JOURNAL OF APPLIED PHYSICS
VOLUME 92, NUMBER 10
15 NOVEMBER 2002
Magnetic flux bifurcation and frequency doubling
in rotated superconductors
A. Badı́aa)
Departamento de Fı́sica de la Materia Condensada-ICMA, CPS University of Zaragoza,
Marı́a de Luna 1, E-50.018 Zaragoza, Spain
C. López
Departamento de Matemática Aplicada, CPS University of Zaragoza, Marı́a de Luna 1,
E-50.018 Zaragoza, Spain
共Received 14 June 2002; accepted 1 September 2002兲
Starting from our variational statement of the general critical state in type II superconductors, we
develop anisotropic current flow simulations in various conditions. The theory is applied to the slab
geometry under rotating applied field, parallel to the surface of the sample. By comparison to the
isotropic case, we show that anisotropy strongly influences the underlying physical phenomena. A
magnetic-flux bifurcation point arises for the isotropic hypothesis. The issue of this point defines a
boundary between two groups of vortex lines, one of which rigidly settles within the sample and
another one, which frictionally rotates relative to the sample. A much more complex scenario arises
in the anisotropic case, for which dynamic fronts must be defined. As a consequence, we predict the
appearance of nonlinear phenomena such as the magnetization frequency doubling. The anharmonic
contribution may be tuned by the applied field modulus. © 2002 American Institute of Physics.
关DOI: 10.1063/1.1516867兴
I. INTRODUCTION
well below H c2 and allows to use the linear relation
B⫽ ␮ 0 H.
Although the main facts have been explained, some observations are still lacking a theoretical understanding. In
particular, as it was indicated in Ref. 2, the appearance of
higher frequency oscillations in the magnetic moment components, as the applied field modulus is increased, cannot be
predicted by the available models. In this article, we will
show that, as it was suggested by the authors of Ref. 2 such
phenomena can be ascribed to anisotropy in the current flow.
In particular, the investigation of the evolutionary field penetration profiles reveals a class of physical phenomena, such
as the appearance of counter-rotating flux tubes within the
sample. These mechanisms are behind the mentioned anharmonic effects.
By the application of our variational statement of the
critical state8 one can describe anisotropic specimens by a
single continuous parameter ␥, which takes the limiting
value of unity for the isotropic case. This will allow to elucidate the physical processes which directly relate to anisotropy.
In brief, our method is based on the minimization of a
cost functional which weighs the magnetic field changes
within the sample. This is done under the constraint J苸⌬
with ⌬ a bounded set. In this work, we have explored two
cases: 共i兲 ⌬ is an elliptic region with a given eccentricity and
共ii兲 ⌬ is a circle with variable radius, according to the magnetic field orientation with respect to some given axis. It will
be shown that both models produce several distinctive features. However, the relevant underlying phenomena behind
the experimental observations are jointly described and, thus,
independent of the specific model in use.
Rotating field experiments have been used by several
groups in order to provide information on crossflow of flux
lines in superconductors.1– 6 Typically, an irreversible type II
superconducting disk is subjected to an applied magnetic
field HS parallel to the flat faces. The disk is slowly rotated
about the axis perpendicular to the surface and the components of magnetic flux parallel and perpendicular to HS
monitored.
The analysis of the experimental data has led to important concepts on the dynamics of nonparallel flux line lattices. In particular, Clem and Pérez-González7 have shown
that most of the observed features can be well described by a
general critical-state model which incorporates both the fluxline cutting and pinning concepts. The underlying idea is that
the electrical current density vector J within the sample cannot exceed two critical values. When the component of J
along the direction of the magnetic field exceeds J c 储 flux-line
cutting between adjacent tilted lines occurs. When the component of J perpendicular to the field exceeds J c⬜ flux depinning occurs. As long as both phenomena take place with
high associated resistivities, one can neglect relaxation processes between the successive metastable equilibrium states.
In practical terms, the critical state approach makes sense if
slow variations of the applied field are imposed. Then, the
induced electric fields are small enough and one can neglect
resistive losses. The second fundamental assumption on
which the theory relies is that the reversible magnetization
contribution to the sample’s response can also be neglected.
This relates to strong pinning specimens well above H c1 and
a兲
Electronic mail: anabadia@posta.unizar.es
0021-8979/2002/92(10)/6110/9/$19.00
6110
© 2002 American Institute of Physics
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J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
A. Badı́a and C. López
In Sec. II we present the mathematical statement of the
anisotropic critical state models. This encloses the timediscretized description of the successive field penetration
profiles in an arbitrary magnetization process. The theory is
then applied to the rotating field experiment 共Sec. III兲. The
results are compared to the already established predictions of
the isotropic model. In Sec. IV we summarize the main findings and discuss possible extensions of the theory.
II. ANISOTROPIC CRITICAL STATE
Consider a type II superconducting infinite slab with applied magnetic field parallel to the surfaces. We take the X
axis perpendicular to the faces and the origin of coordinates
at the midplane. Thus, the induced fields in the superconductor H, J, and E are also parallel to the YZ plane and only
dependent on the coordinate x and the time. Anisotropy will
be modeled as a property related to the Y and Z axes defined
earlier.
In this work, we attempt a phenomenological approach
with highest simplicity, but enough physical content, so as to
predict the experimental facts. Inspired by the isotopic
model, which was shown to be the minimal theory for a
number of observations in rotating and crossed field
experiments,8,9 we propose two different approximations.
First, we will consider that the critical current density
obeys a tensor relation of the kind 兩 J y 兩 ⭐J c , 兩 J z 兩 ⭐ ␥ J c , with
␥ some anisotropy parameter. As a straightforward generalization of the isotropic law 兩 J兩 ⭐J c we introduce the rule
J 2y ⫹
J z2
␥
⭐J 2c .
2
兩 J兩 ⭐J c 共 ␣ 兲 ,
the field penetration profiles corresponding to parallel flux
line lattices either along Y or Z axes may be determined by
Maxwell laws together with Eqs. 共3a兲 or 共3b兲. For H directed
along an arbitrary direction, one needs to make some further
statement on the restrictions upon J. In this section, we investigate the rule that J lies within the elliptic region given
by Eq. 共1兲. This includes the isotropic model as the particular
case for which ␥ ⫽1. Additionally, the one-dimensional
problems of current flow along principal axes are also included as particular cases.
The mathematical statement will be built on previous
work,8 where we showed that the critical state can be understood as a variational problem with a restriction of the kind
J苸⌬ for the current density. Here, ⌬ is defined by Eq. 共1兲.
Now, Ampère’s law, together with the critical current
restriction may be written as
dH y
⫽␥uy
dx
共4a兲
dH z
⫽u z ,
dx
共4b兲
with u a two-dimensional vector within the unit disk D. Notice, in passing, that Eqs. 共4a兲 and 共4b兲 arise when x is measured in units of the slab half-thickness a and H in units of
J c a.
Next, we require the minimization of the functional
C关 Hn⫹1 共 x 兲兴 ⫽ 21
共1兲
It is apparent that Eq. 共1兲 may be expressed as 兩 J兩 苸⌬, ⌬
being an elliptic region with semiaxes J c and ␥ J c . In consequence, this will be named elliptic model hereafter.
The second approximation will introduce the tensorial
characteristic as a dependence of the critical current density
on the magnetic field orientation. Namely, we will use the
relation
共2兲
with ␣ the angle between the magnetic field and a given axis
within the sample. This hypothesis may also be written as
兩 J兩 苸⌬ with ⌬ a circle of variable radius as a function of ␣.
Thus, we will call it pseudoisotropic model.
Let us assume that the superconductor possesses nonisotropic limiting values for the current density along the Y and
Z axes. Just for convenience, these will be considered as the
principal directions of the sample, i.e.,
共 0,0,H z 兲 → 兩 J y 兩 ⭐J c
共3a兲
共 0,H y ,0兲 → 兩 J z 兩 ⭐ ␥ J c ,
共3b兲
with J c the critical current density, which is taken to be constant along this article.
The sample’s response to an external drive may be
straightforwardly obtained by conventional critical state
principles as far as oblique current flow is not induced. Thus,
冕
1
0
兩 Hn⫹1 ⫺Hn 兩 2 dx,
共5兲
which measures the magnetic field changes along successive
time steps, satisfying Eqs. 共4a兲 and 共4b兲. As usual, Hn (x)
stands for the magnetic field profile at the time layer n ␦ t.
Recall that minimization is attained upon defining the associated Hamiltonian
H⫽p y ␥ u y ⫹ p z u z ⫺ 21 共 Hn⫹1 ⫺Hn 兲 2 ,
共6兲
and imposing the algebraic condition
* ,p* ,u* 兲 ⫽maxH共 Hn⫹1
* ,p* ,u兲 .
H共 Hn⫹1
共7兲
u苸D
From the previous equation one can solve for the control
variable u:
u* ⫽
A. Elliptic model
6111
共 ␥ p ⬘y ,p z⬘ 兲
冑␥ 2 p *y 2 ⫹ p z* 2
.
共8兲
Eventually, the Hamiltonian equations, which hold the solution of the problem, read
*
dH n⫹1,y
dx
*
dH n⫹1,z
dx
dp*
y
dx
⫽
⫽
␥ 2 p *y
冑␥ 2 p *y 2 ⫹ p z* 2
p z*
冑␥ 2 p *y 2 ⫹ p z* 2
* ⫺H n,y
⫽H n⫹1,y
共9a兲
共9b兲
共9c兲
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6112
J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
dp z*
dx
* ⫺H n,z .
⫽H n⫹1,z
A. Badı́a and C. López
共9d兲
As usual, this set of equations requires an appropriate number of boundary conditions to be supplied. Continuity of H
across the surface will produce two conditions at x⫽1. The
remaining two will be determined according to the partial or
full penetration regime for the field changes 共see Ref. 8 for
details兲.
Thus, the iterative solution of the system 关Eqs. 共9兲兴 for
successive time steps will produce the field and current penetration profiles for a changing external excitation.
A specific application of the previous formalism will be
given in Sec. III, where an excitation rotating field with constant modulus H S at the sample’s surface is considered for
various values of H S .
B. Pseudoisotropic model
Later we develop the pseudoisotropic hypothesis
兩 J兩 ⭐J c f 共 ␣ 兲 ,
共10兲
in which the current density modulus is constrained by a
certain function of the magnetic field tilt with respect to
some axis within the sample. For definiteness, let us take it
with respect to the Z axis and consider
f 共 ␣ 兲 ⫽ ␥ sin2 ␣ ⫹cos2 ␣ .
共11兲
Notice that the earlier expression leads to the correct limit for
␥ ⫽1 and produces the required particular models for current
flow along principal axes ( ␣ ⫽0,␲ /2). This allows a direct
comparison to the previous model.
Just on following the same steps as before, and in terms
of the same dimensionless units, the critical current restriction may be written as
dH y ␥ H 2y ⫹H z2
⫽
uy
dx
H2
共12a兲
dH z ␥ H 2y ⫹H z2
⫽
uz ,
dx
H2
共12b兲
with u belonging to the unit disk D.
The minimization of the cost functional C关 Hn⫹1 (x) 兴 is
now attained by maximizing the Hamiltonian
H⫽
2
2
␥ H n⫹1,y
⫹H n⫹1,z
2
H n⫹1
1
p"u⫺ 共 Hn⫹1 ⫺Hn 兲 2 .
2
共13兲
This leads to the condition uÄp* /p * and one gets the
Hamiltonian equations
*
dH n⫹1,y
dx
*
dH n⫹1,z
dx
dp *
y
dx
⫽
⫽
* 2 ⫹H n⫹1,z
*2
p*
y ␥ H n⫹1,y
p*
共14a兲
*2
H n⫹1
* 2 ⫹H n⫹1,z
*2
p z* ␥ H n⫹1,y
p*
共14b兲
*2
H n⫹1
* ⫺H n,y ⫺2 p *
⫽H n⫹1,y
* H n⫹1,z
*2
共 ␥ ⫺1 兲 H n⫹1,y
*4
H n⫹1
共14c兲
d p z*
dx
* ⫺H n,z ⫺2 p *
⫽H n⫹1,z
* H n⫹1,y
*2
共 1⫺ ␥ 兲 H n⫹1,z
*4
H n⫹1
. 共14d兲
Again, this first order system must be solved by supplying
appropriate boundary conditions. In the next section, this
will be done for the rotating field experiment.
III. ROTATIONAL PROPERTIES
Here we apply the mathematical models developed earlier to analyze the magnetic response of a superconducting
slab, rotating in a uniform magnetic field perpendicular to
the axis of rotation. Just for convenience we use a reference
frame which rotates with the sample and matches the anisotropy directions. Thus, we have an applied magnetic field at
the surface HS (t)⫽H S (0,sin ␣S ,cos ␣S) with ␣ S ⫽ ␻ 0 t. ␻ 0
stands for a constant angular velocity, which is assumed to
be small enough for relaxation effects to be neglected. The
field penetration profiles H(x,t) will be derived for such a
process. Eventually, the magnetic response of the sample will
be obtained by using averaged values, i.e.: M⬅ 具 H(x) 典
⫺HS .
Although the rotational properties for isotropic samples
have been well studied before, we will make a brief description of the main facts. This will allow to reveal the physical
phenomena with are either hindered or promoted by anisotropy.
A. Isotropic sample rotation
The most outstanding experimental fact, which one can
well describe within the isotropic model is the so-called magentic flux bifurcation. Such a phenomenon is clearly observed in field cooled experiments for which the sample
holds a nonmagnetic initial profile 共0,0,H S ). This will be the
situation considered hereafter. As rotation takes place, the
process consists of the separation of the vortex lines in two
groups, one which rigidly pins to the sample (V R ) and another one, which frictionally rotates relative to the sample
(V F ). The inset in Fig. 1 illustrates this behavior. We have
depicted the characteristic stationary state V shape profile for
the field modulus, as well as several rotation angle profiles
subsequent to the transient period. It is apparent that the
vortices within the region V R are rigidly fixed to the sample
共constant modulus and angle兲, whilst those in region V F frictionally rotate relative to it.
Although Fig. 1 has been obtained by application of our
isotropic model ( ␥ ⫽1), we want to stress that such behavior
can be predicted by other theoretical approaches.2,4,7
As a final observation we recall that the volume occupied by the region V R is progressively reduced as H S increases and vanishes for a characteristic field H S* , which
penetrates to the center with maximum gradient.
The experimental counterpart of the previously described process corresponds to the appearance of a stationary
state with harmonic oscillations in the magnetization components. It is customary to record the magnetic moment com-
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J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
A. Badı́a and C. López
6113
ponents along the longitudinal and transverse directions to
the applied field (M L ,M T ). The vortices within V R contribute as the projections of a constant modulus rotating vector
and the vortices within V F as the projections of a vector with
static components. Thus,
stationary
dc
⫽M L,T
M L,T
共 H S 兲 ⫹M 0 共 H S 兲 sin共 ␻ 0 t⫹ ␾ L,T 兲 .
共15兲
As indicated earlier, the amplitude factor M 0 (H S ) goes to
zero when the region V R disappears. This leads to constant
values M L ,M T , coming from the V F vortices which, in this
case, occupy the whole sample. This transition defines the
disappearance of the flux bifurcation point x 0 . We have also
depicted this behavior in Fig. 1.
B. Anisotropic elliptic model
FIG. 1. Calculated longitudinal magnetization curves of an isotropic slab as
a function of the rotation angle ␣ S relative to the applied field at the surface.
Open symbols show the oscillations for low values of the applied field (H S ),
and filled symbols correspond to high fields. The inset displays the field
modulus H and angle rotation ␣ profiles within the sample for the stationary
regime in the case of low fields. H has reached a fixed V shape. The angle
variation is blocked below the decoupling point x 0 .
Steady state solutions leading to harmonic magnetization
components as given by Eq. 共15兲 will be no longer valid for
anisotropic samples. It will be shown that, in this case, the
magnetic moment of the sample must be described as a combination of the Fourier components ␻ 0 and 2 ␻ 0 . Later, we
discuss the distinctive features of rotation experiments analyzed within the elliptic model framework. For definiteness,
we display the solution of Eqs. 共9兲 in the case ␥ ⫽0.5. The
FIG. 2. Rotation angle ␣ and field modulus H penetration profiles as calculated according to the elliptic model for a field cooled rotation experiment. The
horizontal axis displays the normalized coordinate within the slab 共midplane x⫽0 and surface x⫽1). Corresponding modulus and angle curves have been
plotted in the same column and using a common line style. Decoupling points for which field cancellation occurs and counter-rotation starts have been labeled
(x 0,1 and x 0,2). x A labels the boundary with rigid vortices within the sample. The rotation angle is expressed in radians and the magnetic field modulus in
dimensionless units. The insets display the rotation stages for which penetration profiles were plotted.
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6114
J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
A. Badı́a and C. López
boundary value problem for the nonlinear ordinary differential equations of our system has been solved by a finitedifference approximation.
1. Low-field rotation
First, we analyze the rotation experiment for low values
of the applied field (H S ⫽0.1 and H S ⫽0.25 in our dimensionless units兲. As it will become clear later, this means that
the midplane is not reached by the perturbation.
Again, we choose the modulus and angle representation,
which is most significant for rotation experiments. Figure 2
shows the results in the case H S ⫽0.25 for several intermediate stages when the sample undergoes two turnabouts (0
⭐ ␣ S ⭐4 ␲ ).
As the sample turns from 0 to 2␲, flux consumption
produces the well known behavior observed for isotropic
cases. The rotation angle penetration regime evolves towards
a step shape. Simultaneously, the field modulus becomes
V-shaped, and a decoupling point 关 H(x 0,1⫽0 兴 is defined.
However, when rotation proceeds, the profile H(x) keeps no
longer stationary. On the contrary, a complex structure with
two minima arises. Notice, in addition that flux lines are no
longer blocked for x⬍x 0,1 . The region of rigid vortices is
shifted to 0⬍x⬍x A , with x A as the boundary. Now, the
negative angle values for x A ⬍x⬍x 0,1 correspond to a
counter rotation of the flux tubes, subsequent to the decoupling step.
A second turnabout 共2␲ – 4␲兲 completes the full picture
of a periodic fashion in the physical phenomena described
earlier. Note that the system evolves again towards a V shape
for the modulus and a step-like angle profile, defining a second blocking point x 0,2 . Again, this is accompanied by
counter rotating flux lines for x A ⬍x⬍x 0,2 . We notice that
the flux consumption previous to decoupling is asymmetric
and this is the reason why one gets a shifted cancellation
point and a branch point for the rotation angle.
2. High-field rotation
We consider now the case that H S is large enough for
magnetic field changes to reach the midplane as the sample
rotates. The characteristic behavior has been displayed in
Fig. 3. Basically, ␣ (x) shows a quasilinear penetration profile as rotation proceeds. On the other hand, subsequent to a
transient period, the field modulus evolves in between well
defined boundary profiles. This evolution takes place in a
nearly harmonic stationary regime at the frequency 2 ␻ 0 .
3. Magnetization curves
Finally, we analyze the experimental fingerprint of the
field penetration profiles described above. In Fig. 4 we have
plotted the longitudinal and transverse components of the
averaged magnetization for three values of the applied field
modulus. Notice that, for the lowest field value, M L,T basically display a stationary harmonic behavior at the sample’s
rotation frequency ␻ 0 . As H S increases, one can readily observe a frequency mixing phenomenon. This corresponds to
the competition between the fundamental frequency contribution from the rigid vortices region, and the double fre-
FIG. 3. Rotation angle ␣ and field modulus H penetration profiles as calculated according to the elliptic model for a field cooled rotation experiment.
In this case, the high value of the magnetic field (H S ⫽1) produces a full
penetration regime. Symbols have been used as a guide of the evolutionary
successive profiles.
quency arising from the frictionally rotated vortices. Eventually, for high enough fields, the rigid core disappears and M
becomes a rotating vector at frequency 2 ␻ 0 .
C. Pseudoisotropic model
In this part we present the rotational characteristics of
the anisotropic slab as calculated by the pseudoisotropic
model equations 共Sec. II B兲. For comparison purposes, the
set of Eqs. 共14兲 has been solved for ␥ ⫽0.5 and the same
boundary conditions as in Sec. III B. Again, we provide numerical solutions obtained by a finite-difference algorithm.
1. Low-field rotation
Figure 5 depicts the modulus and angle penetration profiles for the rotation experiment at H S ⫽0.25. Several stages
have been selected as the sample turns from 0 to 4␲. On
purpose, we have chosen the curves for which decoupling
occurs and some nearby steps.
We emphasize that, again, a decoupling point (x 0,1)
arises in the first turn 共0–2␲兲, followed by flux transport and
counter-rotation. Also, a second turnabout 共2␲ – 4␲兲 leads to
a V shape for the modulus profile, as well as a bifurcation
point for the rotation angle. As one could expect, the values
of x A ,x 0,1 , and x 0,2 are close but not coincident with the
corresponding quantities in the elliptic model simulations.
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J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
A. Badı́a and C. López
6115
FIG. 4. Longitudinal and transverse components of the averaged magnetization as a function of the rotation angle ␣ S for a field cooled slab at different values
of the applied magnetic field. Calculations have been made for the current density elliptic model.
2. High-field rotation
The calculated field penetration profiles in this case display similar properties to the elliptic model predictions, but
some differences appear. Basically, the field modulus develops a stationary state behavior, in which also evolves in between an upper and lower boundary. However, these boundaries are not so rigid and some intermediate profiles can
develop a small hump beyond the boundary 共see Fig. 6兲.
Notice also the vertical scale for the field modulus, which
indicates the prediction of a smaller field variation in the
present model 共compare to Fig. 3兲.
3. Magnetization curves
The magnetization tracings M L,T ( ␣ S ) for three values of
the applied magnetic field are shown in Fig. 7. As predicted
in the previous model 共see Fig. 4兲, subsequent to a transient
period, we get nearly harmonic oscillations for low fields, a
frequency mixing phenomenon for intermediate values, and
a double frequency oscillation for high fields. Having fixed
all the parameters of the theory, the two models just differ in
some features of the double frequency response. Notice the
phase shift and smaller amplitude for the pseudoisotropic
curves.
IV. CONCLUSIONS
In this work we have explored the application of our
variational statement of the critical state8,9 to anisotropic systems. Theoretical predictions have been made, which confirm the long suspected relation of nonlinear effects in rotation experiments to anisotropy in the critical current density.
Mathematically, the theory is posed as a minimization
problem under a constraint of the kind J苸⌬ for the current
density. Two different phenomenological selections of ⌬
have been studied: 共i兲 ⌬ is an elliptic region and 共ii兲 ⌬ is a
circle with variable radius, depending on the local magnetic
field orientation relative to some axis within the sample.
Though quite difference in their statement, both models provide quite similar predictions, in good agreement with experiment. We obtain harmonic oscillations in the magnetic
moment when the sample is rotated at low values of the
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6116
J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
A. Badı́a and C. López
FIG. 5. Same as Fig. 2, but calculated according to the pseudoisotropic model.
applied field. These oscillations take place at the sample’s
rotation frequency. Moderate field values, however, involve
frequency mixing phenomena, while high field oscillations
occur at a nearly second harmonic regime.
By comparison of the magnetic field penetration profiles
obtained within our anisotropic models, to the predictions of
the isotropic model 共⌬ is a circle兲, we can reveal the physical
phenomena on which the magnetic properties rely. For field
cooled isotropic systems, a stationary rotational state is established in which two groups of vortices exist: a rigid core
well within the rotating sample and a region of flux tubes
near the surface which frictionally rotate relative to the
sample, but remain at rest with respect to the applied field.
Both regions are connected by a zero field point 共decoupling
point x 0 ). The rigid core contributes as a harmonically oscillating magnetic moment and the rotating flux as a constant
signal in the laboratory reference. On increasing the applied
magnetic field H S , the boundary x 0 is shifted towards the
center of the sample, and the oscillations are gradually reduced until their amplitude becomes null. When anisotropy
in the critical current density is allowed, one gets a quite
different scenario. The rigid core may still exist, but multiple
decoupling points must be defined. Specifically, a modulus
cancellation and angle branching point appears for each turn
of the sample. However, as rotation proceeds, flux transport
occurs and the zero field point disappears until the next turn.
On the other hand, we obtain 2␲ steps in the rotation angle
which are related to flux counter-rotation.
FIG. 6. Same as Fig. 3, but calculated according to the pseudoisotropic
model.
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J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
A. Badı́a and C. López
6117
.
共16b兲
FIG. 7. Same as Fig. 4, but calculated according to the pseudoisotropic model.
As a result of the earlier described events for anisotropic
systems, the frictionally rotating vortices do not contribute as
a constant signal and one gets a superimposed doublefrequency rotation. Eventually, the second harmonic signal at
high fields relates the filed modulus bouncing between well
defined boundaries, instead of the frozen profile for the isotropic case.
From the mathematical point of view, as well as for
physical interpretation purposes, our variational principle is a
convenient, but nonunique formulation of the critical state in
superconductors. For instance, the use of Maxwell equations,
together with appropriate constitutive J(E) laws is a standard
practice. Of particular relevance to the present work is to
mention that an alternative model has been posed in terms of
nonlinear diffusion of electromagnetic fields.10 In that reference 共see Chap. 4兲 the author uses Faraday’s law in the form
ⵜ 2 E⫽ ␮ 0 ⳵ t J(E), for constitutive relations, which in the
critical state and within our selection or coordinate axes read
J y ⫽k 共 1⫹ ⑀ 兲
Ey
冑共 1⫹ ⑀ 兲 E 2y ⫹ 共 1⫺ ⑀ 兲 E z2
,
共16a兲
J z ⫽k 共 1⫺ ⑀ 兲
Ez
冑共 1⫹ ⑀ 兲 E 2y ⫹ 共 1⫺ ⑀ 兲 E z2
Here k and ⑀ are parameters accounting for the superconductor properties. These equations are valid both for isotropic ( ⑀ ⫽0) and anisotropic 共elliptic兲 cases. Later, we show
that this model and ours are equivalent in such situations.
The key point is that the transient electric field, which arises
when the penetration profile evolves from the equilibrium
state Hn (x) to Hn⫹1 (x) may be basically identified with the
momentum p(x) in our theory. p(x) was introduced as a
Lagrange multiplier, but here we give a definite physical interpretation of this variable. Notice that Eqs. 共9c兲 and 共9d兲
are nothing bu the time-discretized form of Faraday’s law in
convenient dimensionless units for E 共defined by
␮ 0 J c a 2 / ␦ t):
dp*
y
dx
* ⫺H n,y ⬅
⫽H n⫹1,y
dE z
⇒E z ⫽ p *
y ,
dx
共17a兲
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6118
J. Appl. Phys., Vol. 92, No. 10, 15 November 2002
dp z*
dx
* ⫺H n,z ⬅⫺
⫽H n⫹1,z
dE y
⇒E y ⫽⫺p z* .
dx
A. Badı́a and C. López
共17b兲
Of course, these equalities hold safe arbitrary constants.
However, is we recall that p must vanish either for free
boundary conditions 关partial penetration, p(x * )⫽0] or when
the midplane is reached by field changes 关full penetration,
p(0)⫽0], these constants must be zero. The physical counterpart of the former relations are the continuity conditions
for the electric field 关leading to E(x * )⫽0 at free boundaries兴
and the antisymmetry with respect to the midpoint 关 E(0)
⫽0 兴 . On substituting Eqs. 共17a兲 and 共17b兲 in the Hamiltonian relations and recalling Ampère’s law, we get the corresponding constitutive equations in our formalism
J y ⫽J c
J z ⫽J c
冑
Ey
,
2
E y ⫹ ␥ 2 E z2
␥ 2E z
冑E 2y ⫹ ␥ 2 E z2
.
共18a兲
共18b兲
The equivalence of this system and Eqs. 共16a兲 and 共16b兲 is
apparent upon using the relations
1⫹ ⑀ ⫽
J 2c
k
2
1⫺ ⑀ ⫽ ␥ 2
k2
共19兲
where g ⫾ are constants related to the sample’s demagnetizing factor, ␤ stands for the rotation axis misalignment relative to the sample’s surface vector, and ␣ S is once more used
for the rotation angle.
Finally, we will comment on some issues in prospect.
This work explores the consequences of using anisotropic
constraints on the current density vector (J苸⌬) within the
critical state of hard superconductors. A reasonable understanding of available experiments has been provided. However, our work is developed at a phenomenological level and
a rule for the selection of ⌬ would be of interest. Two research lines are suggested: 共i兲 from a more fundamental point
of view, we propose the consideration of the flux line lattice
and pinning sites interactions in anisotropic systems with
flux cutting phenomena, and 共ii兲 it would be also desirable to
design experiments which allow a well-posed inverse problem, i.e., such that one can unambiguously reconstruct ⌬ on
the basis of experimental data.
ACKNOWLEDGMENTS
The authors are indebted to Professor J. R. Clem for
helpful discussions and suggestions. Financial support from
Spanish CICYT 共Project Nos. MAT99-1028 and BFM-20001066/C0301兲 is acknowledged.
,
J 2c
M L ⫽⫺H S 关 g ⫹ ⫹g ⫺ 共 sin2 ␤ cos 2 ␣ S ⫺cos2 ␤ 兲兴
.
An important remark from the practical point of view is
that double-frequency oscillations could also be obtained for
isotropic samples, as a result of the rotation axis misalignment. However, this can be considered as a small correction
unless for very noticeable deviations. Basically, the correction depends on the square of the error angle. In order to see
how this factor arises one can start with the relations of Ref.
11 where the longitudinal and transverse magnetic moment
of a flat sample under oblique field have been obtained for
the Meissner state. For instance, after some algebra, the longitudinal magnetization component may be written as
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