Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля
В рамках моделi критичного стану розглянуто питання застосовностi отриманих Клемом i Санчезом спiввiдношень для змiнної (ac) магнiтної сприйнятливостi тонких плiвок надпровiдника 2-го роду у випадку наявностi постiйного магнiтного поля, перпендикулярного площинi плiвки. Обговорено питання “пам’ятi”...
Gespeichert in:
| Datum: | 2010 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainian |
| Veröffentlicht: |
Відділення фізики і астрономії НАН України
2010
|
| Schlagworte: | |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/13431 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля / Д.Г. Ковальчук, М.П. Чорноморець // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 417-423. — Бібліогр.: 11 назв. — укр. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-13431 |
|---|---|
| record_format |
dspace |
| spelling |
Ковальчук, Д.Г. Чорноморець, М.П. 2010-11-08T17:16:22Z 2010-11-08T17:16:22Z 2010 Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля / Д.Г. Ковальчук, М.П. Чорноморець // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 417-423. — Бібліогр.: 11 назв. — укр. 2071-0194 PACS 74.25.Ha, 74.78.-w https://nasplib.isofts.kiev.ua/handle/123456789/13431 537.9 В рамках моделi критичного стану розглянуто питання застосовностi отриманих Клемом i Санчезом спiввiдношень для змiнної (ac) магнiтної сприйнятливостi тонких плiвок надпровiдника 2-го роду у випадку наявностi постiйного магнiтного поля, перпендикулярного площинi плiвки. Обговорено питання “пам’ятi” зразка i вплив передiсторiї змiн магнiтного поля на поточний стан зразка. Показано, що ac компонента магнiтного моменту, а, отже, i амплiтуди гармонiк ac магнiтної сприйнятливостi, встановлюються протягом одного перiоду ac магнiтного поля незалежно вiд передiсторiї. В рамках модели критического состояния рассмотрен вопрос применимости полученных Клемом и Санчезом соотношений для переменной (ac) магнитной восприимчивости тонких пленок сверхпроводника 2-го рода в случае наличия постоянного магнитного поля, перпендикулярного плоскости пленки. Обсужден вопрос “памяти” образца и влияния предыстории изменений магнитного поля на текущее состояние образца. Показано, что ac компонента магнитного момента, а, значит, и амплитуды гармоник ac магнитной восприимчивости, устанавливаются на протяжении одного периода ac магнитного поля независимо от предыстории. The applicability of relations obtained by Clem and Sanchez for the ac magnetic susceptibility of type-II superconductor thin films to the case where an additional constant magnetic field is applied perpendicularly to the film has been analyzed in the framework of the critical state model. The issues concerning the sample “memory” and the influence of the magnetic field change prehistory on the current sample state have been discussed. It has been shown that the ac component of the magnetic moment and, hence, the amplitudes of ac magnetic susceptibility harmonics are established within one period of the ac magnetic field irrespective of the field prehistory. Автори вдячнi С.М. Рябченку за обговорення та кориснi поради. Робота пiдтримана цiльовою темою Президiї НАН України ВЦ/139-38. uk Відділення фізики і астрономії НАН України Тверде тіло Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля Перемагничивание тонкого диска сверхпроводника 2-го рода при наличии постоянного магнитного поля Magnetization Reversal of a Type-II Superconductor Thin Disk under the Action of a Constant Magnetic Field Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля |
| spellingShingle |
Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля Ковальчук, Д.Г. Чорноморець, М.П. Тверде тіло |
| title_short |
Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля |
| title_full |
Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля |
| title_fullStr |
Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля |
| title_full_unstemmed |
Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля |
| title_sort |
перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля |
| author |
Ковальчук, Д.Г. Чорноморець, М.П. |
| author_facet |
Ковальчук, Д.Г. Чорноморець, М.П. |
| topic |
Тверде тіло |
| topic_facet |
Тверде тіло |
| publishDate |
2010 |
| language |
Ukrainian |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Перемагничивание тонкого диска сверхпроводника 2-го рода при наличии постоянного магнитного поля Magnetization Reversal of a Type-II Superconductor Thin Disk under the Action of a Constant Magnetic Field |
| description |
В рамках моделi критичного стану розглянуто питання застосовностi отриманих Клемом i Санчезом спiввiдношень для змiнної (ac) магнiтної сприйнятливостi тонких плiвок надпровiдника 2-го роду у випадку наявностi постiйного магнiтного поля, перпендикулярного площинi плiвки. Обговорено питання “пам’ятi” зразка i вплив передiсторiї змiн магнiтного поля на поточний стан зразка. Показано, що ac компонента магнiтного моменту, а, отже, i амплiтуди гармонiк ac магнiтної сприйнятливостi, встановлюються протягом одного перiоду ac магнiтного поля незалежно вiд передiсторiї.
В рамках модели критического состояния рассмотрен вопрос применимости полученных Клемом и Санчезом соотношений для переменной (ac) магнитной восприимчивости тонких пленок сверхпроводника 2-го рода в случае наличия постоянного магнитного поля, перпендикулярного плоскости пленки. Обсужден вопрос “памяти” образца и влияния предыстории изменений магнитного поля на текущее состояние образца. Показано, что ac компонента магнитного момента, а, значит, и амплитуды гармоник ac магнитной восприимчивости, устанавливаются на протяжении одного периода ac магнитного поля независимо от предыстории.
The applicability of relations obtained by Clem and Sanchez for the ac magnetic susceptibility of type-II superconductor thin films to the case where an additional constant magnetic field is applied perpendicularly to the film has been analyzed in the framework of the critical state model. The issues concerning the sample “memory” and the influence of the magnetic field change prehistory on the current sample state have been discussed. It has been shown that the ac component of the magnetic moment and, hence, the amplitudes of ac magnetic susceptibility harmonics are established within one period of the ac magnetic field irrespective of the field prehistory.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13431 |
| citation_txt |
Перемагнічування тонкого диска надпровідника 2-го роду за наявності постійного магнітного поля / Д.Г. Ковальчук, М.П. Чорноморець // Укр. фіз. журн. — 2010. — Т. 55, № 4. — С. 417-423. — Бібліогр.: 11 назв. — укр. |
| work_keys_str_mv |
AT kovalʹčukdg peremagníčuvannâtonkogodiskanadprovídnika2goroduzanaâvnostípostíinogomagnítnogopolâ AT čornomorecʹmp peremagníčuvannâtonkogodiskanadprovídnika2goroduzanaâvnostípostíinogomagnítnogopolâ AT kovalʹčukdg peremagničivanietonkogodiskasverhprovodnika2gorodaprinaličiipostoânnogomagnitnogopolâ AT čornomorecʹmp peremagničivanietonkogodiskasverhprovodnika2gorodaprinaličiipostoânnogomagnitnogopolâ AT kovalʹčukdg magnetizationreversalofatypeiisuperconductorthindiskundertheactionofaconstantmagneticfield AT čornomorecʹmp magnetizationreversalofatypeiisuperconductorthindiskundertheactionofaconstantmagneticfield |
| first_indexed |
2025-11-25T23:53:50Z |
| last_indexed |
2025-11-25T23:53:50Z |
| _version_ |
1850589028872617984 |
| fulltext |
SOLID MATTER
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 415
MAGNETIZATION REVERSAL
OF A TYPE-II SUPERCONDUCTOR THIN DISK
UNDERTHE ACTIONOF ACONSTANTMAGNETIC FIELD
D.G. KOVALCHUK, M.P. CHORNOMORETS
Institute of Physics, Nat. Acad. of Sci. of Ukraine
(46, Nauky Ave., Kyiv 03680, Ukraine; e-mail: kovalch@ iop. kiev. ua )
PACS 74.25.Ha, 74.78.-w
c©2010
The applicability of relations obtained by Clem and Sanchez for
the ac magnetic susceptibility of type-II superconductor thin films
to the case where an additional constant magnetic field is applied
perpendicularly to the film has been analyzed in the framework of
the critical state model. The issues concerning the sample “mem-
ory” and the influence of the magnetic field change prehistory on
the current sample state have been discussed. It has been shown
that the ac component of the magnetic moment and, hence, the
amplitudes of ac magnetic susceptibility harmonics are established
within one period of the ac magnetic field irrespective of the field
prehistory.
1. Introduction
One of the techniques aimed at determining the criti-
cal current density in high-temperature superconductor
thin films is the noncontact measurement of their mag-
netic susceptibility in an alternating field (the ac mag-
netic susceptibility). It is based on the magnetization
reversal model for a superconductor of the second kind
which was theoretically substantiated in works [1–3]. In
work [3], Clem and Sanchez obtained a relation for the ac
magnetic susceptibility of a thin disk provided that only
an ac magnetic field acts on the disk (the CS-model).
However, the further experimental researches of the crit-
ical current density in thin superconductor films allowed
the scope of the CS relation to be extended upon the
results of measurements in a dc magnetic field as well
[4–9]. In the literature, there are only collateral (indi-
rect) verifications that such an extension is valid. The
authors of work [4] believe that the imposed constant
field does not change the expression for ac magnetic sus-
ceptibility obtained in work [3], if the critical current
density weakly depends on the magnetic field. A simi-
lar conclusion can be drawn from Brandt’s remark [10]
that, in the case of a periodic field with slowly grow-
ing amplitude, a Bean superconductor (i.e. a supercon-
ductor described in the framework of the critical state
model developed by Bean [11]) “reminds” only the last
cycle of magnetization reversal. The presented specu-
lations may intuitively seem plausible; however, their
validity is not evident a priori, and their more rigor-
ous substantiation is required. The absence of such
a substantiation in the literature forces experimenters
to carry out additional checks of the CS-model appli-
cability under that or another experimental condition
[4, 5, 7].
For ordinary magnets, the amplitude of local mag-
netization in a magnetic field is almost uniform over
the specimen volume in the case where the specimen
shape is an ellipsoid of rotation; in this case, the magne-
tization is a characteristic of the medium (substance).
The hysteresis phenomenon consists in an ambiguity
of the local magnetization dependence on the external
field. For a superconductor of the second kind in a
magnetic field, primary is the spatial distribution of the
screening current density on a macroscopic scale (in re-
ality, on the specimen-size scale, whatever the specimen
shape). The current distribution is governed, in par-
ticular, by the entry/exit of Abrikosov vortices; and
the magnetic moment and the volume-averaged mag-
netization are integral characteristics of the specimen,
being the derivatives of the current distribution. As
a consequence, the hysteresis dependence of supercon-
ductor magnetic characteristics on the external mag-
netic field turns out to be a complicated indirect one.
This circumstance makes an answer to the question on
D.G. KOVALCHUK, M.P. CHORNOMORETS
the specimen “memory”, i.e. on the influence of a se-
quence of external conditions (“prehistory”) imposed on
the specimen on its state, nonevident a priori. This
prehistory includes variations of the applied magnetic
field, by starting from the moment, when the speci-
men is in the ZFC (zero field cooling) state, i.e. in
the state obtained by cooling down the specimen in
the zero external magnetic field from the temperature
T > Tc, where Tc is the critical temperature of transi-
tion into the superconducting state. The key parame-
ters of the CS model are the external magnetic field and
the critical current density, the latter being a function
of the temperature. Therefore, the “prehistory” also in-
cludes the sequence of temperature regimes applied to
the specimen after the magnetic field having been im-
posed.
In this context, Brandt’s statement given above is
valid only partly, and an additional explanation is re-
quired. For instance, if the specimen state – i.e. the
spatial distributions of the current density and the mag-
netic field in it – at a definite time moment is meant, then
the Bean specimen, by demonstrating a constant criti-
cal current, always “remembers” the maximal, by the
magnitude, value of the applied magnetic field and its
sign that occurred in the sample prehistory. In general,
depending on the prehistory, the distributions of cur-
rent density and magnetic field over the specimen can be
rather complicated, but a number of parameters which
are determined in the CS model are really invariable, if
a constant magnetic field is imposed.
In this work, the description of the magnetization re-
versal process in a thin disk made up of a type-II super-
conductor in an ac magnetic field which was developed in
works [1–3] has been extended to include the presence of
a dc magnetic field directed perpendicularly to the disk
plane. The matter was considered in the framework of
the same assumptions that were formulated and used in
work [3]. An algorithm for writing down the solutions
for the radial distribution of azimuthal current density in
a thin disk in the case of a multiple change of the exter-
nal magnetic field variation direction (increase/decrease)
has been formulated. Issues concerning what exactly the
specimen “forgets” and under which conditions this takes
place have been discussed. The consideration demon-
strates that, after the temperature and the amplitude of
an ac field have stabilized, the vortex entry/exit depth,
as well as all variable components of specimen character-
istics, including the ac magnetic susceptibility, acquire
their values within a single period of ac field oscillations
irrespective of the field prehistory and whether a dc mag-
netic field is imposed or not.
2. Results and Their Discussion
Consider a type-II superconductor specimen fabricated
in the form of a thin disk of radius R and thickness
d � R. The corresponding London penetration depth
λ < d (or, if λ > d, Λ = λ2/d � R). The specimen is
cooled down in the ZFC regime. A magnetic field which
is parallel to the z-axis is applied perpendicularly to the
specimen plane. The field changes quasistatically. We
assume that the critical current density Jc does not de-
pend on the magnetic field strength. If the external field
h applied to the specimen is low (lower than the corre-
sponding first critical field), the specimen behaves like a
specimen made up of a superconductor of the first kind:
the magnetic field does not penetrate into the specimen
depth (but a near-surface region of the characteristic
thickness λ). The external field is completely compen-
sated in the specimen bulk by the field of the azimuthal
Meissner current, the distribution of which over the disk
radius is [1]
J(ρ) = −4h
πd
ρ√
R2 − ρ2
, (1)
where ρ is the distance from the disk center. Hereafter, a
current creating a magnetic field which is directed along
the z-axis at the specimen center is considered as posi-
tive.
When the external field exceeds the corresponding first
critical field, Hc1s, vortices start to enter the specimen.
The value of the “critical field for a specimen” corre-
sponds to an “internal” field in it. The latter is equal to
the external field minus the demagnetization one, and, in
the case of a thin disk, is much lower than the first crit-
ical field for the disk material, Hc1. In particular, in the
case λ < d� R, we have Hc1s ≈ (d/R)1/2Hc1. Pinning
centers, if they exist in the specimen, pin the vortices.
To describe the vortex entry into the specimen, the crit-
ical state model [11] is widely applied. According to it,
when the applied field increases, vortices enter the spec-
imen and reduce the local current density to the level of
critical current density. The vortex entry depth depends
on the critical current density Jc which is determined
by the conditions of vortex pinning. Hence, at the stage
where the field h monotonously grows from zero, two re-
gions can be distinguished in the specimen. These are
an external ring – in which there are vortices, and the
current density is equal to −Jc – and an internal circle
of radius a(h) – in which there are no vortices, so that
the field is equal to zero. As was shown in work [1], the
416 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
MAGNETIZATION REVERSAL OF TYPE-II SUPERCONDUCTOR
distribution of the current density in this case looks like
J(ρ, h) =
−( 2Jc
π ) tan−1
[
ρ
R
√
R2−a(h)2
a(h)2−ρ2
]
, ρ ≤ a(h),
−Jc, a(h) ≤ ρ ≤ R,
(2)
where
a(x) =
R
cosh(x/Hd)
, Hd =
Jcd
2
. (2a)
The specimen magnetization can be determined using
the current density distribution by the formula
M =
1
V
πd
R∫
0
ρ2J (ρ) dρ, (3a)
where V is the specimen volume.
The current with distribution (2) creates the magne-
tization [1]
M(h) = −χ0hS (h/Hd) , (3b)
where
S (x) =
1
2x
[
cos−1
(
1
coshx
)
+
sinhx
cosh2 x
]
, χ0 =
8R
3πd
.
(3c)
Consider now the case where the field, having reached
the value H0 at the stage of monotonous growth, de-
creases to the current value h. In the course of such a re-
duction of the field in the specimen, vortices change their
distribution. Formally, this process can be described as
an entry of vortices into the specimen, the sign of which
is opposite to the sign of those vortices entered at the
stage of field growth. In the framework of the critical
state model, the entry of those vortices brings about a
change of the current density to the value +Jc in an
external ring, the width of which is equal to the vortex
entry depth. Taking into account that the distribution of
the current density and, therefore, the field in the inter-
nal circle remain invariable at that, the authors of work
[2] showed that the current density distribution at this
stage can be represented as a superposition of two cur-
rents, the both looking like expression (2). One of them
arises owing to the switching-on of the field H0, and the
other is induced by a subsequent variation of the field in
the opposite direction by the value Δh1 = H0 − h:
J1(ρ, h) = J(ρ,H0)− 2J
(
ρ,
Δh1
2
)
. (4)
In this case, the quantity a
(
Δh1
2
)
defines the radius of
a circle, in which the field remains invariable (vortices
do not enter this circle at the stage of field reduction).
This circle, in turn, consists of an internal circle of ra-
dius a(H0), in which the field is absent, and a ring with
internal and external radii a(H0) and a
(
Δh1
2
)
, respec-
tively, where the field distribution was attained at the
previous stage. The coefficient in the second summand
reflects the fact that the current density at the specimen
edge changes by 2Jc, when the field changes its direction
(the field “reversal”). The formula given remains valid as
long as h ≥ −H0. At h = −H0, the current distribution
looks the same as it was at h = H0, but with the opposite
sign. If the field diminishes further, vortices penetrate
more deeply into the specimen than they did at the first
stage. In so doing, the specimen “forgets” about the first
stage, and the current distribution is described by for-
mula (2) with the corresponding substitution of Jc by
−Jc, as if the field at the second stage changed from the
ZFC state rather than +H0.
In the case where the field starts to increase again after
having reached some value H1 > −H0, the correspond-
ing distribution of the current density can be written
down analogously as
J2(ρ, h) = J1(ρ,H1) + 2J
(
ρ,
Δh2
2
)
=
= J(ρ,H0)− 2J
(
ρ,
H0 −H1
2
)
+ 2J
(
ρ,
Δh2
2
)
, (5)
where Δh2 = h −H1 is the difference between the cur-
rent field value and the field value at the last reversal
of a field variation direction. Similarly to the previous
case, the field distribution that arose at the previous
stages remains invariable in the circle of radius a
(
Δh2
2
)
,
whereas the current density in the external ring of width
R−a
(
Δh2
2
)
is critical. Formula (5) also remains correct
until the depth of vortex entry at this stage exceeds the
entry depth at the previous stage, i.e. as long as h ≤ H0.
When the value h = H0 is attained, the last two sum-
mands in Eq. (5) are mutually compensated, so that the
specimen “forgets” about the last two stages of field vari-
ation.
Hence, one can create an algorithm for constructing a
formula which would describe the current density distri-
bution in the general case and at an arbitrary sequence of
quasistatic variations of the field h after the ZFC state.
Let the field, alternately increasing and decreasing in
the course of such variations, attain the values H0, H1,
H2,. . . , HN at the “reversal” points. Introducing the
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 417
D.G. KOVALCHUK, M.P. CHORNOMORETS
notation Ji−1(ρ) for the current distribution over the
specimen at the time moment when the field achieved
the “reversal” point h = Hi, one can write down the
following recurrent formula for the current density dis-
tribution, provided that the field varies monotonously
after this point:
Ji(ρ, h) = Ji−1(ρ)± 2J
(
ρ,
Δih
2
)
, (6)
where Δih = |h − Hi|. In this case, the absolute value
of current density in the external ring of the width
R − a
(
Δih
2
)
is equal to the critical current density, and
the field in the circle of radius a
(
Δih
2
)
remains invari-
able and equal to that created at the previous stages.
At the center, there is a circle, where the field equals
zero. The radius of this circle, a(Hmax), is determined
by the maximal, by the absolute value, field Hmax ap-
plied to the specimen. The sign before the second term
in Eq. (6) is determined by the field variation direction
after the point h = Hi: it is plus, if the field increases,
and minus, if it decreases. This formula remains correct
until the vortex entry depth attained at this stage of field
variation exceeds the corresponding value attained at the
previous stage. If there are no stages with monotonous
field variation, at which the vortex entry depth exceeds
that reached at the previous stage, one may say that
the specimen “remembers” the whole history of imposed
external magnetic fields that took place after the ZFC
state, and formula (6) can be presented as the sum
JN (ρ) = J(ρ,H0) + 2
N∑
i=1
(−1)iJ
(
ρ,
ΔiH
2
)
, (7)
where ΔiH = |Hi −Hi−1|.
If vortices penetrate more deeply into the specimen at
the j-th stage than at the previous one, two summands
(j-th and (j − 1)-th) disappear from this sum (i.e. the
specimen “forgets” about the corresponding stages), and
Hj−2 in the (j−2)-th term is to be substituted byHj . At
the same time, if the vortex penetration depth becomes
larger at some stage than it was at every previous stage,
all terms that describe stages before this one disappear
from the sum. This means that the field maximal by
its absolute value can be taken as H0, and all previous
stages of field variation can be excluded from considera-
tion.
Let a magnetic field h = HDC + hac, where HDC > 0
is a constant field and hac is a current value of an ac field
that oscillates between its peak values ±h0, be applied
to the specimen perpendicularly to its plane. When the
total field achieves the value HDC + h0, a certain distri-
bution of current density, which we denote as Jmax(ρ),
is established in the specimen. Provided that no field
higher than HDC + h0 was applied to the specimen be-
fore, the distribution Jmax(ρ) is determined by formulas
(2) and (2a), in which the vortex-free circle radius is
equal to a(HDC + h0). Otherwise, the current density
distribution Jmax(ρ) depends on the field variation his-
tory before the field reaches the value HDC + h0.
If the total field diminishes further from HDC + h0 to
HDC − h0, the current density distribution is described
by formula (6):
J(ρ, h) = Jmax(ρ)− 2J
(
ρ,
h0 − hac
2
)
. (8a)
At the next stage of field growth from HDC − h0 to
HDC + h0, the current density distribution looks like
J(ρ, h) = Jmax(ρ)− 2J
(
ρ, h0) + 2J(ρ,
h0 + hac
2
)
. (8b)
When the field achieves the value HDC+h0, the last two
summands in Eq. (8b) become mutually compensated,
the current density distribution coincides with Jmax(ρ),
and formula (8a) becomes applicable again at the next
stage where the field decreases. The maximal depth, to
which vortices of different signs will alternately enter at
subsequent hac-oscillations – i.e. the width of the ex-
ternal ring, within limits of which a redistribution of the
field in the specimen will take place – will be determined
by only the varying part of the external field. Namely,
it will be equal to R− a(h0), whereas the magnetic field
within the limits of the circle ρ < a(h0) will remain
invariable. At the same time, the current density distri-
bution will change in the circle ρ < a(h0) too, tracing a
current position of the vortex penetration depth a (see
formula (2)).
Substituting Eqs. (8a) and (8b) into Eq. (3a) and de-
noting the specimen magnetization at HDC + h0 and
HDC−h0 as Mmax and Mmin, respectively, we obtain the
following formula for magnetization at the stage where
the ac field decreases:
M−(h) =
πd
V
R∫
0
ρ2Jmax (ρ) dρ−
−πd
V
R∫
0
ρ22J
(
ρ,
Δ−h
2
)
dρ = Mmax−2M
(
Δ−h
2
)
, (9a)
418 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
MAGNETIZATION REVERSAL OF TYPE-II SUPERCONDUCTOR
where Δ−h = h0 − hac. Accordingly, at the stage where
the ac field increases, we have
M+(h) =
πd
V
R∫
0
ρ2(Jmax (ρ)− 2J(ρ, h0))dρ+
+
πd
V
R∫
0
ρ22J
(
ρ,
Δ+h
2
)
dρ = Mmin +2M
(
Δ+h
2
)
, (9b)
where Δ+h = h0 + hac. Both quantities Δ−h and Δ+h
stand for a difference between the current field value at
the corresponding stage and its value at the last field “re-
versal”, and both do not depend on HDC . The formulas
obtained allow two basic conclusions to be drawn. First,
the identical dependences of a magnetization change on
the field variation at the stages of field decrease and in-
crease leads to the symmetry of a hysteresis loop with
respect to the point
(
HDC ,
Mmin+Mmax
2
)
. Second, the
dependence of the variable part of the magnetization on
only the ac field means that the hysteresis loop shape
does not depend on the dc magnetic field.
In work [3], an interrelation between the critical cur-
rent density and the dependence of complex magnetic
susceptibility harmonics of a thin-disk-shaped specimen
on the ac magnetic field amplitude h0 h(t) = h0 cos(ωt)
directed perpendicularly to the disk plane was consid-
ered. The harmonics of the real and imaginary parts of
the ac magnetic susceptibility are defined by the formu-
las
χ′n =
ω
πh0
T∫
0
M(t) cos(nωt)dt, (10a)
χ′′n =
ω
πh0
T∫
0
M(t) sin(nωt)dt. (10b)
Substituting expressions (9a) and (9b) into them, we
obtain that, owing to the averaging over the period of
the ac field, those components of the magnetic moment
of a specimen that depend on the dc field magnitude do
not make any contribution to harmonic amplitudes, and,
therefore, the latter depend only on the ac field strength.
When applying the CS model to the description of pro-
cesses in specimens, the issue concerning the influence of
a prehistory of variations of the dc and ac components of
the magnetic field, without returning to the ZFC state,
on the ultimate results is of importance. Every change of
the applied field is accompanied by a variation of the az-
imuthal current density distribution in a specimen and,
as a consequence, by a variation of the magnetic moment
of the specimen. As a result, after a number of reversals
of the applied field variation direction, a quite compli-
cated distribution of current density over the specimen,
which is described by formula (7), can emerge.
However, the actual state of the specimen can depend
not only on the prehistory of magnetic field changes, but
also on the temperature by means of the temperature in-
fluence on the critical current density. In the case where
the dc field is constant, and the critical current density
increases – e.g., as a result of temperature reduction –
the external field at the center of a specimen remains
completely compensated by the existing current distri-
bution, and the current density in the specimen does
not exceed the critical value anywhere. Hence, neither
new vortices enter the specimen, nor a redistribution of
vortices takes place in it. Therefore, the growth of the
critical current density is not accompanied by a change of
the current and field distributions in the specimen that
existed at that moment. However, the conditions for the
vortex motion do change, so that formula (7) loses its
validity for the description of subsequent variations of
the magnetic field.
In the case where the temperature grows, provided
that the field is constant, which results in a reduction of
the critical current density, the motion of those vortices
which have entered the specimen at the previous stage of
field variation, becomes restored, and the entry of new
ones continues. Vortices enter the specimen, move to-
ward its center, and expand the external ring formed at
the previous stage until the current density in this ring
falls down to the actual critical value. In this case, the
formulas presented above lose their applicability again.
Though the influence of the prehistory of critical cur-
rent density variations on the specimen state can also
be considered in the framework of the CS-model formal-
ism, this problem, however, is not the subject of this
work. Concerning the issue of the specimen “memory”
at a constant critical current density, one may assert
that, at any variations of the applied field, there will re-
main a vortex-free circle at the center of the specimen,
the radius of which will be determined by the maximal
value of a field applied to the specimen.
The distribution of the current density in the specimen
can be rather complicated. Nevertheless, as was shown
above, when both the ac and dc fields are simultane-
ously imposed perpendicularly to the specimen plane,
the current density distribution, as well as the magnetic
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 419
D.G. KOVALCHUK, M.P. CHORNOMORETS
moment, can be divided into two components. One of
them does not depend on the current value of the ac field
(though it depends both on the magnitude and the pre-
history of the dc field), and the other depends only on
the amplitude and the actual value of the ac field. There-
fore, after the maximal, by magnitude, field – i.e. the
sum of the constant and alternating components – has
been reached, the current density distribution changes
cyclically with a period equal to that of the ac compo-
nent of the applied field. This means that the prehistory
of establishing the dc field or the temperature can affect
the results of determination of the variable part of the
magnetic moment within one period of the ac magnetic
field only.
In view of the fact that the relations of the CS model
are widely used for the interpretation of experimental re-
sults, it should be noted that, in this work, the subject
of consideration was an answer to the theoretical ques-
tion “What will happen, if a constant magnetic field is
imposed on the specimen, provided that the considera-
tion is carried out in the framework of the Bean critical
state model and Clem and Sanchez’s additional assump-
tions are made?” An answer to another question, “Which
accuracy does the CS model provide for the descrip-
tion of processes in real specimens?”, does not basically
depend on adding a dc magnetic field. This question
requires a separate consideration in every experimen-
tal situation, so that it has not been analyzed in this
work.
3. Conclusions
Hence, the Bean specimen always “remembers” the high-
est value of applied magnetic field in its prehistory. Al-
though after multiple changes of the applied field or tem-
perature have been made, the specimen state can be de-
scribed by rather complicated distributions of current
density and magnetic field in it, the magnetic moment
of the specimen can be divided into two components not
later than in the applied ac field period. One of them
depends on the dc field and can partly “remember” the
prehistory. It does not depend on the actual value of
the ac field. The other depends only on the ac field (its
amplitude and the actual value).
In the framework of the critical state model, the rela-
tions obtained by Clem and Sanchez for the ac mag-
netic susceptibility of a thin disk do not change, if
a constant component of the magnetic field directed
perpendicularly to the disk plane is added. Conse-
quently, they can be used in experimental researches
aimed at studying the dependence of the critical cur-
rent density on the magnetic field in cases where
this dependence can be neglected (the interval of to-
tal field variation within the period of ac field oscil-
lations is narrow enough). The reliability of the re-
sults obtained turns out to be the same as that al-
lowed by the CS model for the description of pro-
cesses in real specimens without imposing a constant
field.
The authors are grateful to S.M. Ryabchenko for the
discussion and the useful advice. The work was sup-
ported in the framework of target project VTs/130-38 of
the Presidium of the NAS of Ukraine.
1. P.N. Mikheenko and Yu.E. Kuzovlev, Physica C 204, 229
(1993).
2. J. Zhu, J. Mester, J. Lockhart, and J. Turneaure, Physica
C 212, 216 (1993).
3. J.R. Clem and A. Sanchez, Phys. Rev. B 50, 9355 (1994).
4. E. Mezzetti, R. Gerbaldo, G. Ghigo, L. Gozzelino, B. Mi-
netti, C. Camerlingo, A. Monaco, G. Cuttone, and A. Ro-
velli, Phys. Rev. B 60, 7623 (1999).
5. Yu.V. Fedotov, S.M. Ryabchenko, E.A. Pashitskii,
A.V. Semenov, V.I. Vakaryuk, V.M. Pan, and V.S. Flis,
Fiz. Nizk. Temp. 28, 245 (2002).
6. M.P. Chernomorets, D.G. Kovalchuk, S.M. Ryabchenko,
A.V. Semenov, and E.A. Pashitskii, Fiz. Nizk. Temp. 32,
1096 (2006).
7. A.I. Kosse, A.Yu. Prokhorov, V.A. Khokhlov, G.G. Lev-
chenko, A.V. Semenov, D.G. Kovalchuk, M.P. Cher-
nomorets, and P.N. Mikheenko, Supercond. Sci. Technol.
21, 075015 (2008).
8. B.J. Jönsson-Åkerman, K.V. Rao, and E.H. Brandt,
Phys. Rev. B 60, 14913 (1999).
9. J.J. Åkerman, S.H. Yunand, U.O. Karlsson, and
K.V. Rao, Phys. Rev. B 64, 184520 (2001).
10. E.H. Brandt, Phys Rev. B 55, 14513 (1997).
11. C.P. Bean, Phys. Rev. Lett. 8, 250 (1962).
Received 28.11.08.
Translated from Ukrainian by O.I. Voitenko
ПЕРЕМАГНIЧУВАННЯ ТОНКОГО
ДИСКА НАДПРОВIДНИКА 2-ГО РОДУ
ЗА НАЯВНОСТI ПОСТIЙНОГО МАГНIТНОГО ПОЛЯ
Д.Г. Ковальчук, М.П. Чорноморець
Р е з ю м е
В рамках моделi критичного стану розглянуто питання за-
стосовностi отриманих Клемом i Санчезом спiввiдношень для
змiнної (ac) магнiтної сприйнятливостi тонких плiвок надпро-
вiдника 2-го роду у випадку наявностi постiйного магнiтного
420 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4
MAGNETIZATION REVERSAL OF TYPE-II SUPERCONDUCTOR
поля, перпендикулярного площинi плiвки. Обговорено питання
“пам’ятi” зразка i вплив передiсторiї змiн магнiтного поля на
поточний стан зразка. Показано, що ac компонента магнiтного
моменту, а, отже, i амплiтуди гармонiк ac магнiтної сприйня-
тливостi, встановлюються протягом одного перiоду ac магнi-
тного поля незалежно вiд передiсторiї.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 4 421
|