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Physica B 385–386 (2006) 79–81

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Phonon scattering effects on thermal conductivity of MgB2

Dinesh Varshneya, K.K. Choudharyb,�, M.S. Azada, R.K. Singhc

aSchool of Physics, Vigyan Bhavan, Devi Ahilya University, Khandwa Road Campus, Indore-452017, IndiabDepartment of Physics, S. V. I. T. S., Sanwer Road, Indore-453331, IndiacLakshmi Narain College of Technology, Raisen Road, Bhopal-462021, India

Abstract

A theoretical model is developed to account the anomalies reported for the thermal conductivity (k) of MgB2 superconductor. The

inelastic neutron scattering shows that phonons are important in test material, we begin with the lattice thermal conductivity by

incorporating the scattering of phonons with defects, grain boundaries, electrons as charge carriers, and phonons in the model

Hamiltonian. Lattice k dominates below Tc and is attributed to strong phonon-defects, and impurity scattering mechanism.

Furthermore, k reveals power dependence at low temperatures. On the other hand, the contribution of careers towards k is substantial

and is due to the fact that the carriers are condensed, do not carry entropy. We argue that the response of k in MgB2 is determined by

competition among several scatterings for carriers and a balance between carrier and phonon contributions.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Phonon scattering; Thermal conductivity; Electron–phonon interaction

1. Introduction

The discovery of superconductivity [1] at 39K in MgB2

has produced a notable excitement in the researchers ofcondensed matter physics community. For several reasonsMgB2 is a very unusual superconducting compound.Though it is a non-Copper oxide and a non-C60-basedcompound yet having a Tc of 40K is a distinguishedfeature. Several experiments have been carried out to givean understanding of the high Tc of MgB2. Ramanscattering of light and inelastic neutron scattering havebeen used to probe its phonon structure. Raman spectro-scopy by Goncharov et al. [2] showed that a broadenedRaman band at 620 cm�1 (77meV) is observed. Vibrationsare highly anharmonic consistent with theoretical predic-tions for the E2g inplane B stretching mode. Bohnen et al.[3] indicated the presence of a broad mode at 72meV(580 cm�1). Chen et al. [4] suggests a conventional phonon-mediated pairing mechanism. Inelastic neutron measure-ments [5] gives phonon density of states in which acousticphonon extend in energy to 36meV and there are highly

e front matter r 2006 Elsevier B.V. All rights reserved.

ysb.2006.05.108

ng author. Tel.: +917321 224371; fax: +91 7321 224371.

ss: kkchoudhary1@yahoo.com (K.K. Choudhary).

dissipative optic branches peaking at 54, 78, 89 and97meV. Applying a simple model they get l ¼ 0.9. Thesignificant broadening of the Raman peak arises mainlyfrom the exceptionally strong nonlinear electron–phononcoupling of the E2g mode to partially occupied planarboron s bands near the Fermi surface and its largeanharmonicity [6]. The E2g mode is responsible for thedescription of superconductivity [7].Furthermore, the phonon spectrum can be conveniently

divided into two parts by the interval of low density ofstates between 36 and 97meV [5]. The role of acousticDebye branch characterized by the Debye temperature isimportant for the explanation of thermal transport in thenormal state. Also, in most reports the superconductingenergy gap is claimed to be nodeless, compatible with s

wave pairing. Various types of experiments [8–10], mainlyusing powder or polycrystalline samples and often-surfacesensitive have given evidence for two gaps of differentmagnitude in the quasiparticle excitation spectrum of thissuperconductor.The low density of states feature, i.e., acoustic phonon in

phonon spectrum is significant and it remains to be seenwhether they play a substantial role in the thermaltransport. In this regard, the thermal conductivity (k)

ARTICLE IN PRESSD. Varshney et al. / Physica B 385–386 (2006) 79–8180

measurements are an indispensable tool. As heat is carriedby charge carriers and by phonons, a measurement of kyield valuable information not only about phonons andcarriers, the interaction between them but also aboutimpurities in the system. Salient features on thermaltransport reported follows: k exhibits a power temperaturedependence in the low-temperature domain, k increase witha change of the slope in vicinity of Tc and above, and asaturating nature is seen at high temperatures [11]. Theimportance of impurity scattering is also emphasized [12].

To explore the anomalies observed in k, we begin withthe understanding of scattering rates between heat carrierthemselves and between electrons and grain boundaries,impurities, etc. Apart from the interplay, it is meaningful toidentify the relative magnitudes of these relaxation rates,which lead to anomalous behaviour, and this motivates usfor the present investigation. Analyses of transport resultsreport in the present studies do indeed shed important lighton the nature of phonon–impurity interaction in the testmaterial. For the description of reported behavior ofthermal conductivity in MgB2, we place particular empha-sis on the acoustic phonons as a source of thermaltransport in the model Hamiltonian.

2. The model

We begin with a simple approach where the phononsare described in the Debye model and the charge carriersare treated in an isotropic BCS-like model. The choiceof Debye model is appropriate as the temperature domainof interest lies well below the Debye temperature. Asthe simplest approximation to the problem at hand, theisotropic BCS-like model shall be used to derive qualitativeresults, as we will demonstrate later.

We begin with the model Hamiltonian that follows [13]

H ¼X

p

�paþp ap þX

q

oqbþq bq þXp1;p2

jðp1; p2Þaþp1

ap2

þDXp;q

q_

2roq

� �1=2aþpþq

ap bp þ bþ�q

� �

þR

2N

Xq1;q2

eiðq1þq2ÞRi_oq1

_oq2

4

� �1=2

� bq1� bþ�q1

� �bq2� bþ�q2

� �þHp�p. ð1Þ

In the above, the initial two terms are carrier, andphonon excitation. The third and fourth terms representelectron–impurity and electron–phonon interactions, re-spectively. The fifth term is phonon–impurity interactionand last term stands for the phonon–phonon interaction.The notations in Eq. (1) are: ep is hole free energy, oq isacoustic phonon frequency, j is coupling parameter ofelectron and impurity potential, r is ionic density, D isdeformation-potential constant, R is relative ionic-massdifference [(M00�M)/M00], number of cells is N and Ri

stands for the position of defects. We begin with thephonon contribution.

2.1. Lattice contribution

The thermal conductivity from lattice part follows [13]:

kp ¼kB_

2

2p2vs

Z oD

0

doo2tðoÞðboÞ2ebo

ebo � 1ð Þ2

(2)

with vs is the sound velocity, oD is Debye frequency andb ¼ _=kBT . The relaxation time is proportional to imagin-ary part of phonon self-energy and is written as

1=tðoÞ ¼ 2 ImPðo=vs;oÞ�� ��

¼ 1=tph�d þ 1=tph�e þ 1=tph�gb þ 1=tph�ph, ð3Þ

t being the phonon relaxation rates with defects, grainboundary, impurity and phonons.

2.2. Carrier contribution

The carrier component of k is expressed as [13]

ke ¼p2

3neðTÞ

k2BT

m�teðTÞ (4)

with ne (T) is the concentration of normal charge carriers ofeffective mass m* and te(T) is the temperature-dependentelectron relaxation time. The carrier relaxation rates in thepresent context also accounts for the increase in relaxationrates of carriers due to fluctuations.

3. Results and discussion

Let us first discuss the properties of thermal conductivitydue to phonons (kph). Fig. 1 reveals that as temperature islowered through Tc, quasiparticle excitation condensedinto ground state and they cannot scatter phonons. kphthus, increases exponentially as T increases in the absenceof other scattering mechanism. Although kph experiencesan exponential increase below Tc, the presence of chargecarrier and grain boundaries scattering set a limit on itsgrowth therefore k phonon diminishes as T increasesfurther after Tc. The maximum position depends onrelative magnitudes of phonon scattering with carrier,defect, grain-boundaries and phonon itself. Below 5K,defects become effective phonon scatterers and k exhibits apower temperature behaviour. At even lower temperatures,grain-boundary scattering dominates and the usual Debyecubic temperature behavior appears. Further thermalconductivity develops a broad peak above Tc, about 80Kbefore falling off at lower temperatures. We argue that thisfeature is attributed to the fact that below Tc the phononmean free path increases with the decrease in temperature,as more and more carriers are condensed in the super-conducting state. These are the features contained inEqs. (2) and (3). Following Eq. (4), the carrier contributiontowards thermal conductivity (ke) is numerically analysed

ARTICLE IN PRESS

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

κp

κe

MgB2

Expt. Theo.

κ (m

Wat

t cm

-1 K

-1)

T(K)

Fig. 2. Variation of thermal conductivity with temperature. Closed circles

are the experimental data taken from Ref. [11].

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

MgB2 ph

ph + d + e

ph + d + e + gb

ph + d

κ (m

Wat

t cm

-1 K

-1)

T(K)

Fig. 1. Variation of phonon thermal conductivity as function of

temperature in the presence of various phonon scattering mechanism.

D. Varshney et al. / Physica B 385–386 (2006) 79–81 81

and displayed in Fig. 2. It is evident that ke vanishes at lowtemperature and exhibits power temperature behaviourabove Tc, Nevertheless, it has been found that thecarrier contribution is about 5% of total k below 150K.Both phonon and carrier contributions are clubbedtogether (k ¼ kp+kc) to get the total thermal conductivitywhich is consistent with the experimental reported data [11](see Fig. 2).

4. Conclusions

In summary, we can say that the present analysis cansatisfactorily account for the observed anomalous tem-

perature behaviour of the thermal conductivity. Thenoteworthy findings include: in the low-temperature regionk exhibits power temperature dependence, a peak value ofk at about 80K, and a saturating nature is inferred100pTp300K which is radically different from cuprates.It is stressed that the phonon-defect scattering dominates inMgB2 and their contribution to thermal conduction issignificant. In brief, behaviour of k is determined bycompetition among the several operating scattering ratesfor the heat carriers and a balance between electronicand phonon contributions. In principle, the acousticphonons are the major source of thermal transport inMgB2. Despite the limitations, the present model calcula-tions successfully retrace the reported T dependence ofthermal conductivity k.

Acknowledgement

KKC is thankful to AICTE, New Delhi for financialassistance under CAYT scheme.

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