# Microwave surface resistance of pristine and neutron-irradiated 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

upper-critical-field anisotropy is taken into due account.

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**ABSTRACT:**We report on the magnetic-field-induced variations of the microwave surface resistance, R_s, in a polycrystalline MgB_2 sample, at different values of temperature. We have detected a magnetic hysteresis in R_s, which exhibits an unexpected plateau on decreasing the DC magnetic field below a certain value. In particular, at temperatures near T_c the hysteresis manifests itself only through the presence of the plateau. Although we do not quantitatively justify the anomalous shape of the magnetic hysteresis, we show that the results obtained in the reversible region of the R_s(H) curve can be quite well accounted for by supposing that, in this range of magnetic field, the pi-gap is almost suppressed by the applied field and, consequently, all the pi-band charge carriers are quasiparticles. On this hypothesis, we have calculated R_s(H) supposing that fluxons assume a conventional (single core) structure and the flux dynamics can be described in the framework of conventional models. From the fitting of the experimental results, we determine the values of H_{c2}^pi(T) at temperatures near T_c. In our opinion, the most important result of our investigation is that, at least at temperatures near T_c, the value of the applied field that separates the reversible and irreversible regions of the R_s(H) curves is just H_{c2}^pi(T); a qualitative discussion of the possible reason of this finding is given.Superconductor Science and Technology 04/2009; · 2.76 Impact Factor - SourceAvailable from: Maria Li Vigni[Show abstract] [Hide abstract]

**ABSTRACT:**The magnetic-field-induced variations of the microwave surface resistance have been investigated in a heavily neutron-irradiated MgB2 sample, in which the irradiation has caused the merging of the two gaps into a single value. The experimental results have been analyzed in the framework of the Coffey and Clem model. By fitting the experimental data, we have determined the field dependence of the depinning frequency, omega_0, at different values of the temperature. Although the pinning is not particularly effective, the value of omega_0 obtained at low temperatures is considerably higher than that observed in conventional low-temperature superconductors.Physica C Superconductivity 10/2008; · 0.72 Impact Factor -
##### Article: Study of microwave surface resistance of type‐II superconductors carrying transport current

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**ABSTRACT:**We investigate the microwave surface resistance of an infinitely long type-II 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 field-cooled 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

Page 1

arXiv:0709.0618v2 [cond-mat.supr-con] 8 May 2008

EPJ manuscript No.

(will be inserted by the editor)

Microwave surface resistance of pristine and neutron-irradiated

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, I-90123 Palermo, Italy

2CNR-INFM-LAMIA and Dipartimento di Fisica, Universit` a di Genova, Via Dodecaneso 33, I-16146 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 upper-critical-field 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 two-gap 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, inter-band scattering of quasiparticles is much

smaller than the intra-band one, making the two super-

conducting gaps quite different though they close at the

same Tc[8].

According to the theory of multi-band 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 inter-band 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 point-contact 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 mixed-state prop-

erties of MgB2arises from the unusual flux-line 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 c-axis, 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 point-contact-spectroscopy 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 σ-band quasiparticles local-

Page 2

2M. Bonura et al.: Microwave surface resistance in MgB2

ized in a region of radius ξσand π-band 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 π-band gap, the vortex

cores shrink and the π-band quasiparticles are widespread

in the whole sample [19,22]. This field-induced evolution

of the vortex lattice is expected to affect the vortex-vortex

and vortex-pinning 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 condensed-fluid 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

magnetic-field-induced 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 field-induced 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 π-band superfluid cannot be

neglected, the magnetic-field 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 magnetic-field 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 two-gap 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 two-gap and one-

gap MgB2 superconductors. To our knowledge, the mw

response of neutron-irradiated 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 two-fluid 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 flux-flow 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 point-contact spectroscopy [15] and specific-heat mea-

surements [16], sample P0 shows a clear two-gap 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 normal-state resis-

tivity); its residual normal-state 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 normal-state 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

normal-state resistivity ρ(Tc). Nevertheless, it has been

shown that the irradiation does not affect the variation

of the normal-state 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 thermal-neutron irra-

diation does not affect the grain-boundary properties. On

the other hand, it has been shown that either neutron ir-

radiation or He-ion irradiation [11] do not affect the grain

connectivity, even at high irradiation levels, contrary to

what occurs using heavy-ion 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 intra-band 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 cavity-perturbation technique [27]. A copper cavity,

of cylindrical shape with golden-plated 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

hp-8719D 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 zero-field 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 field-induced variations of Rsin the two samples have

roughly the same magnitude. In sample P0, one can ob-

serve an anomalously enhanced field-induced 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 field-induced 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 field-induced 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 best-fit 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 first-penetration 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

field-induced 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 field-induced

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 two-gap 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

decreasing-field 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 critical-state 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 field-induced 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 field-induced 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. Field-induced 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 best-fit 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 field-induced variations of Rs for samples

P0 (a) and P6 (b), at T/Tc ≈ 0.77. The line in panel (b) is the

best-fit 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 surface-barrier 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 field-induced 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 flux-flow 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 best-fit

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 two-gap MgB2materials [1,

4,5].

H′(T) of Fig. 6c corresponds to the value of the DC

magnetic field at which the decreasing-field branch of the

Rs(H0) curves deviates from the increasing-field 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 best-fit 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 field-induced 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 two-fluid model of superconductiv-

ity [30]. The theory has been developed under two basic

assumptions: i) inter-vortex 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 normal-fluid skin depth at T = Tc, w0(T) is the

fraction of normal electrons at H0= 0; in the Gorter and

Casimir two-fluid model w0(T) = (T/Tc)4.

?δvis the effective complex skin depth arising from the vor-

viscous and restoring-pinning forces, which identities the

depinning frequency ω0.?δvcan be written as

?δ2

δ2

µ0ωη,

with η the viscous-drag 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 restoring-pinning

force. On the contrary, for ω ≫ ω0, the fluxon motion

takes place around the minimum of the pinning-potential

well and, consequently, the restoring-pinning force is nearly

ineffective. So, the contribution of the viscous-drag force

predominates and the induced em current makes fluxons

move in the flux-flow 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

applied-field 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 non-uniform 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 critical-state 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 field-induced 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 increasing-field

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 increasing-field branch

is expected to have a negative concavity. The shape of the

decreasing-field 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 single-gap 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 magnetic-field 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 field-induced

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 best-fit 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 two-fluid 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 normal-fluid 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

best-fit 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 two-fluid 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 best-fit 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 two-fluid 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 two-fluid 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 two-gap 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 best-fit curve; it has been

obtained by Eqs. (1–4) with B = 0, w0(T) = T/Tc and

λ0/δ0 = 0.15. The wide low-T 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 (single-gap) 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 field-induced 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

upper-critical-field anisotropy, γ, because it has been shown

that in the P6 sample γ = 1 [14]. When the critical-state

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 best-fit 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

best-fit 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 increasing-field profiles, the

dashed ones are the decreasing-field profiles at the same

external-field 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 increasing-field 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 decreasing-field 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. Field-induced 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 best-fit 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 flux-flow regime.

The best-fit 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 flux-flow regime.

5.3 Field dependence of Rsin sample P0

It has been shown by several authors that the proper-

ties of the two-gap 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 double-gap 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 magnetic-field scales [21,22]. Because of the different

magnetic-field 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 π-band quasipar-

ticles spread over the sample [22]; on further increasing the

field, ∆πis strongly reduced, the π-quasiparticle contribu-

tion remains almost unchanged while the σ-quasiparticle

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 fluxon-lattice

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 field-induced evolu-

tion of the vortex lattice is expected to affect the vortex-

vortex and vortex-pinning interactions, making the stan-

dard models most likely inadequate to describe the fluxon

dynamics.

Results on the field-induced 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 field-induced 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

field-induced mw dissipation up to fields of the order of

1 T, followed by a slower one at higher fields. Consis-

tently with the sharp field-induced variation of the heat

capacity and thermal conductivity, the enhanced low-field

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 π-quasiparticle contribution.

According to Shibata et al., the enhanced field-induced

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 increasing-field 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 decreasing-field 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 c-axis randomly oriented with respect to

the DC-magnetic-field 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

Ginzburg-Landau relation

Hc2(θ) =

H⊥c

c2

?

γ2cos2(θ) + sin2(θ)

,

where γ = H⊥c

The field-induced 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 flux-flow 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 best-fit curve reported in Fig. 5a has been

obtained with γ = 3.3 and µ0H⊥c

see, the field-induced 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 flux-flow 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 field-induced 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 upper-critical-field 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 field-induced mw

losses are larger than those expected for single-gap su-

perconductors in the mixed state. We suggest that this

surplus of mw losses is due to the additional contribu-

tion of π-band quasiparticles within the vortex cores with

respect to that of one-gap 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 two-gap nature of

MgB2. Furthermore, we remark that Hcrcannot be iden-

tified with the magnetic field at which the π-band 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]. Point-contact spectroscopy and specific-heat mea-

surements, have shown that sample P0 exhibits a clear

two-gap-supercon-ductivity 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 field-induced 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 two-fluid 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 non-uniform 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 two-fluid 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 two-gap models

for the MgB2superconductor.

The Rs(H0) curves in sample P0 have shown several

anomalies, especially at low temperatures, among which

an enhanced field-induced 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 normal-state 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 critical-state models; the most unexpected behavior

concerns the decreasing-field 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 field-induced 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

π-band 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 two-gap 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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