INSTITUTE OF PHYSICS PUBLISHING
SUPERCONDUCTOR SCIENCE AND TECHNOLOGY
Supercond. Sci. Technol. 17 (2004) 1458–1463
PII: S0953-2048(04)76898-0
Two-bandgap magneto-thermal
conductivity of polycrystalline MgB2
Bai-Mei Wu1,2 , Dong-Sheng Yang1 , Wei-Hua Zeng1 , Shi-Yan Li1 ,
Bo Li1 , Rong Fan1 , Xian-Hui Chen1 , Lie-Zhao Cao1 and
Marcel Ausloos2
1
Structural Research Laboratory, Department of Physics, University of Science and
Technology of China, Hefei, Anhui 230026, People’s Republic of China
2
SUPRATECS Institut de Physique B5, Université de Liège, B-4000 Liège, Belgium
Received 25 February 2004, in final form 9 August 2004
Published 2 November 2004
Online at stacks.iop.org/SUST/17/1458
doi:10.1088/0953-2048/17/12/017
Abstract
The thermal conductivity κ(B, T ) of polycrystalline superconducting MgB2
has been studied as a function of temperature (from 4 up to 50 K) and (up to
a rather large 14 T) applied magnetic field. No anomaly in the thermal
conductivity κ(B, T ) is observed around the superconducting transition in
the absence or presence of magnetic fields up to 14 T; at that field value the
superconductivity of MgB2 persisted. The thermal conductivity in zero field
shows a T -linear increase up to 50 K. The thermal conductivity is found to
increase with increasing field at high fields. We interpret the findings as if
there are two subsystems of quasiparticles with different field-dependent
characters in a two- (L- and S-) band superconductor reacting differently
with the vortex structure. The unusual enhancement of κ(B, T ) at low
temperature but higher than a (Bc2S 3 T) critical field is interpreted as a
result of the overlap of the low energy states outside the vortex cores in the
S band. An order of magnitude of the Bc2S temperature dependence is given,
in agreement with recent results by other authors under different
experimental conditions.
1. Introduction
Superconductivity at a remarkably high transition temperature,
i.e. close to 40 K, was discovered in MgB2 [1, 2] and has
attracted much attention3 . MgB2 has a Debye temperature
about 800 K, several times larger than that of Nb3 Sn, as seen
from specific heat measurements [3–7]. The phonon density of
states of MgB2 is known from inelastic neutron scattering [8–
11]. The data indicate that phonons play an important
role in the electron–electron interaction in superconducting
MgB2 . Most experiments in MgB2 , such as the isotope
effect [12, 13], Tc pressure dependence [14, 15] and tunnelling
spectroscopy [16, 17], indicate that the superconductivity of
MgB2 is thus thought to be consistent with a phonon-mediated
BCS electron pairing. Moreover, reports [5–7, 17] have
also shown some evidence for two gaps in the quasiparticle
excitation spectrum of MgB2 [18–20].
3
The reference list cannot be exhaustive; see e.g.
http://www.iitap.iastate.edu/htcu/mgb2preprints.html
0953-2048/04/121458+06$30.00 © 2004 IOP Publishing Ltd
Several physical properties can be further investigated
to pin information on the basic parameters characterizing
models for MgB2 behaviour. The thermal conductivity has
been used widely to study superconductors [21], offering
important clues about the nature of heat carriers and scattering
processes between them, especially in B = 0, and in
unconventional superconductors [22–26]—the more so in the
superconducting state for which traditional electrical probes
such as the electrical resistivity, Hall effect, and thermopower
are inoperative. The study of the thermal conductivity has
already been an interesting probe of the vortex state of
high Tc superconductors [27–37] as well as in MgB2 , in the
absence [38–40] or presence [41–45] of a magnetic field,
sometimes in single crystals.
In this paper we present a new study of the thermal
conductivity κ(B, T ) of polycrystalline MgB2 , as a function
of temperature in both the absence and presence of magnetic
fields in order to probe the anomalous dependences [40, 41] in
light of recent vortex structure considerations [7] and MgB2 ’s
Printed in the UK
1458
Two-bandgap magneto-thermal conductivity of polycrystalline MgB2
0T
1T
4T
7T
10 T
14 T
7
0.05 T
1T
5T
1
0
10
20
30
40
50
T [K]
(H,T) [W/mÆK]
Increasing field
3
-1
0.02 T
10 T
6
4
κ
R [x102µΩ]
5
0T
8
Figure 1. Electrical resistivity of polycrystalline MgB2 versus
temperature in 0, 1, 4, 7, 10, and 14 T applied magnetic fields.
2
complicated order parameter. The investigated temperature
range extends between 4 and 50 K and the applied magnetic
field goes up to 13.9 T. The superconductivity character
of MgB2 persisted over these investigated conditions. The
thermal conductivity almost monotonically increases with
increasing temperature in zero and low magnetic fields,
whereas a significant enhancement of the values of the
thermal conductivity is found for high magnetic fields. No
sharp anomaly in the thermal conductivity κ(B, T ) was
observed around the superconducting transition. An unusual
enhancement of κ(B, T ) at low temperature for a field higher
than some critical field (called Bc2S and 3T ) is interpreted
as a result of the overlap of low energy states outside the
vortex cores in the S band, a critical field in agreement with
recent results by other authors under different experimental
conditions. A model based on two different gaps, i.e. (small)
S and (large) L, two types of vortices, two subsystems of
quasiparticles with different field-dependent characters, and
two critical fields Bc2S and Bc2L in superconducting MgB2
and scattering relaxation times is discussed and argued to be
consistent with these results. The L vortex bound states are
highly confined, due to a narrow core radius, while the S-band
vortex core states are more extended or loosely bound.
2. Sample preparation and measurements
Polycrystalline MgB2 samples were prepared by the
conventional solid state reaction method. A mixed powder of
high purity Mg and B with an appropriate ratio was ground and
pressed into pellets. The pellets were wrapped in a tantalum
foil and sealed in a stainless steel reactor, then heated at 950 ◦ C
for 4 h under an argon gas flow. Repeating the process we got
a high density bulk sample. The phase purity was confirmed
by x-ray diffraction analysis. No impurity phase was observed
within the usual x-ray resolution. For the electrical and thermal
conductivity measurements one 8 × 1.5 × 0.8 mm3 bar was
cut from a bulk sample pellet.
In order to check the sample superconductivity we have
measured the electrical resistivity by the standard four-probe
dc technique with systematic current reversal in the 4–50 K
temperature range and for magnetic fields up to 14 T. Figure 1
shows the temperature dependence of the electrical resistance,
ρ(B, T ), of polycrystalline MgB2 for several magnetic fields
Increasing field
0
0
10
20
30
40
50
T [K]
Figure 2. Thermal conductivity, κ(H, T ), of polycrystalline MgB2
versus temperature T in zero B-field and in 0.02, 0.5, 1.0, 5.0, and
10.0 T applied magnetic fields.
under heating conditions. The electrical resistivity can be
deduced from the sample dimensions through the usual Pouillet
formula. The applied field was 0, 1, 4, 7, 10 and 14 T in turn
applied at low temperature after so-called zero-field cooling.
A narrow superconductivity transition occurs at Tc = 38.1 K in
zero field. With increasing fields, Tc shifts to lower temperature
and the transition width increases as expected. It should be
noted that the superconductivity persisted for such investigated
fields.
The thermal conductivity of the sample was measured
by the longitudinal steady state thermal flow method in the
temperature range 4–50 K and for several magnetic fields
up to 13.9 T. The thermal flow and the magnetic field were
parallel to the long axis of the sample. A thin film chip
resistor was used as an end heater, and thin wire differential
thermocouples for monitoring the thermal gradients on the
sample. The measurements were performed under high
vacuum with two additional shields mounted around the
sample in order to reduce the heat losses due to radiation
at finite temperature [46]. A carbon-doped glass resistance
thermometer was used to measure the sample temperature.
The sensitivity of the thermocouple as well as the resistance
of the thin film resistor were previously calibrated for such
magnetic field conditions. Data were recorded while warming
the sample up at a fixed magnetic field. The whole experiment
was computer controlled [47].
The temperature dependence of the thermal conductivity
of a polycrystalline MgB2 sample, in applied 0.0, 0.02, 0.5,
1.0, 5.0, 10.0, and 13.9 T fields after zero-field cooling, is
shown in figure 2. The thermal conductivity is seen to increase
monotonically with increasing temperature in the investigated
ranges and is about a factor of ten larger than the best metals
at room temperature. The temperature dependence of κ(0, T )
is linear (with a correlation coefficient R = 0.9995) if we let
1459
B-M Wu et al
κ(0, T ) = aT + b, with b small and negative (∼−1) even
at high fields. It should be expected that κ would curl and
remain finite and positive at 0 K. A significant enhancement
in κ(B, T ) is obvious for high fields, with a change between 1
and 5 T, at low temperature, as in [41–45]. All κ(B, T ) over T
curves intersect each other at 19 K, above which temperature
the curves of κ(B, T ) for different fields merge into each other.
Both electrical resistivity and thermal conductivity appear to be
field independent above Tc . However, we have not investigated
the temperature region above 77 K. It should be expected
that the electrical resistance would curl up while the thermal
conductivity should go through a umklapp peak.
3. Discussion of measurements
It should be recalled that the sample is polycrystalline, whence
a warning on possible intergrain and intragrain contributions
is in order from the start. Vortices can be pinned at intergrain
boundaries and do influence the electrical resistivity behaviour
in a field. However, it is widely accepted that the thermal
conductivity intergrain features are quickly killed by a small
magnetic field. In view of the investigated field values the
following discussion thus considers that the features reflect
intragrain properties.
3.1. Zero-field case
As shown in figure 2, the absence of an anomaly in the
thermal conductivity in zero field near the superconducting
transition Tc is first in agreement with other results reported for
polycrystalline [37–39] and single-crystalline MgB2 [41, 42].
Such a smoothness is often attributed to charge carrier and
phonon scattering by variously distributed localized impurities
or extended defects. However, the MgB2 thermal conductivity
markedly differs in behaviour at Tc from that of conventional
superconductors and high temperature superconductors, be
they single- or polycrystalline materials. Indeed the thermal
conductivity of superconducting materials generally shows
an anomalous behaviour at the superconducting transition
temperature, either a decrease or an increase below Tc
depending on the nature of the thermal carrier and their
interactions. In conventional superconductors κ(T ) decreases
below Tc due to the condensation of electrons. In the case
of the high temperature superconductors κ(T ) shows a peak
below Tc that is explained by considering either an increase
in the phonon mean free path due to carrier condensation
or an increase in the quasi-particle relaxation time in the
superconducting state or both. Our results do not show any
such response in the thermal conductivity for MgB2 around Tc
either in the absence or presence of magnetic fields, within our
experimental resolution.
It can be concluded that MgB2 heat conduction is neither
sensitive to some Cooper pairing below Tc nor to a magnetic
field breaking effect. However, the lattice contribution is
substantial in both the superconducting and the normal state
region. Indeed, in zero field, the thermal conductivity κ in a
metal is generally [48] considered to be the sum of electron κe
and phonon contributions κph , i.e.
κ(T ) = κe (T ) + κph (T ).
1460
(1)
Figure 3. Field dependence of MgB2 reduced thermal conductivity
κ(H )/κ(H = 0) at several temperatures.
The electronic contribution to κ(T ) in the superconducting
state has been calculated by Bardeen et al [21]. The
κe (T ) magnitude can be usually estimated from the electrical
resistivity and the Wiedemann–Franz (WF) rule κe = L T /ρ
if the scattering is elastic; L = L 0 , the Lorentz number;
L < L 0 if it is inelastic. Since the value of κe is an upper
limit when obtained by using the WF rule and L 0 , we estimate
that the value of κe is about 20% of the total value of the
thermal conductivity in the normal state, κn e.g. at T = 50 K
for our sample. The best fit value for L (at 50 K) being
smaller (L 1.88×10−8 W K−2 ) than the expected Lorentz
constant (L 0 = 2.45 × 10−8 W K−2 ) calls attention to the
influence of inelastic scattering and to a large phonon (80%)
contribution.
The phonon thermal conductivity is usually [43] represented within the Debye-type relaxation rate approximation
in terms of the phonon frequency spectrum, a relaxation time
τ (ω, T ), and the sound velocity v = D (kB /h̄)(6π 2 n)−1/3 ,
where n is the electronic density. For our analysis, we
use D = 750 K, as deduced from specific heat measurements [4, 9]. The total phonon relaxation rate is usually represented as a sum of terms corresponding to independent scattering mechanisms, dominating in different temperature intervals,
like phonon scattering by sample boundaries, point defects,
phonons, dislocations, and electrons.
3.2. Field dependence
Next, consider the field dependence of the thermal conductivity
κ(B)/κ(B = 0) at several temperatures (figure 3). In
low fields, the thermal conductivity seems insensitive to the
magnitude of the applied magnetic fields whereas it becomes
variably enhanced at higher fields. Notably, the value κ(B)
is higher than that of κ(B = 0) in the field region larger than
0.5 T at low temperature (below 19 K). According to standard
thinking for conventional superconductors the introduction of
a magnetic field would lead to a decrease in heat conductivity
because vortices constitute new scattering centres for heat
carriers. This picture provided a qualitative explanation of the
field-induced decrease in κ(B) observed in the vortex states
of conventional superconductors and high Tc cuprates in the
investigated regions of the field–temperature plane [27–37].
Two-bandgap magneto-thermal conductivity of polycrystalline MgB2
The non-linear field dependence and the field-dependence
enhancement in κ(B) imply that the effect of thermal carriers
scattered by vortices in the mixed state of MgB2 is far from
trivial. At low (0) field the intergrain boundaries surely
control the κ(B) features.
To interpret quantitatively the field dependence of the
thermal conductivity at finite field it is best again to assume
that phonons and electrons contribute independently as heat
carriers [48, 49], such that
κ(B, T ) = κe (B, T ) + κph (B, T ),
(2)
where κe (B, T ) and κph (B, T ) are the electron and phonon
contributions to κ(B, T ), respectively.
Again the mean free path of both electrons and phonons
can change below Tc , because the scattering of phonons
on electrons is reduced when electrons condense into the
superconducting state.
Moreover, the electron–electron
interaction can change when the superconducting gap opens
below Tc and modify the mobility because of the condensate
presence. Therefore, it is not immediately clear a priori which
type of heat carriers and which interactions are responsible
for the behaviour of the thermal conductivity in various
temperature and field ranges.
According to band structure calculations and tunnelling
experiments, there are two distinctive Fermi surfaces: one
is a two-dimensional cylindrical Fermi surface arising from
σ orbitals due to px and p y electrons of B atoms and the other
is a three-dimensional tubular Fermi surface network coming
from π orbitals due to pz electrons of B atoms. They are
weakly hybridized with Mg electron orbitals. Hence these two
Fermi surfaces have different superconducting energy gaps:
respectively, a large bandgap (LBG) L on the 2D Fermi
surface sheets and a small bandgap (SBG) S on the 3D Fermi
surface. The ratio S /L is estimated to be around 0.3–
0.4 [7, 17, 41]. Electrons on these two Fermi surfaces have
different characters since the σ orbital is strongly coupled to the
in-plane B-atom vibration with E2g symmetry but the π orbital
is weakly coupled with this phonon mode.
Consider first the possible electronic contribution and let
us analyse the field dependence of the thermal conductivity at
low temperature (19 K in our study) within a vortex lattice
structure picture for a two-bandgap superconductor. The
single-vortex state has been studied by Nakai et al [7]. Within
a two-band superconducting model [7], the vortex core radius
has to be small for the LBG and large for the SBG electrons.
When increasing the field the vortex core radius widens (up to
the order parameter suppression at some critical field). This
field widening effect should be stronger for the SBG than for
the LBG electrons. Therefore, the LBG vortex bound states
are thought to disappear at some Bc2L , while the SBG extended
and loosely bound vortex core states could disappear at some
other, smaller, Bc2S .
Notice that in the vortex core of the L band, the local
density of states (LDOS) at site r , NL (r, E ∼ O), is thus
highly concentrated, while the NS (r, E ∼ O) LDOS is rather
spread out in the S-band case. In fact, the low energy states
extending from the S-band vortex cores are even expected
to overlap with neighbouring ones. When increasing B, the
overlap is expected to become more pronounced and the LDOS
to reduce to a flat profile, i.e. NS (r, E ∼ O)/Ns (E F ) 0.5
for the S band, where NS (E F ) is the total DOS in the normal
state at the Fermi level. This picture of the vortex structure
in MgB2 is in fine agreement with the field dependence of the
electronic specific heat [5–7].
Let us introduce these notions into the electronic heat
transport property hereby examined. From a scattering point of
view, the electrons in each grain are scattered by other electrons
κe,e , phonons κe,ph and vortices κe,v , e.g.
κe,v (B, T ) = 13 ce (B, T )vF2 τe−v (B, T )
(3)
where ce (B, T ) is the electronic specific heat, vF the
−1
the electron–vortex scattering rate.
Fermi velocity, and τe−v
The temperature and magnetic field dependence of the
electronic specific heat can be obtained from data or from a
phenomenological expression.
We have also to consider two subsystems of quasiparticles
characterized by L and S bandgaps, and scattering
in ‘parallel’. For each considered electronic scattering
contribution, let
κe (B, T ) = κe,L (B, T ) + κe,S (B, T ),
(4)
thereby taking into account both sets of electrons with different
DOS characters. According to a similar analysis on singlecrystalline MgB2 [41] the electrons experiencing the large gap
provide approximately two-thirds of the total electronic heat
conduction, and the dominant part of the interaction between
quasiparticles and low frequency phonons is provided by that
part of the electronic excitation spectrum experiencing the
small gap [25, 41].
It seems reasonable to expect that the field more
easily suppresses the SBG than the LBG. Therefore,
superconductivity is maintained in a field mainly because the
LBG survives, up to some Bc2,L . The Ns (r, E ∼ O) domains
with low energy excitations survive, but for increasing fields
the SBG LDOS reaches a flat profile Ns (r, E ∼ O)/Ns (E F ) 0.5, thereby not contributing to heat conduction. These SBG
electrons become heat carrier prone, inducing above Bc2S an
increase in the electronic thermal conductivity, as seen in
figure 1. This Bc2S seems to be about 3.0 T. Nakai et al
[7] claim that Bc2S and Bc2L are the same because they do
not observe a hump in the specific heat at some intermediary
field, i.e. the lower one. This may be due to the fact that
the specific heat is on one hand a bulk quantity in contrast
to the thermal conductivity rather probing a percolation path.
Moreover, it is sometimes very hard to obtain accurately kinks
and phase transition lines from specific heat data, even with
high precision measurements [49].
Consider the phonon contribution next. Phonons are
accepted to be the dominant heat carriers in MgB2 , especially
down to T / Tc 0.4 [37–39]. The study of the phonon
spectrum in MgB2 [8–11] has clearly shown an anomalous
phonon behaviour at the low energy transfer of −24 meV on
cooling through Tc reflecting a strong relation between these
phonon states and superconductivity origin in MgB2 .
From the thermal conductivity in a magnetic field
behaviour point of view, the phonon (double-type) vortex
structure interaction should be expected to depend also on the
nature of the phonon spectrum, in the sense of the Tewordt–
Wolkhausen picture [22]. The phonons which can inelastically
1461
B-M Wu et al
interact with the bound quasiparticles in the vortex cores are
only those for which the phonon wavelength λ < ξ is of the
size of a vortex core [27]. Thus it seems obvious that due
to the overlapping features of the low energy states the SBG
vortices become less effective scattering centres for phonons as
well. Therefore, the mean free path of low frequency phonons
may also increase, resulting in an increase of the thermal
conduction process at high field, in agreement with the above
data. We conjecture that the quasiparticles associated with the
LBG contribute moderately to the thermal conductivity at high
field and low temperature. In fact interactions involving the
quasiparticles associated with the LBG and thermal phonons
have not been well described. Nevertheless they could contain
an elastic component of the scattering process. Such a picture
seems a fine way of interpreting the decrease of the thermal
conductivity at high field and high temperature (figure 2).
Finally, an alternative view considers that electrons
self-organize into collective textures rather than point-like
entities [50]. Such an image of electronic collective textures
in an applied magnetic field is complex and not immediately
ready for interpreting the (electron or phonon in fact) scattering
process within the framework of a double-vortex structure,
as discussed above. We cannot further reject the possibility
that a field-induced shift of the phonon spectrum itself could
produce effects similar to the temperature. Some softening
of the phonon spectrum indeed exists, as seen in the thermal
expansion coefficient [50–52]. This equivalency, to be
fair, may be also at the origin of the enhanced behaviour
of the field–temperature-dependent thermal conductivity in
superconducting MgB2 .
4. Conclusion
In conclusion, we have investigated the heat transport in MgB2
in both the absence and the presence of magnetic fields (up
to 14 T) in the temperature range between 4 and 50 K.
In these regimes, the superconductivity of MgB2 persisted
as seen from electrical resistivity measurements; no clear
anomaly appeared in the thermal conductivity at Tc . The
thermal conductivity in the absence of field shows a T -linear
increase up to 50 K. In weak magnetic fields the thermal
conductivity is insensitive to the magnitude of the applied field;
an enhanced thermal conductivity appears at higher fields and
low temperature, but not at high temperature. The existence
of two quasiparticle subsystems with different bandgaps and
field-dependent characters explains the results. The field
dependence of the thermal conductivity for such a two-band
superconductor indicates that the enhancement of κ(B, T ) at
low temperature in high fields is a result of the overlap of
the low energy states outside the small bandgap vortex cores,
above a critical Bc2S field. The latter goes from ca. 1 T at 30 K
up to 20 T at 20 K, in view of the investigated range available.
We realize that for a full proof of the above, even a thorough fit
would be meaningless, in view of the huge number of involved
parameters. That, in fact, shows the enormous potential for
investigations—in particular the extremely complex vortex
phase diagram to be expected due to the presence of two
anisotropic gaps. Neutron scattering investigations might be
of interest to confirm the existence of Bc2S. Its exact value
might be obtained from microwave data, muon scattering, or
paraconductivity fluctuation in a field.
1462
Acknowledgments
This work was supported by the National Natural Science
Foundation of China (No 10174070), the Ministry of Science
and Technology of China (NKBRSF-G19990646) and partly
supported by the Belgium National Fund for Scientific
Research (FNRS).
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