Third Harmonic Susceptibility and the Irreversibility Line of Fe/MgB2 Tapes
ABSTRACT We report a study of the pinning properties of Fe/MgB2 tapes performed by means of ac magnetic susceptibility measurements. In particular, the third harmonic response has been measured as a function of the dc magnetic field up to 9 T, at different temperatures, for various frequencies and amplitudes of the ac field. The irreversibility line has been determined using the third harmonic onset criterion for samples made with different ball-milled MgB2 powders. The study of the harmonic response allows us to probe the nature of the vortex pinning and to analyse its influence on the enhancement of the irreversibility field in strongly ball-milled samples. We have also investigated the dynamical regimes governing the vortex motion in the tapes by comparing the experimental curves with numerical simulations of the non-linear diffusion equation for the magnetic field.
Third Harmonic Susceptibility and the Irreversibility Line of
C. Senatore 1,2, N. Clayton 2, P. Lezza 2, M. Polichetti 1, S. Pace 1, R. Flükiger 2
1Dipartimento di Fisica “E.R.Caianiello” and INFM, Università degli Studi di Salerno,
2Département de Physique de la Matière Condensée, Université de Genève, Genève,
We report a study of the pinning properties of Fe/MgB2 tapes performed by means of ac
magnetic susceptibility measurements. In particular, the third harmonic response has been
measured as a function of the dc magnetic field up to 9 T, at different temperatures, for
various frequencies and amplitudes of the ac field. The irreversibility line has been
determined using the third harmonic onset criterion for samples made with different
ball-milled MgB2 powders. The study of the harmonic response allows us to probe the nature
of the vortex pinning and to analyse its influence on the enhancement of the irreversibility
field in strongly ball-milled samples. We have also investigated the dynamical regimes
governing the vortex motion in the tapes by comparing the experimental curves with
numerical simulations of the non-linear diffusion equation for the magnetic field.
Keywords: AC Susceptibility, Fe/MgB2 Tapes, Irreversibility Field, Flux Dynamics
PACS:74.25.Ha, 74.25.Qt, 74.70.Ad, 84.71.Mn
Contacting author: Carmine Senatore (e-mail: Carmine.Senatore@Physics.Unisa.it)
Since the discovery of superconductivity in MgB2 below a critical temperature Tc of 39 K ,
significant progress has been made in order to understand and develop this material to exploit
its high intrinsic performance for power applications, such as magnets and transformers. It is
believed that the supercurrent density in MgB2 is controlled predominantly by flux pinning
rather than by grain boundary connectivity because of the large coherence length in this
material (x0 = 50 Å) . However, the lack of natural defects is responsible for the rapid
decline of the critical current density with increasing magnetic field strength. In view of the
applications mentioned above, soon after the discovery of MgB2, many research groups tried
to demonstrate the feasibility of fabrication of wires and tapes and several processing methods
have been developed in order to obtain high critical current densities . The most studied
approach has been the power-in-tube (PIT) process, where powder is packed into a metal tube
and drawn into a wire. The wire can subsequently be rolled to form a tape. Indeed, very high
critical current densities (Jc » 105÷106 A/cm2) have been reported at 5 K [4-6], but these high
values rapidly decrease with increasing temperature and magnetic field. To improve the
performance of MgB2 wires, it is necessary to find a processing method that can introduce
more pinning centres and also overcome the poor connectivity between the powder grains,
which is mainly caused by porosity. In order to do this, it is essential to understand the flux
pinning mechanism and the irreversibility properties of MgB2.
The third harmonic response of the ac magnetic susceptibility is a highly sensitive tool to
investigate the flux dynamics in relation to the electrical transport properties of the sample
[7,8]. In the Bean critical state model the harmonic generation is attributed to the hysteretic
relationship between the magnetization and the external field due to flux pinning .
However, in order to account for the frequency dependence of the ac magnetic response, the
simultaneous presence of hysteretic and dynamic losses has to be included in the model. In
fact, when the vortex motion is governed by thermally activated flux creep, the I-V curve of a
type-II superconductor exhibits non-linear behaviour, which is a direct consequence of the
current dependence of the pinning energy U, and this leads to the generation of higher
harmonics in the ac susceptibility .
In this paper we investigate the effect of ball-milling the MgB2 powders, and the
consequent reduction of the particle size, on the pinning properties of monofilamentary
Fe/MgB2 tapes by measuring the third harmonics (c3 = c’ 3 + ic”3) of the ac magnetic
susceptibility as a function of the dc magnetic field, for various frequencies and amplitude of
the ac field. We have determined the irreversibility line by using the third harmonic onset
criterion. In particular, we have studied the nature of the enhancement of the irreversibility
field, due to the ball-milling process, by performing numerical simulations of the non-linear
magnetic diffusion equation.
The paper is organized as follows. The sample preparation and the experimental procedure
are described in Sec.2. In Sec.3 we report the numerical method applied to solve the magnetic
diffusion equation. We present in Sec.4 the c3(B) measurements and the irreversibility line for
Fe/MgB2 tapes prepared with different ball-milling times. Finally, Sec.5 reports the
conclusions of this work.
2. Experimental details
The Fe/MgB2 tapes studied in this paper have been fabricated by the PIT technique. MgB2
powder has been introduced into Fe tubes inside a glove box under an Ar atmosphere, which
have then been swaged, drawn and rolled. The Fe tubes used in this experiment had an outer
diameter of 8 mm and an inner diameter of 5 mm. Both ends of the tube were sealed with lead
pieces. After swaging to 3.85 mm and drawing to 2 mm, the wire is deformed by flat rolling
to obtain a 3.9×0.38 mm2 tape. A 920°C final heat treatment is performed for 0.5 h in a pure
In this work we report measurements performed on three different tapes in order to study
the effect of the initial powder size on the pinning properties. One tape has been prepared
directly from the as-purchased powder, i.e. standard commercial MgB2 powder from Alpha-
Aesar with a purity of 98%. Two other tapes have been prepared by ball-milling the starting
powder for 3h and 100h under Ar atmosphere. Granulometry measurements show that the as-
purchased powder contains a large number of agglomerated grains with a wide size
distribution centred at around 60 µm, and a maximum diameter of about 120 µm. After the
ball-milling process the maximum diameter of the grains is around 20 µm, while in the
3h /100h ball-milled powder 35% / 50% of the grains are of sub-micron size.
The ac susceptibility measurements reported in this work have been performed on the
MgB2 cores of three tapes using an home made susceptometer. The magnetic field
dependence of both the first and the third harmonics (c1(B) and c3(B)) has been measured by
sweeping the dc field from 0 to 9 T, at different temperatures (15, 20, 25, 30, 35 K).
Measurements have been performed using different ac field amplitudes (1, 2, 4, 8, 16 G) in
the frequency range 1007 ÷ 5007 Hz. Both the ac and dc fields are applied parallel to the
longest side of the sample.
Because of the sensitivity of the third harmonic response to the phase setting of the
measurements, we have set the phase in a very accurate way. We have measured the
wide-band susceptibility  as a function of the lock-in phase in the range [-180°, 180°] for
each measured frequency at Bac = 1 G and T = 4.2 K, both with and without the sample. We
then calculate the difference between the curves acquired with and without the sample in
order to remove all the contributions due to spurious signals. The amplitude susceptibility ca
is proportional to the sample magnetization when the external field reaches its maximum
value, whereas the remanent susceptibility cr is proportional to the magnetization remaining
in the sample at the zero instantaneous value of the ac field. When the sample is in the
Meissner state, cr is zero and ca reaches its maximum negative value. This allows us to
determine the phase to an accuracy better than 0.1 degree.
3. Numerical method
In order to relate the harmonic susceptibilities to the vortex pinning and dynamics, we have
numerically solved the non-linear diffusion equation which governs the magnetic flux
penetration in a type-II superconductor. In the case of an infinite slab of thickness d in a
parallel field geometry this equation, derived by the Maxwell equations and Ohm’s law, can
be written as:
T , J ,B
with the boundary conditions B(±d/2,t) = Bext(t) = Bdc + Bac sin(2p f t). The diffusion
coefficient r (B,J,T) is the residual resistivity due to thermally activated flux motion. In fact
this leads to an electric field, whose expression is:
J ,T ,BU
We assume that the temperature, magnetic field and current density dependence in the
effective energy barrier UP(B, T, J) may be separated, using for the current dependence the
1T ,BUT , J ,BU
This expression has been derived by assuming that flux creep occurs by bundles of flux lines
jumping between adjacent pinning points .
Equation (1) has been numerically solved by means of Fortran NAG  routines. In order
to obtain the harmonic susceptibilities cn= c’n+ic”n, we have first calculated the
magnetization loop for the applied time-dependent field:
and then its Fourier transforms:
To account for the magnetic field and temperature dependence of the ac susceptibilities we
have to specify the magnetic field and temperature dependence of the critical current density
Jc(B,T) and the thermal activation energy U0(B,T). To this end, we chose the following forms
0 T , 0
BJ T ,BJ
0T , 0
where t represents the reduced temperature T /Tc and the values of the parameters in the
magnetic field dependence are B1 =2.5 10-2 T and B2 =5 10-2T. The upper critical field is :
with Bc2 (T=0)= 20 T .
The numerical calculations of the third harmonic response reported in this paper have been
performed as a function of the ratio U0 (B=0,T=0)/ kBTc for fixed Jc (B=0,T=0)× d, simulating
an enhancement of the pinning and as a function of the value of Jc (B=0,T=0)× d for fixed
U0 (B=0,T=0)/ kBTc, simulating an improvement of the grain connectivity. This has been done
in order to investigate the modification induced in the vortex dynamics by means of the
ball-milling process, as will be shown in the next section.
4. Experimental results and discussion
In Fig.1 we report the magnetic field dependence of the real and imaginary components of the
first harmonic c1 (Fig.1a) and the third harmonic c3 (Fig.1b) measured at T =20 K on the 3 h
ball-milled tape with f =1607 Hz and Bac =16 G. According to the critical state prediction,
reported in the inset, the real part of the first harmonic c’1 shows the superconducting
transition whereas the imaginary part exhibits a peak that corresponds to the losses of the
system. We can also observe the typical structures predicted by the Bean model, i.e. a peak
that rises when the relation Bac=Jc(Bdc)× d holds and goes monotonously to zero in c’3(B) and
a peak followed by a dip in c”3(B). We note that the height of the c3(B) peaks is higher in the
experimental curves than in the curves calculated in the framework of the critical state.
Moreover, in the Bean model the peak in the imaginary part c”3(B) is predicted to be higher
than the one in the real part c’3(B) (inset Fig.1b), whereas in the experiment the opposite is
observed. This behaviour can be ascribed to flux dynamics. In particular, since the flux flow
regime becomes linear when Bdc >> Bac , the only dynamic regime contributing to the signal
of higher harmonics is flux creep. Similar features are encountered in the measurements
performed on the other tapes.
Fig.2 shows the magnetic field dependence of the third harmonic modulus |c3| measured at
different temperatures (T =15, 20, 25, 30 K) on the non-ball-milled tape, with f =1607 Hz and
Bac =4 G. The curves exhibit a pronounced granular behaviour, due to the porosity of the
sample. Because of the poor connectivity between the grains we can observe in the |c3|(B)
curves two well separated peaks. The low field peak is determined by intergranular currents,
likely due to weak Josephson coupling. At higher fields, the intergranular currents become
negligible and the behaviour of the sample is governed by the decoupled grains: the resulting
high field peak corresponds to the currents flowing inside the grains. In Fig.3 the
magnetization loop measured on the same sample at 5 K for a sweep rate of 2 T/min. is
reported; the measurement has been performed using a Vibrating Sample Magnetometer
(VSM). The loop exhibits the typical shape due to flux pinning on a surface barrier and this
further confirms the granular behaviour of the sample. In fact, because of the low electrical
field induced in the sample in a VSM measurement (typically (dB/dt)VSM/(dB/dt)c » 10-1÷10-2),
the grains are completely decoupled. Moreover, the grain surfaces acts as surface barriers and
the lack of strong bulk pinning determines the Z-shape of the hysteresis loop.
Fig.4 shows the irreversibility line of the tapes, whose powders have had different
ball-milling times, determined by using the third harmonic onset criterion. In the presence of
an applied magnetic field Bdc >> Bac the onset of the third harmonic is the result of a dynamic
crossover from a regime in which the system’s response is dominated by flux flow, and
characterized by the absence of harmonic signal to one that is dominated by pinning.
Therefore, the onset field corresponds to the irreversibility field, as defined in the flux creep
The irreversibility line of the non-ball-milled tape in Fig.4 reflects the intragrain peak of
the c3 (B) curves. We notice that, as a consequence of the ball-milling process, the
irreversibility field is strongly enhanced. In particular, the value Birr =9 T is reached at
T =15 K for the non-ball-milled tape, at T =16 K for the 3h ball-milled tape, at T =18 K for
the 100h ball-milled tape.
The ball-milling process reduces the size of the powder grains and this leads to an
improvement of the connectivity. The transport properties of the Fe/MgB2 tapes are
determined by both the strength of the pinning centres and the connectivity of the grains. In
order to study the mechanism responsible for the enhancement of the irreversibility field in
the ball-milled samples we performed numerical simulations changing, separately, the
Jc (B=0,T=0)× d value and the U0 (B=0,T=0)/ kBTc ratio. The |c3|(B) curves calculated for
U0 (B=0,T=0)/ kBTc =750, 1125, 1500 at fixed Jc (B=0,T=0)× d =5 105 A/m are plotted in
Fig.5. The onset of |c3| and, thus, Birr increases with U0 (B=0,T=0)/ kBTc, i.e. with the strength
of the pinning. Moreover, the increase of U0 (B=0,T=0)/ kBTc determines the broadening of
the peak and the increase of its height. On the other hand, when Jc (B=0,T=0)× d is increased
at fixed U0 (B=0,T=0)/ kBTc, simulating an increase of the current density due to the
improvement of the connectivity and but not of pinning, the onset of |c3| is unaffected while
the peak becomes sharper, and its height is reduced (Fig.6).
In Fig.7 we report the |c3|(B) curves measured at T =20 K with Bac =8 G and f =1607 Hz
for the three different tapes. By comparing the experimental curves with the simulated ones
we can associate the shift of the third harmonic onset to higher fields and, thus, the increase of
Birr to the enhancement of the pinning. Since the upper critical field Bc2 and the critical
temperature Tc have been found in the literature to be unaffected by the ball-milling process
[17,18], this enhanced pinning is likely to be confined to the grain surface . Nevertheless
the disappearance of the intragrain peak in the 3h and 100h ball-milled curves can be ascribed
to an improvement of the grain connectivity due to the ball-milling process.
In Fig.8 we report the |c3|(B) curves measured at fixed temperature and ac field (T =20 K,
Bac =4 G) for two frequencies (f = 1007 and 5007 Hz). The frequency behaviour of the third
harmonic response is related to the current density dependence of the pinning energy
UP(B, T, J) and, thus, to the pinning mechanisms [19,20]. Measurements have been performed
on four different samples, namely the commercial powders (Fig.8a), the non-ball-milled tape
(Fig.8b), the 3h ball-milled tape (Fig.8c), the 100h ball-milled tape (Fig.8d). The interesting
feature is the increase of the |c3|(B) peak amplitude with the frequency, common to all the
samples. This means that the ball-milling of the initial powder, the deformations during the
fabrication of the tape, the annealing process and the consequent recrystallization do not
change the pinning mechanism present in the powders, but lead to the an increase of the
strength of the pinning centres. Moreover, we reproduce the frequency behaviour of the
|c3|(B) curves in the numerical simulations (Fig.9) by using the Anderson-Kim pinning model,
corresponding, as shown in Sec.3, to the dependence U(J)∝1-(J/Jc). The parameters used for
the simulated |c3|(B) curves in Fig.9 are U0 (B=0,T=0)/ kBTc =1125, Jc (B=0,T=0)× d =5×105
A/m, T /Tc =0.5, Bac =4 G, f = 1007 and 5007 Hz. The Anderson-Kim model implies the
presence of extended pinning centres, i.e. whose dimensions are large compared with the
inter-flux-line spacing (=1.07(f0/B)½), whereas collective pinning arises from point-size
disorder randomly distributed in the superconductor . The collective pinning mechanism
(U(J)∝( Jc /J)m), that has been found to correctly describe the frequency behaviour of the third
harmonic response for MgB2 bulk samples [8,19], has to be excluded in this case since it leads
to the decrease of the |c3| peak amplitude with the frequency, as reported in Ref..
In this paper, the third harmonic susceptibilities as a function of the dc magnetic field have
been measured on Fe/MgB2 tapes made with different ball-milled powders. We have
determined the irreversibility line by using the third harmonic onset criterion. In order to
probe the nature of the enhancement of the irreversibility in the strongly ball-milled samples
we have performed numerical simulations of the non-linear magnetic diffusion equation. The
ball-milling process produces both the improvement of the connectivity of the grains and an
enhancement of the pinning. The latter effect leads to the increase of the irreversibility field
Birr. Moreover, the frequency behaviour of the |c3|(B) curves suggests that the vortex
dynamics in the measured samples can be described by the Anderson-Kim model, excluding
the presence of the collective pinning mechanism. This demonstrates the extended size of the
We express our thanks to Dr. Bernd Seeber for useful suggestions. Thanks are also due to
Mr. Robert Modoux and Mr. Alexandre Ferreira for their technical support.
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Figure 1: Magnetic field dependence of c’1 and c”1 in (a), c’3 and c”3 in (b) measured at
T =20 K, f =1607 Hz and Bac= 16 G on the 3h ball-milled sample. Insets: For comparison,
c’1 and c”1 (a), c’3 and c”3 (b) as a function of the magnetic field, calculated in the critical
Figure 2: Magnetic field dependence of |c3| measured at f =1607 Hz, Bac= 4 G and T =15, 20,
25, 30 K on the non-ball-milled tape; the curves exhibit two well-separated peaks due to the
granularity of the sample.
Figure 3: Magnetization loop for non-ball-milled tape measured at T =5 K with a sweep rate
of 2 T/min.; the loop shows the typical shape of flux pinning on a surface barrier.
Figure 4: Irreversibility line for the non-ball-milled tape (solid square), the 3h ball-milled
tape (open circle), the 100h ball-milled tape (solid triangle); the curves have been determined
by using the third harmonic onset criterion.
Figure 5: Magnetic field dependence of |c3| calculated for T/Tc= 0.5, f =1607 Hz and
Bac =8 G. The parameters used for the simulations are Jc (B=0,T=0)× d =5×105 A/m and
U0 (B=0,T=0)/ kBTc =750 (solid square), 1125 (open circle), 1500 (solid triangle); the onset of
|c3| and, thus, Birr increases with U0 (B=0,T=0)/ kBTc.
Figure 6: Magnetic field dependence of |c3| calculated for T/Tc= 0.5, f =1607 Hz and
Bac =8 G. The parameters used for the simulations are U0 (B=0,T=0)/ kBTc =750 and
Jc (B=0,T=0)× d =2.5×105 A/m (solid square), 5×105 A/m (open circle), 7.5×105 A/m (solid
triangle); the onset of |c3| does not change on increasing Jc (B=0,T=0)× d.
Figure 7: Magnetic field dependence of |c3| measured at T =20 K, f =1607 Hz and
Bac =8 G for the non-ball-milled tape (solid square), the 3h ball-milled tape (open circle), the
100h ball-milled tape (solid triangle); the shift of the onset to higher fields has to be ascribed
to the enhancement of the pinning.
Figure 8: |c3|(B) curves measured at T =20 K and Bac =4 G for two frequencies (f = 1007 and
5007 Hz). Measurements have been performed on the commercial powders (a), the non-ball-
milled tape (b), the 3h ball-milled tape (c), the 100h ball-milled tape (d); the height of the
modulus peak increases as the frequency increases for all the samples.
Figure 9: Magnetic field dependence of |c3| calculated in the framework of the
Anderson-Kim model for T/Tc= 0.5, Bac =4 G, f =1007 Hz (solid square) and 5007 Hz (open
square); the experimental behaviour in Fig.8 is well reproduced. The parameters used for the
simulations are U0 (B=0,T=0)/ kBTc =750 and Jc (B=0,T=0)× d =5×105 A/m.
3h ball-milled tape
T =20 K
f =1607 Hz
Bac= 4 G
f =1607 Hz
T =15 K
T =20 K
T =25 K
T =30 K
T = 5 K
Sweep Rate = 2 T/min