
0 45 90 135 180 225 270 315 360
0
0.2
0.4
0.6
0.8 c
Circular model
, deg�
H Ha P/ = 0.5
< >My – < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 0.8
< >My
– < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 2.0
< >My
– < >Mz
M
ag
n
et
iz
at
io
n
(
)
H
P
Fig. 4. Curves of the average magnetization components versus
the rotation angle, calculated with the original elliptic critical-
state model.
0 45 90 135 180 225 270 315 360
–0.2
0
0.2
0.4
0.6
0.8 a
Elliptic model
0 45 90 135 180 225 270 315 360
0
0.2
0.4
0.6
0.8 b
Elliptic model
0 45 90 135 180 225 270 315 360
0
0.2
0.4
0.6
0.8 c
Elliptic model
, deg�
H Ha P/ = 0.5
< >My
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 1.05
< >My
– < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 2.0
< >My
– < >Mz
M
ag
n
et
iz
at
io
n
(
)
H
P – < >Mz
, deg�
R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, and F. Pérez-Rodríguez
1196 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11
3.2.3. Extended elliptic model. As was commented in
Sec. 2, both elliptic and circular critical-state models are
particular cases of the extended elliptic one. Therefore, the
results presented in Fig. 3, predicted by the circular model,
can also be calculated by using the new model (Eqs. (21)
and (22)) with =c cJ J⊥ as in Eq. (31) and
= / = /c dr ⊥ρ ρ ρ ρ being equal to one ( =1r ) at > .cJ J
The condition =1r guarantees that the electric field E and
current density J be parallel as it is postulated by Bean’s
critical-state model (11). In addition, graphs in Fig. 4 (origi-
nal elliptic model predictions), which quantitatively repro-
duce experimental measurements (Fig. 2), are also ob-
tained with the extended elliptic critical-state model
(Eqs. (21) and (22)) if = /c cr J J⊥ (i.e. / =c dρ ρ
= /c cJ J⊥ ). According to the parameters ( )cJ B⊥ (31)
and ( )cJ B (32), used for calculating magnetization
curves in Fig. 4, the ratio r is here smaller than 1 ( <1r ).
It is interesting to study the effect of the parameter r ,
controlling the relation between the electric field E and
the current density J at > .cJ J For this reason, we have
calculated magnetization curves (Fig. 5) by applying the
extended elliptic model with the same parameters ( )cJ B⊥
and ( )cJ B as those employed in Fig. 4, but with the pa-
rameter = / = 1.c dr ρ ρ In other words, the magnetization
curves in Fig. 5 correspond to an anisotropic critical-state
model with / < 1,c cJ J⊥ but the parameter = 1,r indi-
Fig. 5. Curves of the average magnetization components versus
the rotation angle, calculated with the extended elliptic critical-
state model using a ratio =1r .
0 45 90 135 180 225 270 315 360
–0.2
0
0.2
0.4
0.6
0.8 a
Extended elliptic model ( = 1)r
0 45 90 135 180 225 270 315 360
0
0.2
0.4
0.6
0.8
Extended elliptic model ( = 1)r
b
0 45 90 135 180 225 270 315 360
0.2
0.4
0.6
0.8
Extended elliptic model ( = 1)r
c
0
, deg�
H Ha P/ = 0.5
< >My
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 1.16
< >My
– < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 2.0
< >My
– < >Mz
M
ag
n
et
iz
at
io
n
(
)
H
P
– < >Mz
, deg�
Fig. 6. Curves of the average magnetization components versus
the rotation angle, calculated with the generalized double critical-
state model.
0 45 90 135 180 225 270 315 360
–0.2
0
0.2
0.4
0.6
0.8
Rectangular model
a
0 45 90 135 180 225 270 315 360
0.2
0.4
0.6
0.8
Rectangular model
b
0 45 90 135 180 225 270 315 360
0.2
0.4
0.6
0.8 c
Rectangular model
0
0
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 0.5
< >My
– < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 1.29
< >My
– < >Mz
, deg�
M
ag
n
et
iz
at
io
n
(
)
H
P
H Ha P/ = 2.0
< >My
– < >Mz
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11 1197
cating that E and J are parallel when > .cJ J From the
comparison of Fig. 5 with 4, we note that magnetization
curves significantly depend upon the parameter r when
the applied magnetic field is large enough ( >a PH H as in
panels (b) and (c)). So, in order the magnetization compo-
nents, < >yM and < >,zM− to have the same value at
large angles of rotation, the applied magnetic field aH for
=1r (Fig. 5,b) should be larger than the field used in
Fig. 4,b. Besides, the value of < >yM and < >zM−
( 0.4 ),PH≈ at sufficiently large angles ,θ turns out to be
smaller than that ( 0.5 )PH≈ predicted by the original el-
liptic model (Fig. 4,b). At = 2.0 ,a PH H there is also a
noticeable difference between magnetization y-components
(compare panels (c) of Figs. 4 and 5).
3.2.4. Rectangular model. For completeness of our study,
we have employed the GDCSM (rectangular model), which
also uses two critical current densities, namely ( )cJ B⊥ and
( ).cJ B The former is determined from the curves of mag-
netization versus the applied field, varying along one direc-
tion only (Fig. 1). In our case, the magnetic dependence of
cJ ⊥ is the same as in Eq. (31). To reproduce the main fea-
tures of the experiment (Fig. 2), the other parameter is cho-
sen as in Eq. (32), but (0) = 1.32 (0)c cJ J ⊥ and = 1.06n
(compare Figs. 2 and 6). Although these values are different
from those used within the elliptic critical-state model, the
parallel critical current density cJ remains being larger
than the perpendicular one .cJ ⊥ It should be noted that the
GDCSM predicts the equality of < >yM and < >zM−
( 0.5 )PH≈ with an external field = 1.29 >a P PH H H at
relatively large rotation angles > 270θ (see Fig. 6,b), in
contrast to the experiment where such a behavior occurs
from 150 .θ ≈ Besides, the numerical calculations for
= 0.5a PH H (panel (a) in Fig. 6) had to be stopped at
338θ ≈ because the solution further diverged.
3.3. Magnetic induction profiles
The fact that the elliptic critical-state model is able to
quantitatively reproduce the experiment, with the use of a
parallel critical current density ( )cJ B larger than the per-
pendicular one ( )cJ B⊥ , illustrates how flux-line cutting
influences on the magnetic behavior of a rotating super-
conductor. To explain the features observed in both expe-
rimental (Fig. 2) and theoretical (Fig. 4) magnetization
curves, we shall analyze the evolution of the profiles for
the magnitude of the magnetic induction ( ),B x the tilt an-
gle ( )xα , and the components ( )yB x′ (29) and ( )zB x′
(30), calculated within the original elliptic flux-line-cutting
critical-state model (Figs. 7–9).
The calculated profiles of the magnetic induction in the
case when the external magnetic field aH has a magnitude
smaller than the penetration field pH ( = 0.5 )a pH H are
shown in Fig. 7. As the angle of rotation is increased, two
Fig. 7. Profiles of the angle α (panel a), magnitude B (panel b) and components yB ′ (Eq. (29), panel c) and zB ′ (Eq. (30), panel d)
of the magnetic induction, calculated with the original elliptic critical-state model at = 0.5 .a PH H
0.2 0.4 0.6 0.8 1.00
45
90
135
180
225
270
315
360
8
7
6
5
4
3
2
1
0
x d/
a
xm2x2x1xm1
0.2 0.4 0.6 0.8 1.0
0
0.1
0.2
0.3
0.4
0.5
0.6
8 7
6
5
4
3
2
1
0b
x d/
0 0.2 0.4 0.6 0.8 1.0
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
c
8
7
6
5
4
3
2
1
0
x/d
0 0.2 0.4 0.6 0.8 1.0
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
x/d
d
8
7
6
5
4
3
2
1
0
B
/
H
�
0
P
xm2x2x1xm1
xm2x2x1xm1
xm2x2x1xm1
–
,
d
eg
�
B
z
P
/
H
�
0
�
B
y
P
/
H
�
0
�
R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, and F. Pérez-Rodríguez
1198 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11
U-shaped minima in the ( )B x profile (panel b) appear
because of the flux consumption (decrement of B) which
results from flux-line cutting [4]. The absolute value of the
tilt angle α increases with θ in the near-surface intervals
10 < mx x≤ and 2 < .mx x d≤ However, in the intervals
1 1< <mx x x and 2 2< < ,mx x x where there is flux con-
sumption, the angle α is slightly modified. In the central
interval, 1 2< < ,x x x neither B or α is altered. When
360 ,θ ≈ the minimum values of B inside the supercon-
ducting disk tend to zero and, as follows from Eq. (8), the
magnitude of the derivative / x∂α ∂ considerably increases
at points corresponding to such minima. Besides, at 1= mx x
and 2= mx x with 1 2( ) = ( ) 0,m mB x B x ≈ the accuracy of
our calculations is low and, therefore, the values 1( )mx−α
and 2( )mx−α turned out to be apparently higher than they
should be (see curve 8 for = 360θ in Fig. 7,a). The com-
ponent zB′ of the magnetic induction, parallel to the ap-
plied magnetic field, decreases near sample surfaces be-
cause of the flux consumption (Fig. 7,d). Nevertheless, the
most important change occurs in the central part of the
sample (in 1 2< < )x x x because of the sample rotation.
So, at = 180θ (curve 4) the component zB′ varies from
0=z aB H′ μ at the surfaces = 0x and =x d to the oppo-
site value 0=z aB H′ −μ in the central region of the sample.
When an entire cycle is finished, zB′ again takes the value
0=z aB H′ μ in the middle of the disk (curve 8). This cyclic
behavior of zB′ is responsible for the “oscillations” of
the magnetization component < > ( )zM θ (panels (a) in
Figs. 2 and 4), being negative for any value of the angle of
rotation > 0θ because 0<z aB H′ μ near surfaces, i.e. in the
intervals 10 < <x x and 2 < < .x x d The component yB′
also oscillates in the middle of the sample as θ is increased
(Fig. 7,c). Such a behavior of yB′ makes the magnetization
y-component < >yM oscillate with θ (Figs. 2,a and 4,a).
As it is seen in Fig. 7,c, there is an increment of yB′ in the
near-surface regions, producing a small positive value for
< >yM (27) after a complete cycle, i.e. at = 360θ (see
Figs. 2,a and 4,a).
Figure 8 exhibits profiles calculated within the elliptic
critical-state model for = 1.05 .a pH H Due to the decrease
of the critical current densities cJ ⊥ (31) and cJ (32) with
the magnitude B of the magnetic induction, the slopes of
the critical profiles for ( )B x and ( )xα near surfaces are
smaller than the slopes observed in the corresponding pro-
files of Fig. 7. Therefore, the central region with unaltered
B and α (see curves 1 in panels (a) and (b) of Fig. 8)
rapidly disappears as the rotation angle θ is increased (see
curves 2 therein). Also, the U -shaped minima of ( )B x
coalesce forming a unique minimum at the center of the
disk. The resulting critical profile ( )B x does not further
change despite the fact that the disk continues rotating (see
curves 5–8 in panel (b)). In this case, ( )zB x′ initially de-
creases (curves 1–2 in Fig. 8,d) inside the sample as θ
varies until it reaches the critical profile (curves 3–8).
Hence, the dependence < > ( )zM θ has a monotonic beha-
vior at > 120θ (see panels (b) in Figs. 2 and 4). On the
other hand, ( )yB x′ increases so that a huge maximum in
the dependence < > ( )yM θ (Figs. 2,b and 4,b) appears at
Fig. 8. Profiles of the angle α (panel a), magnitude B (panel b) and components yB ′ (Eq. (29), panel c) and zB ′ (Eq. (30), panel d) of
the magnetic induction, calculated with the original elliptic critical-state model at = 1.05 .a PH H
0.2 0.4 0.6 0.8 1.0
0
45
90
135
180
225
270
315
360
x/d
8
7
6
5
4
3
2
1
0
a
0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.2
b
43
2 1
0
x/d
0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.2
x/d
c
5–8
4 3
2
0
1
0 0.2 0.4 0.6 0.8 1.0
–1.2
–0.8
–0.4
0
0.4
0.8
1.2
d
x/d
3–8
2
1
0
5–8
–
,
d
eg
� B
/
H
�
0
P
B
z
P
/
H
�
0
�
B
y
P
/
H
�
0
�
Flux-cutting and flux-transport effects in type-II superconductor slabs in a parallel rotating magnetic field
Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11 1199
70 .θ ≈ At large rotation angles ( > 180θ ), the profile
( )yB x′ becomes stationary and < > ( )yM θ is, practically,
a constant function, having a value close to < > .zM−
So, the magnitude of the magnetization, |< >|M , is inde-
pendent of θ when the rotation angle is sufficiently large.
The profiles for the case when the external magnetic field
is large enough, in comparison with the penetration field
PH (as in Fig. 9), have an evolution similar to that pre-
sented in Fig. 8. However, the central regions of unaltered
magnetic induction rapidly disappear as θ is increased
(compare Figs. 8 and 9). This fact is due to noticeable reduc-
tion of the critical current densities cJ ⊥ and cJ with .B
4. Conclusion
We have applied the circular, elliptic, extended-elliptic,
and rectangular critical-state models to study the magnetic
behavior of irreversible type-II superconductors in a paral-
lel rotating magnetic field. The numerical method em-
ployed here is based on the substitution of the vertical law,
relating the electric field E and the current density ,J for
a nonlinear material equation having effective flux-cutting
and flux-flow resistivities in the dissipative region. The
substitution is justified when the applied magnetic field
aH slowly varies either in magnitude or direction, induc-
ing electric fields of sufficiently small magnitude inside the
superconductor. Within the elliptic (circular) critical-state
model such resistivities are not independent of each other
and have a ratio = /r ⊥ρ ρ equal to /c cJ J⊥ (=1 for the
circular model) at J just above its critical value .cJ On the
other hand, within the extended elliptic critical-state model
the ratio r is an independent parameter to be determined.
The rectangular critical-state model also uses two indepen-
dent resistivities, ρ and .⊥ρ However, unlike the other
critical-state models, the GDCSM assumes that flux cutting
and flux depinning do not affect each other.
The comparison of the predictions of the mentioned
critical-state models with experimental measurements of
magnetization for a rotating PbBi disk in a fixed magnetic
field [33] shows that the original critical-state model can
reproduce the main features of the magnetization curves.
The circular and rectangular critical-state models only
achieve a qualitative description of the experiment. The
extended elliptic model, being more general than the origi-
nal elliptic one, has allowed us to study the effect of the
relation between E and J in the dissipative region. How-
ever, additional theoretical and experimental studies are
needed to elucidate on the effects associated with both
flux-cutting and flux-flow resistivities.
This work was partially supported by Consejo Nacional
de Ciencia y Tecnología (CONACYT, Mexico).
Fig. 8. Profiles of the angle α (panel a), magnitude B (panel b) and components yB ′ (Eq. (29), panel c) and zB ′ (Eq. (30), panel d) of
the magnetic induction, calculated with the original elliptic critical-state model at = 2.0 .a PH H
0.2 0.4 0.6 0.8 1.0
0
45
90
135
180
225
270
315
360
x/d
a
8
7
6
5
4
3
2
1
0
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R. Cortés-Maldonado, J.E. Espinosa-Rosales, A.F. Carballo-Sánchez, and F. Pérez-Rodríguez
1200 Low Temperature Physics/Fizika Nizkikh Temperatur, 2011, v. 37, No. 11
1. L.M. Fisher, A.V. Kalinov, I.F. Voloshin, I.V. Baltaga, K.V.
Il’enko, and V.A. Yampol’skii, Solid State Commun. 97, 833
(1996).
2. C.P. Bean, Phys. Rev. Lett. 8, 250 (1962).
3. C.P. Bean, J. Appl. Phys. 41, 2482 (1970).
4. J.R. Clem, Phys. Rev. 26, 2463 (1982).
5. J.R. Clem and A. Pérez-González, Phys. Rev. 30, 5041
(1984).
6. A. Pérez-González and J.R. Clem, Phys. Rev. 31, 7048
(1985).
7. A. Pérez-González and J.R. Clem, Phys. Rev. 32, 2909
(1985).
8. A. Pérez-González and J.R. Clem, J. Appl. Phys. 58, 4326
(1985).
9. D.G. Walmsley, J. Phys. F2, 510 (1972).
10. A.M. Campbell and J.E. Evetts, Adv. Phys. 21, 199 (1972).
11. J.R. Cave and M.A.R. LeBlanc, J. Appl. Phys. 53, 1631
(1982).
12. R. Boyer and M.A.R. LeBlanc, Solid State Commun. 24, 261
(1977).
13. R. Boyer, G. Fillion, and M.A.R. LeBlanc, J. Appl. Phys. 51,
1692 (1980).
14. M.A.R. LeBlanc and J.P. Lorrain, J. Appl. Phys. 55, 4035
(1984).
15. F. Pérez-Rodríguez, A. Pérez-González, J.R. Clem, G.
Gandolfini, and M.A.R. LeBlanc, Phys. Rev. 56, 3473
(1997).
16. A. Silva-Castillo, R.A. Brito-Orta, A. Pérez-González, and
F. Pérez-Rodríguez, Physica C296, 75 (1998).
17. L.M. Fisher, K.V. Il’enko, A.V. Kalinov, M.A.R. LeBlanc,
F. Pérez-Rodríguez, S.E. Savel’ev, I.F. Voloshin, and V.A.
Yampol’skii, Phys. Rev. 61, 15382 (2000).
18. I.F. Voloshin, L.M. Fisher, and V.A. Yampol’skii, Fiz. Nizk.
Temp. 36, 50 (2010) [Low Temp. Phys. 36, 39 (2010)].
19. C. Romero-Salazar and F. Pérez-Rodríguez, Appl. Phys. Lett.
83, 5256 (2003).
20. I.F. Voloshin, A.V. Kalinov, L.M. Fisher, A.V. Aksenov,
and V.A. Yampol’skii, JETP 93, 1105 (2001).
21. C. Romero-Salazar and F. Pérez-Rodríguez, Supercond. Sci.
Technol. 16, 1273 (2003).
22. C. Romero-Salazar and F. Pérez-Rodríguez, Physica C404,
317 (2004).
23. L.M. Fisher, A.V. Kalinov, S.E. Savel’ev, I.F. Voloshin,
V.A. Yampol’skii, M.A. R. LeBlanc, and S. Hirscher,
Physica C278, 169 (1997).
24. L.M. Fisher, A.V. Kalinov, S.E. Savel’ev, I.F. Voloshin, and
V.A. Yampol’skii, Solid State Commun. 103, 313 (1997).
25. C. Romero-Salazar, L.D. Valenzuela-Alacio, A.F. Carballo-
Sánchez, and F. Pérez-Rodríguez, J. Low Temp. Phys. 139,
273 (2005).
26. J.P. Lorrain, M.A.R. LeBlanc, and A. Lachaine, Can. J.
Phys. 57, 1458 (1979).
27. C. Romero-Salazar and O.A. Hernández-Flores, J. Appl.
Phys. 103, 093907 (2008).
28. J.R. Clem, M. Weigand, J.H. Durrell, and A.M. Campbell,
Supercond. Sci. Technol. 24, 062002 (2011).
29. E.H. Brandt and G.P. Mikitik, Phys. Rev. 76, 064526 (2007).
30. G.P. Mikitik, Fiz. Nizk. Temp. 36, 17 (2010) [Low Temp.
Phys. 36, 13 (2010)].
31. J.R. Clem, Phys. Rev. B83, 214511 (2011).
32. A. Bada-Majós, C. López, and H.S. Ruiz, Phys. Rev. 80,
144509 (2009).
33. J. Sekerka, M.Sc. Thesis “Flux Cutting in Semi-reversible
and Irreversible Type II Superconductors”, University of
Ottawa (1989).
34. C. Romero-Salazar and F. Pérez-Rodríguez, J. Non-Cryst.
Sol. 329, 159 (2003).
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