The electromagnetic response of a superconducting ferromagnets
The electromagnetic response of the superconducting ferromagnets RuSr₂Gd₁.₅Ce₀.₅Cu₂O₁₀ (Ru–1222 Gd) in an ac magnetic field of finite amplitude is investigated. Taking into account weak links between granules and magnetization of the magnetic sublattice, it is shown that the response of a sample i...
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nasplib_isofts_kiev_ua-123456789-1209092025-02-09T13:52:14Z The electromagnetic response of a superconducting ferromagnets Leviev, G.I. Tsindlekht, M.I. Sonin, E.B. Felner, I. Свеpхпpоводимость, в том числе высокотемпеpатуpная The electromagnetic response of the superconducting ferromagnets RuSr₂Gd₁.₅Ce₀.₅Cu₂O₁₀ (Ru–1222 Gd) in an ac magnetic field of finite amplitude is investigated. Taking into account weak links between granules and magnetization of the magnetic sublattice, it is shown that the response of a sample in a superconducting state with the fundamental frequency and frequency of the 3rd harmonic can be described by the nonlinear equation for the macroscopic field. Generation of the harmonic at temperatures above superconducting transition corresponds to Rayleigh`s mechanism. Using various regimes of a sample cooling, the internal magnetic field determined by the magnetic sublattice was measured. This is direct evidence of the coexistence of ferromagnetic and superconductive order parameters in high-Tc ruthenocuprates. The work was supported by the Klatchky foundation. The authors are deeply grateful to E. Galstyan for the sample preparation and useful discussions. 2007 Article The electromagnetic response of a superconducting ferromagnets / G.I. Leviev, M.I. Tsindlekht, E.B. Sonin, I.Felner // Физика низких температур. — 2007. — Т. 33, № 08. — С. 844–848. — Бібліогр.: 14 назв. — англ. 0132-6414 PACS: 74.70.Pq, 74.25.Nf https://nasplib.isofts.kiev.ua/handle/123456789/120909 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Leviev, G.I. Tsindlekht, M.I. Sonin, E.B. Felner, I. The electromagnetic response of a superconducting ferromagnets Физика низких температур |
| description |
The electromagnetic response of the superconducting ferromagnets RuSr₂Gd₁.₅Ce₀.₅Cu₂O₁₀ (Ru–1222
Gd) in an ac magnetic field of finite amplitude is investigated. Taking into account weak links between granules
and magnetization of the magnetic sublattice, it is shown that the response of a sample in a superconducting
state with the fundamental frequency and frequency of the 3rd harmonic can be described by the
nonlinear equation for the macroscopic field. Generation of the harmonic at temperatures above superconducting
transition corresponds to Rayleigh`s mechanism. Using various regimes of a sample cooling, the internal
magnetic field determined by the magnetic sublattice was measured. This is direct evidence of the coexistence
of ferromagnetic and superconductive order parameters in high-Tc ruthenocuprates. |
| format |
Article |
| author |
Leviev, G.I. Tsindlekht, M.I. Sonin, E.B. Felner, I. |
| author_facet |
Leviev, G.I. Tsindlekht, M.I. Sonin, E.B. Felner, I. |
| author_sort |
Leviev, G.I. |
| title |
The electromagnetic response of a superconducting ferromagnets |
| title_short |
The electromagnetic response of a superconducting ferromagnets |
| title_full |
The electromagnetic response of a superconducting ferromagnets |
| title_fullStr |
The electromagnetic response of a superconducting ferromagnets |
| title_full_unstemmed |
The electromagnetic response of a superconducting ferromagnets |
| title_sort |
electromagnetic response of a superconducting ferromagnets |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120909 |
| citation_txt |
The electromagnetic response of a superconducting ferromagnets / G.I. Leviev, M.I. Tsindlekht, E.B. Sonin, I.Felner // Физика низких температур. — 2007. — Т. 33, № 08. — С. 844–848. — Бібліогр.: 14 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT levievgi theelectromagneticresponseofasuperconductingferromagnets AT tsindlekhtmi theelectromagneticresponseofasuperconductingferromagnets AT sonineb theelectromagneticresponseofasuperconductingferromagnets AT felneri theelectromagneticresponseofasuperconductingferromagnets AT levievgi electromagneticresponseofasuperconductingferromagnets AT tsindlekhtmi electromagneticresponseofasuperconductingferromagnets AT sonineb electromagneticresponseofasuperconductingferromagnets AT felneri electromagneticresponseofasuperconductingferromagnets |
| first_indexed |
2025-11-26T11:45:04Z |
| last_indexed |
2025-11-26T11:45:04Z |
| _version_ |
1849853225769893888 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 8, p. 844–848
The electromagnetic response of a superconducting
ferromagnets
Grigory I. Leviev, Menachem I. Tsindlekht, Edouard B. Sonin,
and Israel Felner
The Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
E-mail: gileviev@vms.huji.ac.il
Received September 19, 2006, revised December 7, 2006
The electromagnetic response of the superconducting ferromagnets RuSr 2Gd15. Ce 0 5. Cu 2O10 (Ru–1222
Gd) in an ac magnetic field of finite amplitude is investigated. Taking into account weak links between gran-
ules and magnetization of the magnetic sublattice, it is shown that the response of a sample in a supercon-
ducting state with the fundamental frequency and frequency of the 3rd harmonic can be described by the
nonlinear equation for the macroscopic field. Generation of the harmonic at temperatures above supercon-
ducting transition corresponds to Rayleigh`s mechanism. Using various regimes of a sample cooling, the in-
ternal magnetic field determined by the magnetic sublattice was measured. This is direct evidence of the co-
existence of ferromagnetic and superconductive order parameters in high-Tc ruthenocuprates.
PACS: 74.70.Pq Ruthenates;
74.25.Nf Response to electromagnetic fields.
Keywords: electromagnetic response, superconducting ferromagnets, high-Tc ruthenocuprates.
1. Introduction
The nature of the coexistence of the long-range mag-
netic and superconducting order parameters is discussed in-
tensively, since Ginzburg’s work [1]. The origins of the or-
der parameters in ferromagnetic superconductors, magnetic
structure and magnetic properties of such materials are hot
topic in the present time [2]. Magnetic characteristics of
superconducting ferromagnets such as Ru–1222 Gd is ex-
tremely interesting [3]. Those samples exhibit a magnetic
transition at TN � 125–180 K and superconducting transi-
tion at Tc � 25–50 K (T TN c� ). However experimental
study of the low frequency electrodynamics is almost ab-
sent. In the present paper we study the behavior of the
ceramic samples Ru–1222 Gd in the presence of dc and
low-frequency magnetic fields at temperatures both above
and below of the superconducting transition temperature.
Our measurements [4] show that the critical current in the
superconducting state is very small. It leads to the specific
character of the response of the sample with respect to the ac
fields. The response becomes nonlinear already at small am-
plitudes and then — at high enough amplitudes — the linear
modes return. Experimentally in our case nonlinearity also
appearers at T T Tc N� � , but the mechanism in this temper-
ature range is different as the characters of the curves are
completely different. In particular, the generation of all har-
monics is well described by the Rayleigh`s mechanism for
ferromagnetic materials [5,6].
Once a remanent magnetization is induced above Tc
those materials «remember» it even when they are cooled
down to T Tc� , and the internal field which is created,
can induce the so-called spontaneous vortex phase. The
purpose of the present paper is to show how this internal
field can be evaluated through the amplitude of the third
harmonic generation signal.
2. Experimental details
We studied the response of a Ru–1222 Gd sample to
an ac magnetic field. A ceramic sample with dimensions
of 8 2 2� � mm was prepared by a solid-state reaction as
described in Ref. 4. In all experiments described here we
measured the voltage drop induced in a pickup coil,
which is proportional to the time derivative of the magne-
tization M( )t . Our home made experimental setup was
adapted to a commercial MPMS SQUID magnetometer.
An ac field h t( ) at a frequency of �� �2 1 5� . kHz and an
amplitude up to the h0 3� Oe was generated by a copper
© Grigory I. Leviev, Menachem I. Tsindlekht, Edouard B. Sonin, and Israel Felner, 2007
solenoid existing inside the SQUID magnetometer. The
two signals at the fundamental and at the harmonic fre-
quencies were simultaneously recorded. The temperature,
dc magnetic field, and amplitude dependencies of the fun-
damental and harmonic signals have been measured by
the two coils method. In the present paper only the results
of the first and third harmonics will be discussed.
3. Experimental results and discussion
The magnetization of a sample can be expressed by the
macroscopic field inside the sample B( , )r t and the homo-
geneous ac field h h( ) cos( )t t� 0 � to which the sample
was exposed:
M B r h( ) ( ) [ ( , ) ( )]t / V t t dV� � � 1 4�
�
�
�h 0 � � � �n nn t n tcos sin( ) ( ) ,
(1)
where the integration is made over a volume of the sample
V and
� n ,
� n are the in-phase and out-of-phase suscepti-
bilities. Here we expand the magnetization to Fourier se-
ries [7,8]. Generally speaking
� n and
� n depend on: the
temperature T , the external magnetic field H and the am-
plitude of the ac field h0.
3.1. Susceptibility at H = 0
Figure 1 shows the zero field cooled (ZFC) temperature
dependence of the in-phase magnetic susceptibility
�
1
and
the amplitude of the third harmonic A i3 3�
� ��
�
| |, mea-
sured at H � 0.
The temperature dependence of
��( )T is typical for
superconducting ferromagnets [9]. This plot reveals three
transitions: (i) the paramagnetic-antiferromagnetic tran-
sition at TN �125 K, (ii) the most pronounced transition,
which corresponds to peak at Tm � 78 K, and (iii) the tran-
sition into a superconducting state at Tc � 28 K. The na-
ture of the second transition, which is evident in both the
linear and nonlinear responses, is not yet completely clear
and it is discussed elsewhere [10]. Ambiguity is
connected with the magnetic phase between Tm and TN ,
which is characterized by low coercivity. However, the
low-temperature side near the Tm transition definitely
corresponds to a weak ferromagnetic phase, and is dis-
cussed here.
The amplitude dependencies of
�
�
( )h0 and A h3 0�( )
are shown on Fig. 2. The
�� dependence looks like a step
function. Similar behavior is observed for A h3 0�( ). At
T Tc� A h3 0�( ) is different, and its typical dependence is
presented at Fig. 3.
We shall discuss the amplitude dependencies for �1
and A3� in a superconducting state. Let us start with the
microscopic Maxwell`s equation:
curl h r j r( , ) ( , )t
c
t�
4�
, (2)
where r is a radius-vector, t is time, and j is current density.
For macroscopic values it is necessary to average both parts
of this equation over a certain volume, which includes many
grains. The average current density is given by
� � � � � � �j r j M M( , )t c cJ s gcurl curl . (3)
The electromagnetic response of a superconducting ferromagnets
Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 845
200
100
0
T, K
Tm
0 50 100
TN
Tc
A , h = 2 Oe
3 0�
�
1
3
, A
,
ar
b
.
u
n
it
s
�
�
1 0, h = 0.4 Oe
Fig. 1. Temperature dependencies of
�� and A3� at H � 0.
R
e
,
ar
b
.
u
n
it
s
�
1
0
–0.02
–0.04
–0.06
–0.08
0.01 0.1 1.0
h , Oe0
0.01 0.1 1.0
0.20
0.15
0.10
0.05
0
h , Oe0
T = 5 K
ZFC
A
,
ar
b
.
u
n
it
s
3
�
Fig. 2. Amplitude dependencies of
��. Inset: amplitude depen-
dence A3� after ZFC at T � 5 K.
h , Oe0
0 1 2 3
A
,
ar
b
.
u
n
it
s
3
�
1.0
0.5
0
T = 62 K
H = 0
Fig. 3. Amplitude dependencies of A3� after ZFC at T � 62 K.
Here the average current density is the sum of the contri-
butions from Josephson’s junctions jJ , and is propor-
tional to the spin magnetization M s and to the magnetiza-
tion M g caused by superconducting currents inside the
grains. Using the relation � �� ��� � �1 4 4s g , where � is
permeability, � g and � s are the grains and lattice suscep-
tibilities, one obtains the macroscopic Maxwell equation:
curl B r j r( , ) ( , )t
c
tJ�
4��
. (4)
There is no doubt, that in granular superconductors the
nonlinearity in the superconducting state is caused by
weak links between grains through which Josephson’s
currents jJ flow. This current averaged over numbers of
junctions can be expressed by the parameters describing
the granulated media. By combining the current density
expression [11,12] and Maxwell`s equation (4) we obtain
the nonlinear equation for the vector potential A r( , )t ,
which describes the behavior of a sample exposed to an ac
field with finite amplitude:
curl curl expA A A� � �
�
�
�
�
�
�
�
�
�
1
2
2
2 2 2� �
�
�j j j j
A
B
d
dt
, (5)
where � j , B j , � are defined by the average parameters of
the media. These parameters are: � � ��j c / I a2
0
3
0
2� ( ),
B I /cj
2
04� �� , � � � ��� j a / Rc2 2
0
2 22( ), here 0 is a flux
quantum, I is average current of junction, � is number of
junctions per unit volume, a0 is average distance between
grain centers, R is the junction resistance in a resistive
junction model. Parameter � represents a characteristic
time of the problem. The vector potential A permits one
to define the magnetization of the sample M and the sus-
ceptibilities of the sample.
Let us consider: (a) the sample as an infinite slab of
thickness d located at � ! !d/ x d/2 2, (b) the external ac
field h t( ) is along the y axis, and (c) the vector potential
A, electric field E, a current density jJ are all directed to z
axis. Then only derivatives over x and t of the z compo-
nent of the vector potential A z exist. (Further the index z
will be omitted.) Now we obtain:
d A x t
dx
A x t
A x t
Bj j j
2
2 2
2
2 2
1( , )
( , )
( , )
� �
�
�
�
�
�
�
�
�
�
� �
�
�
exp
j
dA x t
dt2
( , )
.
(6)
The solution of Eq. (6) for the vector potential at the fun-
damental frequency looks as follows:
A x t A x i t A x i t( , ) [ ( ) ( ) ( ) ( )]*� � � �
1
2
0 0exp exp� �
� �| | ( )A t0 cos � " .
Using Fourier-transformation, one can get for the ampli-
tude A x0( ) the second order ordinary differential equation:
d A x t
dx
A x A x i A xj j
2
0
2
2
0 0
2
0
( , )
( ) [ ( )] ( )� �� �� ���# . (7)
Function
#( )
| | | |
y
y
B
I
y
Bj j j j
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�exp
2
2 2 0
2
2 22 2� �
I
y
Bj j
1
2
2 22
| |
�
�
�
�
�
�
�
�
�
$
%
&
&
'
(
)
)
takes into account the nonlinearity of the media, which is
connected with Josephson’s currents. Here I 0 and I1 are
modified Bessel functions. At small amplitudes, # ap-
proaches to 1 and this case corresponds to the linear me-
dia. For a sample inside the ac coil, it is convenient to con-
sider a symmetric excitation. So the boundary conditions
for the magnetic induction has the form: B t x d/( )| � � �2
� ��B t B tx d/( )| ( )2 0 cos � . Due to the symmetry, the vec-
tor potential is equal to zero in the middle of the slab
( x � 0). Using Eq. (6) we can obtain the expression for the
vector potential at the boundary A d/0 2( )� :
0
2
2
0
2
1
0
0
2
A d/
A d/
z
y y i dy B
( )
( )
( ( ) )
�
�
�
� �
$
%
&
&
&
'
(
)
)
)
# ��
/2
2
dz
d
j
�
�
.
(8)
The dependence of the complex susceptibility of a
sample on the fundamental frequency �1 is connected to
the vector potential on the boundary:
�
� �1
0
0
1
4
2 2
4
� � �
�A d/
dh
( )
. (9)
With the limit of low and large amplitudes of excitations,
Eq. (7) becomes linear and the susceptibility in these
cases is
�
�
��
� �
1
1
4 2 2
� � �
*
*d
d
th , (10)
where � � �� � �� � ** ( ) ( )� � � ��
j
/
ji i1 11 2 — at low am-
plitudes and � � ��* ( )� �
j
/i 1 2 — at large amplituded. The
transition from low to the large amplitudes occurs at a
field about B j . In our experiments B j � 0 015. Oe. The ra-
tio + between the real part of the susceptibility at low and
high amplitudes is
+
�
�
��
�
�
��
��
�
� �
� �
1 0
1 0
1 2
1
( )
( )
h B
h B
/dj
j
j
.
Using the experimental value of this ratio, it is possible
to evaluate �: � + +� �( )1 / . In our experiment we obtained
+ � 5, and � � 0 8. . Other parameters of the model can be
evaluated as well by using the grains size a0 and resistivity
the sample � in a normal state. Our SEM indicates that
a0
41 10 10� � � �( ) cm, and four probe resistivity measu-
rement at room temperature � res � � �2 5 10 14. in Gauss
846 Fizika Nizkikh Temperatur, 2007, v. 33, No. 8
Grigory I. Leviev, Menachem I. Tsindlekht, Edouard B. Sonin, and Israel Felner
units [4]. Therefore we estimate: � j � � � �� �5 10 5 103 2 cm,
� � �� �10 1011 9 s, j c � �0 3 3. A cm/ 2, R � �01 1. Ohm.
3.2. Third harmonic generation
According to the nonlinear Eq. (5), application of an
ac field at fundamental frequency causes the harmonic
generation. To fit qualitatively the experimental data
shown in 2 it is sufficient to substitute in the Eq. (6) the
calculated field at the fundamental frequency, nonlinearly
dependent on the amplitude of the excitation. The ampli-
tude dependence of the response at T Tc� in the mag-
netic-ordered state does not show a saturation (Fig. 3).
The amplitude of 3rd harmonic exhibits square-law de-
pendence on h0 quite well describe by the Rayleigh mech-
anism [5,6]. Nonlinearity at these temperatures corre-
sponds to oscillatory motion of the domain walls in an ac
field (the hysteretic dependence of a susceptibility on a
magnetic field).
3.3. Influence of an applied magnetic field
The response in the presence of a dc magnetic field de-
pends on the prehistory and the regimes of sample cool-
ing. The physical picture thus is complicated — there is a
magnetization connected to a lattice, the sample is broken
into domains, and at low temperatures magnetization of
the lattice can lead to a spontaneous vortex phase [13].
We used the dependence of the amplitude of the 3rd har-
monic on an external field (Fig. 4) and various cooling re-
gimes to measure internal magnetic field in a sample, re-
maining after turning off the external field. The procedure
of cooling of a sample which we have named internal-field
cooling (IFC) is this: the sample was cooled down to TIFC
at an external magnetic field H IFC (T TIFC N� ). At TIFC ,
the magnetic field was turned off and further cooled down
to T � 5K in zero external field. It appears that by using the
IFC procedure the properties of the superconducting state
were different from those measured after the regular ZFC
process from temperatures above TN . Thus, in the super-
conducting state, the sample senses the internal magnetic
field evolved from the remanent magnetization, which was
formed in the normal (not superconducting) ferromagnetic
phase and then quenched on further cooling.
We measured the signal of the third harmonic at 5 K af-
ter the IFC procedure.
Figure 5 shows the A H IFC3�( ) dependence for
TIFC � 40 and 70 K. It is evident that the field H IFC sup-
presses the third harmonic as well as the external field
after ZFC even though the field H IFC was turned off be-
fore the superconductivity onset. The turn off H IFC at
T � 40 K affects A3� more strongly than for T � 70 K be-
cause the remanent magnetization in the first case is
larger than for the latter.
Figure 6 presents the signal of the third harmonic
A T3�( � 5 K) as a function of TIFC after cooling in
H IFC � 30 Oe. The signal of the third harmonic goes down
for TIFC � 80 K. This demonstrates that the suppression of
the third harmonic response by the internal magnetic field
takes place only if the field cooling continues down to the
weakly ferromagnetic phase with essential coercivity. It is
known that in idealized single-domain superconducting
The electromagnetic response of a superconducting ferromagnets
Fizika Nizkikh Temperatur, 2007, v. 33, No. 8 847
T = 5 K
h = 0.2 Oe0
q = 0.8
H–q
A
,
V
3
�
�
15
10
5
0
H, Oe
0 50 100 150
Fig. 4. Magnetic field dependence of A3� at T � 5 K after ZFC.
T = 70 KIFC
T = 0 KIFC 4
h = 0.2 Oe0
A
,
V
3
�
�
H, Oe
0 50 100 150
10
5
Fig. 5. A T3 5�( )� K as a function of HIFC for TIFC � 40 and 70 K.
h = 0.2 Oe0
H = 30 OeIFC
T , KIFC
A
,
ar
b
.
u
n
it
s
3
�
15
10
5
0
0 20 40 60 80 100
Fig. 6. Amplitude of the third harmonic A3� at H � 0 and T � 5 K
versus TIFC .
ferromagnets the internal magnetic field from spontaneous
magnetization 4�M has the same effect on the phase dia-
gram, i.e., on the magnetic flux penetrating into the sample,
as the external field. This can be generalized in a more real-
istic case of a multi-domain sample with nonzero average
internal field � �4�M . On the basis of this argument we can
use the third harmonic signal versus dc magnetic field de-
pendence as a calibration curve to estimate the magnitude of
the quenched internal magnetic field H I .
Figure 7 presents the dependence of H I on H IFC . The
phenomenon revealed in our experiment is possible if the
domain structure formed in the ferromagnetic phase can
be quenched down to the superconducting state. On the
other hand, as was noted in the pioneer paper by Ginzburg
[1] and confirmed by detailed analysis in [14], supercon-
ductivity should strongly affect the equilibrium domain
structure: its period should grow, and in the Meissner
state any sample is a single domain in equilibrium. But in
our case we deal with a nonequilibrium domain structure,
which is a metastable state possible due to coercivity. The
presence of the quenched internal field in the supercon-
ducting phase clearly demonstrates that the sample is in
the mixed state with many vortices in the bulk. One can
hardly call this state as the spontaneous vortex phase be-
cause the latter refer to the equilibrium state, but we deal
with a metastable state. The nonlinear response is sensi-
tive to the average internal field � �4�M . The absolute
value of the average magnetization � �M is less than the
saturation magnetization M, which can determine the vor-
tex density in a single-domain sample. However, the satu-
ration magnetization may create vortices inside domains.
Since M changes its directions from a domain to a do-
main, we obtain the vortex tangle, which does not contrib-
ute to the average internal field � �4�M , studied here. This
vortex tangle is expected to exist even after the ZFC pro-
cess, and contributes to the initial value of the third har-
monic, which was detected without external or internal
magnetic field. These arguments illustrate that vorticity
(magnetic flux) distribution in real especially ceramic
superconducting ferromagnets can be very complicated.
Genuinely zero-field cooling is practically impossible: if
one cools a sample in zero external field, one cannot
avoid an internal magnetic field from the spontaneous
magnetization even if these fields vanish on average but
still remain inside the domains.
4. Summary
In summary, our measurements of the nonlinear re-
sponse unambiguously demonstrates coexistence of the
superconducring and ferromagnetic order parameters in
Ru-1222 samples below superconducting critical temper-
atures. Coexistence is manifested by the clear effect from
the domain structure quenched from temperature above
the superconducting critical temperatures on supercon-
ducting properties. We have shown that the nonlinear de-
pendence of the response of a sample on a variable mag-
netic field manages to be understood qualitatively in all
ranges of temperatures.
The work was supported by the Klatchky foundation.
The authors are deeply grateful to E. Galstyan for the
sample preparation and useful discussions.
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H , OeIFC
0 50 100 150
H
,
O
e
I
50
40
30
20
10
0
40 K
70 K
Fig. 7. Internal magnetic field versus HIFC for TIFC � 40 K and
70 K.
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