Indian Journal of Pure & Applied Physics Vol. 40, December 2002, pp. 879-886 Effect of plasmons on superconducting transition temperature of Bao.6Ko.4Bi03: A three square well approach Dinesh Varshney', R P Kumhar' , Sanjay Shah' & R K Singht *School of Physics, Vi gyan Bhawan, Devi Ahilya University, Kh andwa Road Campus, Indore 452017 t Department of Physics, PMB Gujarati Science College, Devi Ahilya University, Indore 452 00 I tSchool of Pure and Applied Physics, Guru Ghasidas University, Bilaspur 495009 Received 12 June 200 I; revised December 200 I; accepted 29 Apri I 2002 By assuming thc existence of a low energy plasmon in attractive intcraction, the cooper pairing theory is cxtendcd for cubic pcrovskite Bao.{,Ko.4 BiO,. Three square well model for the three interactions namely, electron-phonon, electron­ plasmon and Coulomb in the calculation of superconducting transition temperat ure (T..) has been employed. The analyti cal solutions for the energy gap equation allow one to visualise the relative interplay of various intcractions. To correlatc th e T.. with various coupling strengths (A'ph , Api and ~ *) curvcs of T,.. are presented with them. The values of the coupling strength and of the Coulomb interaction parameter indicate that, the superconductor is in the intermediate coupling regime. The superconducting transiti on temperature of optimall y doped Ba-K-BiO is estimated as 25 K, for Aph = 0.6, Api = 0. 1 and ~ * = 0. 18 . The low energy plasmons play a key role in raising T .. with the increased Api values. The prcsent analysis points to the importance of both plasmons and optical phonons in determining thc effective clectron-elcctron interaction leading to superconductivity in doped cubic perovskites. 1 Introduction striking feature of this system is, the absence of metal-oxygen planes, which are beli eved to be Among the various potential non-phonon crucial in producing a high transition temperature in mechanisms proposed to expl ai n the dramatic cuprates. The parent compound BaBiO, has the increase in the superconducting transItIOn perovskite structure. It is a charge-density-wave temperatures in cubic perovskites, one possibility is insulator, similar to the under-doped cuprates, which of the electronic collective excitations (plasmon) are anti-ferromagnetic insulators. mechanism. The plasmons make a noticeable contribution to the electron pairing and hi gh-Tc The K-doped compound exhibits superconductivity in cubic bismuth oxide is caused superconductivity in the range 0.37 ~ x ~ 0.5. The by coexistence of the phonon and non-phonon Bi-O-Bi bonds in the K-doped system form an (plasmon) mechanism. Although, no definitive orthorhombic or a simple cubic structure depending 2 exposition of the correct physical picture for this on x • Despite intense interest in three-dimensional cubic perovskite is available despite a large number Ba-K-BiO, the question of the electron-pairing of highly original and intriguing proposals for mechanism that is responsible for superconductivity cuprates, the authors present in this work some at 7~ - 30 K remains open. As the K-doped system is results that they believe, add weight to the plasmon diamagnetic' and there is no evidence for static idea. Specifically, it is aimed to show qualitatively magnetic order\ the magnetic pairing mechanisms that, the plasmons in three-dimensional materials as proposed for cuprates can be excluded for this lead to their substantial role apart from the optical material. Apart from the attractive mechanism, the phonons in an ionic lattice of Bal_xK,BiO,. magnitude of the coupling strengths in this superconductor from different measurements either The high-Tc cubic Bal _xK,BiO:, (Ba-K-BiO) has the hi ghest superconducting transition temperature assists weak, intermediate or strong coupling, and is an important issue to answer. Tc == 30 K at x == 0.4, among metal oxide superconductors not containing copper and much For a theoretical understanding, Bardeen­ attention has been focussed in recent years I . A most Cooper-Schrieffer (BCS) theory for conventional 880 INDIAN J PURE & APPL PHYS, VOL 40, DECEMBER 2002 superconductors5 invokes electron-phonon coupling together with the Raman", photo-emission and 11 for Cooper-pair formation. Optical phonons are reflectance studies '2. have provided the motivation important in an ionic crystal such as, Ba-K-B iO and for the present work. Here, the authors evaluate a Raman scattering of light as well as ine lastic pairing mechani sm in Ba-K-BiO, as arisi ng from the neutron scattering which have been used to probe its coupling of the carriers to both optical phonons and phonon structure. The Raman spectrum of plasmon excitations. Specifically, it is aimed at superconducting B~ II,KIIABiO .h shows a strong peak assessing whether, this mechanism can expl ain the near 43 meV, due to high-energy optical phonons large binding energy and a critical temperature of 30 6 coupled to electronic states • Inelastic neutron K. It is the purpose of this in vesti gation, to scattering experiments7 show that, the phonon determine the relative contributions in the pairing spectrum in B~I (, K( 14 Bi01 comprises two bands near that they coexist, and various limitations pl aced on 30 and 60 meV, which are due to oxygen vibrations. the theory by the authors. It may also be referred at Subsequently, Braden et a/.x found that, the hi ghest this point, to the work of Shirai et al. '4 on bi-oxide, longitudinal optical (LO) branch at - 17 THz is who find a strong frequency re-normalisati on for the associated with Bi-O bond-stretching vibrations. hi ghest optical phonons, due to dopin g. These experimental studies indicate that, in Ba-K­ To obtain analytic results, the authors worked BiO the carriers may be coupled to hi gh-energy with a three, square-well model 011 the basis of oxygen vibrations. available experimental and theoreti cal data. Within the standard mechanism, moderate to strong electron-phonon coupling (A. "" I) is required 2 Model to account for T, - 30 K. Tunnelling spectroscopy It is started by giving a brief description of the can provide a direct measure of the interactions cubic perovskite Ba' xKxBiO,. The parent compound responsible for superconductivity, even though BaBiO, is a diamagnetic insulator. The formal hi gh-quality tunnel junctions on hi g h-1~ materi als charge-state of the Bi ions is +4, a'ld si nce the Bi are difficult to obtain. High-resolution tunnelling atom has five valence electrons (two 6s and three measurements on po lycrystalline Ba,.xKrB iO, (x "" 6p) an average of one valence electron is left on 0.38, T, - 29 K) have been reported by Zasadzinski each Bi ion. The chemical substitution of mono­ el a 1.'). The tunnelling data reveal well-resolved valent alkali K at the divalent Ba sites introduces phonon structures corresponding to optical modes of free charge-carriers (holes). Further, th e contraction the oxygen in the range 40-65 meV. Huang et al.1O and expansion of oxygen octahedra around the Bi have demonstrated that, hi gh-energy optical ions will lead to breathing phonon modes. phonons are involved in the carrier-pairing, with a The gap equation of such an ionic solid coupling constant A. near unity. containing electronic carri ers is now discussed. Among the non-conventional mechanisms, 2(a) Superconducting Gap Equation which have been invoked, for high-T" superconductors, screening by collective charge The transition temperature for the fluctu ations (plasmons)" is shown to be important superconducting state is, well estimated from the by several experiments. Measurements of X-ray gap equation that follows: photo-emission and reflectance have probed the e lectroni c structure of Ba-K-BiO superconductors. ~(r) = g7rTIfd 3r'K(r, r';w) .. ( I) The energy-loss structure of core level spectra 12 w reveals a free-carrier pl asmon at about I eV. From g being the coupling strength and the Kernel K (r the reflectance spectrum" of B ~I." KII4B iO " the plasmon li es near 1.6 eV. This implies that, carrier r';m) is defined as: charge fluctuations are large in the K-doped oxides and contribute significantly to the pairing K (r, r'; w ) = C i (r, r'; W )CJ, (r, r';-w) ... (2) mechanism. C being the usual electron Green's function . Fourier The strong coupling of the carriers to the oxygen transform of Eq. (2) leads to: 9 10 vibrations as shown in the tunnelling spectra . V ARSHNEY el al.: SUPERCONDUCTING TRANSITION TEMPERATURE 88 1 d ' q' "1"1 , , easily classify a dirty tJ. (q) = glrTI f--, d q- K (q, q ;w )tJ.(q ) .. (3) su perconductor. w' (2lr) , In the situation of interest i.e., weak-scattering where K (q, q';w) is defined as: limits, all quantities are replaced, depending on p' in Eq. (I) by their averages over the surfaces of K (q, q'; w ) = I C i (PI' P3 ;w)C J. ( P2' P4 ;-w) constant energies, solving the scattering problem"'. Further, the lifetime effects shall not be considered ... (4) due to retardation of the Coulomb interactions i.e., th e frequency dependence of V in Eq. ( I ), as the The temperature-dependent gap equation within localisation parameter is small in bi smuth oxide the framework of two-particle mass centre, Eq. 0) systems. At finite temperatures, the occupation of following Eliashberg approach I, leads to: 2 the excited one-electron states Ek = (ul + .1 ) 1/2 tJ. " (w,,) = -lrT obeys the Fermi statIstIcs, with the Fermi di stribution function f(Ek, T) and while determining I . V Cp, 17'; w,,' w ;, )C ,,' (W;,)C -p' (-w;, )tJ.,,' (w ',,) th e gap equation, the authors employed it by I}"'(V'I 2 incorporating the non-occupation [1- 2/( ul + .1 )"2 ..