IL

Da

b

a

ARRAA

P7677

KMDES

1

lsbpCaiampopps

vf

(

0d

Journal of Alloys and Compounds 486 (2009) 726–732

Contents lists available at ScienceDirect

Journal of Alloys and Compounds

journa l homepage: www.e lsev ier .com/ locate / ja l l com

nterpretation of temperature-dependent resistivity ofa0.7Ba0.3MnO3 manganites

inesh Varshney a,b,∗, M.W. Shaikh a, I. Mansuri a

School of Physics, Vigyan Bhawan, Devi Ahilya University, Khandwa Road Campus, Indore 452001, IndiaSchool of Instrumentation, USIC Bhawan, Devi Ahilya University, Khandwa road Campus, Indore 452001, India

r t i c l e i n f o

rticle history:eceived 2 April 2009eceived in revised form 2 July 2009ccepted 7 July 2009vailable online 17 July 2009

ACS:5.47.Lx3.70.+h

a b s t r a c t

In this paper, we undertake a quantitative analysis of reported metallic and semiconducting behaviourof resistivity in perovskite manganites La0.7Ba0.3MnO3. An effective inter-ionic interaction potential(EIoIP) with the long-range Coulomb, van der Waals (vdW) interaction and the short-range repulsiveinteraction up to second-neighbour ions within the Hafemeister and Flygare approach is employed toestimate the Debye and Einstein temperature and is consistent with the available experimental data. Thetemperature-dependent resistivity for temperatures less than metal–insulator transition (TMI ∼= 330 K), istheoretically analysed within the framework of the classical electron–phonon model of resistivity, i.e.,the Bloch–Gruneisen (BG) model. Due to inherent acoustic (low-frequency) phonons (ωac) as well as

1.38.−k1.38.Ht

eywords:anganitesebye temperaturelectron–phonon interaction

high-frequency optical phonons (ωop), the contributions to the resistivity have first been estimated anddescribe consistently the reported metallic resistivity behaviour. The condition for metallic conduction,i.e., kF� > 1 and εF� > 1 holds good in Ba-doped manganites. For temperatures, T > TMI, the semiconduct-ing nature is discussed with Mott’s variable range hopping (VRH) model and small polaron conduction(SPC) model. The fitted density of states as revealed from VRH differs drastically from the experimentalvalue. The SPC model consistently retraces the higher temperature resistivity behaviour (T > �D/2). The

tal d

mall polarons comparison of experimen

. Introduction

Hole-doped (Mn3+-rich) manganites R1−xAxMnO3, where R is aanthanide and A is an alkaline-earth cation, have received con-iderable attention in the recent past not only due to rich physicsut also due to its technological importance [1]. R1−xAxMnO3 com-ounds exhibit a metal–insulator transition in the vicinity of theurie temperature (TC), corresponding to the transition betweenferromagnetic metallic phase and a paramagnetic semiconduct-

ng phase. While discussing the correlation between the magneticnd transport properties, ideas such as double exchange (DE)echanism [2] and the Jahn–Teller (JT) formation [3] have been

roposed. The metallic behaviour is usually described in terms

f electron–phonon interaction, however the JT distortion alsolays an important role in the high-temperature paramagnetichase, where the carriers (electrons or holes) are localized asmall polarons due to a strong JT distortion. Henceforth, polaronic

∗ Corresponding author at: School of Physics, Vigyan Bhawan, Devi Ahilya Uni-ersity, Khandwa Road Campus, Indore 452001, India. Tel.: +91 7312467028;ax: +91 7312465689.

E-mail addresses: vdinesh33@rediffmail.com, vdinesh33@gmail.comD. Varshney).

925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.jallcom.2009.07.056

ata appears favourable with the present analysis.© 2009 Elsevier B.V. All rights reserved.

hopping conduction mechanism is effective in describing the semi-conducting behaviour above TC [4].

Raman spectroscopy and neutron inelastic scattering measure-ments had obtained most convincing evidence for the phonons andstructural properties related to phase transition. The first orderRaman spectra of single phase (cubic) La1−xBaxMnO3 with x = 0.3shows a broad hump at 465 cm−1 and the compound is a metal-lic ferromagnet. It is noticed that for x in the range of 0.25–0.35,the structure is cubic with space group Pm3m for which there areno Raman active modes. The broad hump at about 465 cm−1 couldarise from the free carrier contribution to Raman scattering [5].

Furthermore, as concerned to phonon properties and thelow-temperature properties of the materials, heat capacity mea-surements are understood to be an instructive probe. Heat capacitymeasurement of La1−xBaxMnO3 reveals the electronic linear termand a lattice cubic term. Observed Debye temperatures are typi-cal of perovskites, and the carrier density of states is found to beenhanced over free electron and band structure estimates [6].

In general, the electrical resistivity of doped manganites as func-

tions of temperature exhibits three regimes (a) low-temperaturemetallic like conduction with an unexpectedly large absolute valueof the resistivity, (b) an abrupt drop in resistivity associated withmagnetic ordering, and (c) high-temperature activated conduc-tion. Analysis of charge transport in the ferromagnetic metallic

s and

sfep

lp1Kcilsdseyei

revott[dVd

2obspEiirmBtrmman

(asortmihtbs

2

m

D. Varshney et al. / Journal of Alloy

tate is essential in clarifying the specific mechanisms responsibleor the resistivity behaviour. Electron–phonon, electron–electron,lectron–magnon scattering and polaronic effects are the majorroponents of various conceptions in electrical transport.

It has been first pointed out by Alexandrov et al. [7] that theow-temperature metallic state (T < TMI) in doped manganites is aolaronic Fermi liquid. The resistivity data of La1−xCaxMnO3 below00 K is consistently retraced by polaron mechanism. However,ubo and Ohata [8] stressed that the perfect spin polarization ofonduction electrons makes a qualitative change in the scatter-ng processes of charge carriers by electron–magnon interactions,eading to T4.5 dependence in a DE system. In a previous resistivitytudy on La0.67Ca0.33MnO3 the observation of power temperatureependence of resistivity is viewed in terms of electron–electroncattering. We stress that the extra contribution arising from thelectron–electron contribution is required in manganites to anal-se the resistivity behaviour [9]. These results suggest that thelectron–phonon, electron–electron and electron–magnon scatter-ng must be an important cause of resistivity in the metallic state.

On the other hand, in the high-temperature semiconductingegime (T > TMI), the electrical conduction is generally explainedither by small polaronic conduction (SPC) model [10] or by Mott’sariable range hopping (VRH) model [11]. We may refer to the workf Jaime et al. who have suggested that doping at Mn site influenceshe polaronic transport as it causes change in polaron hopping dis-ance and also the polaron concentration [12]. Recently, Ang et al.13] have showed that for T > TMI, the resistivity data of their Co-oped bilayer manganite LaSr2Mn2O7 could be fitted well usingRH model and SPC model and also using thermally activated con-uction (TAC) law.

The present investigations are organized as follows. In Section, we introduce the model and sketch the formalism applied. Latern, we supply technical details to estimate the phonon contri-ution to resistivity for metallic (T < TMI), polaron conduction foremiconducting resistivity (T > TMI) and motivate them by simplehysical arguments before summarizing our results. The Debye andinstein temperatures are obtained following the effective inter-

onic interaction potential (EIoIP) with the long-range Coulombnteraction, van der Waals (vdW) interaction and the short-rangeepulsive interaction up to second-neighbour ions within the Hafe-

eister and Flygare approach in an ionic solid [14]. We employ theloch–Gruneisen method to estimate the independent contribu-ions of acoustic and optical phonons and both. For semi conductingegion we compute resistivity behaviour with both Mott’s VRH

odel and small polaron conduction (SPC). The small polaronodel takes care of coherent motion of charge carriers and involvesrelaxation due to a low-lying optical phonon mode. Details of theumerical analysis and its results are discussed in Section 3.

The main findings for La0.7Ba0.3MnO3 manganites include:a) Deduced Debye and Einstein temperatures from the EIoIPre consistent with the specific heat measurements and Ramanpectroscopy results, (b) the classical electron–phonon modelf resistivity, i.e., the Bloch–Gruneisen (BG) model consistentlyetraces the reported metallic resistivity behaviour in the tempera-ure range [T < TMI ∼= 330 K], (c) the Mott–Ioffe–Regel criterion for

etallic conductivity is valid, kF� > 1, εF� > 1, (d) the VRH models inappropriate for the description of resistivity behaviour in theigh-temperature region, T > TMI and (e) the small polaron conduc-ion model consistently explains the higher temperature resistivityehaviour (T > �D/2). A summary and our main conclusions are pre-ented in Section 4.

. The model

We begin with a brief description of the Ba-doped lanthanumanganites. The parent compound LaMnO3 is best characterized

Compounds 486 (2009) 726–732 727

and the most studied of the manganites. However the overlappingof the manganese and oxygen orbitals depends on the geometricarrangement of the ions within the DE framework. The bandwidthof the conduction band is primarily determined by the overlappingof the Mn and O and the larger the overlap, the wider the band-width. For a given distance between the Mn and O ions the overlapis largest when the Mn–O–Mn bond angle is 180◦. Such structure,in which manganese and six oxygen ions form regular octahedra.Meanwhile if La is replaced with a smaller ion, the octahedra buckleand the bond angle become smaller. Henceforth, bond distancesplay a crucial role in governing the properties of manganites. In thetest material, we anticipate that both acoustic and optical phononsparticipate in the process of electrical conduction.

We begin by writing an effective two-body inter-ionic potentialto express the crystal energy. For the development of an effectiveinter-ionic interaction, we made the following assumptions: thechange in force constants is small; the short-range interactions areeffective up to the second-neighbour ions; and the atoms are heldtogether by harmonic elastic forces without any internal strainswithin the crystal. The crystal energy for a particular lattice sep-aration (r) as:

U(r) = UC (r) + UR(r) + UV (r). (1)

The first term is the Coulomb energy, and follows:

UC (r) = −∑

ij

ZiZje2

rij= −˛mZ2e2

r, (2)

with ˛m as the Madelung constant [15] and rij being the separationdistance between i and j ions.

The short-range overlap repulsive energy is the second term inEq. (1), and is expressed as:

UR(r) = nbˇij exp( ri + rj − rij

�

)+ n′bˇii exp

(2ri − krij

�

)

+ n′bˇjj exp

(2rj − krij

�

), (3)

following Hafemeister and Flygare [16]. The ionic radii are ri andrj, k is the structure factor, n(n′) is the number of nearest (nextnearest) ions, respectively. Further, the notations b and � denotethe hardness and range parameters, respectively. The Pauling coef-ficients, ˇij are defined in terms of valency [Zi (Zj)] and number ofthe outermost electrons [ni (nj)] in the anions (cations) respectivelyas:

ˇij = 1 +(

zi

ni

)+

(zj

nj

). (4)

The last term in Eq. (1) is the van der Waals (vdW) energy,denoted as:

UV (r) = −

⎛⎝∑

ij

cij

r6ij

+∑

ij

dij

r8ij

⎞⎠ (5)

= −(

C

r6+ D

r8

), (5a)

due to dipole–dipole (d–d) and dipole–quadrupole (d–q) inter-actions. The abbreviations C and D represent the overall vdW

coefficients, due to the interactions mentioned in Eq. (1), definedas [15]:

C = cijS6(r) + 12

(cii + cjj)S6(0), (6)

7 s and

a

D

cW

c

a

avaSie

afpd(a

B

uC

C

C

C

wne

A

A

B

B

28 D. Varshney et al. / Journal of Alloy

nd

= dijS8(r) + 12

(dii + djj)S8(0), (7)

ij and dij are the vdW coefficients due to d–d and d–q interactions.e follow the variational method [17] to derive cij and dij as:

ij = 32

eh√me

˛i˛j

[(˛i

Ni

)1/2+

(˛j

Nj

)1/2]−1

, (8a)

nd

dij = 278

h2

m˛i˛j

[(˛i

Ni

)1/2+

(˛j

Nj

)1/2]2

[(˛i

Ni

)+ 20

3

(˛i˛j

NiNj

)1/2

+(

˛j

Nj

)]−1 . (8b)

Here, me is the electron mass, ˛i is the electronic polarisabilitynd Ni denotes the effective number of electrons of the ith ion. Thealues of the overall vdW coefficients are obtained using Eqs. (6)nd (7), and weighted in terms of appropriate lattice sums [S6(0),6(r), S8(0) and S8(r)] [15]. We believe that there is no uncertaintynvolved in the evaluation of cij and dij, due to the fact that thexcitation energies are ignored in Eq. ((8a) and (8b)).

Herein, the above description we shall seek the inter-ionic inter-ction in between a pair such as Mn–O and La/Ba–O. It is clearrom the above descriptions that the present effective inter-ionicotential contains only two free parameters (b and �), which areetermined from the equilibrium conditions:

dU

dr

)r=r0

= 0, (9a)

nd bulk modulus

T = 19kr0

(d2U

dr2

)r=r0

. (9b)

The model parameters obtained from Eq. (9a) and (9b) have beensed to compute the second-order elastic constants (C11, C12 and44) as [14]

11 = e2

4r04

[−5.112Z2

m + A1 + (A2 + B2)2

], (10)

12 = e2

4r04

[0.226Z2

m − B1 + (A2 − 5B2)2

], (11)

44 = e2

4r04

[2.556Z2

m + B1 + (A2 + 3B2)4

], (12)

here (A1, B1) and (A2, B2) are the short-range parameters for theearest and the next nearest neighbours, respectively. These param-ters are defined as

1 = 4r30

e2

[d2

dr2Vij(r)

]r=r0

, (13)

2 = 4(r0√

2)3

e2

[d2

dr2Vii(r) + d2

dr2Vjj(r)

]r=r0

√2

, (14)

4r3 [d

]

1 = 0

e2 drVij(r)

r=r0

, (15)

2 = 4(r0√

2)2

e2

[d

drVii(r) + d

drVjj(r)

]r=r0

√2, (16)

Compounds 486 (2009) 726–732

where Vij(r) is the short-range potentials between the ions, whichfollow

Vij(r) = bˇij exp( ri + rj − rij

�

)+ cijr

−6ij

+ dijr−8ij

. (17)

The elastic force constant � is derived at the equilibrium inter-ionic distance r0 following

� = r0

2[(3.15)2(C11 − C12)(C11 + C12 + 2C44)(C44)]

1/3. (18)

We have thus estimated the elastic force constants in terms ofEIoIP for a pair such as Mn–O and La/Ba–O and then to have the totalof the Ba-doped LaMnO3. As a next step, we shall now estimate theacoustic Debye branch characterized by the Debye temperature �D

and an optical peak defined by the Einstein temperature �E. TheDebye frequency is characterized as a cut off frequency at the Bril-louin zone boundary, and it can be expressed in terms of effectivevalue of ionic mass and elastic force constant for crystal latticeswith two different kinds of atoms such as Mn–O and La/Ba–O, whichwe deal with. The acoustic mode and optical mode frequencies areestimated in an ionic model using a value of effective ion chargeZe = −2e.

We choose an acoustic mass M = (2 M+ + M−) [Mn (O) is symbol-ised by M+(M−)], �* = 2� for each directional oscillation mode to getthe acoustic phonon frequency as [9]

ωD =√

2�∗

M. (19)

Furthermore, when the phonons belong to optic modes,their frequency is determined by the reduced mass as−1 = M(A)−1 + M(B)−1 with A is the anion (La/Ba, Mn) and Bis the cation (O)

ω2LO = � +

, (20)

ω2TO = � −

, (21)

where is the force constant as

= 8�

3(Ze)2

˝, (22)

ωLO (ωTO) symbolized for the longitudinal (transverse) opticalphonon frequency and ˝ the volume of the unit cell.

We shall now turn to understand the scattering ofelectron–phonon for the resistivity in the ferromagnetic metal-lic state (T < TMI). The electron–phonon, electron–electron,electron–magnon scattering and polaronic effects are the majorproponents of various conceptions in electrical transport. Toformulate a specific model, we start with the general expressionfor the temperature-dependent part of the resistivity, followingthe Debye model, which is given by [18]

� ≈(

3

he2v2F

)kBT

Mv2s

∫ 2kF

0

|F(q)|2[

xq3dq

[ex − 1][1 − e−x]

], (23)

with x =�ω/kBT. In the above, F(q) is the Fourier transform of thepotential associated with one lattice site, vF being the Fermi veloc-ity. vs being the sound velocity. Eq. (23) in terms of acoustic phononcontribution yields the Bloch–Gruneisen function of temperaturedependence resistivity:

�ac(T, �D) = 4AacT(T/�D)4∫ �D/T

x5(ex − 1)−1(1 − e−x)−1dx, (24)

0

where, Aac is a constant of proportionality defined as

Aac ∼= 3�2e2kB

k2F v2

s Lhv2F M

. (25)

s and Compounds 486 (2009) 726–732 729

rrshm

�

if

�

tI

dr

�

o

T

wttte

tmtaff[

�

sda

�

iTitt

Table 1van der Waals coefficients of La0.7Ba0.3MnO3 (cij in units of 10−60ergcm6 and dij inunit of 10−76 erg cm8).

c c c C d d d D

D. Varshney et al. / Journal of Alloy

If the Matthiessen rule is obeyed, the resistivity may be rep-esented as a sum � (T) = �0 + �e–ph(T), where �0 is the residualesistivity that does not depend on temperature as electrons alsocatter off impurities, defects and disordered regions. On the otherand, in case of the Einstein type of phonon spectrum (an opticalode) �op(T) may be described as follows:

op(T, �E) = Aop�E2T−1[e�E/T − 1]−1[1 − e−�E/T ]−1, (26)

Aop is defined analogously to Eq. (24). Thus, the phonon resistiv-ty can be conveniently modelled by combining both terms arisingrom acoustic and optical phonons

e−ph(T) = �ac(T, �D) + �op(T, �E). (27)

Finally, the total resistivity is now rewritten as

�(T, �D, �E) = � + �ac(T, �D) + �op(T, �E)

= �0 + 4Aac

(T

�D

)4

T ×∫ �D/T

0

x5(ex − 1)−1(1 − e−x)−1dx

+Aop�2ET−1

[exp

(�E

T

)− 1

]−1[1 − exp

(− �E

T

)]−1

. (28)

We have further analysed the resistivity data of high-emperature region, T > TMI, using model I: VRH model, and modelI: adiabatic SPC model.

For the temperature ranges T > TMI, we have fitted the resistivityata using VRH model. The expression as derived by Mott [11] foresistivity follows

= �oh exp(

T0

T

)0.25. (29)

Here, T0 is defined in terms of the density of state in the vicinityf Fermi energy N(εF) and the localization length a as

0 = 18kBN(εF )a3

, (30)

ith �oh as constant. In doped manganites, as we dealt with,he carriers are localized by random potential fluctuations andhe carriers preferred hopping for the purpose of hoppingransport in between sites lying within a certain range ofnergy.

We shall now switch to a brief description of the resistivity inhe temperature range T > TMI, due to adiabatic SPC model. We must

ention that the most rapid motion of a small polaron occurs whenhe carrier hops each time the configuration of vibrating atoms inn adjacent site coincides with that in the occupied site. Hence-orth, the charge carrier motion within the adiabatic regime isaster than the lattice vibrations and the resistivity for SPC follows10]:

= �osT exp(

Ep

kBT

), (31)

Ep is being the polaron formation energy, kB is Boltzman con-tant. The resistivity coefficient �os in terms of the charge carrierensity (n), the hole (Mn4+) content (x), the electronic charge (e),nd the polaron diffusion constant (D) is

os = kB

n(1 − x)e2D. (32)

As earlier pointed by Raman measurements the La0.7Ba0.3MnO3

s viewed as cubic perovskite structure with (space group Pm3m).hus, the polaron diffusion constant for a typical cubic coordination

s D = a2�/6 where a is being the lattice constant and �, the charac-eristic frequency of the longitudinal optical phonon that carrieshe polaron through the lattice.

ii ij jj ii ij jj

Mn–O 120.19 180.39 270.87 1542.98 131.53 193.36 284.11 1354.63La/Ba–O 29.82 81.03 270.87 806.03 27.89 89.35 284.11 673.96

3. Discussions and analysis of results

For the actual calculation of the transport properties, it is essen-tial to know the realistic values of some physical parametersgoverning the resistive behaviour. Any discussion of the manganitesnecessitates knowledge of the crystal structure, and this is par-ticularly true of the calculations documented here. There can beorthorhombic, rhombohedral and cubic phases in entire range ofdoping both as a function of temperature and as a function of con-centration. In order to obtain the Debye and Einstein temperaturewe need to determine the elastic constants in terms of two freeparameters: range (�) and hardness (b) for a pair such as La/Ba–Oand Mn–O from the knowledge of the equilibrium distance and thebulk modulus following the equilibrium conditions in Eq. ((9a) and(9b)).

While estimating the free parameters, we first deduced thevdW coefficients from the variational method [17] and are listedin Table 1. Deduced values of range (�) and hardness (b) parameterare listed in Table 2. The calculated values of C11, C12 and C44 are11.75, 5.4 and 3.7 (in units of 1012 Dyne cm−2) respectively. The elas-tic force constant � is derived at the equilibrium inter-ionic distancer0 following Eq. (18). The calculated value of � is 4.42 × 105 gm s−2.

With these parameters, the Debye frequency is estimated as29 meV (339 K), and is required for estimation of the acousticphonon contribution in temperature-dependent resistivity. Thededuced value of the Debye temperature is consistent with thereported value from heat capacity measurements (�D = 333 K forx = 0.3) for La1−xBaxMnO3 manganite [23]. The above consistency ofDebye temperature is attributed to the fact that we have derived theelastic constants by considering short- and long-range interactionscontributing to the crystal energy. Thus the assumptions made forformulating EIoIP are valid. However, this value was extracted fromdata at lower temperature (T ≤ 20 K) fitted directly to the Debyefunction. Usually, the Debye temperature is a function of tempera-ture and varies from technique to technique. We may comment thatthe values of the Debye temperature also vary from sample to sam-ple with an average value and standard deviation of �D = �D ± 15 K.

Furthermore, the optical phonon mode is obtained asωLO ≈ 57.6 meV (675 K) and ωTO ≈ 56.2 meV (658 K). We find theforce constant as 1.1 × 104 gm s−2 from Eq. (22). The calculated val-ues of the LO/TO frequencies are consistent with those observed inRaman spectra [5] and are associated with the dynamic Jahn–Tellerdistortion, arising from the local lattice distortion due to the strongelectron–phonon coupling. It is worth mentioning that a direct rela-tionship has been established between the degree of the Jahn–Tellerdistortions of MnO6 octahedra and conductivity and magneticproperties of the structure [24]. This necessarily points to the factthat the optical phonon mode can be correlated with the degree ofthe Jahn–Teller distortion activated modes corresponding to bend-ing and stretching oxygen vibrations of the MnO6 octahedra.

The electrical resistivity behaviour of La0.7Ba0.3MnO3 is dividedin two regions a) the ferromagnetic metallic for T < TMI, and b)the semiconducting for T > TMI. For metallic region we employ theBloch–Gruneisen method to estimate the independent contribu-

tions of acoustic and optical phonons. Fig. 1 illustrates the resultsof temperature dependence of resistivity via the electron–phononinteraction from Eq. (27) with our earlier choice of �D (339 K) and �E

(675 K). The contributions of acoustic and optical phonons towards

730 D. Varshney et al. / Journal of Alloys and Compounds 486 (2009) 726–732

Table 2Input crystal data and model parameters for La0.7Ba0.3MnO3.

Input parameters Model parameters

ri (Å) rj (Å) r0 (Å)

Mn–O 0.66 [19] 1.32 [19] 1.96 [21]La/Ba–O 1.29 [20] 1.32 [19] 2.98 [21]

Fa

riitarm

odwre

Fe

ig. 1. Variation of �e–ph with temperature for La0.7Ba0.3MnO3, the contribution ofcoustic phonons �ac as well of optical phonons �op to the resistivity.

esistivity are shown separately along with the total resistivity. Its inferred from the curve that �ac increases linearly, while �op

ncreases exponentially with the increase in temperature. The con-ributions are summed and the resultant resistivity is exponentialt low-temperatures, and nearly linear at high-temperatures up tooom temperature. In the following, we have used �0 = 1 × 10−6 -

[23] and coefficients (Aac and Aop) are 1.44 × 10−4 and 3.4 × 10−4.Our numerical results on temperature dependence of resistivity

f La0.7Ba0.3MnO3, are plotted in Fig. 2 along with the experimentalata on polycrystalline sample [25]. The model phonon spectrumith characteristic �D (339 K) and �E (675 K) consistently reveals the

eported resistivity behaviour. We must mention that earlier, Zhaot al. [26] have argued that small polaronic transport is the prevalent

ig. 2. Variation of � with temperature T (K) for metallic state. Open circles are thexperimental data taken from Tripathi et al. [25].

BT (GPa) b (10−12 erg) � (10−1 Å)

179.5 [22] 5.27 3.28179.5 [22] 4.65 1.82

conduction mechanism below 100 K, which involves a relaxationdue to a soft optical phonon mode (86 K) that is strongly coupled tothe carriers. In addition, a small contribution from acoustic phononwould give almost perfect fit with a negligible systematic deviation.On the other hand, we use high-energy optical phonons as wellacoustic phonons to retrace the reported resistivity behaviour.

We now address the metallic behaviour of doped mangan-ites. If the high-frequency phonon modes, as deduced are indeedstrongly coupled with charge carriers, the effective mass of the car-riers should be substantially enhanced. Following the Fermi liquidapproximation, the effective mass of the carrier along the conduct-ing Mn–O plane is deduced from electronic specific heat coefficient� , using, m∗ = 3h2��d/k2

B . The parameters employed are d = 6.75 Å[21] and � = 6.1 mJ mol−1 K−2 [23] to get m* = 3.4 me. The two-dimensional charge carrier density is obtained as 2.2 × 1018 m−2

following n2Dd2 = 1. Henceforth, the electron parameters are esti-mated as the Fermi wave vector kF (3.7 × 109 m−1), the plasmafrequency ωp (1.15 eV), Fermi velocity vF (1.27 × 105 ms−1), andthe Fermi energy εF (0.15 eV). However, electronic energy bandstructure calculations [27] derive the average Fermi velocity as7.4 × 105 ms−1, much higher than our estimate from the Fermi liq-uid approximation. Previously, we have stressed that the effectsinduced by electron correlations and mass renormalizations byelectron–electron interactions are crucial in magnetic systems suchas doped manganites [28].

We may comment that in conventional metals theelectron–phonon scattering is mathematically identical to conven-tional impurity scattering, and leads to a resistivity proportionalto (v2

F �) − 1 where � is the mean free path. The mean free pathin this approximation is usually related to the Fermi velocityand is estimated following � = vF �. We find � of about 14 Å forLa0.7Ba0.3MnO3. We follow the Drude relation, �−1 = �0ω2

p/4�,to obtain �−1 = 1.1 × 1014 s−1. We must mention that the residualresistance obtained for nominally the same compositions mayvary significantly for different groups of compounds. Theoreticallyit remains unclear whether �0 only characterizes the sample’squality or if there is an intrinsic component in the residual resis-tivity. The former suggestion is argued in the results of resistivityexperiments [29]. For the Sr-doped manganites, the resistivitydata in the crystalline films yields �0 as low as 10−5 cm and is inthe range of typical metallic conductors [30]. Using the deducedvalue of Fermi energy (0.15 eV) that shows the narrow band andthe scattering rate �, we find product εF � > 1. Thus, the Ba-dopedmanganites (x = 0.3) at low-temperatures are good metal.

We shall now highlight the electron correlations in view ofenhance mass in narrow band materials such as Ba-doped LaMnO3.We notice that larger the electron mass (m* = 3.4 me) in correlatedelectron systems, the smaller the plasma frequency, and hence thereduced zero temperature elastic scattering rate in comparison toconventional metals. It is perhaps worth noticing that, in hole-doped manganites, the scattering rate at low-temperatures is of theorder of 1014 s−1 [9]. Furthermore, the Mott–Ioffe–Regel criterionfor metallic conductivity is valid, as the mean free path is several

times [� ∼= 14 Å] larger than the Mn–O bond length (1.96 Å [21]). Asignificantly enhanced mean free path is an indication of metallicconduction as the product kF� (∼5) seems to be much larger thanunity. Thus, it is appropriate to use the Bloch–Gruneisen expressionin estimating the electron–phonon contributions.

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D. Varshney et al. / Journal of Alloy

We must refer to the work of Egilmez and researchers whoncorporate the effects of the grain boundary-induced lattice dis-rder on the resistivity in Sm0.55Sr0.45MnO3 at temperatures nearhe metal–insulator transition [31]. The low-temperature resistiv-ty data (T ≤ 75 K) of the SmSrMnO was successfully fitted usinghe relation � = �0 + �2 T2 + �5 T5 with �0 as the residual resistivitynd �2 and �5 are the electron–electron and the electron–phononcattering coefficients, respectively. It is further noticed that in thisemperature range, the disorder does not affect the temperatureependence of � however; it causes an increase in the coefficients0, �2 and �5 by two orders of magnitude. Here, in the present

nvestigations for La0.7Ba0.3MnO3 we comment that residual resis-ivity and the electron–phonon resistivity is sufficient to retrace theeported experimental data on polycrystalline sample [25].

In contrast to electron–phonon scattering as the source of resis-ivity in ferromagnetic metallic (FM) state, the angle-resolvedhotoemission spectroscopy data for the bilayer manganitea1.2Sr1.8Mn2O7 identifies a coherent polaronic metallic groundtate below metal–insulator transition [32]. The FM state is a pola-onic metal with a strong anisotropic character of the electronicxcitations, strikingly similar to the pseudogap phase in heavilynderdoped cuprate high-temperature superconductors as Bi 2212.strong mass enhancement and a small renormalization factor is

ound to account for the metallic properties [33]. The temperatureependence of resistivity in the metallic state is intimately relatedo polaronic metallic ground state and the insulator-to-metallictate can be attributed to polaron coherence condensation processcting in concert with the Double exchange mechanism. We mayomment that the present theory finds an enhanced mass of holess carrier (m* = 3.4 me) from the residual resistivity to validate theott–Ioffe–Regel criterion in La0.7Ba0.3MnO3. However, a detailed

nalysis is further required to understand the polaronic metallictate in the ferromagnetic phase and the condensation process anal-gous to underdoped cuprate high-temperature superconductors.e shall address this issue in near future.We shall now switch to discuss the semiconducting behaviour

f resistivity data in the high-temperature region, T > TMI. We haveomputed the resistivity of La0.7Ba0.3MnO3 using VRH and adia-atic small polaron conduction model. Keeping in mind that the

harge carrier motion is faster than the lattice vibrations in thediabatic regime and hence the nearest neighbour hopping of amall polaron leads to mobility with a thermally activated form.he plots of � versus T for La0.7Ba0.3MnO3 above TMI have been fit-

ig. 3. Variation of � for high-temperature region, T > TMI . Open circles are the exper-mental data taken from Tripathi et al. [25]. Solid line is the fitting by small polarononduction model and dashed line is from the variable range hopping model.

Compounds 486 (2009) 726–732 731

ted with VRH model (dashed line) and SPC model (solid line) inFig. 3.

We shall first discuss the VRH model for carrier conduction.The values of fitting parameters obtained from the fit of the high-temperature resistivity (T > TMI) of La0.7Ba0.3MnO3 by VRH modelare �oh = 2.3 × 10−4 m cm and T0 = 1.074 × 107 K. The fitted valueof T0 and of the localization length or the average hopping distance{a = 4.6 Å [34]} yields the density of states at the Fermi level, N(εF),of the order of 1.16 × 1021 eV−1 cm−3. The above sets of values areconsistent with the fitting parameters for Nd0.33Ln0.34Sr0.33MnO3[35].

To have a cross check of the density of states at Fermi level,we use the value of electronic specific heat coefficient � of6.1 mJ/mol/K2 [23]. This � value leads us to have the value ofN(εF) = 0.696 × 1020 eV−1 cm−3. We comment that the value ofdensity of states at Fermi level obtained from semiconducting resis-tivity fit using VRH is higher by an order as obtained from specificheat measurements for La0.7Ba0.3MnO3. This unphysical result leadus to argue that the VRH model is not a proper choice to describethe resistivity behaviour in the high-temperature region, T > TMI forLa0.7Ba0.3MnO3. As originally proposed by Mott and Davies [11], theVRH model is applicable for the temperature region (TMI < T < �D/2)in doped semiconductors, the above is also true for La0.7Ba0.3MnO3manganites with TMI ∼= 330 K and �D ∼= 333 K. In passing, we mustrefer to Viret et al. who earlier stressed that the above discrepancyin density of states at Fermi level might be due to spin-dependentpotential in doped manganites [34]. With these facts, we com-ment that the small polaron conduction model is only plausible forthe higher temperature resistivity behaviour (T > �D/2). The aboveresults are consistent with previous work on La1−xNaxMnO3 (x = 0.1and 0.2) [36].

With a motivation that the VRH model is unsuitable for thedescription of the electrical resistivity data for La0.7Ba0.3MnO3, inthe temperature region T > �D/2, we now switch for its explana-tion following adiabatic small polaron hopping model representedby the equation, � = �osT exp (EP/kBT). In strongly correlated elec-tron systems, electron hopping is always of variable-range type atlow-temperature. As at low-temperatures the thermal energy (kBT)is insufficient that allows electrons to hop to there nearest neigh-bours. However, electrons hop to farther neighbours convenientlyto find a smaller potential difference. Thus there is a competitionbetween the potential energy difference and the distance electronscan hop. On the other hand, at higher temperatures, the electronconduction is due to the activation above the mobility edge.

We have thus followed the adiabatic SPC model for the analysisof resistivity behaviour for high-temperature (T > TMI). We have firstdetermined the diffusion coefficient using the lattice constant andthe longitudinal optical phonon frequency as 9.62 × 10−4 cm2 s−1.At x = 0.3, we find �os = 1.87 × 10−6 cm K−1. The experimentalcurve in the region of T > TMI is retraced successfully by using an Ep

of 156 meV. The best fitted value of Ep is consistent with the earlierreported values of polaron formation energy of about 175 meV forPr0.8Na0.2MnO3 manganite [37] and ranges from 90 to 150 meV inLa1−x NaxMnO3 (0 ≤ x ≤ 0.15) [38]. We end up by stating that the SPCmodel with realistic physical parameters successfully retraces thereported experimental behaviour (T > TMI) of resistivity in Ba-dopedmanganites.

4. Conclusions

The present investigations reports the analysis of experimen-tally known behaviour of electrical resistivity in Ba-doped LaMnO3manganites for metallic and semiconducting state. For T < TMI, i.e., inthe ferromagnetic metallic state, we have use the electron–phononinteraction with the model phonon spectrum consisting of two

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arts: an acoustic branch of Debye type and optical mode withharacteristic Einstein temperature. Deduced values of Debye andinstein temperatures from the EIoIP with the long-range Coulomb,an der Waals (vdW) interaction and the short-range repulsiventeraction up to second-neighbour ions within the Hafemeisternd Flygare approach are consistent with the specific heat andaman spectroscopy measurements. For the sake of simplicity, aingle (longitudinal and transverse) optical phonon mode has beenonsidered, with a flat dispersion relation.

The high-energy optical phonon yields a large contribution tohe resistivity and is attributed to significant optical phonon hard-ning effect on carrier transport. The mean free path is severalimes larger than the Mn–O bond length and the product kF� > 1nd εF� > 1 favours metallic conduction. Hence, it is appropriateo use the Bloch–Gruneisen expression in estimating the elec-ron phonon contributions at T < TMI, and is associated with theynamic Jahn–Teller distortion, arising from the local lattice dis-ortion due to the strong electron–phonon coupling. The resistivityata of semiconducting state at high-temperature region, T > TMI, isnalysed by using both VRH and adiabatic SPC model. Deduced val-es of N(εF) from resistivity fit using VRH is inconsistent as thosebtained from specific heat measurements. Thus, the VRH model

s inappropriate to describe the semiconducting behaviour in theigher temperature regions (T > �D/2). The small polaron conduc-ion model with realistic physical parameters consistently retraceshe semiconducting behaviour. The nearest neighbour hopping ofsmall polaron leads to a mobility with a thermally activated formnd successfully retraced the reported experimental curve in thearamagnetic phase.

Conclusively, we have made an attempt to analyse the reportedlectrical resistivity behaviour in Ba-doped manganites based onlectron–phonon and small polaron conduction by deducing Debye,instein temperature, the diffusion coefficient and the polaron for-ation energy. The developed approach consistently explains the

eported behaviour in the two-temperature region. The schemepted in the present study is so natural that it extracts only thessential contributions to describe the resistivity behaviour. As �D

s about 339 K in this system, the use of Bloch–Gruneisen expres-ion and Debye model with T �D is valid at low-temperatures.lthough we have provided a simple explanation of these effects,

here is a clear need for good theoretical understanding of the resis-ivity behaviour in view of the formation of small polarons may bef magnetic origin in manganites.

cknowledgement

Financial support from M.P.C.S.T. Bhopal is gratefully acknowl-dged.

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