Microwave surface resistance of pristine and neutronirradiated MgB2 samples in magnetic field
ABSTRACT We report on the microwave surface resistance of two
polycrystalline Mg11 B2 samples; one consists of
pristine material, the other has been irradiated at very high neutron
fluence. It has already been reported that in the strongly irradiated
sample the two gaps merge into a single value. The mw surface resistance
has been measured in the linear regime as a function of the temperature and the DC magnetic
field, at increasing and decreasing fields. The results obtained in the
strongly irradiated sample are quite well justified in the framework of a
generalized Coffey and Clem model, in which we take into account the field
distribution inside the sample due to the critical state. The results
obtained in the pristine sample show several anomalies, especially at low
temperatures, which cannot be justified in the framework of standard
models for the fluxon dynamics. Only at temperatures near Tc and for
magnetic fields greater than 0.5Hc2(T) the experimental data can quantitatively be
accounted for by the Coffey and Clem model, provided that the
uppercriticalfield anisotropy is taken into due account.

Article: Effects of Meissner surface current on the microwave surface resistance of typeII superconductors
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ABSTRACT: Influence of the Meissner current on the fieldinduced variations of the microwave (mw) surface resistance in typeII superconductors is demonstrated to be well reproduced by exploiting different critical state models. Rs(B)/Rn versus Ba curves have been obtained using Kim and exponential type Jc(B) considering Meissner current circulating at the surface of the superconducting specimen.Physica C Superconductivity 07/2010; 470:575581. · 0.72 Impact Factor 
Article: Study of microwave surface resistance of type‐II superconductors carrying transport current
[Show abstract] [Hide abstract]
ABSTRACT: We investigate the microwave surface resistance of an infinitely long typeII superconductor in the critical state carrying transport DC current, in the absence and in the presence of external magnetic fields. Calculations include both different cooling processes and various current and field arrangements. Theoretical results in fieldcooled sample show that the microwave surface resistance exhibits an initial slow variation followed by a faster one above a threshold current value. This occurs when the current induced magnetic field generates fluxons of opposite directions on the two sample edges.physica status solidi (b) 12/2010; 248(6):1477  1482. · 1.49 Impact Factor  SourceAvailable from: M. Monni[Show abstract] [Hide abstract]
ABSTRACT: The magneticfieldinduced variations of the microwave surface resistance, Rs, have been investigated in ceramic Mg1−x(LiAl)xB2, with x in the range 0.1–0.4. The measurements have been performed on increasing and decreasing the DC magnetic field, H0, at fixed temperatures. At low temperatures, we have observed a magnetic hysteresis in the Rs(H0) curves in all the investigated samples. On increasing the temperature, the range of H0 in which the hysteretic behavior is visible shrinks; however, in the sample with x=0.1 it is present up to temperatures close to Tc. We show that the field dependence of Rs can be quantitatively justified taking into account the criticalstate effects on the fluxon lattice only in the sample with x=0.4. On the contrary, in the samples with x<0.4 the hysteresis exhibits an unusual shape, similar to that observed in others twogap MgB2 samples, which cannot be justified in the framework of the criticalstate models.Physica C Superconductivity 09/2009; · 0.72 Impact Factor
Page 1
arXiv:0709.0618v2 [condmat.suprcon] 8 May 2008
EPJ manuscript No.
(will be inserted by the editor)
Microwave surface resistance of pristine and neutronirradiated
MgB2samples in magnetic field
M. Bonura1, A. Agliolo Gallitto1, M. Li Vigni1, C. Ferdeghini2, and C. Tarantini2
1CNISM and Dipartimento di Scienze Fisiche e Astronomiche, Universit` a di Palermo, Via Archirafi 36, I90123 Palermo, Italy
2CNRINFMLAMIA and Dipartimento di Fisica, Universit` a di Genova, Via Dodecaneso 33, I16146 Genova, Italy
the date of receipt and acceptance should be inserted later
Abstract. We report on the microwave surface resistance of two polycrystalline Mg11B2 samples; one
consists of pristine material, the other has been irradiated at very high neutron fluence. It has already
been reported that in the strongly irradiated sample the two gaps merge into a single value. The mw
surface resistance has been measured in the linear regime as a function of the temperature and the DC
magnetic field, at increasing and decreasing fields. The results obtained in the strongly irradiated sample
are quite well justified in the framework of a generalized Coffey and Clem model, in which we take into
account the field distribution inside the sample due to the critical state. The results obtained in the pristine
sample show several anomalies, especially at low temperatures, which cannot be justified in the framework
of standard models for the fluxon dynamics. Only at temperatures near Tc and for magnetic fields greater
than 0.5Hc2(T) the experimental data can quantitatively be accounted for by the Coffey and Clem model,
provided that the uppercriticalfield anisotropy is taken into due account.
PACS. 74.25.Ha Magnetic properties – 74.25.Nf Response to electromagnetic fields (nuclear magnetic
resonance, surface impedance, etc.) – 74.25.Op Mixed states, critical fields, and surface sheaths
1 Introduction
It has widely been shown that the superconducting prop
erties of MgB2are strongly related to the twogap struc
ture of its electronic states [1,2,3,4,5,6]. The smaller su
perconducting gap, ∆π, arises from the quite isotropic π
bands and the largerone, ∆σ, from the strongly anisotropic
σ bands [7]. Due to the different parity of the σ and
π bands, interband scattering of quasiparticles is much
smaller than the intraband one, making the two super
conducting gaps quite different though they close at the
same Tc[8].
According to the theory of multiband superconduc
tivity [7], the inclusion of defects would increase the inter
band scattering and, consequently, change the relative mag
nitude of the different gaps. In order to carry out inves
tigation on this topic, essentially two methods have been
used to insert defects and/or disorder in MgB2: chemical
substitution and damage by irradiation [9,10,11,12,13].
In any cases, the inclusion of defects, besides to change
the interband scattering, might increase the upper crit
ical field and the critical current, strongly affecting the
fluxon dynamics. Very recently, the effects of neutron ir
radiation have extensively been investigated on polycrys
talline Mg11B2 [13,14,15,16,17]. It has been shown that
irradiation leads to an improvement in both critical field
and critical current density for an exposure level in the
range 1 ÷ 2 × 1018cm−2. On further increasing the neu
tron fluence, all the superconducting properties, such as
Tc, Hc2, Jc, are strongly suppressed. Furthermore, mea
surements of specific heat, as well as pointcontact spec
troscopy, have shown that in the sample irradiated at the
highest fluence (1.4×1020cm−2) the two gaps merge into
a single value [15,16].
Despite the large amount of experimental and theo
retical work done on MgB2, some arguments are still un
der discussion, such as the effects of the magnetic field
on the superconducting properties [18,19,20]. The main
difficulty to quantitatively discuss the mixedstate prop
erties of MgB2arises from the unusual fluxline properties
due to the different coherence lengths, ξσ and ξπ, asso
ciated with ∆σ and ∆π [21]. Scanning tunnelling spec
troscopy on MgB2single crystals along the caxis, which
probes mainly the π band, has highlighted a core size much
larger than the estimates based on the measured Hc2val
ues, as well as a significant core overlap at fields much
lower than the macroscopic Hc2[22]. Furthermore, mea
surements of neutron scattering from the vortex lattice
have highlighted a spatial rotation of the vortex lattice for
applied magnetic fields in the range 0.5÷1 T [23]. Accord
ing to pointcontactspectroscopy experiments [24,25,26],
these unusual properties have been ascribed to the strong
suppression of the superconductivity in the π band, oc
curring in that field range. At low fields, each vortex has
a composite structure, with σband quasiparticles local
Page 2
2M. Bonura et al.: Microwave surface resistance in MgB2
ized in a region of radius ξσand πband quasiparticles in
a wider region of radius ξπ. In the field range 0.5÷1 T (at
low T), the giant cores start to overlap; when the magnetic
field is large enough to suppress the πband gap, the vortex
cores shrink and the πband quasiparticles are widespread
in the whole sample [19,22]. This fieldinduced evolution
of the vortex lattice is expected to affect the vortexvortex
and vortexpinning interactions, making the description of
the properties involving the presence and motion of flux
ons very difficult.
The investigation of the microwave(mw) surface imped
ance, Zs= Rs+ iXs, in superconductors is a useful tool
for determining several properties of the superconducting
state. In the absence of static magnetic fields, the variation
with the temperature of the condensedfluid density de
termines the temperature dependence of Zs. On the other
hand, the field dependence of Rs in superconductors in
the mixed state is determined by the presence of fluxons,
which bring along normal fluid in their cores, as well as the
fluxon motion [27,28,29,30,31,32]. So, investigation of the
magneticfieldinduced variations of the surface resistance
provides important information on the fluxon dynamics.
Several studies of the mw response of MgB2reported
in the literature have shown that the experimental results
cannot be accounted for in the framework of standard
theories [33,34,35,36,37,38,39]. The temperature depen
dence of the mw conductivity, at zero DC magnetic field,
has been justified considering the coexistence of two dif
ferent superconducting fluids, one related to carriers living
on the σ band and the other to carriers living on the π
band [33,34]. The magnetic field dependence of the surface
resistance has shown an anomalous behavior especially at
low temperatures; several authors have highlighted un
usually enhanced fieldinduced mw losses at applied mag
netic fields much lower than the upper critical field [35,
36,37,38]. Sarti et al. [39], investigating the mw surface
impedance of MgB2 film, have shown that at low fields,
when the contribution of the πband superfluid cannot be
neglected, the magneticfield dependence of the real and
imaginary components of the surface impedance exhibits
several anomalies. Furthermore, a magnetic hysteresis of
unconventional shape has been detected in the Rs(H)
curves [38,40]. All these results have suggested that in a
wide magneticfield range the standard models for fluxon
dynamics fail when applied to MgB2.
In this paper, we report on the microwave surface resis
tance of two of the polycrystalline Mg11B2samples stud
ied in Refs. [13,14,15,16,17]. We have investigated the
unirradiated sample, which clearly shows twogap super
conductivity, and the sample irradiated at the highest neu
tron fluence, in which the two gaps merge into a single
value. The investigation has been carried out with the
aim to compare the results obtained in twogap and one
gap MgB2 superconductors. To our knowledge, the mw
response of neutronirradiated MgB2samples has not yet
been investigated. The mw surface resistance has been
measured as a function of the temperature, in the range
2.5 ÷ 40 K, and the DC magnetic field, from 0 to 1 T, at
increasing and decreasing values. We show that the results
obtained in the strongly irradiated sample can quite well
be justified in the framework of standard models, in the
whole ranges of temperatures and magnetic fields investi
gated. On the contrary, the results obtained in the pris
tine sample cannot thoroughly be justified. In particular,
the Rs(T) behavior at zero field has been accounted for,
in the framework of the twofluid model, assuming a lin
ear temperature dependence of the normal and condensed
fluid densities. At low temperatures, the field dependence
of Rshas shown several anomalies, among which a mag
netic hysteresis having a unexpected shape. At temper
atures near Tc and applied magnetic fields greater than
≈ 0.5Hc2(T), the results are well accounted for in the
framework of the Coffey and Clem model [30], with flux
ons moving in the fluxflow regime, taking into account
the anisotropy of the upper critical field.
2 Experimental apparatus and samples
The microwave surface resistance, Rs, has been investi
gated in two bulk MgB2samples. The procedure for the
preparation and irradiation of the samples is reported
in detail elsewhere [13,16]. The samples have been pre
pared by direct synthesis from Mg (99.999% purity) and
crystalline isotopically enriched11B (99.95% purity), with
a residual10B concentration lower than 0.5%. The use
of isotopically enriched11B makes the penetration depth
of the thermal neutrons greater than the sample thick
ness, guarantying the irradiation effect almost homoge
neous over the sample. Several superconducting properties
of the samples have been reported in Refs. [13,14,15,16,
17]. For simplicity and ease of comparison, we label the
two samples as in Ref. [13], i.e. P0 (pristine Mg11B2) and
P6 (irradiated at the highest neutron fluence). According
to pointcontact spectroscopy [15] and specificheat mea
surements [16], sample P0 shows a clear twogap super
conductivity; in sample P6 the irradiation process at very
high fluence (1.4 × 1020cm−2) determined a merging of
the two gaps into a single value.
Sample P0 has a nearly parallelepiped shape with w ≈
3.1 mm, t ≈ 1.5 mm and h ≈ 3.2 mm; it undergoes a nar
row superconducting transition with Tonset
∆Tc≈ 0.2 K (from 90% to 10% of the normalstate resis
tivity); its residual normalstate resistivity is ρ(40 K) ≈
1.6 µΩ cm and the residual resistivity ratio RRR ≈ 11,
the critical current density at zero magnetic field is Jc0≈
4 × 105A/cm2, and µ0Hc2(5 K) ≈ 15 T; the anisotropy
factor of the upper critical field at T = 5 K is γ ≈ 4.4 [13,
14].
Sample P6 has a nearly parallelepiped shape with w ≈
1.1 mm, t ≈ 0.8 mm and h ≈ 1.4 mm. The main char
acteristic parameters of sample P6 are: Tonset
∆Tc≈ 0.3 K, RRR ≈ 1.1, ρ(40 K) ≈ 130 µΩ cm. The crit
ical current density at T = 5 K and at zero magnetic field
is Jc0≈ 3 × 104A/cm2; it exhibits a monotonic decrease
with the magnetic field, following roughly an exponential
law. The upper critical field is isotropic and its value at
T = 5 K is µ0Hc2≈ 2 T.
c
≈ 39.0 K and
c
≈ 9.1 K,
Page 3
M. Bonura et al.: Microwave surface resistance in MgB2
3
The effects of the neutron irradiation on both super
conducting and normalstate properties of a large series
of Mg11B2 bulk samples, including sample P0 and P6,
have extensively been investigated in Refs. [13,14,17]. On
increasing the neutron fluence, it has been observed a
monotonic decrease of Tcand an increase of the residual
normalstate resistivity ρ(Tc). Nevertheless, it has been
shown that the irradiation does not affect the variation
of the normalstate resistivity ∆ρ = ρ(300 K) − ρ(Tc).
As suggested by Rowell [41], just ∆ρ is a parameter that
gives information on the grain connectivity. The results
reported in Ref. [13] show that ∆ρ remains of the order
of 10 µΩ cm over the whole range of irradiation level; in
particular, in sample P0 ∆ρ = 16 µΩ cm and in sample
P6 ∆ρ = 12 µΩ cm, indicating that thermalneutron irra
diation does not affect the grainboundary properties. On
the other hand, it has been shown that either neutron ir
radiation or Heion irradiation [11] do not affect the grain
connectivity, even at high irradiation levels, contrary to
what occurs using heavyion irradiation [12]. Recent stud
ies by transmission electron microscopy have highlighted
that neutron irradiation in these samples creates nano
metric amorphous regions (mean diameter ∼ 4 nm) in
the crystal lattice, whose density scales with the neutron
dose [17]. Studies on the field dependence of the critical
current density have shown that at moderate neutron
fluence levels (≤ 1019cm−2) such defects introduce new
pinning centers, leading to an improvement of the criti
cal current density; on the contrary, for neutron fluence
higher than 1019cm−2(as for sample P6) these nanomet
ric defects do not act as pinning centers because they are
smaller than the coherence length [17]. Moreover, these
studies have shown that in the pristine and the heavily ir
radiated samples the pinning mechanism is ruled by grain
boundaries. The defects induced by neutron irradiation
act as inter and intraband scattering centers; the intra
band scattering causes a reduction of the electron mean
free path and is responsible for the growth of the normal
state resistivity. The reduction of Tc has been ascribed
to both the scattering processes and the smearing of the
electron density of states near the Fermi surface [11,13].
Although in the two samples ∆Tcis roughly the same,
in sample P6, due to the reduced Tc value, ∆Tc/Tc ≈
0.03, affecting noticeably the temperature dependence of
the mw surface resistance near Tc. On the other hand,
from AC susceptibility measurements at 100 kHz, we have
found that the first derivative of the real part of the AC
susceptibility can be described by a Gaussian distribution
function of Tc, centered at Tc0= 8.5 ± 0.1 K with σTc=
0.2±0.05 K. In the following, we will use this distribution
function to quantitatively discuss the results obtained in
sample P6. On the contrary, for sample P0, ∆Tc/Tcis one
order of magnitude smaller, not noticeably affecting the
Rs(T) curve.
The mw surface resistance has been measured using
the cavityperturbation technique [27]. A copper cavity,
of cylindrical shape with goldenplated walls, is tuned in
the TE011mode resonating at ω/2π ≈ 9.6 GHz. The sam
ple is located in the center of the cavity, by a sapphire
rod, where the mw magnetic field is maximum. The cav
ity is placed between the poles of an electromagnet which
generates DC magnetic fields up to µ0H0≈ 1 T. Two ad
ditional coils, independently fed, allow compensating the
residual field and working at low magnetic fields. A liquid
helium cryostat and a temperature controller allow work
ing either at fixed temperatures or at temperature varying
with a constant rate. The sample and the field geometries
are shown in Fig. 1a; the DC magnetic field is perpendic
ular to the mw magnetic field, Hω. When the sample is in
the mixed state, the induced mw current causes a tilt mo
tion of the whole vortex lattice [31]; Fig. 1b schematically
shows the motion of a flux line.
?ω
Hω
H0
FL
FL
(a) (b)
t
h
w
Fig. 1. (a) Field and current geometry at the sample surface.
(b) Schematic representation of the motion of a flux line.
The surface resistance of the sample is given by
?
where QLis the quality factor of the cavity loaded with the
sample, QU that of the empty cavity and Γ the geometry
factor of the sample.
The quality factor of the cavity has been measured by an
hp8719D Network Analyzer. The surface resistance has
been measured as a function of the temperature, at fixed
values of the DC magnetic field, and as a function of the
field, at fixed temperatures. All the measurements have
been performed at very low input power; the estimated
amplitude of the mw magnetic field in the region in which
the sample is located is of the order of 0.1 µT.
Rs= Γ
1
QL
−
1
QU
?
,
3 Experimental results
Figure 2 shows the temperature dependence of the sur
face resistance in the pristine (a) and irradiated (b) MgB2
samples, at different values of the DC magnetic field. In
order to disregard the geometry factor, and compare the
results in samples of different dimensions, we have normal
ized the data to the value of the surface resistance in the
normal state, Rn, at T = Tonset
c
obtained according to the following procedure: the sam
ple was zerofield cooled (ZFC) down to low temperature,
. The results have been
Page 4
4M. Bonura et al.: Microwave surface resistance in MgB2
then H0was set at a given value and kept constant during
the time the measurement has been performed.
On increasing H0, the Rs(T) curves broaden and shift
towards lower temperatures; however, the effects of the
applied magnetic field is different in the two samples. Al
though the value of Hc2 of sample P0 at low tempera
tures is one order of magnitude larger than that of P6,
the fieldinduced variations of Rsin the two samples have
roughly the same magnitude. In sample P0, one can ob
serve an anomalously enhanced fieldinduced broadening
of the Rs(T) curve, which extends down to the lowest tem
perature. On the contrary, in sample P6 the larger shift
of the Rs(T) curve induced by H0is expected because of
the lower Hc2value.
?
??
??
??
?
?
?
?
??
??
??
?
?
?
Fig. 2. Normalized values of the surface resistance as a func
tion of the temperature, obtained in the two samples, at dif
ferent values of the DC field. Rn is the surface resistance at
T = Tc.
The fieldinduced variations of Rs have been investi
gated for different values of the temperature. For each
measurement, the sample was ZFC down to the desired
temperature; the DC magnetic field was increased up to
a certain value and, successively, decreased down to zero.
Figures 3, 4 and 5 show the fieldinduced variations of Rs
for the two samples, at different temperatures. In all the
figures, ∆Rs(H0) ≡ Rs(H0,T) − Rres, where Rresis the
residual mw surface resistance at T = 2.5 K and H0= 0;
moreover, the data are normalized to the maximum varia
tion, ∆Rmax
s
≡ Rn− Rres. The continuous lines reported
in the figures are the bestfit curves obtained by the model
reported in Sec. 4.
In both samples, Rs does not show any variation as
long as the magnetic field reaches a certain value, depend
ing on T, that identifies the firstpenetration field, Hp. For
H0> Hp, vortices start to penetrate the sample and, con
sequently, Rsincreases.
Figure 3 refers to the results obtained at T = 4.2 K.
At this temperature, in both samples the Rs(H0) curves
exhibit a magnetic hysteresis, which disappears for H0
higher than a certain value, indicated in the figure as H′.
The inset in panel (b) shows a minor hysteresis loop ob
tained by sweeping H0 from 0 to 0.25 T and back. The
fieldinduced variations of Rs in sample P0 show some
anomalies. Firstly, the application of a magnetic field of
≈ 1 T, which is about Hc2/15, causes a Rs variation of
≈ 35% of the maximum variation. These fieldinduced
variations of Rs are much greater than those expected
from the models reported in the literature [30,31,32] and
detected in other superconductors [28,29]. A comparison
with the results of panel (b) shows that in sample P6 a Rs
variation of the same order is obtained for the same value
of H0, even though, in this case, 1 T is about Hc2/2. Re
sults similar to those obtained in P0 have been observed in
other MgB2samples, produced by different methods and,
therefore, seems to be a peculiarity of MgB2 [35,37,38,
40]. The finding that in sample P6 we have not observed
this anomalous result strongly suggests that the enhanced
Rsvariation is due to the twogap superconductivity.
A magnetic hysteresis of Rsis expected in supercon
ducting samples in the critical state; it is ascribable to the
different magnetic induction at increasing and decreasing
DC fields [42]. Most likely, the different amplitude of the
hysteresis loop obtained in the two samples is due to the
different values of the critical current density; a smaller
hysteresis is observed in sample P6 because of the smaller
Jcvalue. However, as it is visible in Fig. 3, also the shape
of the hysteresis loop is different in the two samples. The
decreasingfield branch of the Rs(H0) curve in sample P6
has a negative concavity down to Hp, as expected [42].
On the contrary, in sample P0 one can observe a plateau,
in the field range 0 ÷ 0.2 T, which cannot be justified in
the framework of the criticalstate models, considering the
measured field dependence of Jc[13]. We would like to re
mark that this result has been obtained in all of the MgB2
samples we have investigated [38,40].
Figure 4 shows the fieldinduced variations of Rs, for
sample P0 (a), at T = 30 K, and for sample P6 (b), at T =
7 K; for both samples, T/Tc≈ 0.77. In the Rs(H0) curve
of sample P0 the hysteresis is still present, probably due
to the high value of Jcat this temperature, and still has an
anomalous shape. We remark that in sample P0 we have
observed magnetic hysteresis of Rs up to T/Tc ≈ 0.95,
while in sample P6 the hysteresis becomes undetectable
at T/Tc? 0.55.
Figure 5 shows the fieldinduced variations of Rs at
temperatures near Tc, where in both samples the Rs(H0)
Page 5
M. Bonura et al.: Microwave surface resistance in MgB2
5
?
?
?
?
?
?
?
?
?
?
?
Fig. 3. Fieldinduced variations of Rs for samples P0 (a) and
P6 (b), at T = 4.2 K. ∆Rs(H0) ≡ Rs(H0,T)−Rres, where Rres
is the residual mw surface resistance at T = 2.5 K and H0 = 0;
∆Rmax
s
≡ Rn− Rres. The line is the bestfit curve obtained,
as explained in Sec. 5, with µ0Hc2 = 1.71 T, ω0/ω = 0.67 and
the field dependence of the critical current density reported in
Ref. [13]. The inset shows a minor hysteresis loop obtained by
sweeping H0 from 0 to 0.25 T and back.
curve is reversible. As will be shown in Sec. 5, at tem
peratures near Tcthe field dependence of the mw surface
resistance can be accounted for by standard models also
for sample P0, provided that the anisotropy of the upper
critical field is taken into account.
From isothermal Rs(H0) curves, obtained at different
temperatures, we have deduced the temperature depen
dence of the characteristic fields, Hp, Hc2and H′. In Fig. 6
we report the values of Hp, Hc2and H′as a function of
the reduced temperature, T/Tonset
The inset in panel (b) shows Hc2(T) of sample P6 in an
enlarged scale.
From Fig. 6a, one can see that Hp of sample P0 ex
hibits a linear temperature dependence down to low tem
peratures, consistently with results reported by different
authors in bulk [43,44] and crystalline [45,46] MgB2sam
ples. The extrapolated value at T = 0 is about 55 mT;
so, considering the demagnetization effect, the estimated
value of the lower critical field is Hc1(0) ≈ 70 mT. This
value, although consistent with the lower critical field re
ported for MgB2crystals by some authors [46], is slightly
c
, for the two samples.
?
?
?
?
?
?
?
?
?
Fig. 4. Normalized fieldinduced variations of Rs for samples
P0 (a) and P6 (b), at T/Tc ≈ 0.77. The line in panel (b) is the
bestfit curve of the data, obtained as described in the text by
using the field dependence of the depinning frequency reported
in Fig. 9.
larger than that reported for bulk samples, which ranges
from 15 to 45 mT. We suggest that this is ascribable to
weak surfacebarrier effects.
In sample P6 we have obtained Hpvalues smaller, but
of the same order, than those of sample P0; this may
be due to the irradiation effects. Indeed, the authors of
Ref. [18], from magnetization measurements in neutron
irradiated MgB2 crystals, have observed that the lower
critical field reduces monotonically on increasing the flu
ence.
The values of Hc2(T) indicated in Fig. 6b as open tri
angles have been deduced measuring the magnetic field at
which Rsreaches the normal state value, Rn. At the tem
peratures in which the upper critical field of sample P6 is
higher than the maximum magnetic field achievable with
our experimental apparatus (≈ 1 T), the Hc2(T) values
(full triangles in the figure) have been obtained as best
fit parameters, using the model reported in Sec. 4. On the
contrary,since the results obtained in sample P0 cannot be
accounted for by the model (except at temperature close
to Tc), we report only the values we have directly deduced
from the experimental results. For T ≥ 5 K, the values we
obtained for Hc2(T), in both samples, agree with those re
ported in Ref. [13] (the authors do not report Hc2at lower
Page 6
6M. Bonura et al.: Microwave surface resistance in MgB2
?
?
?
?
?
?
?
?
Fig. 5. Normalized fieldinduced variations of Rs for samples
P0 (a) and P6 (b), at temperatures close to Tc. Lines are best
fit curves, obtained by the model of Sec. 4 considering fluxons
move in the fluxflow regime. The line of panel (a) has been
obtained taking into account the anisotropy of the upper criti
cal field, as described in the text, with γ = 3.3; for sample P0,
we have set γ = 1 consistently with the results of Ref. [14].
temperatures). Our results give complementary informa
tion about the temperature dependence of Hc2of sample
P6 at low temperatures. The continuous line in Fig. 6b
has been obtained by fitting the data of sample P6 with
Hc2(T) = Hc20[1−(T/Tc)α]; we have obtained, as bestfit
parameters, µ0Hc20 = (2.2 ± 0.2) T, α = 1.9 ± 0.3 and
Tc= (8.9 ± 0.2) K. This temperature dependence of the
upper critical field is consistent with that expected in con
ventional superconductors. On the contrary, for sample P0
we observed an upward curvature of Hc2(T), clearly visible
in the inset, characteristic of twogap MgB2materials [1,
4,5].
H′(T) of Fig. 6c corresponds to the value of the DC
magnetic field at which the decreasingfield branch of the
Rs(H0) curves deviates from the increasingfield branch.
The zero values (without error bar) mean that the hystere
sis is not detectable at the corresponding temperatures.
Consistently with the lower value of the critical current
density, H′(T) is smaller in sample P6 than in P0. We
would like to remark that the values of H′(T) could differ
from the irreversibility field deduced from magnetization
measurements. Indeed, it has been shown that, in sam
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Fig. 6. Temperature dependence of the characteristic fields,
Hp, Hc2 and H′, for the two samples. In panel (b): the inset
shows the Hc2(T) values of sample P6 in an enlarged scale; the
continuous line is the bestfit curve of the experimental data,
obtained for sample P6, as described in the text.
ples of finite dimensions, the application of an AC mag
netic field normal to the DC field can induce the fluxon
lattice to relax toward an uniform distribution [47]. Fur
thermore, measurements we have performed in different
superconducting samples have pointed out that, for sam
ples of millimetric size, the sensitivity of our experimental
apparatus does not allow resolving magnetic hysteresis of
Rs when Jc < 104A/cm2. By considering the values of
Jcreported in Ref. [13], at T = 4.2 K we should obtain
µ0H′≈ 0.2 T for sample P6 and µ0H′≈ 3.5 T for sam
Page 7
M. Bonura et al.: Microwave surface resistance in MgB2
7
ple P0. From Fig. 6c, one can see that this expectation is
verified in sample P6; on the contrary, the value of H′in
sample P0 is about one order of magnitude smaller than
the expected one.
4 The model
Microwave losses induced by static magnetic fields have
been investigated by several authors [27,28,29,30,31,32,
42,48]. At low temperatures and for applied magnetic fields
lower enough than the upper critical field, the main con
tribution arises from the fluxon motion; however, it has
been pointed out that a noticeable contribution can arise
from the presence of normal fluid, especially at temper
atures near Tcand for magnetic fields of the same order
of Hc2(T). The majority of the models assume an uni
form distribution of fluxons inside the sample; so, they
disregard the effects of the critical state. Very recently,
we have investigated the fieldinduced variations of the
mw surface resistance in superconductors in the critical
state [42,48], and have accounted for the magnetic hys
teresis in the Rs(H0) curves.
In the London local limit, the surface resistance is pro
portional to the imaginary part of the complex penetra
tion depth,?λ, of the em field:
Rs= −µ0ω Im[?λ(ω,B,T)].
different approximations [30,31]. Coffey and Clem (CC)
have elaborated a comprehensive theory for the electro
magnetic response of superconductors in the mixed state,
in the frameworkof the twofluid model of superconductiv
ity [30]. The theory has been developed under two basic
assumptions: i) intervortex spacing much less than the
field penetration depth; ii) uniform vortex distribution in
the sample. With these assumptions vortices generate a
magnetic induction field, B, uniform in the sample. This
approximation is valid for H0> 2Hc1whenever the fluxon
distribution can be considered uniform within the AC pen
etration depth.
In the linear approximation, Hω ≪
expected from the CC model is given by
?
1 − 2iλ2(B,T)/δ2
(1)
The complex penetration depth has been calculated in
H0,?λ(ω,B,T)
v(ω,B,T)
nf(ω,B,T),
?λ(ω,B,T) =
λ2(B,T) + (i/2)?δ2
(2)
with
λ(B,T) =
λ0
?[1 − w0(T)][1 − B/Bc2(T)],
δ0(ω)
?1 − [1 − w0(T)][1 − B/Bc2(T)],
(3)
δnf(ω,B,T) =
(4)
where λ0 is the London penetration depth at T = 0, δ0
is the normalfluid skin depth at T = Tc, w0(T) is the
fraction of normal electrons at H0= 0; in the Gorter and
Casimir twofluid model w0(T) = (T/Tc)4.
?δvis the effective complex skin depth arising from the vor
viscous and restoringpinning forces, which identities the
depinning frequency ω0.?δvcan be written as
?δ2
δ2
µ0ωη,
with η the viscousdrag coefficient and φ0the quantum of
flux.
When the frequency of the em wave, ω, is much lower than
ω0, the fluxon motion is ruled by the restoringpinning
force. On the contrary, for ω ≫ ω0, the fluxon motion
takes place around the minimum of the pinningpotential
well and, consequently, the restoringpinning force is nearly
ineffective. So, the contribution of the viscousdrag force
predominates and the induced em current makes fluxons
move in the fluxflow regime. In this case, enhanced field
induced energy losses are expected.
As it is clear from Eqs. (1–4), it is expected that the
features of the Rs(H0) curves strongly depend on the
appliedfield dependence of B. On the other hand, the CC
theory is strictly valid when B is uniform inside the sam
ple; in particular for H0≫ Hc1, the Rs(H0) curves can be
described setting B = µ0H0. When fluxons are in the crit
ical state, the assumption of uniform B is no longer valid
and the CC theory does not correctly describe the field
induced variations of Rs. As a consequence, the hysteresis
in the Rs(H0) curve cannot be justified by Eqs. (1–4). In
our field geometry (see Fig. 1a), the effects of the non
uniform B distribution on Rs are particularly enhanced
because in the two surfaces of the sample normal to the
external magnetic field the mw current and fields pen
etrate along the fluxon axis and, consequently, the mw
losses involve the whole vortex lattice. However, in this
case, one can easily take into account the nonuniform B
distribution by calculating a proper averaged value of Rs
over the whole sample as follows [42,48]
?
where Σ is the sample surface, S is its area and r identi
fies the surface element.
The pinning effects are particularly enhanced at temper
atures smaller enough than Tc, where the dissipations are
essentially due to vortex motion. So, the main contribu
tion to Rscomes from the sample regions in which fluxons
experience the Lorentz force due to the mw current, i.e.
where H0×Jω?= 0. Furthermore, in order to take into due
account the criticalstate effects by Eq. (7), it is essential
to know the B profile inside the sample, determined by
Jc(B).
Recently, using this method, we have investigated the
effects of the critical state on the fieldinduced variation
of Rs, at increasing and decreasing fields [42,48]. We have
shown that the parameter that mainly determines the pe
culiarities of the Rs(H0) curve is the full penetration field,
tex motion; it depends on the relative magnitude of the
1
v
=
1
δ2
f
?
1 + iω0
ω
?
,(5)
where
f=2Bφ0
(6)
Rs=1
S
Σ
Rs(B(r))dS ,(7)
Page 8
8M. Bonura et al.: Microwave surface resistance in MgB2
H∗. Firstly, the width of the hysteresis is directly related
to the value of H∗; samples of small size and/or small Jc
are expected to exhibit weak hysteretic behavior. Further
more, H∗determines the shape of the hysteresis loop as
well. On increasing the external field from zero up to H∗,
more and more sample regions contribute to the mw losses;
this gives rise to a positive curvature of the increasingfield
branch of the Rs(H0) curve. For H0> H∗, in the whole
sample the local magnetic induction depends about lin
early on the external field and the increasingfield branch
is expected to have a negative concavity. The shape of the
decreasingfield branch is strictly related to the shape of
the magnetization curve; it should exhibit a negative con
cavity, with a monotonic reduction of Rsin the whole field
range swept.
5 Discussion
As we have shown in Sec. 3, the Rs(H0,T) curves exhibit
different peculiarities in the unirradiated sample (P0) and
the strongly irradiated sample (P6). The model described
in Sec. 4 fully justifies the experimental results obtained
in sample P6, which exhibits a singlegap superconductiv
ity [15,16]. On the contrary, the results obtained in sample
P0 cannot be justified in the framework of the same model,
either for the magneticfield dependence or for the tem
perature dependence of the surface resistance, even at zero
DC field. Only the results obtained at temperatures close
to Tccan be justified, provided that the anisotropy of the
upper critical field is taken into due account. In the fol
lowing, firstly we will discuss the temperature dependence
of the mw surface resistance in the absence of DC mag
netic fields; successively, we will discuss the fieldinduced
variations of Rs.
5.1 Temperature dependence of Rsin zero magnetic
field
Figure 7 shows the normalized values of the surface resis
tance at H0= 0 as a function of the reduced temperature
for both samples. The Rs(T) curve of sample P6 shows
a wide transition, broadened in a roughly symmetric way
with respect to the middle point at Rs/Rn = 0.5. This
behavior can be ascribed to the Tc distribution over the
sample. On the contrary, in sample P0 one can notice a
sharp variation of Rs(T), at temperatures near Tc, and a
wide tail, extending from T/Tc≈ 0.9 down to T/Tc≈ 0.7,
which cannot be ascribed to the Tcdistribution. The lines
in the figure are bestfit curves; they have been obtained
with different procedures for the two samples.
The results obtained from the model discussed in Sec. 4
setting B = 0 in Eqs. (1–4) converge to those of the em
response of superconductors in the Meissner state, in the
framework of the twofluid model. In this case, the tem
perature dependence of Rs/Rnis determined, apart from
the Tc distribution over the sample, by the temperature
dependence of the normalfluid density, w0(T), and the ra
tio λ0/δ0. In order to fit the experimental data obtained
?
?
?
?
?
?
?
Fig. 7. Normalized values of the mw surface resistance as a
function of the reduced temperature, obtained in the two sam
ples at H0 = 0. Symbols are experimental data; lines are the
bestfit curves obtained as described in the text.
in sample P6, we have assumed w0(T) = (T/Tc)4, consis
tently with the Gorter and Casimir twofluid model, and
have used Eqs. (1–4) with B = 0. We have averaged the
expected curve over a gaussian distribution function of Tc
with Tc0= 8.5 K and σ = 0.2 K (see Sec. 2) and have used
λ0/δ0as fitting parameter. The bestfit curve, dashed line
in Fig. 7, has been obtained with λ0/δ0= 0.14; however,
we have found that the expected curve is little sensitive
to variations of λ0/δ0, except at low temperatures, where
the measured Rs is limited by the sensitivity of our ex
perimental apparatus. In particular, by varying Tc0 and
σ within the experimental uncertainty, good agreement is
obtained with λ0/δ0values ranging from 0.04 to 0.15. This
occurs because the Tc distribution broadens the Rs(T)
curve, hiding the λ0/δ0effects.
Unlike for sample P6, the results of Fig. 7 obtained
in P0 cannot be justified in the framework of the Gorter
and Casimir twofluid model, using reasonable values of
λ0/δ0. On the other hand, different authors [2,6,49] have
shown that the temperature dependence of the field pen
etration depth in MgB2 cannot be accounted for by ei
ther the Gorter and Casimir twofluid model or the stan
dard BCS theory. A linear temperature dependence of the
condensed fluid density, in a wide range of temperatures
below Tc, has been reported, which has been justified in
the framework of twogap models for the MgB2supercon
ductor [6,49]. Prompted by these considerations, we have
hypothesized a linear temperature dependence of w0. The
continuous line in Fig. 7 is the bestfit curve; it has been
obtained by Eqs. (1–4) with B = 0, w0(T) = T/Tc and
λ0/δ0 = 0.15. The wide lowT tail is essentially deter
mined by the linear temperature dependence of w0. The
sensitivity achievable by our experimental apparatus does
not allow determining the small variations of Rs(T) for
T/Tc? 0.5; so, from Rs(T) measurements no indication
about the temperature dependence of the densities of the
normal and condensed fluids at low temperatures can be
Page 9
M. Bonura et al.: Microwave surface resistance in MgB2
9
obtained. However, the linear temperature dependence of
the lower critical field we obtained (see Fig. 6a) strongly
suggests that w0linearly depends on T down to low tem
peratures.
5.2 Field dependence of Rsin sample P6
In conventional (singlegap) superconductors, it is expected
that the field dependence of the mw surface impedance is
described by the model reported in Sec. 4. In this frame
work, in order to calculate the expected fieldinduced vari
ations, by Eqs. (1–4), the essential parameters are the
value of λ0/δ0, Hc2(T), the depinning frequency, ω0, and
its field dependence. It is not necessary to consider the
uppercriticalfield anisotropy, γ, because it has been shown
that in the P6 sample γ = 1 [14]. When the criticalstate
effects cannot be neglected, in order to use Eq. (7), it is
also essential to know the profile of the induction field
determined by the field dependence of the critical current
density. The value of λ0/δ0has been determined by fitting
the Rs(T) curve at H0= 0; the critical current density and
its field dependence are reported in Ref. [13]. The values
of Hc2(T) at T ≥ 5 K are reported in Ref. [13], and/or
deduced from our experimental data; at T < 5 K, Hc2has
to be considered as parameter. It is worth noting that the
large uncertainty of λ0/δ0, we obtained for this sample,
does not affect the bestfit curves because this parameter
essentially determines the normalized ∆Rs(H0) value at
B = 0.
As one can see from Fig. 3, at T = 4.2 K the Rs(H0)
curve exhibits a magnetic hysteresis, indicating that the
effects of the critical state are not negligible. In order to
use Eq. (7), we have calculated the B profile in the sample
using the field dependence of the critical current, Jc(B),
reported in Ref. [13] and we have set the induction field at
the edges of the sample as B = µ0(H0−Hp); we have taken
Hc2and ω0as fitting parameters. The line of Fig. 3b is the
bestfit curve; it has been obtained with µ0Hc2= 1.71 T
and ω0/ω = 0.67 independent of H0. For the sake of clar
ity, in Fig. 8 we report the results obtained by sweeping the
DC magnetic field from 0 to 0.25 T and back, along with
the expected curve. The inset shows the B profile along the
width of the sample at half height, determined by Jc(B);
the continuous lines are the increasingfield profiles, the
dashed ones are the decreasingfield profiles at the same
externalfield values. As one can see, taking into account
the field distribution inside the sample, the experimen
tal results are quite well justified in the framework of the
model discussed in Sec. 4. In the increasingfield branch, a
change of concavity is well visible at µ0(H0−Hp) ≈ 0.04 T,
consistently with the expected value of the full penetra
tion field; the decreasingfield branch exhibits a negative
concavity in the whole range of fields.
Following the procedure above described, from the best
fit of the experimental data of the isothermal Rs(H0)
curves at T < 6 K we have obtained the Hc2(T) values
indicated as full triangles in Fig. 6b.
When the Rs(H0) curves do not show hysteresis, the
effects of the critical state are negligible and the induction
B (T)
0.0
0.1
0.2
0
w/2
0.0 0.10.2
0.0
0.1
0.2
0H0
0Hp(T)
∆Rs(H0)/∆Rmax
s
SampleP6
T=4.2K
Fig. 8. Fieldinduced variations of the mw resistance, obtained
in sample P6 by sweeping the magnetic field from 0 to 0.25 T
and back. The line is the bestfit curve obtained, as explained
in the text, with µ0Hc2 = 1.71 T, ω0/ω = 0.67 and the field
dependence of the critical current density reported in Ref. [13].
The inset shows the B profile at increasing (—) and decreasing
( ) fields; w is the width of the sample.
field, B, can be considered uniform. In this case, we have
used the following approximate expression for the magne
tization:
M = −Hp+
Hp
Hc2− Hp(H0− Hp);
and, consequently
B = µ0
?
1 +
Hp
Hc2− Hp
?
(H0− Hp).
Several calculations have shown that, in order to fit the
experimental data at T ≥ 7 K, it is essential to consider
the Tc distribution over the sample. So, we have aver
aged the expected Rs(H0) curves [calculated by Eqs. (1–
4)] over the Tcdistribution (see Sec. 2). We have used for
Hp(T) and Hc2(T) the values deduced from the experi
mental data, letting them vary within the experimental
uncertainty, and have considered the depinning frequency
as parameter. The lines of Figs. 4 and 5 have been ob
tained by this procedure.
By fitting the results at T = 7 K, we have obtained the
field dependence of ω0/ω reported in Fig. 9. The roughly
constant value of ω0/ω we obtained up to µ0H0≈ 0.3 T
indicates that in this field range individual vortex pinning
occurs; on further increasing the magnetic field, the in
teraction between fluxons becomes important, collective
vortex pinning sets in and, consequently, the depinning
frequency decreases. The data obtained at µ0H0? 0.65 T
are well fitted setting ω0/ω = 0 in Eq. (5); this means
that, at T = 7 K and µ0H0 ? 0.65 T, the induced mw
current makes fluxons move in the fluxflow regime.
The bestfit curve of Fig. 5 has been obtained with
ω0/ω = 0, as expected. Indeed, at temperature very close
Page 10
10M. Bonura et al.: Microwave surface resistance in MgB2
?
?
?
??
Fig. 9. Magnetic field dependence of the depinning frequency,
obtained for sample P6 by fitting the experimental results re
ported in Fig. 4b.
to Tc, the pinning effects are weak and the induced mw
current makes fluxons move in the fluxflow regime.
5.3 Field dependence of Rsin sample P0
It has been shown by several authors that the proper
ties of the twogap MgB2 superconductor in the mixed
state cannot be accounted for by standard theories [1,3,
18,19,20,35,36,39]. It is by now accepted that this is re
lated to the doublegap nature of MgB2that is responsi
ble for an unusual vortex structure. Indeed, it has been
highlighted, both experimentally and theoretically, that
the vortex cores are characterized by two different spatial
and magneticfield scales [21,22]. Because of the different
magneticfield dependence of the two gaps, on varying the
field, the structure of the vortex lattice changes in an un
usual way. At low magnetic fields, quasiparticles by π and
σ bands are trapped within the vortex core, even if on
different spatial scales because of the different coherence
lengths ξπand ξσ; in the field range 0.5 ÷ 1 T, though σ
band quasiparticles remain localized, the πband quasipar
ticles spread over the sample [22]; on further increasing the
field, ∆πis strongly reduced, the πquasiparticle contribu
tion remains almost unchanged while the σquasiparticle
contribution continues to increase with about the same
rate up to the macroscopic Hc2. So, a further charac
teristic field is needed for determining the fluxonlattice
properties of MgB2; the existence of this crossover field,
often indicated as Hπ
c2, has been highlighted in several
experiments [1,3,22,23,24,25]. The fieldinduced evolu
tion of the vortex lattice is expected to affect the vortex
vortex and vortexpinning interactions, making the stan
dard models most likely inadequate to describe the fluxon
dynamics.
Results on the fieldinduced variations of Rsin MgB2
have been reported by some authors [35,36,37,38,39,40,
50]; most of them have highlighted several anomalies, which
cannot be explained in the framework of standard mod
els for fluxon dynamics. In particular, it has been high
lighted unusually enhanced fieldinduced mw losses at ap
plied magnetic field much lower than Hc2. Only Zaitsev et
al. [50] have explained the frequency and field dependence
of the mw surface resistance of MgB2films in the frame
work of standard models. The results we have obtained
in sample P0 are similar to those reported by Shibata et
al. [35], who investigated the field dependence of the sur
face impedance in MgB2single crystal in a wide range of
DC magnetic fields (up to 14 T). At low temperatures,
the authors have observed an initial fast variation of the
fieldinduced mw dissipation up to fields of the order of
1 T, followed by a slower one at higher fields. Consis
tently with the sharp fieldinduced variation of the heat
capacity and thermal conductivity, the enhanced lowfield
variation of the mw losses has been ascribed to the high
increase of π quasiparticles in the vortex cores. At higher
fields, the variation is slower because of the saturation of
the πquasiparticle contribution.
According to Shibata et al., the enhanced fieldinduced
variation we observed in sample P0 can be qualitatively
ascribed to the strong reduction of ∆πin the field range we
have investigated. However, the observed Rs(H0) curves
differ from the expected ones in both the intensity and the
shape. Here, we discuss the shape of the Rs(H0) curves.
In a wide range of temperatures below Tc, we have ob
served a magnetic hysteresis, which should be related to
the different magnetic induction at increasing and decreas
ing fields, due to the critical state. As discussed in Sec. 4,
the increasingfield branch of the Rs(H0) curve should
exhibit a change of concavity, from positive to negative,
when the external magnetic field reaches the full pene
tration field, H∗. By considering the sample width and
the value of Jcat T = 5 K reported for sample P0 [13],
the expected value of H∗is ≈ 2.6 T. Nevertheless, we
observe a negative concavity of the Rs(H0) increasing
field branch in the whole range of fields investigated (see
Fig. 3a), even if the maximum value of the applied field is
well below H∗. The decreasingfield branch should show
a monotonic reduction of Rs down to low fields. In con
trast, for H0< H′, we observe an initial weak reduction
followed by a plateau, from µ0H0≈ 0.2 T down to zero.
The presence of this plateau is puzzling because it would
suggest that the trapped flux does not change anymore on
decreasing the field below ∼ 0.2 T, although this value is
four times larger that Hp.
Another anomalous result concerns the range of mag
netic fields in which we observe the hysteretic behavior.
As we have already mentioned, we have experienced that
for samples of millimetric size the sensitivity of our exper
imental apparatus allows detecting hysteresis in Rs(H0)
for Jc? 104A/cm2. From Fig. 9 of Ref. [13], one can de
duce that, in sample P0, such condition occurs at µ0H0∼
4 T; so, we should detect hysteresis in the whole range of
fields we have investigated. On the contrary, we obtained
H′(4.2 K) ∼ 0.5 T, one order of magnitude lower than the
expected value.
Page 11
M. Bonura et al.: Microwave surface resistance in MgB2
11
We would like to remark that these anomalies have
been observed in all the bulk MgB2samples (unirradiated)
we have investigated, no matter the preparation method
and the components (11B or10B) used in the synthesis
process [38,40]. The finding that in sample P6 the ex
perimental results are fully justified by the used model,
strongly suggests that these anomalies are strictly related
to the presence of the two superconducting gaps.
At temperatures close to Tcand for H0? 0.5Hc2(T),
the experimental results can be accounted for by the model
discussed in Sec. 4, provided that the anisotropy of the
upper critical field is taken into due account. Following
Ref. [14], to take into account the anisotropy, we have as
sumed that the polycrystalline sample is constituted by
grains with the caxis randomly oriented with respect to
the DCmagneticfield direction; so, the distribution of
their orientations follows a sin(θ) law, being θ the angle
between H0and ˆ c. Furthermore, we have used for the an
gular dependence of the upper critical field the anisotropic
GinzburgLandau relation
Hc2(θ) =
H⊥c
c2
?
γ2cos2(θ) + sin2(θ)
,
where γ = H⊥c
The fieldinduced variations of Rs observed at T =
38 K (reported in Fig. 5a) do not exhibit hysteresis; so, in
this case, B can be considered uniform. Furthermore, at
temperatures near Tc, one can reasonably suppose fluxons
move in the fluxflow regime. In this condition, the ex
pected Rs(H0,Hc2(θ)) curve depends on λ0/δ0(obtained
by fitting the Rs(T) curve at H0 = 0), H⊥c
the other hand, the Hc2values deduced from the isother
mal Rs(H0) curves (see Fig. 6b) coincide with the mag
netic field at which the whole sample goes to the normal
state, i.e. H⊥c
c2. In order to fit the results at T = 38 K, we
have averaged the expected curve [calculated by Eqs. (1–
4)] over a sin(θ) distribution, have used for H⊥c
of the magnetic field at which Rs/Rn= 1, letting it vary
within the experimental uncertainty, have taken γ as free
parameter. At this value of temperature, the experimen
tal results can be accounted for using γ = 3.3 ± 0.5. In
particular, the bestfit curve reported in Fig. 5a has been
obtained with γ = 3.3 and µ0H⊥c
see, the fieldinduced variation of Rsis well described by
the model of Sec. 4. We think that this occurs because
at this temperature the superfluid fraction of the π band
is strongly suppressed at low magnetic fields, the flux line
gets a conventional structure and the fluxon dynamics can
be described by standard models.
Prompted by the results obtained at T = 38 K, we
have tried to fit the experimental data obtained in the
temperature range 34 ÷ 37 K by the same method (for
these temperatures, the upper critical field has been di
rectly deduced from the Rs(H0) curves). Since in this
temperature range we have detected magnetic hystere
sis, we have considered only the reversible part of the
Rs(H0) curve. In order to fit the data, we have hypothe
sized fluxons move in the fluxflow regime, have considered
c2/H?c
c2is the anisotropy factor.
c2 and γ. On
c2the value
c2= 145 mT. As one can
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Fig. 10. Normalized fieldinduced variations of Rs for sam
ple P0, at different temperatures near Tc. Open symbols are
the results obtained at increasing H0, full symbols those at
decreasing H0. The Rs(H0) curve at T = 38 K is reversible.
The lines are the expected curves, obtained with γ = 2.6 as
described in the text. The inset shows the temperature depen
dence of the magnetic field Hcr, above which the experimental
data can be justified in the framework of the model of Sec. 4.
for H⊥c
ting them vary within the experimental uncertainty, and
have taken γ as fitting parameter. We have found that
at high fields the experimental results can be fitted us
ing γ = 2.6 ± 0.2. Fig. 10 shows a comparison between
the expected curves, obtained with γ = 2.6, and the ex
perimental data for T = 34 ÷ 38 K; open symbols are the
results obtained at increasing H0, full symbols those at de
creasing H0. As one can see, the expected Rs(H0) curve at
T = 38 K so obtained poorly agrees with the experimen
tal data at low fields; on the contrary, the line of Fig. 5a,
which has been obtained with γ = 3.3, fits the data in the
whole range of magnetic fields. However, the value γ = 2.6
is closer to the uppercriticalfield anisotropy reported in
the literature for MgB2at temperatures near Tc[5,23,45,
46].
The results of Fig. 10 show that for H0 greater than
a certain threshold value, depending on T, the data can
be justified in the framework of the model describing the
fluxon dynamics of conventional vortex lattice. The tem
c2(T) the values reported in the inset of Fig. 6b, let
Page 12
12M. Bonura et al.: Microwave surface resistance in MgB2
perature dependence of the threshold field, Hcr, is re
ported in the inset. For H0 < Hcr the fieldinduced mw
losses are larger than those expected for singlegap su
perconductors in the mixed state. We suggest that this
surplus of mw losses is due to the additional contribu
tion of πband quasiparticles within the vortex cores with
respect to that of onegap superconductors; the finding
that Hcrdecreases on increasing T seems to support this
hypothesis. It is easy to see that the Hcr(T) values coin
cide, within the experimental uncertainty, with 0.5Hc2(T)
(see the inset of Fig. 6). Presently, it is not clear why
just above 0.5Hc2 the results can be justified by a stan
dard model, which does not consider the twogap nature of
MgB2. Furthermore, we remark that Hcrcannot be iden
tified with the magnetic field at which the πband super
fluid is suppressed; indeed several authors have reported
Hπ
c2[1,3,22,23,24,25].
c2∼ 0.1H⊥c
6 Conclusions
We have investigated the microwave surface resistance at
9.6 GHz of two polycrystalline Mg11B2samples prepared
by direct synthesis from Mg (99.999% purity) and crys
talline isotopically enriched11B (99.95% purity). That la
belled as P0 consists of pristine material; the other, la
belled as P6, has been exposed to neutron irradiation
at very high fluence. Several superconducting properties
of these samples have been reported in Refs. [13,14,15,
16,17]. Pointcontact spectroscopy and specificheat mea
surements, have shown that sample P0 exhibits a clear
twogapsuperconductivity behavior; in sample P6 the ir
radiation process determined a merging of the two gaps
into a single value. To our knowledge, the mw response of
neutron irradiated MgB2samples has not yet been inves
tigated.
The mw surface resistance has been measured as a
function of the temperature and the DC magnetic field. By
measuring the fieldinduced variations of Rsat increasing
and decreasing fields we have detected a magnetic hystere
sis ascribable to the critical state of the fluxons lattice.
The range of temperatures in which the hysteretic behav
ior has been observed is different for the two samples;
in the irradiated sample the hysteresis is undetectable at
T/Tc ? 0.55 while in the unirradiated sample it is de
tectable up to T/Tc≈ 0.95.
The results obtained in the irradiated sample have
been quite well justified in the framework of the Coffey
and Clem model with the normal fluid density following
the Gorter and Casimir twofluid model. In order to ac
count for the hysteretic behavior, we have used a gen
eralized Coffey and Clem model in which we take into
account the nonuniform fluxon distribution due to the
critical state.
The peculiarities of the mw surface resistance of sam
ple P0 differ from those observed in sample P6, in both the
temperature and the field dependencies. The Rs(T) curve
obtained at zero field shows a wide tail, from T/Tc≈ 0.9
down to T/Tc ≈ 0.7, which cannot be justified in the
framework of the Gorter and Casimir twofluid model. We
have shown that, in order to account for this behavior, it is
essential to hypothesize a linear temperature dependence
of the normal and condensed fluid densities. Such finding
agrees with the experimental temperature dependence of
the penetration depth reported in the literature, which
have been justified in the framework of twogap models
for the MgB2superconductor.
The Rs(H0) curves in sample P0 have shown several
anomalies, especially at low temperatures, among which
an enhanced fieldinduced variation and a magnetic hys
teresis of unconventional shape. At low temperatures, a
magnetic field H0 ≈ Hc2/15 causes a Rs variation of
≈ 35% of the normalstate value. We remark that in sam
ple P6 a variation of the same order of magnitude is ob
tained for H0 ≈ Hc2/2. The shape of the magnetic hys
teresis, which has been observed in a wide range of tem
peratures below Tc, cannot be justified in the framework
of the criticalstate models; the most unexpected behavior
concerns the decreasingfield branch, in which we observed
a plateau extending from µ0H0∼ 0.2 T down to zero. The
presence of this plateau is puzzling because it would sug
gest that the trapped flux does not change anymore on
decreasing the field below 0.2 T, although this value is
four times larger than the first penetration field.
The investigation at temperatures near Tc has high
lighted that, in the range T = 34 ÷ 38 K, the results
obtained in sample P0 for H0 ? 0.5Hc2can be justified
in the framework of the Coffey and Clem model taking
into account the anisotropy of the upper critical field. We
suggest that this occurs because at these field values the
superfluid fraction of the π band is strongly suppressed,
the flux line gets a conventional structure and the fluxon
dynamics can be described by standard models.
The enhanced fieldinduced variation of Rs, observed
at low T in the whole range of fields investigated as well as
at T ∼ Tcfor H0? 0.5Hc2, may be qualitatively ascribed
to the presence and motion of the giant cores due to the
πband quasiparticles. On the contrary, the origin of the
anomalous shape of the Rs(H0) curve is so far not un
derstood. We would like to remark that the results we ob
tained in sample P0 are very similar to those, not reported
here, we have obtained in several MgB2samples (unirra
diated), no matter the preparation method and the com
ponents (11B or10B) used in the synthesis process. The
comparison between the results obtained in the two sam
ples here investigated strongly suggest that the anomalies
in the Rs(H0) curves are related to the unusual structure
of fluxons due to the two superconducting gaps. According
to what suggested by different authors, our results confirm
that the standard models are inadequate to describe the
fluxon dynamics in twogap MgB2. Further investigation
is necessary for understanding how to take into account
the complex vortex structure in describing the fluxon dy
namics in MgB2.
Acknowledgements
The authors are very glad to thank D. Daghero, G. Ghigo,
R. S. Gonnelli and M. Putti for their interest to this work
Page 13
M. Bonura et al.: Microwave surface resistance in MgB2
13
and helpful suggestions; G. Lapis and G. Napoli for tech
nical assistance.
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