Explanation of Temperature-Dependent Resistivity of Sm-Doped Cuprates by Electron–Phonon...

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Journal of Superconductivity: Incorporating Novel Magnetism (JOSC) pp586-josc-378954 August 13, 2002 17:3 Style file version June 22, 2002

Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 15, No. 4, August 2002 ( C© 2002)

Explanation of Temperature-Dependent Resistivityof Sm-Doped Cuprates by Electron–Phonon Scattering

Dinesh Varshney,1 K. K. Choudhary,1 and R. K. Singh2

Received and accepted 2 May 2002

The resistivity of electron-doped cuprate Sm1.85Ce0.15CuO4−δ is theoretically analyzed withinthe framework of electron–phonon i.e., Bloch–Gruneisen (BG) model of resistivity. Char-acteristic temperatures as the Debye temperature and the Einstein temperature were firstderived from an overlap repulsive potential. The optical phonons of the oxygen-breathingmode yield a relatively larger contribution to the resistivity compared to the contributionof acoustic phonons above 220 K. While to that, below this temperature, acoustic phononis a major cause of resistivity. Estimated contribution to resistivity by considering bothphonons i.e., ωac (acoustic phonons) and ωop (optical phonons), along with the zero limitedresistivity, when subtracted from single crystal data, infers a quadratic temperature depen-dence over most of the temperature range (25 ≤ T ≤ 300). Power temperature dependenceof ρdiff.{=[ρexp. − (ρ0 + ρe-ph(=ρac + ρop))]} points the contribution of electron–electron in-elastic scattering. The present analysis allows us to infer that the single crystal experimentaldata is well approximated within the framework of BG electron–phonon model of resistivity.Further calculations of superconducting transition temperature and isotope effect exponentfrom Kresin’s strong coupling theory indicates that the electron–phonon interaction plays animportant role in the attractive pairing mechanism.

KEY WORDS: cuprate superconductors; resistivity; electron–phonon scattering.

1. INTRODUCTION

The normal state properties of high temperature(high-Tc) superconductors are anomalous and mayprovide the most important clue to their physics. Thepairing mechanism responsible for superconductivityin high-Tc cuprates is still unknown. A review of pro-posals to create high-Tc superconductors by phononand nonphonon mechanisms covers many conflictingopinions [1]. After more than a decade of intensivestudies we are still far from a complete understand-ing of the physical properties observed in high-Tc

cuprates. The Ln2−xCexCuO4−δ (Ln = Nd, Sm, andPr) series, which is often called “electron-doped” de-spite a Hall coefficient that changes sign as tempera-

1School of Physics, Vigyan Bhawan, Devi Ahilya University,Khandwa Road Campus, Indore 452017, Madhya Pradesh, India.

2M. P. Bhoj (Open) University, Shivaji Nagar, Bhopal 462016,Madhya Pradesh, India.

ture is increased, exhibits properties that are reminis-cent of standard BCS (Bardeen–Cooper–Schrieffer)approach. Further the test material has significant im-plications for revealing the attractive pairing mecha-nism as well for the normal state transport process incuprates [2].

The crystal structure of electron-doped cuprates,Ln2−xCexCuO4−δ (Ln = Nd, Sm, and Pr) (NCCO)is similar to that of their hole-doped counterparts:La2−xSrxCuO4 (LSCO) and YBa2Cu3O7−δ (YBCO).The only difference between the two structures is thateach copper atom is bonded to four oxygen atoms inNCCO, whereas the copper is surrounded by an oc-tahedral of oxygen’s in LSCO [3]. Superconductivityin electron-doped cuprate systems occurs in a verynarrow doping domain 0.14 ≤ x ≤ 0.18, and smallchanges in oxygen stoichiometry (δ ≈ 0.02) have adramatic effect on electrical transport properties,for example, Nd1.85Ce0.15CuO4−δ , δ = 0.02 is a super-conductor with Tc ≈ 25 K and Nd1.85Ce0.15CuO4−δ ,

281

0896-1107/02/0800-0281/0 C© 2002 Plenum Publishing Corporation

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282 Varshney, Choudhary, and Singh

δ = 0.0 shows semiconductor characteristics [2].However, the doping range of superconductivityin LSCO is rather broader 0.05 < x < 0.3 [4]. Thehighest Tc values in the hole-doped cuprates areover five times those in the highest Tc electron-doped cuprates. Therefore, the hole-doped super-conductors have been studied intensively while theanalysis of electron-doped cuprates remains largelyunclear.

The Tc values of electron-doped cuprates arein the range up to 25 K; their in-plane resistivity isquadratic in temperature rather than the linear Tanomaly of hole doped cuprates [5]. Relatively longcoherence length and large mass anisotropy [6] (γ =mc/mab) are other unique and conventional proper-ties. All these spectral features point that electron-doped cuprates may be in a different category fromthe high-Tc cuprates, and the compatibility with stan-dard BCS theory indicates that electron pairing viaexchange of phonons is a likely source of super-conductivity in the electron-doped cuprates. Tsueiand Kirtley presented very powerful and conclu-sive evidence from a tricrystal phase sensitive ex-periment that electron-doped cuprates have d-wavesymmetry [7].

We may also refer to a recent work of Liu andChen, who explain d-wave symmetry by taking thetwo-local-spin-mediated-interaction (TLSMI). Theauthors show that d-wave makes a small contributionto the gap equation. While to that the superconduc-tivity caused by both the TLSMI and the electron–phonon interaction might have a higher value of Tc

than that caused by just one interaction [8]. In viewof the earlier studies, electron-doped cuprates are thepotential candidate for detailed investigations to elu-cidate the transport mechanism in particular the elec-trical resistivity and superconducting variables as wedeal with in present investigations.

In metals, the investigation of the Raman spectrais used as a standard tool to extract the informationabout lattice modes. Conventional metal supercon-ductors rest on the BCS theory [9], which universallyshows that electron–phonon mechanism creates elec-tron pairs. Raman spectroscopy is a powerful tech-nique in investigating the phonon profiles as well theelectron scattering relevant to the phonon–electroninteraction in the superconductors. For an ionic lat-tice such as Ln–Ce–CuO, Raman scattering measure-ments have been successfully employed to probe itsphonon structure as arising in the CuO2 planes [10–12]. The longitudinal optical (LO) and the transverse

optical (TO) phonon frequencies were determinedfrom a Kramers–Kronig analysis of the infrared re-flection spectra.

The phonon spectrum [10] of Sm1.85Ce0.15CuO4

yields phonons of frequencies: 220 cm−1 {R A1g},332 cm−1 {O(2) B1g}, 485 cm−1 {O(2) Eg}, and586 cm−1 {A∗1g}, respectively. The last mode, near586 cm−1, does not belong to any of the regularRaman phonons expected from the symmetry anal-ysis. The other detailed optical studies on electron-doped cuprates [11,12] of lattice mode reveal thesimilar features. The features in the phonon spectrumindicate that the lattice modes are coupled to theelectrons as carriers in the CuO2 planes. Therefore,electron–phonon interactions are expected to play animportant role in the electron–electron pairing mech-anism in this material.

Conventional evidence for significant electron–phonon coupling in the slope of temperature-dependent resistivity is masked in the cuprates bya much larger contribution that may originate fromelectron collisions. However, the measured slopes ofresistivity for various cuprates at optimized dopinglevel do not yield any correlation with their transition-temperature values. Energy band structure calcula-tions [13] yield values of electron–phonon couplingstrength about unity, which would be enough to ac-count for the Tc value of 25 K. The phonon disper-sion curves measured by neutron scattering [14] andphonon line-widths seen in Raman spectra [10] areconsistent with these estimates.

The characteristic linear temperature depen-dence of the resistivity in hole doped cuprates asLSCO and YBCO has received widespread atten-tion. However, for single crystals of Sm1.85Ce0.15CuO4

metal-like temperatures-dependent in-plane and out-of-plane resistivities were reported [15] and are rad-ically different from hole-doped cuprates. It is worthwhile to refer to an early work of Early et al., whofound double superconducting transition in super-conducting Sm1.85Ce0.15CuO4 and attribute it to thegranularity of the polycrystalline samples [16]. Tsueiet al. have reported the resistivity data of single crys-tal [5] of electron-doped cuprate superconductors andfound a nonlinearity in the temperature-dependentresistivity behavior.

The resistivity behavior follows power temper-ature dependence only over a narrow temperaturedomain, and better agreement is observed using aT2 ln(T) term, which, it is argued, was evidence oftwo-dimensional (2D) electron–electron scattering.

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Resistivity of Sm-Doped Cuprates by Electron–Phonon Scattering 283

The resistivity behavior of Nd1.85Ce0.15CuO4 waslater on interpreted by Tralshawala et al. using theelectron–phonon spectral function [17] from tun-nelling measurements. Deduced result on resistivityusing tunnelling α2 F(ω) for the estimation of thephonon ρ(T), and the resulting T2 behavior of non-phonon term in a wide temperature range is of con-ventional three-dimensional (3D) electron–electronscattering and differs from the prediction of Tsueiet al. [5].

In most of the optimized hole doped cuprates thelarge magnitude and strong temperature dependenceof resistivity anisotropy suggest that the holes as car-riers are tightly confined to the planes, and that thein-plane and the out-of-plane conductions obey dis-tinctly different scattering mechanisms, as a signatureof 2D transport. However, as the doping increases, i.e.,in the overdoped region, the superconductivity sup-presses and the resistivity anisotropy reduces an in-dicative of anisotropic 3D transport. The dimensionalcrossover of charge transport from 2D to anisotropic3D occurs in a composition range where superconduc-tivity disappears [18]. The dimensionality of chargetransport is also a significant problem, which we donot discuss in the present work.

The transport properties of theSm1.85Ce0.15CuO4 single crystals are further in-vestigated [19] under hydrostatic pressure up to2 GPa. The pressure does not change the temper-ature dependence of the normal state resistivity,which at high temperature in both directions followsρ ∼ A(P)T2 law, but the coefficient A(P) lowers aspressure is increased. These experimental results[5,15–19] on resistivity motivate us to perform atheoretical analysis on the temperature-dependentbehavior of resistivity in Sm1.85Ce0.15CuO4 cupratesuperconductors.

Usually, the temperature dependence of resis-tivity reflects an overall electron–phonon couplingstrength, more or less at high temperatures as wellas certain details of the electron–phonon scatteringmostly at low temperatures. However, the electron–phonon coupling strength determines the supercon-ducting Tc, hence temperature-dependent behavior ofresistivity and superconducting Tc are intimately re-lated, and we will see whether we found a reasonablevalue of superconducting Tc from the present analysis.

The plan of the present investigation is as fol-lows. In Section 2, we first estimate the Debye tem-perature and the Einstein temperature following theinverse-power overlap repulsion for nearest neighbor

interactions in an ionic solid. We follow Bloch–Gruneisen method to estimate the independent con-tributions of acoustic and optical phonons. We fur-ther estimate superconducting Tc and isotope effectfollowing Kresin’s strong coupling theory. The de-tails of the numerical analysis and its results are dis-cussed in Section 3. The main findings include thefollowing: (i) optical phonons of the oxygen-breathingmode yield a relatively larger contribution to the re-sistivity than do the acoustic phonons, (ii) power tem-perature dependence of ρdiff.{=[ρexp. − (ρ0 + ρe-ph)]}is interpreted in terms of 3D electron–electron in-elastic scattering, and (iii) consistent value of super-conducting Tc and isotope effect is deduced withinKresin’s strong coupling theory. A summary and ourmain conclusions are presented in Section 4.

2. THE MODEL

We start by giving a brief description of the lay-ered structure of Sm2−xCexCuO4. The crystal struc-ture of Ce-doped Sm2CuO4 superconductor is a mod-ified tetragonal structure which is composed of 2Dsheets of CuO2 layers with no apical oxygen atomsabove and below the copper atoms, resulting in copperatoms with square coordination. The stacking layersequence of this system can be described by suchlayers Sm2−xCexO, Sm2−xCexO, CuO2, Sm2−xCexO,Sm2−xCexO . . .The x − y plane containing a and baxes is taken to lie in the CuO2 layer, with the c axislying along the z direction.

The fundamental aspect of the superconductingoxide is inhibited in the CuO2 plane, and the elec-trons moving in this conducting plane yields super-conductivity. If the phonon field forms an electronpair, the phonon modes relating to CuO2 structurewill be responsible for the most. In principle the longi-tudinal modes will contribute to the deformation po-tential leading to a superconducting state. However,because of screening the transverse optical phononmodes contribute effectively to the potential [20]. Thefrequency ωTO is determined by the reduced mass asµ−1 = M(Cu)−1 + M(O)−1. While to that when thephonons belong to acoustic modes, their frequencyis determined by the total mass in a unit cell of theCuO2 net plane: M = M(Cu)+ M(O). In the test ma-terial, we anticipate that both the acoustic and the op-tical phonons participate in the process of electricalconduction.

We begin by discussing the phonon mechanism.

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2.1. Estimation of Debye Temperatureand Einstein Temperature

The acoustic-mode and optical-mode frequen-cies shall be estimated in an ionic model using a valueof effective ion charge Ze = −2e. The Coulomb inter-actions among the adjacent ions in an ionic crystal interms of inverse-power overlap repulsion as [21]

8(r) = −(Ze)2[

1r− f

r s

], (1)

where f is the repulsion force parameter between theion cores. The elastic force constant κ is convenientlyderived from 8(r) at the equilibrium interionic dis-tance r0 by

κ =(∂28

∂r2

)r0

= (Ze)2[

s − 1r3

0

]. (2)

Here, s is the index number of the overlap repulsivepotential. In the case of CuO 2D unit cell includesone Cu2+ ion, and two O2− ions, and each two bondingassociated with the x, y, zthree modes of polarization,respectively. Then we use acoustic mass M′ = (M+ +2M−), κ∗ = 2κ for each directional oscillation modeto get the acoustic phonon frequency as

ωD =√

2κ∗

M′= 2(Ze)

√(s − 1)

M′1r3

0

, (3)

where M+ being the mass of Cu+, while to that M−symbolizes for O−. The Debye frequency is character-ized as a cutoff frequency at the Brillouin zone bound-ary, and it can be expressed in terms of effective valueof ionic mass and elastic force constant for crystal lat-tices with two different kinds of atoms as Cu and O,as we dealt with. Such a model also yields the longi-tudinal and transverse optical phonon mode as

ω2LO =

κ + ηµ(M)

(4)

and

ω2TO =

κ − ηµ(M)

, (5)

where µ(M)−1 [=(M−1Cu + M−1

O )] being the reducedmass of CuO pair as 12.77 amu, η is the force con-stant as

η = 8π3

(Ze)2

Ä. (6)

HereÄ symbolized for the volume of the unit cell. Wenow turn to estimate the zero limited temperature-

dependent resistivity for Sm1.85Ce0.15CuO4

superconductors.

2.2. Zero-Temperature-Limited Resistivity

In usual metals the electron–phonon scatteringis a major source of temperature-dependent resis-tivity and can describe the normal state transportproperties. However, apart from electron–phononscattering, other scattering mechanisms such as elec-trons scatter off impurities, defects, grain bound-aries, and disordered regions lead to a temperature-independent contribution. Let us begin with theestimation of the temperature-independent contribu-tion to resistivity.

Knowledge of zero-temperature elastic scatter-ing rate and plasma frequency will allow us tohave an independent estimation of zero-temperature-limited resistivity. The zero-temperature scatteringrates are related through the upper critical magneticfield Hc2 (0). Following the two-square-well analy-sis of Eliashberg theory, Carbotte [22] suggested thatthe strong coupling corrections are important and arescaling factor of 1+ λ appears in the modified BCSresults. The Matsubara gap function, which is relatedwith upper critical magnetic field, yields

1+ λλ− µ∗ = 2π

TTc

Nc∑m=0

1

χ−1m (ωm)− (2τ )−1

. (7)

Here, ωm is Matsubara frequency, within the stan-dard two-square-well model and is ωm = ωm(1+ λ)+(2τ )−1(sgnωm), λbeing the electron–phonon couplingstrength with cut off at Nc and τ the scattering time.In this approximation Nc follows [22]

Nc = 12

[ ωπT+ 1

], (8)

µ∗ is the renormalized Coulomb repulsive parameter,and the factor χm appearing in Eq. (7) is

χm(ωm) = 2√ξ ∗

∫ ∞0

exp(−q2) tan−1(φ) dq (9)

with

φ = q√ξ ∗[

(2m+ 1)π TTc

]+ [ 12τ ∗] . (10)

The upper critical magnetic field is related through

ξ ∗ = 12

eH∗c2v∗2F . (11)

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The physical quantities appearing in Eqs. (7)–(11) in-volve renormalized values as

H∗c2 =Hc2

(1+ λ)Tc, (12)

v∗F =vF√

(1+ λ)Tc, (13)

and impurity scattering time

τ ∗ = τ

(1+ λ)Tc. (14)

These equations differ from the BCS limit, as therenormalizations in ξ ∗, v∗F, H∗c2, and τ ∗ are introduced.These expressions are valid for any impurity concen-tration described in Eqs. (7)–(11) by scattering time.In the present analysis Pauli limiting has been ne-glected as an approximation [23], because of rela-tively small value of dHc2/dT[1/(1+ λ)] in electrondoped cuprates. In principle the above approach de-scribes quantitatively the renormalization of the phys-ical properties due to electron–phonon interactionand is therefore reduced by 1+ λ.

The zero-temperature-limited resistivity is ex-pressed as

ρ(0) = 4πτ ∗−1

ω2p

. (15)

From above it is noticed that the determination ofscattering rate essentially needs the Coulomb repul-sive parameter, electron–phonon coupling strength,Fermi velocity, plasma frequency, and upper criti-cal magnetic field. This allows one to estimate thezero-temperature-limited resistivity independently.We proceed to include the temperature-dependentresistivity for Sm1.85Ce0.15CuO4 superconductors.

2.3. Normal State Resistivity

To formulate a specific model, we start with thegeneral expression for the temperature-dependentpart of the resistivity [24] given by

ρ = 3πhe2v2

F

∫ 2kF

0|v(q)|2〈|S(q)|2〉

(1

2kF

)4

q3 dq. (16)

v(q) is the Fourier transform of the potential associ-ated with one lattice site, S(q) is the structure factor,and following the Debye model it takes the followingform:

|S(q)|2 ≈ kBTMv2

s

f (hω/kBT), (17)

f (x) = x2[ex − 1]−1[1− e−x]−1, (18)

f (x) represents the statistical factor. Thus theresistivity expression leads to

ρ ≈(

3he2v2

F

)kBTMv2

s

∫ 2kF

0|v(q)|2

×[

( hω/kBT)2q3 dq(exp( hω/kBT)− 1)(1− exp(−hω/kBT))

],

(19)

vs being the sound velocity. Equation (19) in termsof acoustic phonon contribution yields the Bloch–Gruneisen function of temperature-dependenceresistivity:

ρac(T, θD) = 4Aac(T/θD)4 × T∫ θD/T

0x5(ex − 1)−1

× (1− e−x)−1 dx, (20)

where x = hω/kBT · Aac is a constant of proportion-ality defined as

Aac∼= 3π2 e2kB

k2Fv2

s Lhv2F M

. (21)

In view of inelastic neutron scattering measurements,the phonon spectrum can be conveniently separatedinto two parts of phonon density of states [14]. There-fore it is natural to choose a model phonon spectrumconsisting of two parts: an acoustic Debye branchcharacterized by the Debye temperature θD and anoptical peak defined by the Einstein temperature θE.If the Matthiessen rule is obeyed, the resistivity maybe represented as a sum ρ(T) = ρ0 + ρe-ph(T), whereρ0 is the residual resistivity that does not depend ontemperature as described earlier. On the other hand,in case of the Einstein type of phonon spectrum (anoptical mode) ρop(T) may be described as follows:

ρop(T, θD) = Aopθ2ET−1[exp(θE/T)− 1]−1

× [1− exp(−θE/T)]−1. (22)

Aop is defined analogously to Eq. (21). Finally, thephonon resistivity can be conveniently modeled as

ρe-ph(T) = ρac(T, θD)+ ρop(T, θE). (23)

Finally, the total resistivity is now rewritten as

ρ(T, θDθE) = ρ0 + ρac(T, θD)+ ρop(T, θE)

= ρ0 + 4Aac(T/θD)4T

×∫ θD/T

0x5(ex − 1)−1(1− e−x)−1 dx

+ Aopθ2ET−1[exp(θE/T)− 1]−1

× [1− exp(−θE/T)]−1. (24)

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We shall use the values of various physical param-eters in next section to estimate the zero limitedresistivity and temperature-dependent contributionin high-Tc Sm1.85Ce0.15CuO4 superconductors. To testthe validity of the parameters deduced, we estimatethe superconducting state parameters with the strongcoupling theory.

3. RESULTS AND DISCUSSION

To estimate the Debye temperature and Einsteintemperature, we use s = 10 and the in-plane Cu----Odistance as 1.98 A [2], yielding κ = 13.6× 104 g s−2

and η = 4.1× 104 g s−2. For most ionic crystals, theindex number of the repulsive potential has been re-ported to be s = 6–8 [25]. However, for copper oxidesa reasonable value of the repulsive index is about 10[26]. With these parameters the Debye frequency isestimated as 38.5 meV (447 K) and is consistent withthe specific heat measurement [27,28]. Further theoptical phonon mode is obtained as ωLO

∼= 60 meV(696 K) and ωTO

∼= 44 meV (510 K). The calculatedvalues of the LO/TO frequency are consistent with themeasured values of the A2u optical phonons from theRaman spectra of Nd1.85Ce0.15CuO4 superconductors[10–12].

We use the experimental value of Hc2(0) = 80T [29] and Tc = 25 K [2], respectively. We have ear-lier developed an effective 2D effective interactionpotential for electron doped cuprates by consider-ing both inter- and intralayer interactions [20]. Thephysics borne in mind is the considerations of twoelectrons interacting with boson fields and repellingeach other through a Coulomb force within the CuO2

plane. In a layered stacking sequence of 2D conduct-ing planes well separated by an average spacing d thecondition of optimized pairing yields the 2D electrondensity following the relation, ncd2 = 1.

The effective mass of the electron along theconducting plane is deduced from electronic spe-cific heat coefficient γ , using m∗ = 3h2γd/πk2

B. Theparameters required as d = 6 A [1] and γ = 6 mJmol−1 K−2 [28] to get m∗ = 3me. Thus the elec-tron parameters are estimated as the Fermi velocityvF(=1.6× 107 cm s−1) andωp(=2.3 eV) Electronic en-ergy band structure calculations [30] derive the aver-age Fermi velocity as 4.9× 107 cm s−1 and the plasmafrequency of about 4 eV. The values of plasma fre-quency hωP = 1.07 eV and effective mass m∗ = 1.9me

were obtained from the optical reflectivity spectra ofsuperconducting electron doped cuprates [31]. From

the static and dynamic part of the effective interac-tion potential [20] one obtains screening parameterµ∗ (=0.2) and electron–phonon coupling strength λ(=1.5), respectively.

With the above-deduced model parameters, thezero-temperature elastic scattering rate is obtainedas 2.56× 1014 s−1. It is attributed to the fact thatthe larger the electron mass, the smaller the plasmafrequency and hence the reduced zero-temperatureelastic scattering rate. We further estimate the zero-temperature mean free path, L= vFτ ≈ 6.24 A,which is much smaller then the zero-temperature co-herence length of 80 A [32]. The Mott–Ioffe–Regelcriterion for metallic conductivity is valid, as themean free path is three times larger than Cu----Obond length (∼2 A). A significantly enhanced meanfree path is an indicative of metallic conduction asthe product kFL(∼2.57) seems to be much largerthan unity. Zero-temperature limited resistivity (ρ0 =0.23 mÄcm) as deduced from elastic scattering rateand plasma frequency is consistent with single crystaldata of 0.25 mÄcm [15].

Figure 1 illustrates the results of temperaturedependence of resistivity via the ordinary electron–phonon interaction from Eq. (26) with our choice ofθD(=447 K) and θE(ωTO = 510 K). The contributionsof acoustic and optical phonons towards resistivityare shown separately along with the total resistivity.It is inferred from the curve that ρac increases lin-early, while to that ρop increases exponentially withthe increase in temperature. Both the contributionsare clubbed and the resultant resistivity is exponentialat low temperatures, and nearly linear at high temper-atures till room temperature.

Our numerical results on temperature depen-dence of resistivity of Sm1.85Ce0.15CuO4 are plottedin Fig. 2 along with the single crystal data [15]. It isnoticed from the plot that the estimated ρ is lowerthan the reported data from Tc to near room temper-ature. Deduced values of temperature dependent ρfrom Eq. (25) appears low as ρ0 and ωp values are theconstraints for the present analysis. Thus estimatedmodel parameters [λ, µ∗, vF, ωp, τ, and ρ0] representsa good set of parameters for the estimation of normalstate resistivity in high-Tc Sm–Ce–CuO superconduc-tors. Nevertheless, the role of electron–phonon inter-action is better exploited and found prominent in theinterpretation of normal state transport parameters.

The difference in between the measured ρ andcalculated ρdiff.{=ρexp. − [ρ0 + ρe-ph(=ρac + ρop)]} isplotted in Fig. 3. A power temperature dependenceof ρdiff. is inferred at low (∼25 K) as well near to

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Fig. 1. Variation of ρe-ph with temperature, the contribution of acoustic phonons ρac as well of opticalphonons ρop to the resistivity.

room temperature. The quadratic temperature con-tribution for resistivity is an indication of conven-tional electron–electron scattering. The feature ofquadratic temperature dependence of ρdiff. is consis-tent with the earlier reporting [17]. The additionalterm due to electron–electron contribution was re-quired in understanding the resistivity behavior, as

Fig. 2. Variation of ρ with temperature T (K). Open circles are the experimental data taken from Penget al. [15] (1993).

extensive attempts to fit the data with residual resis-tivity and phonon resistivity were unsuccessful. It isnoteworthy to comment that in conventional metals,the electron–electron contribution to the resistivitycan at best be seen only at very low temperature dueto its small magnitude when a comparison is madewith phonon contribution.

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Fig. 3. Variation of ρdiff.[=ρexp − (ρ0 + ρe-ph)] (mÄcm) with T2(104 K2).

The existence of quadratic temperature depen-dence of resistivity over a wide temperature inter-val permits one to believe that the electron–electronscattering is also significant in determining the resis-tivity in electron doped cuprates. It is worth whileto refer to an earlier work of Thompson [33], whopostulated the power temperature dependence ofelectroresistivity in TiS2 as a consequence of low car-rier concentration. Correlating this concept, electrondoped cuprates have definitely low carrier concentra-tion (∼1021 cm−3) and may result in a large enhance-ment of power temperature dependence of in-planeresistivity even up to room temperature. In this man-ner, one can shed further light by correlating the mag-nitude of electron–electron contribution to resistivitywith plasma frequency.

If the quadratic temperature dependence of re-sistivity is a cause of 3D electron–electron scattering(umklapp), then its magnitude should depend on thecarrier concentration n as n−5/3 or equivalently onthe plasma frequency as ω−10/3

P . Within this scenario,we estimate ρee of about 0.16 mÄcm (as illustratedin Fig. 3) with ω−10/3

P as 0.07 (eV)−10/3 for ρ0 valueof 0.23 mÄcm in the vicinity of room temperature.Most of the high-Tc hole doped cuprates do not showthis signature. We believe that this might be due totwofold reasons: higher carrier concentration of holedoped cuprates, as carrier concentration is sensitive topower temperature dependence, and the other is the

different topology of Fermi surface in electron andhole doped cuprates.

The phonon spectrum in view of inelastic neutronscattering measurements consists of two parts, and theelectron–phonon coupling strength is conveniently di-vided into two terms, λac and λop, which are intimatelyrelated with Eliashberg functionα2 F(ω). At high tem-peratures ρac(op)(T) ∝ λac(op) and hence it is conve-nient to write λac/λop = Aac/Aop. We find the ratioAop/Aac as 1.55 leading roughly λac

∼= 0.51, λop∼= 0.8,

and λ(=λac + λop) ∼= 1.31. We proceed to evaluatethe transition temperature Tc for optimized electrondoped cuprates. In the regime λ > 1 the strong cou-pling theory [34] applies yielding

Tc = 0.25θE[exp(2/λeff)− 1]−1/2 (25)

where

λeff = (λ− µ∗)[1+ 2µ∗ + λµ∗t(λ)]−1 (26)

and

t(λ) = 1.5 exp(−0.28λ) (27)

with µ∗ = 0.24. We find λeff ≈ 0.59 and Tc∼= 24 K,

consistent with the measured [2] values Tc = 22–25 Kin electron doped cuprate superconductors.

In Fig. 4, we show the result for Tc as a func-tion of µ∗ up to 0.50 for a set of parameters λ =1.31 and θE = 510 K. Kresin’s strong coupling ex-pression for Tc from Eq. (25) clearly demonstrates

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Fig. 4. Variation of Tc (K) with µ∗.

that Tc is strongly influenced by the Coulomb re-pulsive parameter and is higher for small values ofµ∗. The net result is that the increased electron–electron repulsive contribution along with both acous-tic and optical phonons producing an attractive in-teraction is reduced and attributes to suppressed Tc.This study is essentially based on the application ofKresin’s strong coupling theory for electron dopedcuprates; we have the feeling that such studies de-scribe the dependence of Tc on different couplingstrengths which has not been tested carefully and isdefinitive to impose any constraints on theoreticalapproaches.

An important feature of Eq. (25) is displayedin Fig. 5. To obtain some specific results, we chooseµ∗ = 0.24 in the metallic density regime for Sm–Ce–CuO. It is noticed from the curve that Tc increasessteeply with the enhanced value of λ. For the aboveset of parameters, if the present system is stronglycoupled, then one can achieve the higher values ofTc. These values of λ and µ∗ are quite reasonable forconventional BCS superconductors.

We now focus on the relationship betweenλ and µ∗ within the strong coupling theory thatincorporates electron–electron, electron–acousticphonon, and electron–optical phonon interactions.Although phonons are capable of yielding high-Tc

for reasonable coupling strengths, conventional

BCS/Eliashberg theory predicts the combination ofhigh-Tc and one half value of isotope effect formonoatomic systems. In electron doped cuprates, iso-tope effect is suppressed from one half value (α ≤0.15), which excludes the bare phonon mechanism.We plot the variation of λ with µ∗ in Fig. 6. For lowervalues of λ(≤0.5), µ∗ yields unphysical values for aset of parameters (Tc = 24 K and θE = 510 K). Neg-ative µ∗ implies a constant attractive interaction. Wecomment that to some extent a rough picture of renor-malized Coulomb repulsive parameter is reflected inthese oxide superconductors with strong strength ofelectron–phonon coupling.

Phenomenological expression, Tc ∝ M−α ex-plains the isotope effect exponent, where M is theionic mass. Since the shift 1Tc induced by isotopicsubstitution is small compared to Tc, the exponent αcan be written as α = −M1Tc/(Tc1M). It is given by[34]

α = 12

(1− µ∗2

λeff[1− exp(−2/λeff)]

×[

1λ− µ∗ +

2+ λt(λ)3+ λt(λ)

]). (28)

Substitutingµ∗ = 0.24, λ = 1.31 into Eq. (28), one ob-tains αtotal = 0.4. Hence the Coulomb screening pa-rameter (large value of µ∗(=0.24) in comparison to

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Fig. 5. Variation of Tc (K) with λ.

usual metals) essentially reduces the isotope effect inthis strongly coupled superconductors.

Figure 7 shows the variation of calculated totalisotope effect from Eq. (28) with screening parame-ter. It is noticed that the BCS one half value is re-

Fig. 6. Variation of µ∗ with λ.

covered for µ∗ = 0.0. With the increase in µ∗ val-ues the isotope effect exponent decreases for λ =1.31. To assess further the role of electron–phononin the pairing mechanism, we require the total iso-tope effect. However, a reliable copper isotope effect

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Fig. 7. Variation of α with µ∗.

is difficult to measure for this system, but one canestimate αCu by assuming that the observed ratioαCu/αO ∼ 1.0–1.4 in the hole doped LSCO cuprates[35] will essentially be applicable to this system. Thejustification lies in the fact that the ratio αCu/αO de-pends on the weights of oxygen and copper domi-nated phonon modes that are insensitive to λ andµ∗. Furthermore, the relative weights will be simi-lar in these cuprates due to similar Cu----O bondsor similar covalency. From the observed αO

∼= 0.15,and the ratio αCu/αO ∼ 1.0–1.4, we find αCu ∼ 0.15–0.21 and αtotal = 0.30–0.36 in electron doped cuprates,consistent with the calculated result from Eq. (28).It is useful to refer to an earlier work of Bill et al.,who show that the Coulomb interactions, multiatomiccompounds, anharmonicity, and nonphononic mech-anisms strongly effect the value of isotope effectexponent [36].

4. CONCLUSION

Addition of trivalent rare-earth element Sm bytetravalent Ce in the parent electron doped cupratematerial, which can be viewed as an ionic solid con-taining well separated CuO2 layers, introduces elec-tron carriers in these layers. For the sake of sim-plicity a single (longitudinal and transverse) opticalphonon mode has been considered with a flat disper-

sion relation. We succeeded in explaining the nor-mal state resistivity and other associated transportparameters by the electron–phonon interaction. Wehave first estimated the zero-temperature elastic scat-tering rate with the use of parameters (λ, µ∗, vF, τ ,and ωp) from the developed approach and the uppercritical magnetic field from magnetization measure-ment. Estimated zero-temperature-limited resistivityis consistent with the single crystal data. Further-more, the zero-temperature mean free path is foundto be smaller than the zero-temperature coherencelength.

The mean free path is several times larger thanCu----O bond length and product kFL> 1 favorsthe metallic conduction. Hence, the use of Bloch–Gruneisen expression in estimating the electron–phonon contributions is appropriate. It is noticed thatcontribution from acoustic and optical phonons to-gether with the zero-temperature-limited resistivityis smaller than the reported data on the single crys-tal. The scattering rate at low temperatures is of theorder of 1015 s−1 in hole doped high-Tc cuprates. Forthe high-Tc electron doped cuprates, we obtained re-duced scattering rate (1014 s−1), which is a naturaloutcome of the present analysis due to mass enhance-ment of active electrons in the conducting copper ox-ide layer. We have thus shown that the temperaturedependence of resistivity in electron-doped cuprates

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can be conveniently described by the usual electron–phonon interaction. In view of inelastic neutronscattering data, the phonon spectrum is conve-niently separated into two parts, acoustic and opti-cal phonons of oxygen-breathing modes. The high-energy optical phonon yields a large contributionto the resistivity. In turn, large value of electron–phonon coupling strength is no doubt from opticalphonons.

When the subtracted data is plotted as a func-tion of T2, a clear straight line is depicted from lowtemperature to the vicinity of room temperature. Theobservation of T2 dependence points towards theelectron–electron scattering. The additional term dueto electron–electron contribution was required in un-derstanding the resistivity behavior, as extensive at-tempts to fit the data with residual resistivity andphonon resistivity were unsuccessful. The magnitudeof the resistivity is high for a metallic system, indi-cating a small density of carriers participating in theelectrical conduction.

The zero-temperature-limited resistivity is alsoa large fraction of the total resistivity at room tem-perature, which suggests a large amount of impurityscattering is present. The large residual scattering isalso susceptible to the phonon drag, and as a conse-quence it decreases even in the single crystals. Thispresence of strong elastic scattering of electrons dueto disorder may play a role in reducing the temper-ature dependence due to scattering by phonons be-low the T5 behavior seen in good metals with smallelastic scattering, or the power temperature behaviormay arise from electron–electron scattering in Fermiliquid. We have thus demonstrated that apart fromelectron–phonon, electron–electron scattering isimportant.

On the assumption that the superconductingproperties of electron-doped cuprates are of electron–phonon origin, the values of coupling strengthsare obtained. Correspondingly, the superconduct-ing transition temperature and isotope effect ex-ponent that is estimated following Kresin’s strongcoupling approach is consistent with the reportedvalue. To summarize, we have presented a theo-retical analysis of the normal state resistivity andallied physical parameters responsible for super-conducting pairing mechanism in electron dopedcuprate superconductors. Our results on the nor-mal state resistivity and associated parameters ofelectron doped cuprates conclusively show thatthey are consistently explained by electron–phononinteraction.

ACKNOWLEDGMENTS

Financial assistance from University GrantsCommission, New Delhi, India, is gratefullyacknowledged.

REFERENCES

1. V. Z. Kresin, H. Morawitz, and S. A. Wolf, Mechanisms of Con-ventional and High Tc Superconductivity (Oxford UniversityPress, New York, 1993).

2. Y. Tokura, H. Takagi, and S. Uchida, Nature 337, 345 (1989);H. Takagi, S. Uchida, and Y. Tokura, Phys. Rev. Lett. 62, 1197(1989).

3. H. Muller-Buschbaum and W. Z. Wollschlager, Anorg. Allg.Chem. 414, 76 (1975); Anorg. Allg. Chem. 428, 120 (1975).

4. T. Nakano, N. Momono, M. Oda, and M. Ido, J. Phys. Soc. Jpn.67, 2622 (1998).

5. C. C. Tsuei, A. Gupta, and G. Koren, Physica C 161, 415 (1989).6. O. Beom-Hoan and J. T. Markert, Phys. Rev. B 47, 8373 (1993).7. C. C. Tsuei and J. R. Kirtley, Phys. Rev. Lett. 85, 182 (2000).8. Fu-sui. Liu and Wan-fang. Chen, J. Cond. Matt. Phys. 12, 8475

(2000).9. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108,

1175 (1957).10. E. T. Heyen, R. Liu, M. Cardona, S. Pinol, R. J. Melville, D McK

Paul, E. Moran, and M. A. Alario-Franco, Phys. Rev. B 43, 2857(1991).

11. Ji-Guang Zhang, Xiang-Xin Bi, E. McRae, P. C. Eklund, B. C.Sales, and M. Mostoller, Phys. Rev. B 43, 5389 (1991).

12. C. C. Homes, B. P. Clayman, J. L. Peng, and R. L. Greene, Phys.Rev. B 56, 5525 (1997).

13. O. K. Andersen, A. I. Liechtenstein, O. Jepsen, and F. Paulsen,J. Phys. Chem. Solids 52, 1411 (1995).

14. L. Pintschovius and W. Reichardt, in Neutron Scattering in lay-ered Copper-Oxide Superconductors, A. Furrer, ed. (KluwerAcademic, Dordrecht, The Netherlands, 1998), pp. 165–223.

15. J. L. Peng, Z. Y. Li, and R. L. Greene, Physica C 177, 79 (1991).16. E. A. Early, A. Almasan, R. F. Jardim, and M. B. Maple, Phys.

Rev. B 47, 433 (1993).17. N. Tralshawala, J. F. Zasadzinski, L. Coffey, and Q. Huang, Phys.

Rev. B 44, 12102 (1991).18. K. Levin, J. H. Kim, J. P. Lu, and Q. Si, Physica C 175, 449

(1991).19. M. A. Crusellas, J. Fontcuberta, S. Pinol, J. Beille, and T. Grenet,

Phys. Rev. B 48, 615 (1993).20. D. Varshney, G. S. Patel, and R. K. Singh, J. Low Temp. Phys.

120, 315 (2000).21. M. Born and K. Huang, Dynamical Theory of Crystal Lattices

(Oxford University Press, London 1966); G. Venkatraman andV. C. Sahni, Rev. Mod. Phys. 42, 409 (1970).

22. J. P. Carbotte, Rev. Mod. Phys. 62, 1102 (1990).23. G. Deutscher, O. Entin-Wohlman, S. Fishman, and Y. Shapira,

Phys. Rev. B 21, 5041 (1980).24. G. Grimvall, The Electron–Phonon Interaction in Metals (North

Holland, Netherlands, 1981).25. M. P. Tosi and F. G. Fumi, J. Phys. Chem. Solids 25, 45 (1964).26. D. Varshney and M. P. Tosi, J. Supercond. 13, 593 (2000).27. S. J. Collocott, R. Driver, and C. Andrikidis, Phys. Rev. B 45,

945 (1992).28. C. Marcenat, R. Calemczuk, A. F. Khoder, E. Bonjour, C.

Marin, and J. Y. Henry, Physica C 216, 443 (1993).29. M. Klauda, J. P. Strobel, M. Lippert, G. Saemann-Ischenko, W.

Gerhauser, and H. W. Newmuller, Physica C 165, 251 (1990).

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30. N. Hamada, S. Massidda, J. Yu, and A. J. Freeman, Phys. Rev.B 42, 6238 (1990).

31. K. Hirochi, S. Hayashi, H. Adachi, T. Mitsuyu, T. Hirao, K.Stetsune, and K. Wasa, Physica C 160, 273 (1989).

32. A. Andreone, A. Cassinese, A. Di Chiara, R. Vaglio, A. Gupta,and E. Sarnelli, Phys. Rev. B 49, 6392 (1994).

33. A. H. Thompson, Phys. Rev. Lett. 35, 1786 (1975).

34. V. Z. Kresin, Phys. Lett. A 122, 434 (1987); Bull. Am. Phys. Soc.32, 796 (1987).

35. J. P. Franck, S. Harker, and J. H. Brewer, Phys. Rev. Lett. 71,283 (1993).

36. A. Bill, V. Z. Kresin, and S. A. Wolf, in Pair Correlation’s inMany–Fermion Systems V. Z. Kresin, ed. (Oxford UniversityPress, New York, 1998), p. 25.

Explanation of Temperature-Dependent Resistivity of Sm-Doped Cuprates by Electron–Phonon...

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