A THEORETICAL ESTIMATION OF TEMPERATURE DEPENDENT RESISTIVITY OF ALKALI METAL DOPED FLUORIDE

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1 Int. J. Chem. Sci.: 1(), 14, ISSN X A HEOREICAL ESIMAION O EMPERAURE DEPENDEN RESISIVIY O ALKALI MEAL DOPED LUORIDE L. P. MISHRA a and L. K. MISHRA * Department of Physics, Magadh University, BODH GAYA 8434 (Bihar) INDIA a P.G. Department of Chemistry, Vinoba Bhave University, HAZARIBAG 8549 (Jharkhand) INDIA ABSRAC Using the theoretical formulism developed by Pintschovius 16, we have presented a method of evaluation of tempeture dependent resistivity ρ e-ph (electron-phonon), ρ er (Inter molecular Phonon), ρ (Int molecular Phonon), total resistivity ρ () and ρ diff (Difference between theoretical-experimental) for K 3 C 6 fluoride. Our theoretical results are in good agreement with experimental data and also with other theoretical workers. Key words: Normal state resistivity, Electron-phonon, Inter-molecular phonon, Int-molecular phonon, Alkali metal doped fluoride, Electron doped cuptes, Dimensionally crossover. INRODUCION In this paper, we have presented the method of evaluation of tempeture dependent resistivity due to electron-phonon intection (ρ e-ph ), tempeture variation of total resistivity ρ () and tempeture variation of ρ diff (measured ρ-calculated ρ) for K 3 C 6 fluoride. As we all know that electrical resistivity ρ can be caused by various scattering mechanism. An important mechanism is the electron-phonon intections. In this mechanism electrons are scattered under the simultaneous excitations of phonons. he equally important mechanism is electron-electron intection. he contribution of these two mechanisms 1, goes to zero as. In case of metal, electron-phonon scattering is the major source of tempeture dependent resistivity. he other scattering mechanisms are electron impurities scattering due to defects, gin and disorder regions. Unfortunately, all these mechanisms give tempeture independent contributions. or the evaluation of resistivity of alkali metal doped fluoride * Author for correspondence; muphysicslkm@gmail.com

2 376 L. P. Mish and L. K. Mish: A heoretical Estimation of empeture. one uses zero tempeture scattering te related to upper critical magnetic field H c () 3, modified BCS theory and the use of Matsuba gap function 4, which uses to square well potential model. In these calculations the zero tempeture resistivity is expressed as ρ = πτ ω (1) * 1 () 4 p Here ω p is the plasma frequency τ is the impurity scattering time. In this calculation, the zero tempeture resistivity has been calculated by keeping the pameter 5 C = K V = 1.9 x 1 7 cm/sec ω p = 1. ev H C () = 49 esla τ = 1.3 x 1-14 sec -1 hese all pameters are taken from the reference 6,7. Our theoretical evaluated results of tempeture variation resistivity ρ e-eh (due to electron-phonon), ρ er (due to intermolecular phonon) and ρ (due to int molecular phonon), we have taken the intermolecular frequency ω er = 9K and intmolecular frequency ω = 1455K. Our theoretical results show that ρ er increases linearly with and ρ increases exponentially with *. When these resistivities are combined together to calculated resultant resistivity ρ e-eh then it was found that ρ e-eh varies exponentially with low and linearly with high. Our other calculation ρ diff shows that it has a power dependence of tempeture as low and it becomes almost satuted at high. Mathematical formula used in the evaluation Now, one can estimate zero tempeture limited resistivity with the help of zero tempeture elastic scattering te and plasma frequency. he zero tempeture scattering te is related through the upper critical magnetic field H C (). ollowing the two square well analysis of Elishberg theory, Carbotte 8 suggested that the strong coupling correlations are important and a rescaling factor (1 + λ) appears in the modified BCS results. he Mastuba gap function, which is related with uppend the factor critical magnetic field yields λ λ μ * = π c Nc m= χ m (ω m 1 ) (τ ) ()

3 Int. J. Chem. Sci.: 1(), ωm Being the Mastuba frequency, within the standard two-square-well model is ω m = ω m (1 + λ) + (τ) -1 (sgn ω m ) λ the electron-phonon coupling strength with cut off at N C and the scattering time. In this approximation, N C follows N C = ( 1 / ) [( ω / π ) + 1] (3) μ * being the normalized Coulomb repulsive pameter and the factor χ m appearing in () with, ( ) = exp( ) tan ( ) ξ χm ωm q φ dq q ξ * φ = [ (m + 1) π ] + 1 τ * C (4) he upper critical magnetic field is related through 1 * * * ξ = ehcν (5) he physical quantities appearing in (1) (5) involve renormalized value as H * C H C = (6) (1+ λ) C ν ν * = (7) (1+ λ) C and impurity scattering time. τ * τ = (8) (1+ λ) C he above derived equation differ from the BCS limit, as the renormalizations in ξ*, * * ν, HC and τ * are introduced. hese expressions are valid for any impurity concenttion described in (1) - (5) by scattering time. In the present analysis, Pauli limit has been neglected as an approximation 9, due to relatively small value of dh C 1 d / [(1 + λ)] in alkali metal

4 378 L. P. Mish and L. K. Mish: A heoretical Estimation of empeture. intercalated fullerenes. In principle, the above approach describes quantitatively the renormalization of the physical properties due to electron-phonon intection and is therefore reduced by 1+λ. he zero tempeture-limited resistivity is expressed as ρ() = 4π τ 1 ωp rom the above, it is noticed that the determination of scattering te essentially needs the Coulomb repulsive pameter, electron- phonon coupling strength, ermi velocity, plasma frequency and upper critical magnetic field. his allows one to estimate the zero tempeture-limited resistivity independently. One uses earlier deduced values 5 to estimate the zero tempeture elastic scattering te which is consistent as those derived from the superconducting fluctuation measurements 1. It is attributed to the fact that the larger the electron mass, the smaller the plasma frequency and hence the zero tempeture elastic scattering te. One has earlier estimated the zero tempeture mean free path, l about 3.4 nm which is highly sensitive for carrier scattering. One further finds that the product K l (~ 17), seems to be much larger than unity indicates the metallic chacteristics. It is worth to mention that the product, ε τ» 1, in the test material refers to the fact that the doped fluorides fall in the weak scattering limit. his is however, consistent with the s wave superconductors. With these pameters, one estimates zero tempeture limited resistivity (ρ =.4 mω cm) consistent with the single crystal result 1. Normal state resistivity o formulae a specific model, one starts with the genel expression for the tempeture dependent part of the resistivity 11 3π k ρ ν(q) S(q) 1 3 = q dq eν (9) h k V(q) is the ourier tnsform of the potential associated with one lattice site and S(q) being the structure factor. ollowing the Debye model it takes the following form - 4 k S(q) B = f(x) (1) Mν S

5 Int. J. Chem. Sci.: 1(), f(x) being the statistical factor and to - f(x) = x[e x -1] -1 [1-e -x ] -1 (11) hus the resistivity expression leads to k 3 3 K xq dq ρ ( ) vq ( ) [ ] x x hev Mv ( e )(1 e ) β s (1) v s the sound velocity. Equation (1) in terms of intermolecular phonon contributation yields the Bloch Gruneisen function of tempeture dependence resistivity v. Where x = θer 4 5 x ρ er = (θer ) = 4Aer ( ) x x (e ) (1 ex) (13) θer h ω.a k being a constant of proportionality defined as - er B A er [ K ν L ν M] 1 (3π e kb) h (14) In views of inelastic neutron scattering measurements, the phonon spectrum can be conveniently sepated into two parts of phonon density of states 1,13. It is natul to choose a model phonon spectrum consisting of two parts; an intermolecular phonon frequency, ω (θ er ) and an int molecular phonon frequency, ω (φ ). If the Matthiessen rule is obeyed, the resistivity may be represented as a sum ρ() = ρ + ρ e-ph (), where ρ is the residual resistivity that does not depend on tempeture. On the other hand, in case of the int molecular phonon spectrum, ρ () may be described as follows ρ 1 1 [ exp ( θ ) ] [ 1 exp ( θ ] 1 = = (15) (,θ ) A θ ) A is defined analogously to (14). inally the phonon resistivity reads - ρ e-ph () = ρ er (, θ er ) + ρ (, θ ) (16) Henceforth, the total resistivity is now written as ρ (, θ er, θ ) = ρ o + ρ er (, θ er ) + ρ (, θ ) = ρ o A + 4A er ( θ er θ 4 er 5 x x ) x (e ) (1 e ) dx + (17) θ [exp(θ / ) ] [1 exp(θ / )

6 38 L. P. Mish and L. K. Mish: A heoretical Estimation of empeture. hough this is a purely phenomenological expression, it seems to provide a reasonable description of the available experimental data. One also studied the tempeture variation of different of resistivity. ρ diff = [ρ exp (ρ o + ρ e ph {ρ er + ρ })] (18) where ρ o Is residual resistivity ρ e-ph Resistivity due to electron-phonon ρ er Resistivity due to intermolecular phonon ρ Resistivity due to int molecular phonon he resistivity has been studied theoretically by an approximate solution to the Boltzmann equation for the case of electron-phonon scattering mechanism 14 ρ() = 8π ω k p ωmax β hωα dω Cosh( τr (ω) ) h ω k 1 β (19) where ω p is the plasma frequency. In the above equation the tnsport coupling function can be replaced by the electron - phonon coupling for superconductivity 15 α (ω( = 1 ωνλ νδ (ω ων ) () ν Due to the factor in the denominator in equation (19) phonon with frequency much larger than do not contribute. However, they give a linear contribution in, if their energy is much smaller than k β. RESULS AND DISCUSSION In this paper, using the theoretical formalism developed by Pintschovicus 16, we have presented the method of evaluation of tempeture variation resistivity of alkali metal doped fluoride. We have evaluated tempeture variation resistivity due to electron-phonon (ρ e-ph ), intermolecular phonon (ρ er ) and int molecular phonon (ρ ), the results shown in table. In this evaluation, we have used the value of intermolecular phonon ω re = 9K and int molecular phonon frequency ω =1455K. Our theoretical result shows that ρ er increasing

7 Int. J. Chem. Sci.: 1(), linearly with tempeture and ρ increasing exponentially with tempeture. When both combined together then resultant resistivity ρ e-ph varies exponentially at low tempeture and linearly at high tempeture. We have also calculated the total resistivity ρ () and compared our results with the experimental data. Our estimated values of ρ () are lower than the measured values. In this estimation, we have used the model pameter λ, µ*, ν, ω P, τ and ρ which could be the reason of the lower valued of ρ(). We have also evaluated the tempeture variation of ρ diff for K 3 C 6 fluoride. rom our evaluated results, it appears that ρ diff has power tempeture dependence at low and almost satuted values at high. he quadtic tempeture dependence at low is an indication of conventional electron-electron scattering. his is similar to electron doped cuptes 17. he departure from dependence of ρ diff is due to dimensionality cross over 18. hus in this paper, we conclude that for the estimation of tempeture dependence resistivity in K 3 C 6. he three scales of energy ω er, ω and ω c (coulomb intection) are of in same order and these are fully utilized in a model to estimate ρ which is sum of ρ, ρ e-ph and ρ e-e. Some recent 19-5 studies on fluorides also reveal similar type of behavior. able 1: Pameter used in the calculation c = K V = 1.91 x 1 7 cm s -1 ω p = 1. ev HC () = 49 esla ω er = 9 K ω re = 1455 K ρ o =.47 (mωcm) able : Evaluated results of tempeture variation of ρ e-ph, ρ er and ρ for K 3 C 6 fluoride (K) ρ e-ph (mωcm) ρ er (mωcm) ρ (mωcm)

8 38 L. P. Mish and L. K. Mish: A heoretical Estimation of empeture. able 3: Evaluated results of tempeture variation of total resistivity ρ() of K 3 C 6 (K) heory ρ () (mωcm) Expt able 4: Evaluated results of tempeture variation of ρ diff = [ρ exp ρ o + ρ e-ph (= ρ er + ρ )] for alkali metal doped fluoride K 3 C 6 emp. ρ diff = [ρ exp ρ o + ρ e-ph (= ρ er + ρ )] (1 4 K ) heoretical (mωcm) Linear fit (mωcm) REERENCES 1. B. Burk, V. H. Crepsi, A. Zettl and M. L. Cohen, Phys. Rev. Lett. (PRL), 7, 376 (1994).. E. Cappelluti, C. Grimaldi, L. Pietronero and S. stssler, Phys. Rev. Lett.(PRL), 85, 4771 (). 3. E. Cappelluti, C. Grimaldi, L. Pietronero, S. Stssler and G. A. Ummrino, Eur. Phys. JB1, 383 (1).

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