INSTITUTE OF PHYSICS PUBLISHING
SUPERCONDUCTOR SCIENCE AND TECHNOLOGY
Supercond. Sci. Technol. 15 (2002) 247–253
PII: S0953-2048(02)28622-4
Harmonic susceptibilities of a bulk
superconductor MgB2 at low
magnetic fields
Ali Gencer
Department of Physics, Faculty of Sciences, Ankara University, 06100-Tandoğan,
Ankara, Turkey
E-mail: gencer@science.ankara.edu.tr
Received 7 September 2001, in final form 19 November 2001
Published 11 January 2002
Online at stacks.iop.org/SUST/15/247
Abstract
Harmonic ac susceptibilities, χn = χn + i χn , and ac magnetization of a
metallic granular superconductor of MgB2 have been measured as a function
of temperature, ac field amplitude and additional small dc field in the
temperature range between 15 and 45 K. Temperature- and
amplitude-dependent measurements are similar to those obtained for a
typical high-temperature superconductor. In addition, MgB2 has a very
narrow transition temperature width and high critical current densities. The
experimental results have been compared to the numerical solutions of a
model based on Bean’s critical state. Odd harmonic susceptibilities (n =
3, 5 and 7) are in good agreement with those obtained numerically from the
model. In contrast, the measured even harmonic susceptibilities exhibit a
peak-like behaviour below Tc, with only small magnitudes of about
10−4 (SI). The results are interpreted in terms of flux penetration to the bulk
of superconductor MgB2.
1. Introduction
Magnesium diboride, MgB2, has been known for almost half
a century [1] and, in addition to its use in making alloys, the
production of high-temperature refractory ceramics has been a
potential field of its applications [2]. Despite having a simple
hexagonal structure like AlB2, it was recently discovered that
it becomes a superconductor at 39 K [3]. This discovery
triggered a worldwide, intensive series of research works (see,
for example, [3–7] and the references therein, related to MgB2)
on the various aspects of the superconducting characteristics
of MgB2. The compound has been found to be particularly
attractive, because, unlike many HTS materials, the external
field-dependences of the Tc and Jc values of connectivity
between the grains of MgB2 do not seem to be discouraging.
As a matter of fact, MgB2 appears to be promising for infield applications up to several teslas [4], whereas, tedious
treatments are required for a granular copper oxide-containing
HTS material due to the weak connectivity between grains
under the influence of magnetic fields.
AC harmonic susceptibility measurements χn = χn + iχn
(n is an integer number) have been used in both the
0953-2048/02/020247+07$30.00 © 2002 IOP Publishing Ltd
characterization and the study of flux line dynamics of
superconductors [8–18]. It is well known that the in-phase
component, χ1 , of the fundamental ac susceptibility (n = 1)
is associated with flux expulsion of the sample and the out-ofphase component, χ1 , is related to ac losses. The theory of
ac losses has been extensively studied by Clem [14]. Critical
state like models have been very successful in accounting for
the main feature of the χ (T , Hac , Hdc ) (see, for example,
[15–17]). In the original Bean’s critical-state model [18],
the fundamental out-of-phase component, χ1 , is related to ac
2
losses as χ1 = AH /µ0 πHac
. Here AH is the area enclosed by
the hysteretic M–H loop being a measure of ac loss in an ac
cycle of H.
Fundamental ac susceptibility and its higher harmonics
are the manifestations of flux redistribution in the
superconducting state during the ac cycle of the magnetic field.
For a linear M–H response (mostly occurs at small ac fields
and low temperatures), the out-of-phase component χ1 reads
zero, indicating non-existence of ac loss, while the in-phase
component χ1 is −1 (SI) indicating complete flux exclusion
from the interior of the sample. When χ1 is –1 and χ1 is
zero, higher harmonic susceptibility components are all zero
Printed in the UK
247
A Gencer
at low temperatures. As Tc is approached from below, the
shielding currents will eventually exceed Jc and a non-linear
M(H ) becomes irreversible. In a real experiment, not only odd
harmonic components but also even harmonic components
are generated [15]. Although the physical interpretation
of the fundamental susceptibility is well established, higher
harmonic susceptibilities need further investigation as to shed
light on the flux dynamics. It is important to understand
flux dynamics since the superconductors have potential use
in electronics applications.
In this paper, we report on the measurements of
fundamental and harmonic susceptibilities (n = 1 to 7) of
a bulk MgB2 superconductor, together with their numerical
calculations for a disc geometry when the Bean model applies.
The experimental results are compared with the numerical
solutions to relate our results to other works in this field.
ac field and Hac is the ac field amplitude. However, in a
real experiment there may be some slight phase shifts which
need adjustment for an accurate separation of the components
before the measurement. Individual phase settings for all the
harmonics of interest were carried out at a fixed temperature
of 20 K, where the M(H ) loop is found to be reversible and
linear for the used ac field amplitudes. The process of phase
adjustment requires a linear sample with no ac losses. We
proceed as follows: let us assume that nth harmonic of a
frequency f is required, the system is phased with an output
and a reference of the PSD, both set at frequency nf . We
then specify the operating frequency to be nf and specify the
fundamental harmonic susceptibility. This process yields a
system phase angle θn for the nth harmonic susceptibility.
2. Experimental details
In general, there are various models of flux dynamics in the
superconductor. Most of the models are based on calculations
of a length scale for flux penetration. The length scale of
effective flux penetration is determined by magnetic field
strength in relation to either the sample size or the skin depth.
The first type of determination is common in critical state
models, while the latter is associated with ‘eddy-current’ type
models (see, for example, [23, 24]) based on the original
model of Maxwell and Strongin [19]. Generation of harmonic
susceptibilities is first introduced by the original Bean’s critical
state model and in its generalized forms, in which the critical
current density is assumed to be independent of the local
magnetic field. Amplitude-dependent curves of harmonic
susceptibilities (only odd ones) can be deduced from the
original Bean model, while it fails to explain the timedependent effects. On the other hand, ‘eddy-current’ type
models could explain the frequency dependence, while it is
inadequate for explaining the field dependence. Meanwhile,
more sophisticated models involve the calculation of ac
susceptibility using numerical solutions of the flux creep
equation (see, for example, [25–29] and the references therein)
and a diffusion-like equation derived from Maxwell equations
[22], for the cases of experimental interest [30, 31] in recent
years to account for both time and ac amplitude dependences
and develop a theoretical model for the description of flux
pinning and dynamics in high-temperature superconductors.
A more realistic model was developed by Brandt [12] to
calculate magnetic field, current profiles, magnetic moment
and ac susceptibility of superconductors with finite length in a
perpendicular magnetic field.
In our measurements for MgB2, we have observed very
small frequency dependence in the range 1 Hz to 1 kHz. The
measurements exhibit field-dependent behaviour especially
at temperatures very close to Tc. In this study, we limit
ourselves to the disc geometry and assume that Jc can be
treated as independent of the applied field for simplicity of
the theoretical treatment. We assume that the ignorance of the
field dependence on Jc within the model calculations is a good
approximation to account for the main features observed in
the measurements. Such an approximation would have been
invalid for weakly connected granular superconductors simply
because of the weak-link limited critical current density, which
has a sensitive magnetic field dependence.
A polycrystalline sample of MgB2 was prepared by using a
conventional solid-state reaction method. The preparation
details and structure properties can be found in [7]. According
to the work of [7], the sample has a single phase with an
average grain size of about 200 µm. The sample appears to
be dense, hard and strong. Lakeshore 7130 with a standard
mutual inductance bridge system using a phase sensitive
detector (PSD) with an input filter was used to measure inphase component χn and out-of-phase component χn of ac
susceptibility. For fundamental response (n = 1), a lowpass filter was used to reject higher harmonic susceptibilities.
For high harmonic response (n > 1), a band-pass filter was
used to measure only the harmonic susceptibility of interest,
while a wide-band input filter was used for ac magnetization
to account for all the harmonics. In the measurements, the
phase adjustment was carried out at low temperatures when the
superconductor is in an almost linear state with zero ac loss. At
a properly adjusted phase angle θ , the out-of-phase component
of fundamental susceptibility, χ1 (T , Hac ), reads zero for a
sample exhibiting linear response to an ac field. Individual
phase setting is required for each measuring frequency, while
it is not necessary for each measuring ac field with the same
frequency. At the correct phase settings, the experimental data
are recorded and analysed using the phase angle θ .
For an ac magnetic field of H (t) = Hac sin(ωt) subjected
to a superconductor, the magnetization of the sample can be
generalized to be of the form
M(t) = Hac
∞
[χn sin(nωt) + χn cos(nωt)]
(1)
n=1
in terms of Fourier coefficients χn and χn . In theory, for
ac susceptibility, the PSD measures voltages vn and vn ,
respectively, at peak and zero ac field excursions. From the
measured voltages, ac susceptibility coefficients with phase
angles set to 0◦ and 90◦ can easily be shown as
χn =
αvn
nV Hac f
χn =
αvn
.
nV Hac f
(2)
Here, α is a geometry-dependent calibration coefficient, V is
the volume of the sample, f is the frequency of the applied
248
3. Model calculations
Harmonic susceptibilities of MgB2 at low magnetic fields
For a superconducting disc subjected to a magnetic field
perpendicular to its surface along the z-axis, the magnetization
could be given as
0.4
Mz = −χ0 Ha S(Ha /Hp )
0.0
(3)
-0.2
χ /χ (SI)
1 0
where Ha is the applied field and Hp is the least magnetic field
penetrating to the centre of the disc. The function S(x) is given
by
sinh x
1
1
cos−1
+
(4)
S(x) =
2x
cosh x
cosh2 x
χ"
0.2
f=1 kHz
H ac (Oe)
-0.4
-0.6
as in [20]. Here, χ0 = 8R/3πd is a value for a complete
flux shielding of a disc with thickness d and radius R, which
exhibits perfect diamagnetism. If the applied ac field Hac
modulates between −Hm and +Hm, the magnetization for the
descending and ascending portions can be given, respectively,
as in [16].
1
1:2
2:5
3:10
4:15
5:20
χ '
1
5
4
3
2
-0.8
1
-1.0
-1.2
34
35
36
37
38
39
Tem perature (K)
(a)
M↓z = −χ0 Hm S(Hm /Hp ) + χ0 (Hm − Ha )S((Hm − Ha )/2Hp )
(5a)
0.4
M↑z = χ0 Hm S(Hm /Hp ) − χ0 (Hm + Ha )S((Hm + Ha )/2Hp ).
(5b)
where θ ranges between 0 and 2π. There exists an analytic
solution to the integrals of equation (6), when x 1 or x 1.
Within the model for fields on the scale of Hp, there is no
analytic solution and we seek numerical solutions with the
Gausskronrod method on a Personnel Computer PC. Note that
there exists an analytical solution in the cylindrical geometry.
4. Results and discussion
The measurements of the fundamental susceptibility versus
temperature are given in figure 1(a) for ac fields of 2, 5, 10,
15 and 20 Oe (rms) between 34 and 39 K at a frequency of
1 kHz. All the measurements reported here are normalized at
30 K. Figure 1(a) is a characteristic behaviour of fundamental
susceptibility, expected from a superconductor. The onset
of the critical temperature is found to be 38.4 K as shown
for all the fields used in figure 1(a). The curves for χ1 and
χ1 versus temperature display a typical single-step process
which reflects the flux penetration into the bulk of MgB2.
The single-step changes indicate the presence of a strong
coupling between the grains of MgB2. In contrast, a weakly
connected granular superconductor shows a distinguishable
two-step process during transition, attributed to the grains
and the coupling between the grains. Note that the transition
width increases with ac field amplitude. As the temperature
decreases from above Tc until the temperature, Tp, at which
χ 1 peaks, the lowest field needed to penetrate to the centre,
0.2
0.0
-0.2
χ /χ
1 0
With the appropriate manipulation, it can be shown that
the harmonic susceptibilities for H = Hac sin(ωt) could be
calculated via Fourier analysis as
2χ0 π/2
χn =
(1 − sin θ )S[(x/2)(1 − sin θ )] sin nθ dθ
π −π/2
(6)
2χ0 π/2
χn =
{−S(x) + (1 − sin θ )S[(x/2)(1 − sin θ )]}
π −π/2
× cos nθ dθ
)
χ"/χ
χ'/χ
-0.4
0
0
-0.6
-0.8
-1.0
-1.2
0.1
1.0
10.0
100.0
H ac /H p
(b)
Figure 1. Fundamental ac susceptibility shown for various ac field
amplitudes. (a) Measurements of in-phase and out-of-phase
components of the fundamental ac susceptibility versus temperature
are given for Hac = 2, 5, 10, 15 and 20 Oe (rms) at f = 1 kHz.
(b) Frequency independent numerically calculated fundamental ac
susceptibility versus Hac/Hp. Note that the peak in χ1 occurs at the
ratio Hac/Hp = 1.942.
Hp, becomes larger. At these temperatures the measuring ac
field amplitude is sufficiently large enough to penetrate to
the centre of the sample. When the Hac is on the scale of Hp,
the full flux penetration occurs. In the measurements, the peak
temperatures Tp are 38.10, 38.02, 37.90, 37.91 and 37.84 K for
ac fields of 2, 5, 10, 15 and 20 Oe, respectively. Corresponding
heights of the peaks in χ1 are 0.17, 0.19, 0.20, 0.21 and 0.21.
On further cooling below Tp , the superconducting sample starts
to shield the applied ac field and flux changes are only confined
to the outer regions of the sample. Therefore, the shifts of the
curves to lower temperatures with increasing ac field amplitude
are understandable.
In figure 1(b), the results of numerical solutions to
equation (6) for χ (Hac/Hp) and χ (Hac/Hp) are given.
249
A Gencer
3.e-3
3.0e-4
2.e-3
-3.0e-4
-6.0e-4
-9.0e-4
Hdc=15 (Oe)
χ /χ (SI)
2 0
χ /χ0 (SI)
2
0.0e+0
f=140 Hz
Hac=2 (Oe)
f = 140 Hz
Hac = 2 Oe
Hdc=0
1.e-3
0e+0
χ '/χ
χ '/χ
2
0
-1.e-3
χ "/χ
2
2
0
2
0
χ "/χ
0
0.0e+0
3.0e-4
-1.5e-4
1.5e-4
χ /χ (SI)
4 0
χ /χ (SI)
4 0
-1.2e-3
-3.0e-4
χ '/χ
4
-4.5e-4
4
χ '/χ
-1.5e-4
0
χ "/χ
0.0e+0
4
0
χ "/χ
0
4
-6.0e-4
-3.0e-4
0e+0
2.5e-5
0
-1.e-4
χ /χ (SI)
6 0
χ /χ (SI)
6 0
0.0e+0
-2.e-4
χ '/χ
6
6
-5.0e-5
χ '/χ
6
0
χ "/ χ
-3.e-4
-2.5e-5
0
χ "/χ
-7.5e-5
6
0
0
-1.0e-4
36
37
38
39
Tem perature (K)
(a)
36
37
38
39
Temp erature (K)
(b)
Figure 2. Even harmonic susceptibilities versus temperature are shown for Hac = 2 Oe (rms) (a) without dc field and (b) with dc field of 15
Oe at f = 140 Hz.
The horizontal axis is chosen as a logarithmic axis to
mimic the experimental results in common with the higher
harmonic susceptibility measurements. The numerical results
are semi-quantitatively in agreement with the experiments.
However, exact comparison is impossible because of the
250
unknown temperature dependence of Hp which increases with
decreasing temperature during the measurement. We also find
from numerical solutions in figure 1(b) that the peak in χ1
occurs at the ratio of Hac/Hp = 1.942 with the maximum
height 0.241 as in agreement with the result of the work in [16].
Harmonic susceptibilities of MgB2 at low magnetic fields
8.e-2
3.e-2
χ '/χ
Hac=10 (Oe)
χ "/χ
3
3
χ /χ (SI)
3 0
χ /χ (SI)
3 0
2.e-2
f=140 Hz
χ '/χ
1.e-2
3
0
3
0
χ "/χ
0
0
4.e-2
0e+0
0e+0
-1.e-2
8.0e-3
0e+0
χ /χ (SI)
5 0
χ /χ (SI)
5 0
2.e-3
-2.e-3
χ '/χ
5
-4.e-3
χ '/χ
0
5
0
-1.6e-2
χ7/χ0(SI)
4.e-3
0
5
0
7
χ '/χ
7
0
χ /χ (SI)
7 0
χ '/χ
χ "/χ
0
-1.e-3
-2.e-3
38
39
Temperature (K)
(a)
7
2.e-3
0e+0
37
0
χ "/χ
0e+0
36
5
χ "/χ
6.e-3
7
1.e-3
-8.0e-3
χ "/χ
3.e-3
2.e-3
0.0e+0
0.01
0.10
0
1.00
10.00 100.001000.00
Hac/Hp
(b)
Figure 3. (a) Experimental results of odd higher harmonic susceptibilities (n = 3, 5 and 7) versus temperature are shown for Hac = 10 Oe
(rms) and f = 140 Hz. (b) Numerical solutions to equation (6) are plotted as to mimic the measurements in (a).
At this ratio and above, flux penetration to the centre of the
sample occurs. At the full flux penetration, as first pointed out
by Norris [21], Hp = Jc d/2, where d is the thickness of the
disc geometry. The critical current density from the magnetic
measurements is nearly 107 A m−2 at 37.84 K about 0.5 K
below Tc. The fields used for figure 1(a) are rather large and
the shift of the peak temperature Tp in χ1 with increasing ac
field amplitude is small. The calculated critical current density
can be assumed as unchanged within 0.2 K around 37.8 K.
In figure 2(a), the measurements of the even harmonic susceptibilities versus temperature are given with Hac = 2 Oe, f =
140 Hz and additional dc field of 15 Oe plotted in figure 2(b).
Although the magnitudes of the even harmonics are as small
as about 10−4 (SI), they exhibit peak-like dependences with
251
A Gencer
252
4.e-3
(a)
f = 133 Hz
T= 38.10 K
3.e-3
Magnetization (a.u)
2.e-3
1.e-3
0e+0
-1.e-3
-2.e-3
H
ac
=5 (Oe)
H
=10 (Oe)
ac
H =15 (Oe)
ac
H =20 (Oe)
ac
-3.e-3
-4.e-3
-40
-30
-20
-10
0
10
20
30
40
H ac (Oe)
6.e-2
f=133 Hz
(b)
H ac =20 (Oe)
4.e-2
Magnetization (a.u)
temperature. It is expected that even harmonic susceptibilities
for a sample with a linear M–H and/or a symmetrical M–H
must read zero. At temperatures below 37 K for the given ac
field amplitudes plotted in figure 2, the magnetic response is
approximately sinusoidal to the sinusoidal changing ac field,
i.e. M is almost linear in H. At temperatures between 37 K and
Tc for the given ac fields, there happen to be little distortions
in the symmetricity of the response during the measuring ac
cycle. This would give rise to the non-linear behaviour of
magnetization together with the irreversible flux penetration
towards the centre. Flux redistribution is dependent on the
magnetic prehistory of the sample which would give rise to
non-symmetric magnetization [22]. Hence, the magnitudes of
the even harmonic susceptibilities become larger to give peaklike dependence as the critical temperature Tc is approached
from below. The application of dc fields would also distort the
symmetricity to make the even harmonic susceptibilities more
pronounced. This is true for the second harmonics, while the
measurements of the fourth and sixth harmonics show dramatic change by the application of dc field. The numerical
results for the even harmonics give zero in the zero dc field.
Therefore, comparison would not have been made. Model
results give non-zero solutions only for the odd harmonic
susceptibilities.
In figure 3(a), the measurements of the odd harmonic
susceptibilities (n = 3, 5 and 7) versus temperature are given
for Hac = 10 Oe and f = 140 Hz. The two components of
χ 3 give peak-like dependences with increasing temperature
up to Tc. The height of the in-phase component χ3 is
smaller than that of the out-of-phase component, χ3 , which
occurs at a higher temperature below Tc. The components
of the fifth harmonic susceptibility χ 5 exhibit peak-like
dependences in the negative direction with magnitudes roughly
ten times less than that of the χ 3. Note that the peaks
occur at higher temperatures and the magnitudes of the peaks
become smaller with increasing degree of harmonics. The
magnitudes of the peaks of χ 7 are even more smaller. This
fact is seen both in the experimental and in the numerically
calculated results. The peaks in the numerical calculations
occur at increasing ratios of Hac/Hp in the subsequent odd
harmonics. Harmonic susceptibilities would be considered
as Fourier coefficients and their successive contribution to
the total magnetization becomes smaller and smaller with
increasing order of harmonics. Our model results are in
agreement with our measurements and also with the model
results of [12, 13, 16]. To the best of our knowledge,
the experimental results of higher harmonic susceptibilities
reported here on MgB2 are newly presented for publication
and we have had no chance of comparison to the experimental
works of others. A semi-quantitative agreement is observed
between the results of odd harmonic susceptibilities (n =
3, 5 and 7). However, an exact comparison would have
been made by knowing the explicit form of the temperature
dependence of Hp and geometry-dependent demagnetizing
effects.
In figure 4(a), hysteretic ac magnetizations for ac fields
of 5, 10, 15 and 20 Oe (rms) are shown for 133 Hz at a fixed
temperature of 38.10 K. Even if the M–H loop is small for
5 Oe, the sample is fully penetrated (see also figure 1(a)).
For larger fields of 10, 15 and 20 Oe, there also exists a
2.e-2
0e+0
-2.e-2
T=37.0 K
T=37.4 K
T=37.6 K
T=37.8 K
T=38.0 K
T=38. 2 K
-4.e-2
-6.e-2
-40
-30
-20
-10
0
10
20
30
40
H ac (Oe)
Figure 4. (a) AC Magnetization at a fixed temperature of 38.10 K
for ac fields of 5, 10, 15 and 20 Oe (rms) at f = 133 Hz. (b) AC
magnetization at various temperatures for ac field of 20 Oe (rms)
and f = 133 Hz.
full flux penetration. AC loss per cycle becomes larger
with increasing Hac. The loop area also increases with Hac.
Note that the magnetization being proportional to the induced
critical currents, is magnetic field-dependent. This would
imply a field-dependent critical current density at temperatures
close to Tc. In figure 4(a), the magnetization at zero field
excursion is larger than the magnetization at the peak field. The
in-phase and out-of-phase components for ac susceptibility
measurements with a wide-band input filter are proportional to
the magnetization at peak and zero field, respectively (compare
figure 1(a) and figure 4(a)).
We have also plotted the ac magnetization for the same
field of 20 Oe (rms) at a few fixed temperatures in figure 4(b)
with f = 133 Hz. The magnetization becomes larger with
decreasing temperature as seen in the figure. At 38.2 K, the
hysteresis loop is the smallest indicating the small magnitude
of critical currents. For 38 K measurements, the area of
the loop becomes larger with increasing critical currents.
Harmonic susceptibilities of MgB2 at low magnetic fields
The hysteresis loop for 37.8 K is the largest in the
measurements. For this loop, the measuring ac field is on
the scale of Hp with the estimated critical current density
of 8.7×103 A cm−2 from Jc = 3M/R. This value is in
line with the one calculated from ac susceptibility while it
is about ten times smaller than the values obtained at 35 K
in [6]. As the temperature is lowered as for 37.6 , 37.4 and
37 K measurements, the measuring ac field of 20 Oe becomes
less than Hp and the flux penetration is confined in the outer
regions of the sample. The critical currents would be much
higher than the estimated value upon lowering temperature.
The magnetization measurements are in agreement with the
susceptibility measurements reported above. The results
given in figure 4(b) for low temperatures below 38 K are
qualitatively consistent with the Bean model, but the results
in figure 4(a) would necessitate the B dependence of Jc for a
better fit to the hysteretic loops observed in the measurements
at high temperatures above 38 K. This issue together with
even harmonics is our current interest and the results are in
preparation for consideration of publication in future.
In summary, we have made a low-field magnetic
characterization on a good quality bulk superconductor of
MgB2. Our experimental results for non-vanishing odd
harmonic susceptibilities are in agreement with our numerical
solutions consistent with the works of [12, 13, 16]. Although
the measured even harmonic susceptibilities are very small in
magnitude, the Bean critical state model can be assumed as
valid to account for the observed temperature and amplitude
dependences in the susceptibility measurements of MgB2.
Acknowledgments
The author is thankful to Dr X L Wang and Professor S X Dou
at Wollongong University for the supply of the sample
and also grateful to Drs H Yılmaz, A Aydınuraz, E Aksu,
N Güçlü and A Kılıç for their help with simulations and
stimulating discussions. This work has been financially
supported by Ankara University Research Fund under a
contract number 2000-07-05-001.
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