Bull. Mater. Sci., Vol. 36, No. 7, December 2013, pp. 1255–1260. c Indian Academy of Sciences.

Phonon, magnon and electron contributions to low temperature specific heat in metallic state of La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3 manganites

DINESH VARSHNEY1,∗, IRFAN MANSURI1,2 and E KHAN1 1Materials Science Laboratory, School of Physics, Devi Ahilya University, Khandwa Road Campus, Indore 452 001, India 2Deparment of Physics, Indore Institute of Science and Technology, Pithampur Road, Rau, Indore 453 331, India

MS received 19 June 2012; revised 21 August 2012

Abstract. The reported specific heat C (T) data of the perovskite manganites, La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3, is theoretically investigated in the temperature domain 3 ≤ T ≤ 50 K. Calculations of C (T) have been made within the three-component scheme: one is the and the others are boson ( and magnon) contributions. Lattice specific heat is well estimated from the Debye temperature for La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3 manganites. Fermion component as the electronic specific heat coefficient is deduced using the band structure calculations. Later on, following double-exchange mechanism the role of magnon is assessed towards spe- cific heat and found that at much low temperature, specific heat shows almost T3/2 dependence on the temperature. The present investigation allows us to believe that electron correlations are essential to enhance the over simple Fermi-liquid approximation in the metallic phase of both the manganite systems. The present numerical analysis of specific heat shows similar results as those revealed from experiments.

Keywords. Oxide materials; crystal structure; specific heat; phonon.

1. Introduction many properties are attributed to ‘electron–phonon’ interac- tion. As electron–phonon interaction is one of the most rele- The colossal magnetoresistance (CMR) manganites are com- vant contributions in determining the conduction mecha- pounds of chemical formula, R1−x DxMnO3, with R being nism in these materials (Goto et al 2004;Dabrowskiet al the rare-earth materials and D the doping materials such as 2005). The strong coupling between the electron and lattice alkaline earth, transition metals or non-metals, etc. CMR is affected by the ratio between trivalent and divalent ions at effect is caused by the temperature dependent phase transi- A site or the average ionic radius of the ions on A site. Thus, tion from a paramagnetic insulator to a ferromagnetic metal. thermal properties of this material may be taken as a starting Thus, for temperature slightly above this , point to a consistent understanding of more complex physical an applied magnetic field not only restores the magnetic properties of the doped compounds. order as it does in all types of ferromagnetic materials but Specific heat measurements are understood to be an also gives rise to metallic conductivity. The coincidence of instructive probe in describing low temperature properties of the ferromagnetic– paramagnetic phase transition with the the materials. Among various manganites, hole-doped sys- metal–insulator transition is generally believed to be associ- tem are widely analysed experimentally due to different elec- ated with the double-exchange mechanism. Besides a tech- tronic and magnetic structure with carrier density. The spe- nological interest in CMR effect, doped manganites are fas- cific heat for La1−x SrxMnO3-doped system reflects the first- cinating materials in that they exhibit a wide variety of order phase transition and the phase transition are observed unusual physical properties, which makes them considerably below Curie temperature. Based on the temperature and field more complex than conventional transition metal ferromag- dependent measured specific heats, it affected the two basic nets (Tokura and Nagosa 2000; Salamon and Jaime 2001; thermodynamic parameters, one being the magneto-caloric Das and Dey 2007). effect, i.e. the adiabatic change in temperature and the other, The systematic study of thermodynamic properties of isothermal change in , induced by the change in solids has become one of the fascinating fields of condensed external magnetic field. The magnetic transition from low- matter physics in the recent years. Thermodynamics play an temperature ferromagnetic to the paramagnetic phase was important role in explaining the behaviour of manganites, as clearly observed (Szewczyk et al 2005). The observed specific heat hump near TC is associated with magnetic ordering and broadening of the peak is pre- sumably due to magnetic inhomogeneities and/or chemi- ∗ Author for correspondence ([email protected]) cal disorders in the half-doped La0·5−x LnxCa0·5−ySry MnO3

1255 1256 Dinesh Varshney, Irfan Mansuri and E Khan

(Ln = Pr, Nd, Sm) samples (Mansuri et al 2011). The spe- structure calculations in order to estimate the electronic spe- cific heat of the divalent-doped manganites is also reported cific heat. In the case of manganites, separation of spin- and it has argued that the phase transition is of second order wave contribution from the lattice effect is difficult without and can be well modelled as a Peierls transition in a dis- an accurate estimation of the Debye temperature. Also, it ordered material. The heat capacity data shows two transi- is worth to seek the possible role of and magnons tions with the transition at higher T exhibiting a much larger in the low temperature specific heat of manganites that has change in entropy than the lower transition. The magnetiza- immediate important meanings to reveal the phenomenon of tion data infers FM transition at higher temperature and AFM colossal magnetoresistance. transition at lower temperature (Cox et al 2007). The present investigation is organized as follows. In On the other hand, for the strongly correlated system, there § 2, we introduce the model and sketch the formalism may be another contribution arising from the renormalization applied. As a first step, we estimate the Debye temperature of the effective e–e interactions, which can modify density of following the inverse-power overlap repulsion for nearest- states at the . In order to get more insight into neighbour interactions in an ionic solid. We follow the Debye e–e interaction, Xu et al 2006 tried to separate its contribu- method to estimate the specific heat contributions of phonons tion from the specific heat for La2/3Ca1/3MnO3 manganite. and magnons. Within this scheme for phonons, electrons and The unique transport behaviour in manganites at low tem- magnons specific heat, we have computed the low tempe- perature can be understood by taking into account both e–e rature specific heat of La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3 interaction and weak spin disorder scattering including the manganites. In § 3, we return to a discussion of results spin polarization and grain boundary tunneling. This kind of obtained by way of heat capacity curves. A summary of the electron–electron interaction is a general characteristics and results is presented in § 4. We believe in giving detailed infor- verifies the strong correlated interaction between electrons in mation on physical parameters in correlated electron system, manganites (Xu et al 2006). as manganites can be understood by low temperature specific Subsequently, heat capacity of La0·78Pb0·22MnO3 in the heat behaviour. temperature range 0·5 ≤ T ≤ 25 K is measured and found that samples having ferromagnetic insulating or antiferro- magnetic ordering exhibits spin-wave signature in specific 2. Lattice specific heat heat data (Ghivelder et al 2001). Salient features of the reported specific heat measurements include a linear term The acoustic mode frequency shall be estimated in an ionic Ze =− e (γT) at low temperature. The contribution to electronic model using a value of effective ion charge 2 .The term (γT) was extracted from the total heat capacity and is Coulomb interactions among the adjacent ions in an ionic argued that this contribution is mainly due to the electronic crystal in terms of inverse-power overlap repulsion is given heat capacity and emphasized that the specific heat can be as (Varshney et al 2002) resolved into a γT contribution with a finite value of γ and   1 f a lattice contribution. (r) =−(Ze)2 − , (1) r rs On the theoretical side, explanation of specific heat behaviour for a wide doping concentration range has, there- where f being the repulsion force parameter between the ion fore, been of considerable interest. The motivation is two- cores. The elastic force constant κ is conveniently derived fold: (i) to reveal the importance of the Coulomb correlations from (r) at the equilibrium inter-ionic distance r0 following which is significant in controlling the transport properties in     ∂2 s − 1 manganites and (ii) to understand the role of magnons as κ = = (Ze)2 . (2) ∂r2 r3 well as electron–phonon in specific heat of the ferromagnetic r0 0 metal or insulator. It is indeed essential to find out if these Here, s is the index number of the overlap repulsive poten- novel materials get their metal to insulator transition from  tial. Then we use acoustic mass, M = (2 M+ + M−) [Mn an extremely strong interaction within the double-exchange (O) is symbolized by M+(M−)], κ∗=2κ for each direc- mechanism and electron-lattice interaction or if any entirely tional oscillation mode to get the acoustic phonon frequency new excitation has to be invoked. as No systematic theoretical efforts addressing the issues of  Coulomb correlations, spin wave impact and lattice contri- ω = 2κ ∗ , D M  bution as well as the competition among these processes  (3) have been made so far, and in the present investigation, we = (Ze) (s−1) 1 . 2 M  r3 try to understand them and retrace the reported specific heat 0 behaviour. We begin with the assumptions and motivate them In most of manganites, the specific heat is dominated by by simple physical arguments before taking up details of quantized lattice vibrations (phonons) and is well estimated numerical results obtained. The idea we had in mind was to by conventional Debye theory. As phonons are bosons and focus attention on the subsisting mass-enhancement mecha- waves of all frequencies are in the range of 0 <ω<ωm nisms in the ferromagnetic metallic regime. We employ the (where ωm is the maximum phonon frequency) they could density of states as deduced from electronic energy band propagate through the crystal. The internal energy of the Phonon, magnon and electron contributions to low temperature 1257 crystal when the maximum energy of the phonons in the The internal energy of unit volume due to magnon spin crystal, x = ω/kBT,is(Tari2003): wave in thermal equilibrium at temperature T is given by (Kittel 1987)   θD/T 3 3 T x   xm U = U + Nk T x, 3/2 3/2 0 9 B x d (4) kBT kBT x θD (e − 1) U = dx, (9) magnon 2 x 0 4π Dstiff (e − 1) 0 The specific heat from lattice contribution is obtained by x =[ + D k2]/k temperature derivative as where stiff BT. The specific heat is usually obtained by temperature derivative for long-wavelength spin θ /T   D waves, which are the dominant excitations at low tempera- T 3 x4ex C = Nk T x, ture, as Phonon 9 B x 2 d (5) θD (e − 1) 0   ∞ k k T 3/2 x5/2ex C = B B dx. (10) θ magnon 2 2 where D is the Debye temperature and N the number of 4π Dstiff (ex − 1) atoms in a unit cell. We shall now switch on to estimation of 0 electronic contribution. Henceforth, the total contribution to specific heat is

2.1 Electronic specific heat Ctot = Cphonon + Cele + Cmagnon, (11)

The internal energy of electron gas is expressed as where first term on the right side, Cphonon, is the phonon 3 contribution and varies as βT [T << θD/10], the coeffi-    / 2π 3mV 3n 1 3 cient β is related to the Debye temperature θ . Second term, U = U + (k T )2 , D 0 B (6) C γT 32 π ele, is the fermionic contribution , the linear coeffi- cient (γ ) can be related to the density of states at the Fermi m being the mass of electron as carriers and n denotes charge surface (Varshney and Kaurav 2006). Last term, Cmagnon, density in volume V. is associated with ferromagnetic spin-wave excitations and The temperature derivative of the internal energy yields vary as δT 3/2 in the low temperature regime. Terms with the specific heat and is higher powers of T than T3 are anharmonic corrections to the lattice contribution and are ignored for the sake of sim- π 2 plicity in the present study. Within the three-component C = k2 N (E ) T = γT, (7) ele 3 B F scheme [phonon,electron and magnon], we have estimated the specific heat of La · Sr · MnO and Er · Y · MnO (E ) 0 85 0 15 3 0 8 0 2 3 where N F is the density of states at and manganites. The calculated results along with discussion are γ denotes the Sommerfield constant. It is evident from (6) presented in the following section. that the electronic specific heat is influenced by the effec- tive mass of the carriers as well as by the carrier concentra- tion. Electron correlation effects are well known to be impor- 3. Results and discussion tant for the transition-metal oxides and the importance of renormalization in mass due to electron–phonon interaction Any discussion of the manganites necessitates knowledge (Carbotte 1990) and electron correlations for the perovskites of the crystal structure and this is particularly true of is assessed in the later section. the calculations reviewed here. We begin with the discu- ssion of phononic behaviour of specific heat. There can be 2.2 Spin wave specific heat orthorhombic, rhombohedral and cubic phases in the entire range of doping both as a function of temperature and as a Spin particles as magnon obey Bose statistics and allow us function of concentration. While calculating the Debye tem- for the calculation of the low-temperature thermal proper- perature, we take s = 8 and the in-plane Mn{O distance ties of magnetic materials. Only long-wavelength spin waves r0 = 2 ·76 [3·061] Å (Urshibara et al 1995; Sekhar et al are excited at low temperature and the dispersion relation for 2005), yielding κ = 3·068 [2·2252]×105 gs−2 due to the magnon in a spin system fact that the chemical pressure induces different bond length with composition. Keeping the above fact in mind, the Debye 2 ω(k) = + Dstiffk . (8) temperature from (3) is estimated as 386 {415} K and is essential for the estimation of phonon contribution in specific Here, the spin-wave excitation at zero field is assumed and heat for La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3 manganites. Dstiff (or D) is the spin stiffness, which is a linear combination It is useful to point out that for correlated electron systems of the exchange integrals. is being the anisotropy spin- as cuprates and manganites, index number of the repulsive wavegap (Varshney and Kaurav 2004). potential has been reported to be s = 8 (Varshney et al 2002). 1258 Dinesh Varshney, Irfan Mansuri and E Khan

The Debye temperatures (θD), 386 K and 415 K, yield β = −4 −1 Er0.8 Y0.2 MnO3 0·169 and β = 0·14 mJK mol for La0·85Sr0·15MnO3 Cexperiment and Er0·8Y0·2MnO3, respectively and are consistent with the C reported data (Uhlenbruck et al 1998; Tokura and Tomioka 40 theoretical fit

1999). However, we do not claim the process to be rigo- ) 1 rous, but a consistent agreement following overlap repul- – Cmagnon

-K 30 sion is obtained on the Debye temperature as those revealed 1 from specific heat data. Usually, the Debye temperature is a – function of temperature and varies from technique to tech- J- mol nique. Values of the Debye temperature also vary from sam- 20 C C ( electron ple to sample with an average value and standard deviation of θD = θD± 15 K. The spin-wave stiffness coefficient is D = JSa2 10 evaluated following 2 ,whereS (average value of C the spin) and J (magnetic exchange coupling parameter) are phonon directly related as S = (4 − x)/2andJ = TckB/4S(S + 1) 10 20 30 40 50 (Perring et al 1996)togetD values as 27 and 44 meV Å. T (K) The specific heat behaviour of La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3 are discerned in figures 1 and 2, respectively. Figure 2. Variation of specific heat with temperature. Solid cir- Here, we have used (5), (7)and(10) for getting Cphonon, cles are experimental data taken from Sekhar et al (2005). Celectron and Cmagnon contributions. The lattice specific heat follows a conventional T3 characteristic and is noticed from the curve that the specific heat from lattice contribution is lesser to that of magnon specific heat below 9 K. On the other yield consistent interpretation of specific heat in ferromag- hand, above 9 K, the magnon contribution is reduced in com- netic insulator (meaning insulating material in ferromagnetic parison to the lattice specific heat. Both the contributions are states). clubbed and the resultant behaviour is plotted along with the In an attempt to understand the observed metallic experimental data for La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3 behaviour in the low temperature domain, it is important to (Okuda et al 1998; Sekhar et al 2005) in figures 1 and 2. explore the possible role of mass renormalization of carri- We find consistent results of heat capacity as those revealed ers due to differed values of γ obtained from electron energy from experiments. The consistency is attributed to the proper band structure calculation and specific heat measurements. estimation of Debye temperature from inverse power over- For La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3, the manganite has lap repulsion as well as other free parameters (spin-wave ferromagnetic ordering. Perhaps the dilemma in determining stiffness, average value of the spin and magnetic exchange the magnetic specific heat of manganites is that it represents coupling parameter). Henceforth, a T3 contribution from the the spin-wave stiffness and anisotropy related with spin-wave 3/2 lattice and a T contribution from the spin wave motion gap. The stiffness is proportional to Tc and S,whereTc is the Curie temperature and S the average value of the spin whereas, anisotropy gap has been determined from neutron- scattering results and is 2·5 ± 0·5meVforLa0·7Pb0·3MnO3 La Sr MnO 0.85 0.15 3 (Okuda et al 1998). By the above intrinsic fashion fo- 0.16 llowed by these compositions, the experimental data can be C fitted qualitatively by the Debye temperature of 386 and experiment Cmagnon ) − − C 415 K, which yields β = 0·169 and 0·14 mJK 4 mol 1 for -1 0.12 theoretical fit

-K La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3, respectively. -1 We shall now address importance of the Coulomb corre- 0.08 lations in revealing the transport properties in manganites. ) = C Using the density of states at the Fermi surface N(EF C (J - mol electron 6·9 × 1037 eV−1 mol−1 and 6·88 × 1037 eV−1 mol−1 from 0.04 electron energy band structure calculations (Pickett and Singh 1996)forLa· Sr · MnO and Er · Y · MnO , C 0 85 0 15 3 0 8 0 2 3 phonon respectively for both ferromagnetic cubic perovskite and . 0 00 ferromagnetic Pnma structure, we find γ = 0·5and1·5mJ 45678910K−2 mol−1. It is instructive to mention that the band struc- T (K) ture studies do not incorporate the Coulomb correlations and electron–phonon interaction. Difference in γb and γexp ne- Figure 1. Variation of specific heat with temperature. Solid cir- cessarily points to the importance of mass renormalization cles are experimental data taken from Okuda et al (1998). of carriers and indeed significant in manganites. In fact, the Phonon, magnon and electron contributions to low temperature 1259 effective charge carrier mass is sensitive to the nature of electron–phonon coupling localizes the conduction band interaction in Fermi-liquid description. However, in CMR electrons, as polarons become stronger well above the tran- materials the electron–electron interactions are apparently sition temperature. Furthermore, nature of polaron changes important. eg electrons also interact with each other via the across Tc and a crossover from a polaronic regime to a Fermi Coulomb repulsion. Precisely, the enhancement of γ due to liquid-like regime, in which the electron–phonon interaction electron correlations in manganites is, therefore, necessarily is insufficient to localize electrons when T tends to zero to be incorporated. (Millis et al 1995, 1996). The importance of renormalization in mass due to elec- Besides the Coulomb correlations and electron–phonon tron correlations and electron–phonon interaction for the interaction, another possibility for the carrier mass renor- perovskites is evaluated following malization arose due to the presence of spin wave in the metallic system and is caused by spin-wave scattering at γ m∗ exp = . low temperature. It is worth referring to an earlier work of γ m (12) b e Fulde and Jensen (1983), who argued that electron mass, is also enhanced by interaction with spin wave in manga- Here m*andme terms represent the effective mass and mass nites. It was established that La0·67Ca0·33MnO3 is half- of electron, respectively. On the other hand, if we incorpo- metallic (Pickett and Singh 1996) at low temperature, which rate the electron–phonon coupling, electronic specific heat means that the conduction is limited to a single Mn up- coefficient will take the following form spin channel of majority carriers. Mn down-spin channels π 2 are localized, consequently, the normal emission and adsorp- γ = k2 N (E )( + λ) , B F 1 (13) tion of spin waves at finite temperatures are forbidden, since 3 there is no conducting state at low energy to scatter into spin λ being the electron–phonon coupling strength and is cha- flips. Low temperature spin–wave scattering for the up-spin racteristic of a material. Naturally, the electronic specific heat channel is primarily due to phonon or to spin conserving constant is directly proportional to the density of states of the electron–electron processes. electron at the Fermi level. It is worth to stress that under- Figures 1 and 2 illustrate variation of temperature standing of the electronic specific heat coefficient (γ ) is quite dependence of specific heat for La0·85Sr0·15MnO3 and relevant as it is a direct measurement of fundamental pro- Er0·8Y0·2MnO3, respectively. The contributions of phonon perty of metallic materials. While estimating effective mass, and to specific heat are shown separately along following (12), we find γ = 7· 5γb and 5·8 γb is either due to with the total specific heat. It is inferred from the plots that, Coulomb correlations or electron–phonon/magnon interac- at low temperature domain phonons play a significant role; tion for La0·85Sr0·15MnO3 and Er0·8Y0·2MnO3, respectively. electronic specific heat varies linearly with temperature and Similar enhancement in γ is reported earlier and is mainly less to phononic component. It is noteworthy to comment from correlation effects in La1−xSrx TiO3 system and it that, in manganites, the electron contribution to the specific becomes metallic by Sr doping (Okuda et al 1998). heat can at best be seen only at very low temperature due to The Coulomb correlations effect for mass renormali- its small magnitude when a comparison is made with phonon zation is important in metal oxides. We may refer to contribution. an earlier work of Snyder and researchers, who have reported the quadratic temperature dependent resistivity of La0·67Ca0·33(Sr0·33)MnO3, which points to electron–electron 4. Conclusions interaction (Snyder et al 1996). Furthermore, the nonli- nearity in the temperature-dependent resistivity behaviour Heat capacity is generally seen as a useful thermodynami- is attributed to electron–electron scattering. The resistivity cal probe in exploring the role of phonon, electron and spin behaviour follows power temperature dependence over a low wave excitations. In order to simulate the actual situation temperature domain and a better argument is observed using occurring in the temperature dependent behaviour of spe- T2 term, which it is argued as an evidence of electron– cific heat in manganites, we have outlined the theory and electron scattering. analysed the available experimental data by assuming three In passing, we must mention that the mass renormalization channels to heat capacity: phonon, electron and magnon. is caused by all those parameters which are not considered We follow the within harmonic approximation in band structure calculations such as electron–phonon cou- in low temperature regime and used the Debye temperature pling, electron–electron correlations, spin fluctuations, etc. It following the inverse-power overlap repulsion for nearest- is not reasonable to totally exclude electron–phonon contri- neighbour interaction. While computing specific heats we bution to mass renormalization. Regarding the Coulomb co- use spin-wave stiffness, average value of the spin and mag- rrelations, mass renormalization is possibly due to electron– netic exchange coupling as free parameters. Developing this phonon interaction in manganites, although small. Indeed, scheme, the present investigation deals with a quantitative the charge carriers interact with each other and with dis- description of temperature dependent behaviour of specific tortions of the surrounding lattice, i.e. with phonons. The heat in manganites. 1260 Dinesh Varshney, Irfan Mansuri and E Khan

With the above-deduced values of θD, the change in θD Dabrowski B, Kolesnik S, Baszczuk A, Chmaissem O, Maxwell T with δ indicates a reduction of the lattice stiffness or increase and Mais J 2005 Solid State Chem. 178 629 3 Das S and Dey T K 2007 J. Alloys Compd. 440 30 of T -term in the specific heat. Values of θD from the present calculation are comparable to those revealed from heat Fulde P and Jensen J 1983 Phys. Rev. B27 9085 capacity data. Furthermore, specific heat increases and shows Ghivelder L, Freitas R S, Rapp R E, Chaves F A B, Gospodinov M and Gusmao M A 2001 J. Magn. Magn. Mater. 226Ð230 almost T3/2 dependence on temperature and are attributed to 845 spin-wave contribution. We found that a simple analysis head Goto T, Kimura T, Lawes G, Ramirez A P and Tokura Y 2004 Phys. 3 of the specific heat into T contribution from the lattice and Rev. Lett. 92 257201 3/2 T contribution from the spin-wave excitation yields the Kittel C 1987 Quantum theory of solids (New York: John Wiley & correct interpretation of ferromagnetic insulator. Sons, Inc.) The appropriateness of the present analysis depends on the Mansuri I, Varshney Dinesh, Kaurav N, Lu C L and Kuo Y K understanding of band-structure estimation of the density of 2011 J. Magn. Magn. Mater. 323 316 states. Fermionic component as the electronic specific heat Millis A J, Littlewood P B and Shrainman B I 1995 Phys. Rev. Lett. coefficient is deduced following the density of states from 74 5144 electronic energy band-structure calculation. We have incor- Millis A J, Shrainman B I and Mueller R 1996 Phys.Rev.Lett.75 porated the Coulomb correlations and electron–phonon cou- 175 Okuda T, Asamitsu A, Tomioka Y, Kimura T, Taguchi Y and pling strength while estimating the electronic specific heat γ Tokura Y 1998 Phys.Rev.Lett.81 3203 coefficient and we believe that both of these have impor- Perring T G, Aeppli G, Hayden S M, Carter S A, Remeika J P and tant implication in describing the electronic channel to the Cheong S W 1996 Phys. Rev. Lett. 77 711 heat capacity in doped manganites. To an end, retraces of Pickett W E and Singh D J 1996 Phys. Rev. B53 1146 the heat capacity experimental curve in La0·85Sr0·15MnO3 Salamon M B and Jaime M 2001 Rev. Mod. Phys. 73 583 and Er0·8Y0·2MnO3is attributed to the assumption that the Sekhar M C, Lee S, Choi G, Lee C and Park J G 2005 Phys. Rev. phonons in the temperature domain 3 ≤ T ≤ 50 K are ther- B51 14103 mally mobile in the ferromagnetic phase of manganites and Snyder G J, Hiskes R, DiCarolis S, Beasley M R and Geballe T H 1996 Phys. Rev. B53 14434 are usually the major components of specific heat. As θD is about 386 and 415 K in manganites, the Debye model with Szewczyk A, Gutowska M, Dabrowski B, Plackowski T, Danilova T << θ /10 is meaningful at low temperatures. It is relevant N P and Gaidukov Y P 2005 Phys. Rev. B71 224432 D Tari A 2003 The Specific Heat of Matter at Low Temperature to incorporate realistic physical parameters based on experi- (London: Imperial College Press) mental observation that discerned a clear picture of the low Tokura Y and Nagosa N 2000 Science 288 468 temperature transport properties in manganites. Tokura Y and Tomioka Y 1999 J. Magn. Magn. Mater. 200 1 Xu Y, Zhang J, Cao G, Jing C and Cao S 2006 Phys. Rev. B73 Acknowledgement 224410 Uhlenbruck S, Büchner B, Gross R, Freimuth Maria de Leon Financial assistance from the Madhya Pradesh Council of Guevara A and Revcolevschiv A 1998 Phys. Rev. B57 Science and Technology, Bhopal, is gratefully acknowl- R5571 edged. Urshibara A, Moritomo Y, Arima T, Asamitsu A, Kido G and Tokura T 1995 Phys. Rev. B51 14103 Varshney Dinesh and Kaurav N 2004 Eur. Phys. J. B37 301 References 2004 Varshney Dinesh and Kaurav N 2006 Int. J. Modern Phys. B20 Carbotte J P 1990 Rev. Mod. Phys. 62 1027 4785 Cox S, Lashley J C, Rosten E, Singleton J, Williams A J and Varshney Dinesh, Choudhary K K and Singh R K 2002 Supercond. Littlewood P B 2007 J. Phys., Condens. Matter 19 192201 Sci. Technol. 15 1119