ER

DS

a

ARRA

KACDP

1

rPpwmhhmanitd

fmmD

0d

Materials Chemistry and Physics 129 (2011) 896– 904

Contents lists available at ScienceDirect

Materials Chemistry and Physics

jo u rn al hom epage : www.elsev ier .com/ locate /matchemphys

lectrical resistivity of the hole doped La0.8Sr0.2MnO3 manganites:ole of electron–electron/phonon/magnon interactions

inesh Varshney ∗, N. Dodiyachool of Physics, Vigyan Bhavan, Devi Ahilya University, Khandwa Road Campus, Indore 452001, India

r t i c l e i n f o

rticle history:eceived 19 October 2010eceived in revised form 2 May 2011ccepted 13 May 2011

eywords:. Magnetic materials. Computational techniques. Electrical propertieshonon

a b s t r a c t

In this work, a quantitative analysis of reported metallic and insulating behaviour of resistivity in per-ovskite manganites La0.8Sr0.2MnO3 is established. An effective inter-ionic interaction potential (EIoIP)with the long-range Coulomb, van der Waals (vdW) interaction and short-range repulsive interac-tion up to second-neighbour ions within the Hafemeister and Flygare approach was employed toestimate the Debye and Einstein temperature and was found to be consistent with the available exper-imental data. The electrical resistivity data in low temperature regime (T < TMI) were theoreticallyanalyzed within the framework of the classical electron–phonon model of resistivity, for example,the Bloch–Gruneisen (BG) model. The Bloch–Gruneisen (BG) model and terms T2, T4.5 simplify theelectron–phonon, electron–electron and electron–magnon scattering processes. On the other hand,

in high temperature regime (T > TMI) the insulating nature is discussed with Mott’s variable rangehopping (VRH) model and small polaron conduction (SPC) model. For T > TMI SPC model is more appro-priate than the VRH model. The SPC model consistently retraces the higher temperature resistivitybehaviour (T > �D/2). The metallic and semiconducting resistivity behaviours of La0.8Sr0.2MnO3 mangan-ites are analyzed, to the knowledge, for the first time highlighting the importance of electron–phonon,electron–electron, electron–magnon interactions and small polaron conduction.

. Introduction

Perovskite materials of the family R1−xAxMnO3 (where R is aare-earth metal: La, Nd, Pr, and A is an alkaline earth: Sr, Ca, Ba,b) have gained eminence in the recent past as these materialsossess a variety of phases with different behaviours consistentith a strongly correlated electronic system. The partial replace-ent of rare-earth element with a alkaline earth element introduce

oles in the parent LaMnO3 systems which results in disorder, andence with changing temperature, magnetic field and pressure, theanganites exhibit a range of magnetic behaviours, conductivity

nd insulating phases, charge and spin ordering, and accompa-ying transitions between these states. The most interesting part

s the coincident resistive and magnetic transition referred ashe CMR transition for the approximate range (0.2 ≤ x ≤ 0.5) ofoping [1].

Theoretically, Zener’s double-exchange (DE) model is often pre-erred framework within which one can describe the ferromagnetic

etallic state and CMR transition. To gain additional insight into theechanism responsible for the CMR transition, it is stressed that theE model is inadequate to discuss the resistivity in the insulating

∗ Corresponding author. Tel.: +91 731 2467028; fax: +91 731 2465689.E-mail address: vdinesh33@rediffmail.com (D. Varshney).

254-0584/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.matchemphys.2011.05.034

© 2011 Elsevier B.V. All rights reserved.

phase [2]. In the high-temperature region it has been argued thatthe carriers are coupled to the phonon system through a strongJahn–Teller distortion in manganites. Note that various experi-mental data suggest that these manganites material divides intodifferent regions having varying hole densities is inhomogeneous,with coexisting regions of charge-ordered and charge-disorderedphases at intermediate temperatures [3,4].

In this context it has become a key issue whether the double-exchange model or some other new mechanism has mainlydepicted the physics of manganites. Although microscopic calcu-lations recommend that a second mechanism such as a strongpolaronic effect should be implicated to explain the basic physics[5,6]. The existence of polaronic charge carriers in the paramagneticstate of manganites has compelling depicted by many experiments[7,8]. Thus, it is still an open question whether the double-exchange(DE)—or polaronic charge carriers dominates in the doped man-ganites and the transport measurements explicitly the resistivityresponse is an impressive technique to answer it.

The electrical resistivity of the pristine Pr2/3(Ba1−xCsx)1/3MnO3(PBMO) documents two metal–insulator transitions which aresystematically shifted to lower temperatures with Cs-doping. An

upturn in resistivity behaviour below 50 K in PBMO manganiteswere explained due to the combined effect of weak localization,electron–electron and electron–phonon scattering [9]. Further-more, the electrical resistivity above TMI is governed by the small

hemist

pytiefio(

tcitoacttas(i

itpii(tHitscqr

fisim[ritsto

Lwtdodsttsoar

(t

(

(

(

(

(

D. Varshney, N. Dodiya / Materials C

olaron hopping due to a non-adiabatic process. An analogous anal-sis at low temperature well below TMI was also put forward inhe La2/3Ca1/3MnO3 manganites [10]. The minimum of resistivitys strongly dependent on applied magnetic field and proves thexistence of intrinsic weak magnetic disorder. Furthermore at higheld, disappearing of Kondo term; the electronic resistivity followsnly T1/2 dependence characteristic of enhanced electron–electrone–e) interactions and carrier localization.

Earlier, Rozenberg [11] has commented on wrong resistivity fit-ing of manganites at low temperature by Chen et al. [10] as thelaim seems to be doubtful due to the fundamental inapplicabil-ty of the existence of low temperature minimum results fromhe existence of intrinsic spin disorder with a weak magnetic dis-rder characteristic caused from antiferromagnetism backgroundnd ferromagnetic ground state for the analysis of low-temperatureonductivity in polycrystalline manganites. It is further emphasizedhat the quantum effects are inapplicable, in principle, for descrip-ion of low-temperature conductivity in structurally, chemicallynd magnetically disordered bulk polycrystalline manganites. Thepin-dependent tunneling and scattering of carriers between grainsgrain boundary effect) accounts reasonably for the resistivity min-ma [11].

Analysis of charge transport in the ferromagnetic metallic states essential in clarifying the specific mechanisms responsible forhe CMR effect. Electron–electron, electron–magnon scattering andolaronic effects are the major proponents of various conceptions

n electrical transport. The resistivity behaviour in doped mangan-tes is noticed as temperature independent at low temperatures<20 K) [12]. On the other hand, in high temperature regime (>50 K)he resistivity exhibits strong power temperature dependence.owever large value of T2 coefficient of resistivity is incompat-

ble with expected electron–electron scattering. Thus, excludinghe electron–electron scattering as the conduction mechanism andingle magnon scattering is better anticipated at long wavelengthsan qualitatively demonstrate the observed data, but there is nouantitative agreement between the calculated and experimentalesults.

Alternatively, the results of low temperature resistivity of thinlms of lanthanum manganites have been analyzed with spin-wavecattering mechanism, contributing T3.5 dependence of the resistiv-ty [13]. The origin of this T3.5 dependence is accepted for a Ferro

agnet at low temperatures identifying the spin-wave interactions14]. Note that according to spin-wave scattering mechanism theesistivity � varies as T3.5, since mean free path for spin collisionss proportional to T3.5. Taking these features as signatures of resis-ivity, it is set out to examine the resistivity behaviour in terms ofpin-wave interactions. However, in the present paper T4.5 varia-ions partially account for the observed temperature dependencef the resistivity that is of particular interest.

The particular compound chosen for this study isa0.8Sr0.2MnO3, because it can easily be overdoped. In manyays, it is the ideal manganite material. It is thus intended

o provide evidence in favor of the electrical conduction to beominated due to phonons indeed. In a previous resistivity studyn La0.67Ca0.33MnO3 and the observation of power temperatureependence of resistivity is viewed in terms of electron–electroncattering. It is stressed that the extra contribution arising fromhe electron–electron contribution is required in manganiteso analyze the resistivity behaviour [15]. Deviations from thetandard T2 dependence have also been observed in a numberf strongly correlated electron systems, and results of resistivityre presented in terms of an additional scattering contribution to

esistivity by spin-wave interactions.

On the other hand, in the high-temperature insulating regimeT > TMI), the electrical conduction is generally explained either byhe SPC model [16] or by Mott’s VRH model [17]. It is shown that

ry and Physics 129 (2011) 896– 904 897

doping at Mn site influences the polaronic transport as it causeschange in polaron hopping distance and also the polaron concen-tration [18]. To test this idea, Ang et al. have considered Co-dopedbilayer manganite LaSr2Mn2O7 and showed that for T > TMI, theresistivity data could be fit well using the VRH model and the SPCmodel and also by using the thermally activated conduction (TAC)law [19]. The purpose of what follows is to improve the under-standing of the metallic and semiconducting resistivity behaviourin hole doped manganites. Specifically, a T2 (electron–electron) andT4.5 (electron–magnon) terms are introduced and expression for theelectron–phonon interaction is derived for the correct descriptionof resistivity with minimal use of fitting parameters.

The present investigations are organized as follows. In Section2, a model is introduced and sketch the formalism applied. Lateron, technical details are supplied to estimate the phonon contri-bution to resistivity for metallic (T < TMI), polaron conduction forinsulating resistivity (T > TMI) and motivate them by simple physicalarguments before summarizing the results. The Debye and Ein-stein temperatures are obtained following the effective interionicinteraction potential (EIoIP) with the long-range Coulomb inter-action, vdW interaction and the short-range repulsive interactionup to second-neighbour ions within the Hafemeister and Fly-gare approach in an ionic solid [20]. The Bloch–Gruneisen methodis employed to estimate both the independent contributions ofacoustic and optical phonons and their combined effects. For thesemi-conducting region we compute resistivity behaviour withboth Mott’s VRH model and SPC model. The SPC model takes care ofcoherent motion of charge carriers and involves a relaxation due toa low-lying optical phonon mode. Details of the numerical analysisand its results are discussed in Section 3.

The main findings for La0.8Sr0.2MnO3 manganites include:

a) the estimated Debye and Einstein temperatures from the EIoIPare consistent with the specific heat measurements and Ramanspectroscopy results,

b) the classical electron–phonon model of resistivity, for exam-ple., the Bloch–Gruneisen (BG) model consistently retraces thereported metallic resistivity behaviour in the temperature range[T < TMI ∼= 210 K],

c) the T2 and T4.5 terms accounting for electron–electron andelectron–magnon interactions are essential for the correctdescription of resistivity,

d) the La0.8Sr0.2MnO3 is a good metal in the temperature regionT < TMI and Mott Ioffe–Regel criterion for metallic conductivityis valid, kFl > 1, εF� > 1,

e) the VRH model is inappropriate for the description of resistivitybehaviour in the high-temperature region, T > TMI, and

(f) the small polaron conduction model consistently explains thehigher temperature resistivity behaviour (T > �D/2).

A summary and the main conclusions are presented in Section4.

2. Method of computation

A brief description of the Sr-doped lanthanum manganites is first sketched. Theparent compound LaMnO3 is best characterized and the most studied of the man-ganites. However the overlapping of the manganese and oxygen orbital depends onthe geometric arrangement of the ions within the DE framework. The bandwidth ofthe conduction band is primarily determined by the overlapping of the Mn and Oand the larger the overlap, the wider the bandwidth. For a given distance betweenthe Mn and O ions the overlap is largest when the Mn–O–Mn bond angle is 180◦ .Structure, in which manganese and six oxygen ions form regular octahedral, mean-while if La is replaced with a smaller ion (such as Sr and Ba), the octahedra buckle

and the bond angle becomes smaller. Henceforth, bond distances play a crucial rolein governing the properties of manganites. In the test material, we anticipate thatboth acoustic and optical phonons participate in the process of electrical conduction.

The understanding of the dynamical properties of materials requires the formu-lation of an effective interionic potential. To begin with, the following assumptions

8 hemist

aeme

U

U

˛aa

U

ffn

o

ˇ

U

=

dvm

C

a

D

cc

c

a

d

HeasaE

apw(a

B

c

C

98 D. Varshney, N. Dodiya / Materials C

re made: the change in force constants is small; the short-range interactions areffective up to the second-neighbour ions; and the atoms are held together by har-onic elastic forces without any internal strains within the crystal. The crystal

nergy for a particular lattice separation (r) is thus expressed as:

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

The first term is the Coulomb energy, and follows:

C (r) = −∑

ij

ZiZje2

rij= − ˛mZ2e2

r, (2)

m is the Madelung constant [21] and rij being the separation distance between ind j ions. The short-range overlap repulsive energy is the second term in Eq. (1),nd is

R(r)=nbˇij exp

(ri+rj − rij

�

)+ n′bˇii exp

(2ri − krij

�

)+ n′bˇjj exp

(2rj − krij

�

),

(3)

ollowing Hafemeister and Flygare [20]. The ionic radii are ri and rj , k is the structureactor, n (n′) is the number of nearest (next nearest) ions, respectively. Further, theotations b and � denote the hardness and range parameters, respectively.

The Pauling coefficients, ˇij are defined in terms of valency [Zi (Zj)] and numberf the outermost electrons [ni (nj)] in the anions (cations), respectively as:

ij = 1 +(

Zi

ni

)+(

Zj

nj

). (4)

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

V (r) = −

(∑ij

cij

r6ij

+∑

ij

dij

r8ij

), (5)

−(

C

r6+ D

r8

), (6)

ue to dipole–dipole (d–d) and dipole–quadruple (d–q) interactions. The abbre-iations C and D represent the overall vdW coefficients, due to the interactionsentioned in Eq. (1), defined as [21]:

= cijS6(r) + 12

(cii + cjj)S6(0), (7)

nd

= dijS8(r) + 12

(dii + djj)S8(0), (8)

ij and dij are the vdW coefficients due to d–d and d–q interactions. The coefficientsij and dij are derived following variational method [22] as:

ij = 32

eh√me

˛i˛j

[(˛i

Ni

)1/2

+(

˛j

Nj

)1/2]−1

, (9)

nd

ij = 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

(10)

ere me is the electron mass, ˛i is the electronic polarizability and Ni denotes theffective number of electrons of the ith ion. The values of the overall vdW coefficientsre obtained using Eqs. (7) and (8), and weighted in terms of appropriate latticeums [S6 (0), S6(r), S8 (0) and S8(r)] [21]. The individual vdW coefficients cij and dij

re obtained with certainty and accuracy as the excitation energies are ignored inqs. (9) and (10).

Herein, the above description we shall seek the interionic interaction in between pair such as Mn–O and La/Sr–O. It is clear from the above descriptions that theresent effective interionic potential contains only two free parameters (b and �),hich are determined from the equilibrium conditions:

dU

dr

)r=r0

= 0, (11)

nd bulk modulus

T = 19kr0

(d2U

dr2

)r=r0

. (12)

The model parameters obtained from Eqs. (11) and (12) have been used toompute the second order elastic constants (C11, C12 and C44) as [23]

11 = e2

4r04

[−5.112Z2

m + A1 + A2 + B2

2

], (13)

ry and Physics 129 (2011) 896– 904

C12 = e2

4r04

[0.226Z2

m − B1 + A2 − 5B2

2

], (14)

C44 = e2

4r04

[2.556Z2

m + B1 + A2 + 3B2

4

], (15)

where (A1, B1) and (A2, B2) are the short-range parameters for the nearest and thenext nearest neighbours, respectively. These parameters are defined as

A1 =4r3

0

e2

[d2

dr2Vij(r)

]r=r0

, (16)

A2 = 4(r0√

2)3

e2

[d2

dr2Vii(r) + d2

dr2Vjj(r)

]r=r0

√2

, (17)

B1 =4r3

0

e2

[d

drVij(r)

]r=r0

, (18)

B2 = 4(r0√

2)2

e2

[d

drVii(r) + d

drVjj(r)

]r=r0

√2

, (19)

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

Vij(r) = bˇij exp

(ri + rj − rij

�

)+ cijr

−6ij

+ dijr−8ij

. (20)

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

following

� = r0

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

1/3. (21)

Thus, the elastic force constants are estimated in terms of the developed EIoIPfor a pair such as Mn–O and La/Sr–O and then to have the total elastic force con-stants of the Sr doped LaMnO3. This continuum model thus takes care of the clearphysical binding in doped manganites. It is stressed that the simple model basedon the effective potential can describe those cohesive properties of such solids thatdepend on van der Waals interactions. However, the true potential must recognizethe correct charge distribution and the relative orientations of the interacting atomsin manganites which is a complicated task.

The acoustic Debye branch characterized by the Debye temperature �D and anoptical peak defined by the Einstein temperature �E are estimated as a next step.The Debye frequency is characterized as a cut off frequency at the Brillouin zoneboundary. It can be expressed in terms of effective value of ionic mass and elasticforce constant for crystal lattices with two different kinds of atoms such as Mn–Oand La/Sr–O, which we deal with. The acoustic-mode and optical-mode frequenciesare estimated in an ionic model using a value of effective ion charge Ze = −2e.

For the sake of simplicity, we have choose an acoustic mass M = (2M+ + M−) [Mn(O) which is symbolized by M+ (M−)], �* = 2� for each directional oscillation modeto get the acoustic phonon frequency as [15]

ωD =√

2�∗

M, (22)

In this regard, for a pair wise potential, the phonons are optic in origin and thefrequency is determined by the reduced mass of the pair as −1 = M(A)−1 + M(B)−1

where A is the anion (La/Sr, Mn) and B is the cation (O)

ω2LO = � + ˛

, (23)

ω2TO = � − ˛

, (24)

where is the force constant as

˛ = 8�

3(Ze)2

˝, (25)

ωLO (ωTO) symbolized for the longitudinal (transverse) optical phonon frequencyand the volume of the unit cell.

It is known that in the metallic state, the electron–phonon, electron–electron,electron–magnon scattering and polaronic effects are the major proponents of var-ious conceptions in electrical transport [24]. It begins with the description of thescattering of electron–phonon for the resistivity in the ferromagnetic metallic state.To formulate a specific model, the temperature dependent part of the metallic resis-tivity, following the Debye model, is( ) ∫ 2kF ∣ ∣ [ ]

� ≈ 3

he2v2F

kBT

Mv2s 0

∣F(q)∣2 xq3dq

[ex − 1][1 − e−x], (26)

with x = �ω/kBT. In the above, F(q) is the Fourier transform of the potential associatedwith one lattice site, vF being the Fermi velocity, and vs being the sound velocity.

D. Varshney, N. Dodiya / Materials Chemistry and Physics 129 (2011) 896– 904 899

Table 1van der Waals coefficients of La0.8Sr0.2MnO3 (cij in units of 10−60 erg cm6 and dij in unit of 10−76 erg cm8).

cii cij cjj C dii dij djj D

56.5914

Ef

�

w

A

rddo

�

A

m

�

�

haarf

wv

T

as

Maaca

�

Ec

�

Htcao

ab

La/Sr–O 16.28 38.03 100.97

Mn–O 120.18 108.30 100.97

q. (26) shows in terms of acoustic phonon contribution yields the Bloch–Gruneisenunction of temperature dependence resistivity:

ac(T, �D) = 4AacT

(T

�D

)4∫ �D/T

0

x5(ex − 1)−1(1 − e−x)−1dx, (27)

here Aac is a constant of proportionality defined as

ac ∼= 3�2e2kB

k2Fv2

s Lhv2FM

. (28)

As the resistivity is additive, if the Matthiessen rule is obeyed, the resistivity isepresented as a sum � (T) = �0 + �e–ph(T), where �0 is the residual resistivity thatoes not depend on temperature as electrons also scatter off impurities, defects andisordered regions. However, in case of the Einstein type of phonon spectrum (anptical mode) �op (T) may be described as follows:

op(T, �E) = Aop�E2T−1[e�E /T − 1]−1[1 − e−�E /T ]−1, (29)

op is defined analogously to Eq. (28).Thus, the phonon resistivity in the ferromagnetic metallic state is conveniently

odeled by combining both terms arising from acoustic and optical phonons

e–ph(T) = �ac(T, �D) + �op(T, �E). (30)

The total resistivity follows:

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

= �0 + 4Aac(T/�D)4T ×∫ �D/T

0

x5(ex − 1)−1(1 − e−x)−1dx + Aop�2E T−1

× [exp(�E/T) − 1]−1[1 − exp(−�E/T)]−1. (31)

In order to analyze the resistivity data of high temperature region, T > TMI , weave made computation following Model I: variable range hopping as VRH model,nd Model II: adiabatic small polaron conduction as SPC model. In the high temper-ture range, for example, in the paramagnetic insulating state, we have fitted theesistivity data using VRH model. The expression as derived by Mott for conductivityollows

= oh exp

(−T0

T

)0.25

. (32)

ith oh as a constant and T0 is being defined in terms of the density of state in theicinity of Fermi energy N(εF) and the localization length a as

0 = 18kBN(εF )a3

, (33)

In doped manganites the carriers are localized by random potential fluctuationsnd the carriers preferred hopping for the purpose of hopping transport in betweenites lying within a certain range of energy.

Further, the description of resistivity in the temperature range T > TMI , due toodel II is first briefly outlined. It is worth mentioning that the most rapid motion of

small polaron occurs when the carrier hops each time the configuration of vibratingtoms in an adjacent site coincides with that in the occupied site. Henceforth, theharge carrier motion within the adiabatic regime is faster than the lattice vibrationsnd the resistivity for SPC follows:

= �osT exp

(Ep

kBT

), (34)

p is being the polaron formation energy; kB is Boltzmann’s constant. The resistivityoefficient �os is given by

os = kB

n(1 − x)e2D. (35)

ere n is the charge carrier density (∼1020 cm−3), x is the hole (Mn4+) content, e,he electronic charge, and D is the polaron diffusion constant. The polaron diffusiononstant for a typical cubic coordination can be given explicitly as D = a2�/6 where

is being the lattice constant and �, the characteristic frequency of the longitudinalptical phonon that carries the polaron through the lattice.

The parent LaMnO3 at high temperatures is an insulator with a cubic structurend becomes tetragonal due to distortions at low-temperatures. Doping at a La sitey Ca, Sr, Ba and so forth leads to a decrease not only in the structural phase transition

8 19.12 40.34 76.25 286.13 131.53 101.93 76.245 709

temperature but also in the overall behaviour of resistivity. However, for optimizeddoped x ∼= 0.3, the material is still insulating (at about room temperature and higher)and the resistivity is much higher than the Mott limit. The substitution at the La siteby divalent ions changes the valence of the Mn site as outer Mn d-orbital showstwo fold degeneration and the result is Jahn–Teller (breathing type) distortion ofthe oxygen octahedra focused at about each of the Mn occupied (unoccupied) sites.The energy required for the formation of a local lattice distortion is about 0.6 eV persite. Identifying a strong electron–phonon coupling and hence polaronic transportis essential in the doped materials. At high temperatures, the Jahn–Teller distortionsare decorrelated but do not disappear [25].

3. Discussion and analysis of result

Any discussion of the mixed-valent oxides necessitates theknowledge of the structural aspects, and this is particularly trueof the calculations reviewed here. Also applying the availableinformation’s on the developed theory inevitably entails certaincomplications and one has to find suitable data that varies fromtechnique to technique. Special attention is paid in this approachto address the issue whether long range or short range interactionsare at the origin of the transport properties of the divalent Sr dopedmanganites. The effective interionic potential is thus constructedin an easily generalizable manner with realistic value of structuralparameters, which actually control the resistivity behaviour. Thevalues of Debye and Einstein temperature have been computedusing the values of the two model parameters, namely, range (�),and hardness (b), for a pair such as La/Sr–O and Mn–O, which havebeen evaluated from the experimental values of equilibrium dis-tance and bulk modulus.

The values of the overall vdW coefficients C and D involvedin Eqs. (7) and (8) have been evaluated from the well-knownSlater–Kirkwood variational methods [22] which are listed inTable 1. It is perhaps worth remarking that we have deduced thevalues of hardness b and range � are listed in Table 2. The cal-culated values of C11, C12 and C44 are 7.178, 3.331 and 2.242 (inunits of 1012 dyn cm−2), respectively. The elastic force constant �is derived at the equilibrium inter-ionic distance r0. The calculatedvalue of � is 2.426 × 105 g s−2. Due to lack of other theoretical cal-culations the deduced value of model force constant could not becompared. Nevertheless, the evaluation of the elastic force con-stant gives a valuable guide to the behaviour of vdW type forcesin predicting the model phonon energies. Yet the goal is not todetermine the manner in which such interactions leads to thedescription of phonon frequencies but to discuss the electrical resis-tivity behaviour.

Let us first focus on the results of Debye and Einstein tem-peratures required for estimation of the acoustic (optical) phononcontribution to the temperature-dependent metallic resistivity inthe ferromagnetic metallic state. The deduced value of the Debyetemperature (�D = 382 K for x = 0.2) for La1−xSrxMnO3 manganitesis consistent with the reported value from heat capacity measure-ments [30,31]. For the x − 0.2, the linear term in heat capacityresults a � value of about 3.3 mJ K−2 mole−1 corresponding to �D

of 400 K. However, this value was extracted from data at lowertemperature (T ≤ 20 K) fitted directly to the Debye function [30,31].The consistency of calculated and measured Debye temperature is

attributed to the fact that we have derived the elastic constantsby considering short- and long-range interactions contributing tothe crystal energy. Thus the assumptions made for formulatingEIoIP are valid. Usually, the Debye temperature is a function of

900 D. Varshney, N. Dodiya / Materials Chemistry and Physics 129 (2011) 896– 904

Table 2Input crystal data and model parameters for La0.8Sr0.2MnO3.

Input parameters Model parameters

ri (A) rj (A) r0 (A) BT (GPa) b (10−12 erg) � (10−1 A)

28]

27]

tms�

ωviJttoipwso

msaemdcwtf

pMaianecract

eit[baTttwuaa

it

with residual resistivity and phonon resistivity were unsuccess-ful. It is noteworthy to comment that in conventional metals, theelectron–electron contribution to the resistivity can at best be seen

16014012010080604020

4

5

6

7

8

9

Experimental Data Theoretical ρ0 + ρe-ph

La0.8Sr 0.2MnO3

ρ ( Ω

- cm

)

La/Sr–O 1.36 [26] 1.18 [26] 2.510 [Mn–O 0.65 [26] 1.40 [26] 1.963 [

emperature and varies from technique to technique. It is thus com-ented that the values of the Debye temperature also vary from

ample to sample with an average value and standard deviation ofD = �D ± 15 K.

Further the optical phonon frequencies are computed asLO ≈ 54.96 meV (643 K) and ωTO ≈ 52.54 meV (614 K). Deducedalues of the LO/TO frequencies are consistent with those observedn Raman spectra [32] and are associated with the dynamicahn–Teller distortion, arising from the local lattice distortion dueo the strong electron–phonon coupling. It is worth mentioninghat a direct relationship has been established between the degreef the Jahn–Teller distortions of MnO6 octahedra and conductiv-ty and magnetic properties of the structure [7]. This necessarilyoints to the fact that the optical phonon mode can be correlatedith the degree of the Jahn–Teller distortion activated modes corre-

ponding to bending and stretching oxygen vibrations of the MnO6ctahedra.

Nevertheless, at least in some manganites, the two-orbitalodel based on Wannier functions predicts the electronic states

uch as charge ordering. It is fair to note that the Wannier functionpproach of the electronic problem is useful for the description oflectron dynamics following semi-classical theory as well as theagnetic interactions in solids [33]. Because of the difficulties in

ealing with localized wave functions, we do not intended to dis-uss the electron dynamics as well the magnetic interactions, bute aimed in determining the acoustic (optical) phonon frequency

o estimate the electron–phonon contribution of resistivity in theerromagnetic metallic phase.

In what follows, the developed effective interionic interactionotential takes care of the interactions in between a pair such asn–O and La/Sr–O. The interactions thus are attractive Coulomb,

nd van der Waals (vdW) as well as short-range overlap repulsiventeraction following Hafemeister and Flygare type potential. Thedvantage of using this potential is that it takes care of number ofearest (next nearest) ions, the valency, and number of the out-rmost electrons in the anions (cations), respectively. Thus it takesare of the structural parameters that yield an approximately a cor-ect description of the interactions between a pair such as Mn–Ond La/Sr–O. Henceforth; we are able to estimate the acoustic (opti-al) phonon frequency consistent with the Raman measurementso estimate the electron–phonon contribution of resistivity.

Nevertheless, the mean field evaluation of the elastic param-ters gives a very valuable guide to the behaviour of molecularnteractions with emphasis on Mn–O bond lengths to evaluatehe Mn–Mn and Mn–O force constants for the lattice distortions25]. On the other hand, in the present model we have consideredoth Mn–O and La/Sr–O bond lengths to obtain the Mn–O, La/Sr–Ond total force constants for strong electron–phonon interaction.he formulated effective interionic interaction potential includeshe long-range Coulomb, van der Waals (vdW) interaction andhe short-range repulsive interaction up to second neighbour ionsithin the Hafemeister and Flygare approach. The resulting val-es of the total force constants in between a pair such as Mn–Ond La/Sr–O leads to Debye and Einstein temperatures parameters

gree well with those determined from experimental data.

The electrical resistivity behaviour of La0.8Sr0.2MnO3 is dividedn two regions (a) the ferromagnetic metallic for T < TMI and (b)he insulating for T > TMI. For metallic region, we employ the

140 [29] 0.9418 3.54140 [29] 4.244 3.5

Bloch–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 along with the experimental data on polycrystallinesample [34] from Eq. (31) with the earlier choice of �D (382 K)and �E (643 K). In the following calculations, we have used resid-ual resistivity �0 ≈ 0.349 cm and coefficients (Aac and Aop) are23 × 10−3 and 8 × 10−3. It is noticed that the electron–phonon alongwith residual resistivity consistently retraces the metallic resistiv-ity behaviour of La0.8Sr0.2MnO3 manganites. Given the numbers forcoefficients (Aac and Aop) that emerge from the analysis, it seemsfair to conclude that the contribution from acoustic and opticalphonons are a significant, perhaps even the dominant mechanismas compared to electron–electron and electron–magnon in dopedmanganites that we shall discuss later on.

It is noticed that the electron–phonon along with residualresistivity partially retraces the metallic resistivity behaviourof La0.8Sr0.2MnO3 manganites and other temperature-dependentmechanisms such as electron–electron and electron–magnonscattering should also be invoked. Fig. 2 shows the differencebetween the measured � and calculated �diff. [=�exp − {�0 + �e–ph(=�ac + �op)}] is plotted as functions of power temperature. Aquadratic temperature dependence of �diff. is depicted up to about170 K. The quadratic temperature contribution for resistivity isan indication of conventional electron–electron scattering. Thequadratic temperature dependence of �diff. is consistent with theearlier argument made by Urushibara et al. [35].

So far, we have discussed only how electron–phonon inter-action retraces approximately the resistivity behaviour, yet thegoal is to determine the manner in which other interactionsmodify the electrical resistivity. The additional term due toelectron–electron contribution was required in understandingthe resistivity behaviour, as extensive attempts to fit the data

Temperature (K)

Fig. 1. Electrical resistivity vs. temperature of La0.8Sr0.2MnO3 manganites. The solidline represents best fit to the equation. � = �0 + �e–ph (=�ac + �op). Hollow circlesrepresent the experimental data [34].

D. Varshney, N. Dodiya / Materials Chemistry and Physics 129 (2011) 896– 904 901

21

3.5

4.0

4.5

5.0La0.8Sr0.2MnO 3

Theoritical points____ Linear fit

ρ diff [

= ρ

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(Ω c

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Experimental Data Theoretical fitted by VRH

T (10 K )

ig. 2. Variation of �diff. [=�exp − {�0 + �e–ph (=�ac + �op)}] with T2 of La0.8Sr0.2MnO3.

nly at very low temperatures, due to its small magnitude in com-arison with the phonon contribution.

The existence of quadratic temperature dependence of resistiv-ty over a wide temperature interval permits one to believe thathe electron–electron scattering is also significant in determin-ng the resistivity in manganites. This infers that electron–electroncattering is also important in the resistivity behaviour of Stron-ium substituted manganites. Slight deviations need an additionalcattering mechanism for consistent explanation of the metallicehaviour of resistivity. At high temperatures, electron scattering

s isotropic due to the predominance of electron–phonon scatter-ng, which is nearly isotropic at high temperatures. Fig. 3 showshe fitting in terms of T4.5 behaviour. Thus, the transport mecha-ism in the near transition temperature region can be attributedo the electron–electron and electron–magnon scattering, whichurther demonstrates that the metallic region lies in the FM phase.or the present temperature region, the contribution of a phonons usually ignored due to the weak vibration of crystal lattice.

For hole-doped manganites Snyder et al. stressed that an addi-ional T4.5 contribution as a result of electron–magnon scatteringrocess is essential [16]. In the recent past, we have also noticed thathe spin-wave scattering in the FM phase is important in discussing

he electrical resistivity behaviour of hole doped La0.67Ca0.33MnO3,a0.8Ca0.2MnO3 and electron-doped La0.8Ce0.2MnO3 manganites15]. It is admitted that in the higher temperature limit, theifference can be predicted linearly with T4.5 in accord with

987654321

4

5

Theoritical points____ Linear fit

La0.8Sr 0.2MnO 3

ρ diff [

= ρ

exp -

( ρ

0 + ρ

e - p

h ) ]

(Ω c

m)

T 4.5 (109 K 3)

ig. 3. Variation of �diff. [=�exp − {�0 + �e–ph (=�ac + �op)}] with T4.5 of La0.8Sr0.2MnO3.

Fig. 4. Plot of ln() vs. T−1/4 of La0.8Sr0.2MnO3 manganites. The solid line representsbest fit to the equation = 0 exp(−T0/T)1/4. Hollow circles represent the experi-mental data [34].

the electron–magnon scattering in the double exchange process.The feature of T4.5 temperature dependence of �diff. is consis-tent with the quantum theory of two-magnon scattering [36]and is valid for half-metallic ferromagnets. Consequently, besideselectron–phonon and electron–electron interaction, another pos-sibility for the changes in carrier density arose due to the presenceof spin wave in the metallic system, and is caused by spin wavescattering.

In view of the above, the metallic behaviour of doped man-ganites shall be addressed. If the high-frequency phonon modes,as deduced are indeed strongly coupled with charge carriers, theeffective mass of the carriers should be substantially enhanced.Following the Fermi liquid approximation, the effective mass ofthe carrier along the conducting Mn–O plane is deduced fromelectronic specific heat coefficient � using, m∗ = 3h2��d/k2

B . Theparameters employed are d = 6.684 A [27] and � = 3.3 mJ mol−1 K−2

[37] to get m* = 2.089me. The two-dimensional charge carrierdensity is obtained as 2.239 × 1018 m−2 following n2Dd2 = 1. Hence-forth, the electron parameters are estimated as the Fermi wavevector kF (3.75 × 109 m−1), the plasma frequency ωp (1.486 eV),Fermi velocity vF (2.1 × 105 m s−1), and the Fermi energy εF

(0.255 eV). However, electronic energy band structure calculations[38] derive the average Fermi velocity as 7.4 × 105 m s−1, muchhigher than the estimate from the Fermi liquid approximation.Previously, we have stressed that the effects induced by electroncorrelations and mass renormalizations by electron–electron inter-actions are crucial in magnetic systems such as doped manganites[39,40].

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

F l)−1

where l is the mean free path. The mean free path inthis approximation is usually related to the Fermi velocity and isestimated following l = vF�. Thus, l of about 8 A for La0.8Sr0.2MnO3 isobtained. Following the Drude relation, �−1 = �0ω2

p/4�, is obtained�−1 = 2.571 × 1014s−1. It is customary to mention that the residualresistance obtained for nominally the same compositions may varysignificantly for different groups of compounds. Theoretically itremains unclear whether �0 only characterizes the sample’s quality

or if there is an intrinsic component in the residual resistivity. Theformer suggestion is argued in the results of resistivity experiments[41]. For the Ba-doped manganites, the resistivity data in the crys-talline films yields �0 as low as 10−6 cm and is in the range of typical

9 hemistry and Physics 129 (2011) 896– 904

m(wl

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0.0070.0060.0050.0040.003-4.5

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-2.5La0.8Sr0.2MnO3

ln(ρ

/T)

T-1 (K-1 )

Experimental Data Theoretical fitted by SPC

02 D. Varshney, N. Dodiya / Materials C

etallic conductors [42]. Using the deduced value of Fermi energy0.255 eV) that shows the narrow band and the scattering rate �,e find product εF� > 1. Thus, the Sr-doped manganites (x = 0.2) at

ow-temperatures are good metal.It is useful to highlight the electron correlations in view of

nhance mass in narrowband materials such as Sr-doped LaMnO3.he larger the electron mass (m* = 2.089me) in correlated electronystems, the smaller the plasma frequency, and hence the reducedero temperature elastic scattering rate in comparison to conven-ional metals. It is perhaps worth noticing that, in hole doped

anganites; the scattering rate at low-temperatures is of the orderf 1014s−1 [39,40]. Furthermore, the Mott–Ioffe–Regel criterion foretallic conductivity is valid, as the mean free path is times [l = 8 A]

arger than the Mn–O bond length (1.9638 A [27]). A significantlynhanced mean free path is an indication of metallic conduction ashe product kFl (∼3) seems to be much larger than unity. Thus, it isppropriate to use the Bloch–Gruneisen expression in estimatinghe electron–phonon contributions.

It is worth to 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 [43]. The low temperature resistiv-ty data (T ≤ 75 K) of the SmSrMnO was successfully fitted usinghe relation � = �0 + �2T2 + �5T5 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.

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. The FM state is a pola-onic metal with a strong anisotropic character of the electronicxcitations, strikingly similar to the pseudo gap phases in heavilynder doped cuprate high temperature superconductors such asi2Sr2CaCu2O8+ı (Bi2212). A strong mass enhancement and a smallenormalization factor obtained are accountable for the metallicroperties [44]. The temperature dependence of resistivity in theetallic state is intimately related to polaronic metallic ground

tate and the insulator-to-metallic state can be attributed to theolaron coherence condensation process acting in concert with theouble exchange mechanism. It is thus commented that the presentheory finds an enhanced mass of holes as carriers [m* = 2.089me]or La0.2Sr0.2MnO3 from the residual resistivity to validate the

ott–Ioffe–Regel criterion. However, a detailed analysis is furtherequired to understand the polaronic metallic state in the ferro-agnetic phase and the condensation process analogous to under

oped cuprate high temperature superconductors. The above shalle addressed in near future.

As a next step, the insulating behaviour of resistivity data inhe high temperature region and paramagnetic insulating stateT > TMI) is discussed. The temperature dependent resistivity ofa1−xSrxMnO3 (x = 0.2) is computed using both VRH and adiabaticPC model. Keeping in mind that the charge carrier motion is fasterhan the lattice vibrations in the adiabatic regime and hence theearest-neighbour hopping of a small Polaron leads to mobilityith a thermally activated form. Fig. 4 shows the plots of ln � ∼ T1/4

evealing good concurrence with the VRH model [Eq. (32)].It is meaningful to check the validity of the VRH model for car-

ier conduction. The values of fitting parameters obtained from

he fit of the high temperature resistivity of La0.2Sr0.2MnO3 fol-owing VRH model are �oh = 1 × 10−2 cm and T0 = 5.506 × 105 K.he fitted value of T0 and of the localization length or the aver-ge hopping distance {a = 4.6 A [45]} yields the density of states at

Fig. 5. Variation of ln((/T) vs. inverse temperature (T−1) of La0.8Sr0.2MnO3 mangan-ites. The solid line represents the best fit to the equation � = �osT exp(Ep/KBT). Hollowcircles represent the experimental data [34].

the Fermi level, N(εF), of the order of 3.898 × 1021 eV−1 cm−3. Theabove sets of values are consistent with the fitting parameters forNd0.33Ln0.34Sr0.33MnO3 [46].

To have a cross check of the density of states at the Fermilevel, we use the value of electronic specific heat coefficient �of 3.3 mJ mol−1 K−2 [37]. This � value leads us to have the valueof N(εF) = 4.335 × 1019 eV−1 cm−3. It is comment that the value ofdensity of states at the Fermi level obtained from semi conduct-ing resistivity fit using VRH is higher than obtained from specificheat measurements for La0.2Sr0.2MnO3. 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 forLa0.2Sr0.2MnO3. As originally proposed by Mott and Davies [47], theVRH model is applicable for the temperature region (TMI < T < �D/2)in doped semiconductors, the above is also true for La0.2Sr0.2MnO3manganites with TMI = 210 K and �D = 382 K. Viret et al. earlierstressed that the above discrepancy in density of states at the Fermilevel might be due to spin-dependent potential in doped mangan-ites [45]. With these facts, we comment that the small polaronconduction model is only plausible for the higher temperatureresistivity behaviour (T > �D). The above results are consistent withprevious work on La1−xNaxMnO3 (x = 0.1 and 0.2) [48].

Usually, in strongly correlated electron systems such as mixedvalent manganites, the carrier hopping is always of a variable-range type at low temperature. As at low temperatures the thermalenergy (kBT) is insufficient to allow electrons to hop to their nearestneighbours. However, electrons conveniently hop to farther to finda smaller potential difference. Thus there is a competition betweenthe potential energy difference and the distance electrons can hop.On the other hand, at higher temperatures, the electron conductionis due to the activation above the mobility edge.

With a motivation that the VRH model is unsuitable for thedescription of the electrical resistivity data for La0.2Sr0.2MnO3, inthe temperature region T > �D/2, we now switch for its explanationfollowing adiabatic small polaron hopping model. In strongly corre-lated electron systems, electron hopping is always of variable-rangetype at low-temperature. As at low-temperatures the thermalenergy (kBT) is insufficient that allows electrons to hop to therenearest neighbours. However, electrons hop to farther neighboursconveniently to find a smaller potential difference. Thus there is a

competition between the potential energy difference and the dis-tance electrons can hop. On the other hand, at higher temperatures,the electron conduction is due to the activation above the mobilityedge.

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D. Varshney, N. Dodiya / Materials C

Following the adiabatic SPC model for the analysis of resistivityehaviour for high-temperature (T > TMI). The diffusion coefficient

s first determined using the lattice constant and the longitudi-al optical phonon frequency as 4.3 × 10−2 cm2 s−1. At x = 0.2, wend �os = 6.786 × 10−4 cm K−1. To elucidate the nature of the chargeransport mechanism in this high temperature region, ln(�/T) haseen plotted as a function of 1/T in Fig. 5. The good linear fitseveal that the conduction in the PM insulating region obeys themall polaron hopping model. The experimental curve in the regionf T > TMI is retraced successfully by using an Ep of 88 meV. Theest-fitted value of Ep is consistent with the earlier reported val-es of polaron formation energy of about 82 meV for La0.825Sr0.175nO3 manganites [49]. It is appropriate to state that the SPCodel with realistic physical parameters successfully retraces the

eported experimental behaviour (T > TMI) of resistivity in Sr-dopedanganites.

. Conclusions

The present investigations report the analysis of experimentallynown behaviour of electrical resistivity in Sr doped manganitesor metallic and insulating state. For T < TMI for example, in the fer-omagnetic metallic state, we have followed the electron–phononnteraction with the model phonon spectrum consisting of twoarts: an acoustic branch of Debye type and optical mode withharacteristic Einstein temperature. Debye and Einstein tempera-ures from the EIoIP with the long-range Coulomb, vdW interactionnd the short-range repulsive interaction up to second neighbourons within the Hafemeister and Flygare approach is deduced. Theroposed model may, in spite of its simple structure, provide aonsistent account for the experimental fact, such as: Debye andinstein temperatures consistent with the specific heat and Ramanpectroscopy measurements. It is emphasized that the conclusionsave been established only within the framework of single (longi-udinal and transverse) optical phonon mode with a flat dispersionelation.

The high-energy optical phonon yields a large contributiono the resistivity and is attributed to significant optical phononardening effect on carrier transport. From such a fitting, signa-ures of different types of interaction terms as electron–phonon,lectron–electron and electron–magnon are found to govern theransport mechanism in the low-temperature phase (T < TMI).aken by itself, this could be attributed to the fact that bothlectron–electron and electron–magnon scattering mechanism aremportant in doped manganites. The temperature dependent resis-ivity data in this region, where the local ferromagnetic orders almost complete, have thus been successfully retraced with

= �0 + �e–ph + �e–e(T2) + �e–mag (T4.5).It is thus argued that apart from the resistivity due to

omain, grain-boundary and the electron–phonon scattering, thelectron–electron and electron–magnon scattering are essen-ial for a complete description of the metallic behaviour ofa0.8Sr0.2MnO3 manganites. The mean free path is larger than then–O bond length and the product kFl > 1 and εF� > 1 favors metal-

ic conduction. Thus we can comment with confidence that it isppropriate to use the Bloch–Gruneisen expression in estimatinghe electron–phonon contributions at T < TMI, and is associated withhe dynamic Jahn–Teller distortion, arising from the local latticeistortion due to the strong electron–phonon coupling.

The resistivity data of insulating state at high temperatureegion, T > TMI, is analyzed by using both VRH and adiabatic SPC

odel. Deduced values of N(εF) from resistivity fit using VRH is

nconsistent as those obtained from specific heat measurements. natural remark from N(εF) observations is that the VRH model

s inappropriate to describe the semi conducting behaviour in the

[

[

[

ry and Physics 129 (2011) 896– 904 903

higher temperature regions (T > �D/2). The small polaron conduc-tion model with realistic physical parameters consistently retracesthe insulating behaviour. The nearest neighbour hopping of a smallpolaron leads to a mobility with a thermally activated form andsuccessfully retraced the reported experimental curve in the para-magnetic phase.

To an end, the ferromagnetic metallic and paramagnetic semi-conducting resistivity behaviour of La0.8Sr0.2MnO3 manganitesis investigated based on electron–phonon, electron–electron andelectron–magnon as well small polaron conduction by deduc-ing Debye, Einstein temperature, the diffusion coefficient and thepolaron formation energy. The developed approach consistentlyexplains the reported behaviour in the low temperature regime(T < TMI) as well in high temperature regime (T > TMI). The schemeopted in the present study is so natural that it extracts only theessential contributions to describe the resistivity behaviour. As �D

is about 382 K in this system, the use of Bloch–Gruneisen expressionand Debye model with T < �D is valid at low-temperatures. Althoughwe have provided a simple explanation of these effects, there isa clear need for good theoretical understanding of the resistivitybehaviour in view of the formation of small polarons may be ofmagnetic origin in manganites.

Acknowledgement

The authors are thankful to M. P. C. S. T. Bhopal for financialassistance.

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