Physica C 297 Ž1998. 153–160
Comparison of the pinning and the bulk currents in the critical
state of a type-II superconductor
V.M. Krasnov ) , V.V. Ryazanov
Institute of Solid State Physics, Russian Academy of Sciences, 142432 ChernogoloÕka, Russia
Received 5 November 1997
Abstract
The bulk current density distribution in the critical state of a type-II superconductor is studied for different pinning
strengths and external magnetic fields. The calculations were made within the extended critical state model for a three-axis
ellipsoid, taking into account the equilibrium vortex lattice magnetization caused by the vortex interaction. It is shown that
the average current density, Jav , could be considerably different from the critical pinning current density, Jc , for the
magnetic fields not much larger than the lower critical field Hc1. The difference between Jav and Jc result in additional
curvature of the local magnetic field profiles and modifies the total moment of the sample which might be important for the
analysis of various magnetization experiments. q 1998 Elsevier Science B.V.
PACS: 74.60.Jg; 74.60.Ge; 74.60.Ec
Keywords: Critical state; Type-II superconductivity; Pinning; Abricosov vortex lattice magnetization
1. Introduction
A study of the critical state in type-II superconductors provides important information about the
current carrying facilities of superconducting materials for their application in practice and about basic
physical properties of superconductors. The discovery of high-Tc superconductors ŽHTSC. has revived
the interest in the nature of the mixed state in type-II
superconductors, where the superconductivity co-exists with the inhomogeneous magnetic field inside
)
Corresponding author. Present address: Department of Physics,
Chalmers University of Technology, S-41296 Goteborg,
Sweden.
¨
Tel.: q46 31 7723397; Fax: q46 31 7723471; E-mail:
krasnov@fy.chalmers.se
the materials. The inevitable presence of defects in
HTSC gives rise to a large pinning of magnetic flux
lines due to a short superconducting coherence length.
This enriches the variety of different inhomogeneous
vortex states, e.g. the critical state. Investigations of
the Abricosov vortex lattice ŽAVL. transitions and
the flux creep mechanisms have demanded the development of local methods for magnetic measurements of reversible and irreversible properties of the
inhomogeneous AVL.
One of the most crucial parameters of the critical
state is the critical current density, Jc , which is
determined by the pinning of Abricosov vortices on
inhomogeneities and by interaction of vortices with
each other within the AVL. The knowledge of the
critical current behaviour yields direct information
about pinning and flux creep mechanisms and about
0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 9 2 1 - 4 5 3 4 Ž 9 7 . 0 1 8 6 7 - 4
154
V.M. KrasnoÕ, V.V. RyazanoÕr Physica C 297 (1998) 153–160
the AVL phase diagram Žfor review see Refs. w1,2x..
Among the experimental methods for the study of
the critical state we mention Ži. global magnetization
methods, in which the integrated magnetic moment
of the whole sample is measured and Žii. local
magnetization methods, in which a local distribution
of the magnetic induction in the sample is obtained.
Such methods include the magneto-optic w3–7x and
the local Hall-sensor w7x techniques. The local magnetic field profiles depend on the magnetic field
dependence of Jc , the sample geometry and equilibrium parameters of the superconductor such as the
lower critical field, Hc1 , the Ginzburg–Landau parameter, k , etc. An explicit critical state model
taking into account all those factors is required for
the evaluation of the critical current density from
magnetization data of type-II superconductors.
Historically, the first critical state model for a
superconducting cylinder or slab with zero demagnetization factor, D s 0, was formulated by Bean w8,9x.
Later the Bean critical state model was extended for
different sample geometry with non-zero demagnetization factor such as a three-axis ellipsoid w10,11x
disk w12x, ring, strip, rectangle, etc., see e.g. Refs.
w2,13,14x and references therein. In the Bean model
the bulk current density flowing in superconductor is
equal to Jc . Although the Bean model qualitatively
describes experimental data, it is essentially macroscopic and does not regard the vortex structure of the
mixed state in type-II superconductor. As a result the
Bean model does not take into account the equilibrium AVL magnetization due to repulsion of vortices. A microscopic critical state model explicitly
taking into account the AVL structure and pinning
effect could of course avoid this problem. An example of the microscopic one and two-dimensional
critical state simulations can be found in Refs.
w15,16x. However, microscopic simulations in the
three-dimensional case corresponding to real experimental situation could require enormous computational efforts. On the other hand, the vortex interaction and equilibrium AVL magnetization could be
introduced within the macroscopic critical state
model. Such modification of the critical state model
was done shortly after the Bean model appeared w17x,
see also Refs. w18–20x. Due to the vortex structure of
the mixed state, the type-II superconductor cannot be
described by a constitutive equation with some mag-
netic permeability. The average current density in
general is different from the local current density at
the vortex origin, which defines the Lorentz force
acting on the vortex. As a result, the average current
density in the critical state is different from Jc ,
Jav s Jc q J L ,
Ž 1.
where J L is the so called ‘lattice magnetization
current’ density due to the vortex interaction in the
AVL. The magnetic moment associated with J L is
referred to as the ‘lattice magnetization’, M L , and
represents the equilibrium AVL magnetization. In
the previous papers w10,11x we have already shown
that the equilibrium AVL magnetization could be
important for the analysis of the magnetic field
profiles in hard type-II superconducters. Though the
AVL magnetization effect in the critical state of
type-II superconductors is well known, nowadays it
is typically ignored in the literature related to the
study of magnetic field penetration and flux creep.
Sometimes a phenomenological dependence of the
critical pinning current Jc Ž B . on the magnetic inductance, B, is introduced in order to describe the
non-linear B Ž x . profiles. Such is for example the
Kim model w21,22x, where Jc ; Ž B q B0 .y1 . Indeed
the Jc could depend on the AVL density, e.g. due to
transition from the individual to collective pinning
w1x. However, as can be seen from Eq. Ž1., the
profiles B Ž x . could be non-linear even for constant
Jc due to contribution of the lattice magnetization
current, which has highly non-linear magnetic field
dependence J LŽ B .. Thus, it is necessary to distinguish between the magnetic field dependencies of Jc
and J L in order to study the pinning mechanisms in
type-II superconductors.
A phenomenon similar to the critical state of
type-II superconductor exists also in long spatially
non-uniform Josephson junctions ŽJJ. w23x. Indeed
long JJ behave in many aspects as type-II superconductors. The magnetic field penetrates into long JJ in
the form of Josephson vortices-fluxons. In spatially
non-uniform JJ the fluxon energy depends on the
position in the JJ. A self-energy gradient of the
fluxon creates a force preventing the fluxon entrance
into the JJ, which is similar to the pinning force in
type-II superconductor. The magnetic field penetrates the spatially non-uniform JJ in a form of
magnetic domain consisting of a fluxon chain with
V.M. KrasnoÕ, V.V. RyazanoÕr Physica C 297 (1998) 153–160
variable fluxon density. This domain is a clear analogy to the critical state of type-II superconductors.
An advantage of JJ is that they can be described by a
rather simple sine-Gordon equation taking into account the fluxon structure of the ‘mixed state’ in
long JJ. Numerical simulation of the magnetic domain representing the critical state in spatially nonuniform long JJ is presented in Ref. w23x.
In this paper we study the average current density
distribution in the critical state of type-II superconductor in comparison with the critical pinning current density for different pinning strengths and external magnetic fields. The calculations were made
within the extended critical state model for a threeaxis ellipsoid w10,11x, taking into account the equilibrium AVL magnetization caused by the vortex interaction. It is shown that the average current density
could be considerably different from Jc for the
magnetic fields not much larger than Hc1 . The difference between Jav and Jc results in additional curvature of the local magnetic field profiles and modifies
the total magnetic moment of the sample. The paper
is organized as follows. In Section 2 we introduce
necessary quantities and formulate the modified
macroscopic critical state model taking into account
the AVL structure of the mixed state in type-II
superconductor. Section 3 describes the extension of
the critical state model for a three-axis ellipsoid.
Section 4 presents the calculations of the bulk current density in the critical state of type-II superconductor as a function of magnetic field and the critical
pinning current and contains a discussion of our
results and their importance for the analysis of magnetization data of type-II superconductors.
155
oscillating function of coordinate. In the London
limit of high-k superconductors it can be written as:
hŽ r . s
F0
2pl
2
Ý K0
ri
žl/
,
Ž 2.
where F 0 is the flux quantum, l is the London
penetration depth and K 0 is the modified Bessel
function. h is related to the microscopic current
density, j, via the Maxwell equation
rot h s
4p
c
j,
Ž 3.
where c is the velocity of light.
Žii. B — The magnetic induction equal to the
average value of h,
B Ž r . s² h Ž r . :S .
Ž 4.
B is a macroscopic quantity measured in local magnetization experiments. The area of averaging, S,
depends on the resolution of experiments and is
typically of the order of several micrometers. B is
related to the average current density, Jav s ² j :, via
the Maxwell equation
rot B s
4p
c
Jav .
Ž 5.
Žiii. H — The magnetic field intensity. This is a
measurable macroscopic quantity defined via the
magnetic moment M:
H s B y 4p M.
Ž 6.
In the absence of externally applied currents
rot H s 0.
Ž 7.
Živ. H ) — The local ‘thermodynamic magnetic
field’ intensity, defined as:
EF
H ) Ž B. s
2. Modified critical state model
In this section we will recall how the Bean critical
state model could be modified to take into account
the AVL magnetization w10,11,17–20x. To include
the vortex interaction into the macroscopic critical
state model it is necessary to introduce the following
quantities:
Ži. h — The microscopic magnetic field intensity.
h is the sum of magnetic fields of vortices and is an
EB
,
Ž 8.
where F is the local free energy density of superconductor. In equilibrium case Ži.e. without pinning.
H ) is equal to H. H ) determines the equilibrium
lattice magnetization, M L , of the AVL w24x
M L s Ž B y H ) . r4p .
Ž 9.
This is also valid in the presence of pinning if the
local quasi-equilibrium state is achieved in every
point of the sample. The dependence M LŽ H ) . for
k s 10 is shown in Fig. 1. The characteristic feature
V.M. KrasnoÕ, V.V. RyazanoÕr Physica C 297 (1998) 153–160
156
and h. The driving force is determined by the microscopic current density j at the vortex origin ŽEq. Ž3..
which is not equal to Jav ŽEq. Ž5...
In the critical state the driving force is balanced
by the pinning force, FD s yFP , which is defined
via the critical pinning current density
FP s BJcrc.
Ž 16 .
Using Eqs. Ž14. and Ž16. we can formulate the
modified critical state model in the general form
Fig. 1. Equilibrium Abricosov vortex lattice magnetization curve
for k s10. The inset shows the derivative d M L rd H ) , which
defines the AVL magnetization current.
rot H ) s
4p
c
Jc .
Ž 17 .
of M LŽ H ) . is a sharp change at H ) s Hc1. From
Eqs. Ž1., Ž5. and Ž9. it follows
The difference from the original Bean model is that
Jc determines the profile of the thermodynamic field
H ) , but not B. The profile of the magnetic induction
rot M L s JL rc.
BsBŽ H ) . ,
Ž 10 .
Ž 18 .
)
With a help of H , the vortex interaction in the
AVL could be taken into account within the macroscopic approach. Following Ref. w17x we consider a
volume V containing a constant amount of vortices.
In equilibrium the Gibbs free energy,
BH )
GsFy
,
4p
reaches its minimum so that
dG s ySdT y
Bd H
Ž 11 .
)
q Vd p s 0.
Ž 12 .
4p
Here S, T and p are the entropy, temperature and
pressure, respectively. Taking into account that the
net driving force is equal to
FD s yV
dp
,
dx
from Eq. Ž12. we obtain
FD s y
Ž 13 .
B dH)
B dB
Mi s Ž H ) y H . r4p .
,
Ž 15 .
4p d x
which is due to the fact that type-II superconductors
do not have a constitutive equation connecting B
Ž 19 .
Finally, we should determine the bulk current
density in the sample. From Eq. Ž1. it is seen that the
average bulk current is different from the critical
pinning current on the quantity of the AVL magnetization current. From Eqs. Ž9. and Ž17. it follows that
JL
Jc
.
Ž 14 .
4p d x
We emphasize that the driving force is different from
the Lorentz force
FL s y
should be determined from Eqs. Ž8. and Ž9. using the
equilibrium AVL magnetization curve M LŽ H ) ., see
Fig. 1. The total magnetic moment is defined by Eq.
Ž6. and is a sum of the AVL magnetization M L and
the moment created by the pinning current Mi
s 4p
d ML
dH)
.
Ž 20 .
In the inset of Fig. 1 the behaviour of the derivative
M LX s d M Lrd H ) is shown. From Fig. 1 it is seen
that the value of J L and consequently the difference
between the bulk and the pinning current is significant in magnetic fields up to several Hc1 . From Eq.
Ž20. it follows that the local difference between the
bulk and the pinning current is entirely defined by
the thermodynamic parameters of the superconductor. However the integrated difference J L y Jc over
the sample volume depends also on the sample geometry due to the dependence of profiles H ) Ž x .,
Eq. Ž17..
V.M. KrasnoÕ, V.V. RyazanoÕr Physica C 297 (1998) 153–160
3. Extension for a three-axis ellipsoid
In the experimental situation for the case of HTSC
the sample has a flat geometry with a large demagnetization factor in the magnetic field perpendicular to
the ab-plane. In order to study the influence of the
sample geometry we will consider an extended critical state model for a three-axis ellipsoid w10,11x,
taking into account the vortex interaction and retaining the simplicity of calculations. For more details
see Refs. w10,11x. In our calculations we will assume
that the pinning strength and Jc are constant
throughout the sample.
We consider a three-axis ellipsoid Ž a, b, c .,
where a ) b ) c are the semi-axes along the x-, yand z-axes, respectively, with the demagnetization
factor D 0 along the z-axis. A solenoid generates an
external magnetic field Hs along the z-axis. For
Hs ) Hc1Ž1 y D 0 . vortices enter the sample to the
depth r along x- and y-axes at which the local
thermodynamic field drops to Hc1 . The remaining
central region is in the Meissner state, B s 0. We
approximate this region by an ellipsoid Ž a y r, b y r,
c . with demagnetization factor DŽ r .. According to
Eq. Ž7. H is constant in the sample and is equal to
w11x
H s Hs q Hc1 D Ž r . .
157
Taking into account that at the edge of the Meissner
region H ) s Hc1 we obtain the profile H ) Ž x . in
the field penetration region:
H ) Ž x . s Hc1 q Hi Ž r . y Hi Ž x . .
Ž 25 .
The profile H ) Ž x . is non-linear due to non-linear
term DŽ j . in Eq. Ž24.. The profile of the magnetic
Ž 21 .
Eq. Ž21. takes into account the decrease of the
effective demagnetization factor with flux penetration into ellipsoid. The profile of H ) can be easily
calculated from Eq. Ž17.. First, we note that a shielding current with density Jc flowing between ellipsoids Ž a, b, c . and Ž a y d r, b y d r, c . generates a
uniform magnetic field d Hi inside the smaller ellipsoid. The magnetic moment per unit volume of such
current is equal to
d Mi s y
9p
32 c
Jc d r s
d Hi
4p Ž 1 y D .
.
Ž 22 .
From Eq. Ž22. we obtain the magnetic moment and
the shielding field of the pinning current w10x:
Mi Ž r . s y
Hi Ž r . s y
9p
32 c
Jc r 1 y
9p 2 Jc
8c
aqb
2 ab
rq
r2
3ab
,
Ž 23 .
r
H0 Ž 1 y D Ž j . . d j .
Ž 24 .
Fig. 2. Calculated profiles of Ža. H ) Ž y ., B Ž y . and Žb. J LŽ y .
along the mean semi-axis, b, Ž y-axis. of a three-axis ellipsoid
Ž0.055 cm, 0.026 cm, 0.007 cm. are shown for different external
magnetic fields: Hs r Hc1 s 0.30, 0.43, 0.65, 1.25, 1.45; curves
1–5, respectively, and for constant Jc s10 4 Arcm2 . The profiles
corresponding to a particular Hs are shown by the same type of
line.
158
V.M. KrasnoÕ, V.V. RyazanoÕr Physica C 297 (1998) 153–160
induction B Ž x . is obtained from Eq. Ž18.. At the full
penetration field, Hp ,
Hp s Hc1 q Hi Ž b . ,
Ž 26 .
the flux fronts meet at the centre of the sample Žsee
curve 4 in Fig. 2b.. This relation was obtained from
Eqs. Ž21. and Ž25. taking into account that the
effective demagnetization factor D decreases with
the flux penetration into the sample and becomes
negligible at high enough magnetic fields. From Eqs.
Ž6., Ž9. and Ž19. it follows that the magnetic moment
of the sample is a sum of the moment from the
shielding pinning current and the AVL magnetization:
M s Mi q M L .
Ž 27 .
We note, that according to the numerical analysis
in Ref. w25x the shape of the Meissner region differs
from ellipsoid for H ; Hp , corresponding to the full
flux penetration into the sample. This however does
not influence significantly neither the shape of the
flux profiles nor the magnetic moment with respect
to that obtained within our model Žwhat we actually
do in our model is: we find self-consistently the flux
profile penetration depth, r, assuming that the
Meissner region can be described by some constant
demagnetization factor.. Indeed, this field region is
small for the case a, b 4 c and moreover though the
demagnetization factor is not constant throughout the
Meissner region, it becomes negligibly small, DŽ x,
y, z . < 1 for H ; Hp and according to Eq. Ž24. it
does not influence the shape of the profile of H ) .
4. Results and discussion
In Fig. 2 the calculated profiles of Ža. H ) Ž y .,
B Ž y . and Žb. JLŽ y . along the mean semi-axis, b,
Ž y-axis. of a three-axis ellipsoid are shown for different external magnetic fields: HsrHc1 s 0.30, 0.43,
0.65, 1.25, 1.45, curves 1–5, respectively, and for
constant Jc s 10 4 Arcm2 . The sample geometry
corresponds to the case of TlŽ2212. HTSC single
crystal studied in Ref. w10x: a s 0.055 cm, b s 0.026
cm, c s 0.007 cm and D 0 s 0.78 for Hs along the
z-axis. Parameters of the superconductor are: Hc1 s
500 Gs, k s 50. The profiles corresponding to a
particular Hs are shown by the same type of line. In
Fig. 3. Calculated profiles of Ža. H ) Ž y ., B Ž y . and Žb. J LŽ y . are
shown for the 50% flux penetration along the mean semi-axis, b,
into the same sample as in Fig. 2 for different critical pinning
currents: Jc s10 3 , 10 4 and 10 5 ŽArcm2 ., curves 1, 2 and 3,
respectively. In the inset of Žb. the profiles of the ratio JL r Jc are
shown.
Fig. 3 the profiles of Ža. H ) Ž y ., B Ž y . and Žb.
J LŽ y . are shown for the 50% flux penetration and for
different critical pinning currents: Jc s 10 3 , 10 4 and
10 5 ŽArcm2 ., curves 1, 2 and 3, respectively. The
sample parameters are the same as in Fig. 2. In the
inset of Fig. 3b the profiles of the ratio JLrJc are
shown.
From Figs. 2 and 3 it is seen that a sharp drop of
the magnetic induction and a sharp increase of the
V.M. KrasnoÕ, V.V. RyazanoÕr Physica C 297 (1998) 153–160
current density occurs at the edge of the Meissner
region. The current growth is associated with the
vortex current from the last row of the AVL. Similar
drop of magnetic induction occurs at the sample
surface due to the surface Meissner current. From
Fig. 2b and Fig. 3b and from Eq. Ž1. it is seen that
the bulk current in the critical state of type-II superconductor can be considerably different from the
critical pinning current. Fig. 2b shows that the AVL
magnetization current decreases with increasing
magnetic field. As it follows from Eq. Ž20. and from
the inset in Fig. 1, the J L remains significant for
H ) up to several Hc1 . The corresponding range of
the applied external magnetic field Hs does not
depend on the demagnetization factor and is of the
order of a full penetration field, Eq. Ž26.. Fig. 3b
illustrates the dependence of J L on the pinning
strength. From Fig. 3b it is seen that the value of J L
decreases with Jc . This is quite natural since in the
uniform case, Jc s 0, the average bulk current is
equal to zero. On the other hand, the relative value
of JL defined by the ratio J LrJc decreases with
increasing the pinning strength for the same flux
penetration into the sample, as can be seen from the
inset in Fig. 3b. This is caused by the increase of
H ) with increasing Jc , see Fig. 3a, due to the Hi
term in Eq. Ž25..
Finally, we discuss when the AVL magnetization
current should be taken into account in the analysis
of magnetization data of type-II superconductors.
This is important for the study of the pinning strength
for various types of inhomogeneities e.g. columnar
defects, twin boundaries etc. In many cases the
irreversible part of magnetization can not be directly
extracted from the experimental data and model calculations are needed for estimation of the critical
current value. The temperature and magnetic field
dependence of Jc and its temporal relaxation due to
flux creep is important for the study of the AVL
phase diagram and vortex lattice transitions. In those
cases it is necessary to distinguish between the magnetic field dependencies of J L and Jc . As we have
shown above, the range of applied magnetic fields
for which JL should be taken into account is of the
order of the penetration field Hp , Eq. Ž26., which is
typically of the order of kG, see e.g. an estimation
for TlŽ2212. HTSC in Refs. w10,11x. Thus, the range
of Hs for which AVL magnetization is considerable
159
is of the order of several kG. Among the experimental methods for which the difference between Jav
and Jc could be important we mention:
Ži. Local magnetization measurements. As it is
shown in Fig. 2b and Fig. 3b the AVL magnetization
causes an additional curvature of the flux profiles
especially in the vicinity of the Meissner region and
at the surface of the sample. The magnetic fields
used in local magnetization experiments are typically
well within the range of importance of JL . Experimentally the drop in B Ž x . was observed magnetooptically in Refs. w26,27x.
Žii. Global magnetization measurements. From Eq.
Ž27. it follows that the magnetic moment consists of
the irreversible part, Mi , and the reversible part,
M L . The M L dominates in the case of small pinning,
e.g. for HTSC above the irreversibility line. The
reversible part contains the information about thermodynamic parameters of superconductor and can be
used for extrapolation of such important quantities as
Hc1 . Such analysis for TlŽ2212. HTSC was made in
Ref. w11x.
Žiii. Magnetic relaxation experiments. This is a
powerful tool for the study of various mechanisms of
flux creep and pinning and the AVL phase diagram.
Due to the flux creep the irreversible magnetization
Mi and the critical pinning current Jc logarithmically decreases with time. On the other hand the
AVL magnetization does not vanish with time. As a
result the relaxation of the total magnetic moment
M Ž t . could be very different from the relaxation of
Mi Ž t ..
In conclusion, we have shown that the bulk current density in the critical state of type-II superconductor is not equal to the critical pinning current
density and moreover could be considerably different
from Jc due to existence of the additional AVL
magnetization current J L . The magnetic field profiles and the current density distributions were calculated for different pinning strengths and external
magnetic fields. The calculations were made within
the extended critical state model for a three-axis
ellipsoid, taking into account the equilibrium vortex
lattice magnetization due to the vortex interaction. It
is shown that the difference between Jav and Jc is
considerable for the magnetic fields of the order of
several Hc1 . The difference between Jav and Jc
results in additional curvature of the local magnetic
160
V.M. KrasnoÕ, V.V. RyazanoÕr Physica C 297 (1998) 153–160
field profiles and modifies the total moment of the
sample which might be important for the analysis of
various magnetization experiments.
Acknowledgements
The work was supported in part by the Russian
Scientific Program for High-Tc Superconductivity
ŽGrant No. 96073. and Russian Foundation for Basic
Research ŽGrant No. 96-02-19319.
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