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
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 ... (5) + £IF, T)] of th e corresponding pair-states. It is now V (17, 17 /; w,,,w',,) being the effective electron-electron useful to rewrite Eq. (5) as:
~ . , interaction . The notation C" ,{ W) is the usual electron , . tJ. -.;w - +tJ. - Green's function with momentum 17' and the tJ. =-fdwQ(w ,w ) W tanh 'v W ~ '1 2T electron Matsubara frequency Wn = (211+ I )nT. 2-.; w - +tJ.,v -
In cubic perovs kite Bao(' ~ J~ Bi01 ' the inelastic . .. (6) scattering at low temperatures T < T,. == 30 K does with Q (w, w') = N (w' ) V(w, w'). Here, th e not playa significant role. Therefore, the potential summation is replaced by integration: scattering by inherent imperfections conserve the energy, but because of the chaotic distributions of ... (7) scatterers the electron momenta 17' are not conserved in the scattering process. So, the momenta 17 are no longer good quantum numbers. As regards cubic The kerne l Q( w,w') of integral Eq. (6)
Bao.(, Ko4 BiO, are concerned, the zero temperature corresponds to the coupling constant A = N(O) Vof mean, free path (23 A) is smaller than zero the BCS theory' . Replacing the product of the gap temperature coherence length (53 A), that can function tJ. /O and the kernel in Eq. (6), by the product
Q(ro,O) = Q(O,ro)
-roc ro
Fi g. I - Three sq uare-well model kernel Q (w, w') as a funclion of energies INDIAN.I PURE & APPL PHYS , VOL 40, DECEMBER 2002
of th e averages of th ese quantities, taken separately. superconducting propert ies on the in teracti Gns and The kernel is approximated in each frequency range electron density of states. 'e', ' pI 'and ' pI! ' of bi smuth oxides by square-wells 2(b) Effective Coupling Constant of half- width s. roughl y equai to th e cut-off energi es co" co", an d w"',, respecti ve ly (see Fi g. I ). These Using Eq. (6) in Eq. (8). one obtains th e values wil l be used from ex periments later, to following set of equ ations for 6" 6",, and 6", within illu strat e their ph ysic al meanlllgs for th e three-square model for cubic Ba-K-BiO for th e superconducti vit y. order parameter. near T, as :
- 6"" = Q"" 2"" 6"" + Q,>I 2", 6", + Q,.2, 6,
CD' - 6", = Q "~I 2"" 6"" + Q", 21" 6", + Q,. Z, ,6" - 6,. = Q , Z"" 6"" + Q, 2, >1 6", + Q,. Z,. 6 ,. ... (9)
CDC QC QC QC ]n th e above equ ati ons:
= III [w", /w,'" ] CD p] Qp] Qp] QC 2",
Q . . ( 10) CD]p 1 ph Qph QC Z integral s are calcu lated in th e logarithmic CD approx i mati on.
0 (Oph CD p] CD C The superconciu cting transition temperature 7: and its effective coup lin g constant shall be obtained from th e condition that, the determinant of the Eq. (9) must vanish: rig. :: - Enc: rgy pl:lI1 C:s I'm thrc:c squ arc- wd l model Q,Z , kc rnc:1 () ( I). (I)') I+ Q""Z"" Q" ,Z", I+ Q", 2 ", Q,Z,. = 0 The sy mmetric in tegral kern els Q( w.w' ) = Q",Z"" Q(w'. co) arc potted in Fig. 2, on the frequency plane Q,.2"" Q, Z ", I+Q,Z ,. (w.w'). whe re: ... ( I I ) Q I'"~ = - /""'" - A", + !l. Soluti on of th e determ inant yields: Q "~I = - /""" + p p ... ( 12) Q " = ,LI T, = 1.13w "" [ex (- _ I l~ }'''" IJ . .. (8) wi th The three square-well method of solutions of integral Eq. (8 ) leads to analytic resu lts and gives less accurate resu lts , than other models, as the form ... ( 13) of effecti ve interacti on po tential is not taken into account. On th e other hand, it clearl y shows how, the size and shape of kern el Q (w,w'), i. e., v
temperature is recovered within the two-well theori es using, paIring th rough excitati ons with repulsive-attractive model (CD,. > CDIII = CDIIII ). In frequ encies exceeding or comparable to Fermi contrast to the two-we ll repul sive-attract ive model, energy and is an unsolved problem. If a boson is a th e two well attracti ve-attracti ve ve rsion (CD, > CDld> candidate for superconducti vity, one must ex pect p * CDIIII ), dealing in the present scenari o, will provide an < P = 0.5 and of course A> p* additional constraint for th e rati o of the To obtain some spec ifi c results, p = 0.2 is characteri stic frequency scales CDl1riCD1111 chosen in the metallic density regime for Ba-K-BiO o
has been obtained from dynami c susceptibility parameters, if the present system is strongly measurements on superconducting Ba-K-BiO coupled, then, one can achi eve the hi gher values of powder, and indicative of strong couplingl x. Specific Te· heat measure ments of ceramic Ba-K-BiO sampl e determined 'A" II to be 0.9, correspond to intermediate coupling l'} The 'A/, II valu e as calcul ated from the 50 magneti c and resisti vity measurements") of critical BaO.6KOABi03 parameters of Baofi2 Ko.1 xBiOl sin gle crystals was 45 0.48-0.96. However, th e estimati on of \,II in Ba) ., ApI =: 0.1 Aph = 0.6 K,BiO, (0.0 ~ x ~ 0 .5) from the structural phase 21 ,...... 40 diagram reveals a valu e of 0.95 at x = 0.4. ~ '-'u 22 Marsiglio et al. , in analysing the imaginary f- 35 part of th e optical conductivity of Ba-K-BiO, argued th at, e lectron phonon coupling mu st be weak, 'A"II == 30 0 .2, to explain the data. Kaufmann et aF1 anal ysed
the optical refl ecti vity data for Bal.,K,Bi01 in the far- in frared region, whi c h suggested a moderate coupling with 'A = 0.7 in the superconducting state. 2 Recent density functional calculations -1 of Ba-K BiO yie ld an average value 'A = 0.29. Although, earli er studies point to the importance of phonon Fig. :1 - Variati on of J..I wi th T, coupling mechanism alone as the dominant, but later optical conductivity measurements and density fu ncti onal theory raise serious doubts on the 40r---~----~----r---' conventi ona l e lectron-phonon pairing force. The BaO.6KO.4Bi03 numerical results for th e transition te mperature are presented in the fo ll owin g plots for various 30 ApI =: 0.1 ~ = 0.2 conditions. In Fig. 3,th e result for T, is shown as a function of 11 up to 0.30, for a set of parameters 'A/, II = 0.6 and 'A/,I = 0 . 1. The analytic expression for Te from Eq. ( 12) c learl y demonstrates that, T, is strongly influenced by the Coulomb-repulsive parameter and 10 is hi ghe r for small values of 11. Further, T,. drops with the increase in 11 values as a consequence of th e unconventional plasmon-optical phonon-pairing o~--~----~----~--~----~--~ 0.50 0.55 0.60 0.65 mechani sm. T hese valu es of 'A" Ii and 11 are quite reasonable for conventional BCS superconductors. The present study, therefore, clearly points the ambiguity in scattered, experimentally reported 'A/'Ii Fig. 4 - V ariat ion of A ph with T. I7 2 valu es . -1, with a pre-assumption of the participation The effect of the e lectron-plasmon coupling of weakly-coupled pl asmons as an additional force strength on T,. has a lso been investigated. F ig. 5 of pairing. shows the vari ation of T, for different valu es of 'A/,I An important feature of Eq. ( 12) is displayed in varyin g from 0.0 to 0.4. In pl otting this curve, we Fig. 4. ~ . is plotted as a function of 'A"II up to 0.65 take 'A/,II = 0.6 and 11 = 0.2. T, is sensiti ve to \ " even for a set of parameters 11 = 0.2 and \ JI = 0. 1. The for weak coupling (Fig. 5). If we stan wi th a pure va lu e of T, increases steeply with the enhanced phonon mechani sm within two square-well mode l va lue of 'A/,II (Fig. 4). For the above set of and later on adding a correction te rm with weakly- V ARSHNEY el al.:SUPERCONDUCTING TRANSITION TEMPERATURE 885
coupled plasmons, then, in this situation one can Apil in a usual way and consistent with the easily get enhanced Tt values. Even though in the conventional superconductors. present three square-we ll approach, Cooper theory has been extended by introducing an attractive plasmon term in an ad hoc way, the authors, Bi0 1. 50 Bao.6Ko.6 3 therefore, do not claim to possess a rigorous result for the coupling strength parameters. However, the TC = 25 K A~ = 0.1 essence of collective excitations can be further 0.75 verified to see whether, by this way, one can enh ance T, and do they play a crucial role in the ~ 000 attractive pairing mechanism.
50r---~--~--~---r--__--~ __~ __~ ·0.75 BaO.6K0.4Bi03
45 Aph =0.6 11 =0 .2 ·1.50 :--~-...l-_~_.....L_~_--I._--.J 0.5 0.6 0.7 0.8 Q '-' 40 f-U
Pig. 6 - Variation of Aph with J-l ' 35 4 Conclusions
Parent BaBiOJ is a diamagnetic insulator and 307---~~~--~---L--~ __~ __~ __~ chemical substitution of mono-valent metal K at the 00 0.1 0.2 0.3 0.4 divalent Ba site in BaBiOJ , which can be viewed as ApI an ionic solid, introduces free holes for the purpose of current carriers. In the present investigation, the Fig. 5 - Variation of Api with Tr authors evaluate a three square-well model and treat The authors now focu s on the relationship in the attractive pairing mechanism for between A" II and 11* within the three square-well superconductivity in such a 3D cubic perovskite mode l for Ba-K-BiO superconductors. Although, material as arising from Coulomb interactions, by optical phonons and plasmons. The main focus has phonons are capable of yi elding high-Tc for reasonable coupling strengths, conventional been on relating the model to physical parameters of BCS/Eliashberg theory predicts the combination of the superconducting state, I.e. the transition hi gh-T, and one-half value of isotope effect for temperature T,. mono-atomic systems. In Ba-K-BiO, isotope effect In the present work, the authors have is suppressed from one half value (a vary between generalized the Cooper pair expression for two-well 0.22 and 0.4) which excludes the purely phonon repulsive-attractive model. To account for high-T. in mechanism. While solving the Eliashberg equation cubic bismuth oxides, they assume that, there is within the three square-well model, the authors plot definitely some other non-phonic mechanism is the variation of "-t,1I with l1*in Fig. 6. For lower involved. They derive a three square-well mode l values of A"II (~ 0.6), 11* yields un-physical values with Coulomb, electron-phonon and electron for a set of parameters (7: = 25 K and A,,, = 0. 1) . plasmon interactions within the framework of Negative 11* implies a constant attractive Eliashberg theory which essentially points to the interaction. Correct picture of re-normalised relative contributions of the various interactions in Coul omb-repulsive parameter is reflected in these yielding a high-T,. value. The present analysis ox ide superconductors with at least an intermediate reveals that, Tr: strongly depends on the Coulomb strength of electron-phonon coupling. It is repulsive parameter and its reduced value leads to emphasised that, even if the weak coupling of high-Tc value as a consequence of the plasmon plasmons is included, the result for 11* is affected by optical phonon-pairing mechanism. 886 INDIAN.J PUR E & APPL PHYS. VOL 40. DECEMB ER 2002
Also, T, in creases steepl y with th e enhanced 1:1 (II .. Phvsim C. 15X ( 19X9) 5 19.
va lu e of either A/'Ii or A/,, va lue. One cann ot escape in 10 Huang Q. Zasaclzinski .I F, T ra las halVa la N . Gray K E "I stating th at. apart from intermediately-coupl ed (II. , N(l/lIrl'. 347 ( 1990) 369, phonons, weak ly-coupl ed plasmons mu st be a viable I I Kreslil V Z. Phr.\" Nc\ ' n, 35 ( 19 (7) 87 16: Ru va lds .I , PhI''\" mec hani sm fo r cubic perovskit e bismuth ox id e Rt'I' n. 35 ( 1(87) X869; Ashkcn zai J, Kuper C .I & T yk R. supercondu ctors. The strength of attrac ti ve Solid 51 COIIIIIIIIII. 63 ( 19 X7 ) 1145: Varshn cy Dincs h &. co ll ective exc itati ons in other low dimensional Tosi M P . .I Phrs Ch('lll Solids. 6 1 (2000) 6X 3 and rckrcnces Ihcrei n. copper ox id e supercondu ctors has been investi ga ted. In conclu sion, supercondu ct ivity in .I' -wave 12 Wagcncr T J. Meyer III H M. Hill D M , I-Iu Y ('/ III .. Phrs R('I' /3 . 40 ( 19(9) 45 32. Ball" K\l1 Bi0 1 is successfull y ex plained by a combined plasmon-opt ica l phonon-pairing 13 Rowvic I, Kim J H. Harris J S (.11' ). Hel lillan ESc/ (/1 .. mechani sm. PhI'S RI'I' 13 . 46 ( 1992) I I X2. 14 Shi rai M . Suzuki N_ & MOlizuk i K . .1 Pin's CO lldells Mall, Acknowledgement 2 ( 1990) 3553
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