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Possible explanation of high-Tc in some 2D cuprate superconductors
Pergamon
PII: S0022-3697(96)00145-X
J. Phys. Chem SolidsVol 58, No. 7, pp. 1153-1159, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/97 $17.00 + 0.00
POSSIBLE EXPLANATION OF HIGH-Tc IN SOME 2D CUPRATE SUPERCONDUCTORS A. K. M. A. I S L A M * a n d S. H. N A Q I B f Jawaharlal Neheru Centre for Advanced Scientific Research, Indian Institute of Science Campus, Bangalore, India, ~'Department of Physics, Rajshahi University, Rajshahi-6205, Bangladesh
(Received 12 June 1996) Abstract--The present paper deals with the superconducting state of a prototype of high transition temperature superconductor (SC) Lal.s5Sr.lsCUO4 with strong 2D features and relatively low carrier concentration. The study is carried out within the Eliashberg framework by considering a two-component total vertex consisting of electron pairing mediated by quantized lattice waves as well as plasmons. For the latter part the low-frequency ionic-plasmon mode has been assumed to play an important role in preference to electronic-plasmon in bringing about superconductivity. We then derive an expression for the transition temperature Tc applicable to layered SCs. A comparison of the calculated Te and other properties with experiments provides support to this type of generalised joint phonon-plasmon mechanism in LSCO. An extension of this formalism to still higher Tc-materials, e.g. in YBCO, is encouraging. The possibility of interlayer hopping and the consequences of inclusion of non-adiabatic effects in interactions beyond Migdal's theorem are also discussed. © 1997 Elsevier Science Ltd. All rights reserved.
Keywords: A. 2D cuprate SCs, D. high Te mechanism, D. non-adiabatic effects, D. pressure effect D. resistivity behaviour.
1. INTRODUCTION Nearly a decade after the discovery of the first cuprate superconductors (Ses) [1, 2] the origin of superconductivity in these and other recently discovered high-Tc (HTC) compounds remains elusive and no general concensus has been reached [3-27]. The larger isotope effect in highly doped La2_xSrxCu04 and a small but still non-zero effect in other perovskites still possess problems. Together with a small isotope effect (especially in cuprates with the higher To) the observation led to the idea of non-phonon mechanism of high Tc [3]. On the other hand there are a number of experimental facts including the considerable phonon softening at the superconducting transition which indicate a pronounced electron-phonon (e-ph) interaction in some of these cuprate superconductors [4]. A variety of pairing mechanisms and superconducting states can be proposed in view of a phase diagram as depicted in Ref. [11]. These involve e - p h or chargetransfer models in which the orbital pairing has s-wave or anisotropic s-wave symmetry, models based upon the antiferromagnetic (AF) nature of the undoped system which involve dx2y: pairing or other possibilities, strongly correlated semion gauge models and so on (see Ref. [11] for citations). As results from
*Permanent address: Department of Physics, Rajshahi University, Rajshahi-6205, Bangladesh.
phase-sensitive experiments accumulate (see Refs [6, 7] for a summary), the nature of the wavefunction in HTCSC, is somewhat clearer, but the pieces do not yet all fit. Some of the recent experiments are consistent with a 'd-wave picture' in which the electrons are paired in a state with l = 2. There are also significant measurements which say that it is controversial or give evidence for s-wave pairing [8, 9]. The tunnelling experiments done by Sun and Dynes [8] tend to rule out d-wave gap anisotropy. The order parameter symmetry in layered cuprate SCs is also one of the most important issues currently under debate [10-12]. Despite favouring evidence for dx2_y~state in BSCCO and others there are still experimental uncertainties in YBCO [12, 13]. There is clear evidence for the conventional BCS-like pairing in edoped HTC cuprates [14], but experimental results on hole-doped materials still remain largely controversial (see Ref. [15]). Moreover the gap anisotropy even in hole-doped materials is low and not as high as commonly believed [15]. The relevance of Eliashberg theory for the description of HTCSC has been considered by many authors (see [18]). Varelogiannis [18] claims that exotic features can be reproduced and understood in the context of s-wave, conventional, strong coupling M i g d a l E l i a s h b e r g - N a m b u (MEN) theory of superconductivity [10]. Characteristic anomalies in the tunnelling and photoemission data of Bi2Sr2CaCu202 like
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a diplike structure and a second peak structure can be obtained within Eliashberg theory [18]. The gap ratio values and the disagreement of the gap measurement by infrared spectroscopy with the other gap measurements in YBaECU307 has also been shown to be understood in the context of conventional strongcoupling theory [18]. The author further remarks that his approach using s-wave pairing is successful and he clearly disagrees with the conclusion of Ref. [26] that these structures are evidence of d-wave. The behaviour of the low energy dynamics of Y BaECU307 can be taken as evidence of conventional superconductivity [23]. The experimental implications of thermal-difference reflections spectroscopy in Tl-, Bibased compounds in addition to YBCO has been discussed thoroughly by Holcombe et al. [21]. They observed that both the temperature and energy dependence of the structure of the R s / R N spectra may be adequately described within Eliashberg theory with an e-boson coupling function with low and high energy components. Thus the situation in HTCSC, unlike heavy fermion SCs (as UPt3 or UBel3 where l = 2 pairing is evident) has been naggingly controversial, both experimentally and from the point of view of theory [10]. In spite of the progress on the material aspects of HTSCs, there remain widely divergent views regarding the nature of both the normal and the SC states as well as the origin of the pairing mechanism responsible for the superconductivity (see Refs [11, 12]). Numerous pieces of circumstantial evidence have been combined with heuristic theoretical arguments to make a compelling case that pairs are bound with non-phonon 'glue' (see Ref. [7]). In addition to the conventional e-ph interaction serious calculations explaining the whole spectrum of interactions are being explained using excitons, spin-fluctuation, holes, spin bags, valence fluctuations, acoustic plasmons, marginal Fermi Liquid, RVB and so on [12]. The proposed models for nonconventional SCs are actually motivated by anomalies in normal state properties. According to Benedetti et al. [24] the extension of these concepts to the superconducting state is not obvious and the conventional Eliashberg theory remains the only framework for the analysis of the superconducting properties. But some important questions that have been raised (e.g. the symmetry of the pairing, or the scattering rate behaviour near To) still remain open. Such questions can only be clarified when contradictions in the experimental results disappear [22]. To summarize, a fully consistent picture of the experimental data is still lacking at present and fresh efforts are needed to reduce the possibilities further and to understand the pairing mechanism and the Fermi or non-Fermi liquid nature of the normal state.
Hence despite intensive studies of the possible involvement of alternate microscopic mechanism, the e-ph interaction remains the only proven underlying cause of superconductivity, and it may play an important role or even crucial role in the basic mechanism of HTCSCs (see Ref. [12]). In the background of all these controversies one continues to discuss HTCSCs using the conventional or generalized electron-boson mechanisms. Several researchers [29-38] made a number of theoretical studies, some immediately after the discovery of the layered HTCSCs. Kresin [29] derived a formula for T¢ (appficable in the strong coupling regime) for LSCO. The analysis made by Shiina and Nakamura [39] shows that for materials with T¢ _> 30 K use of McMillan or Allen-Dynes formulae leads to an incorrect result. Weber [38] calculated the e-ph spectral function ct2F(w) for LSCO. The analysis yields a value of coupling strength A = 2.5 and a single-spin electronic density of states (DOS) which is at least half of the band structure value. This points towards a smaller A value in LSCO, which could be achieved through hardening of the e-ph spectrum with more coupling to the high frequency modes, or it could mean that we have a joint phonon-excitonic mechanism. For LSCO the e-ph mechanism is present-but the question is to what extent? The scenario is more puzzling when we consider the well-known material YBazCu307 where a pure theoretical consideration gives A ,-~ 8.6 with Coulomb pseudopotential #¢ = 0.13 (see Ref. [32]). Here the oxygen isotope effect range from 0.03 to 0.17 with a probable average of 13ox= 0.05 [40-42]. Measurements of isotope effect on Cu and Ba sites show they are very small and often below resolution. The reasons for the apparent absence, or at least smallness of the isotope effect in HTCSC, and what this means for the e-ph interaction are still debated. There may be several factors which can suppress isotope effects. Both a calculation of Allen et al. [43] and isotopic measurements show that LDF theory overestimates Drude frequency and suggest the need for additional pairing mechanisms besides e-ph, particularly for YBCO. Franck et al. [41] investigated Cu- and O-isotopic effects in LSCO. Nickel et al. [42] studied parital 180 substitution in YBCO. The results show that large parts of the phonon spectrum influence the transition temperature, de Wette and co-workers (see Ref. [28]) have published lattice dynamical calculations of the phonon spectrum in LSCO and YBa2Cu3OT_~(6=0-1). The spectra for SC and non-SCs (6 = 1) show marked shifting and difference (in addition to difference in structure) in an expected way. All these results show that large parts of the phonon spectra influence the transition temperature--an effect of strong e-ph interaction.
High-Tc in 2D cupratc superconductors
In this paper we discuss afresh the role of a joint phonon and plasmon mechanism in the 2D layered La2_xSrxCuO 4 ( x = 0 . 1 5 ) material. This type of mechanism could give a fairly consistent picture for all SCs--both conventional and HTCSC. The conventional ones fall in the pure phonon limit, while LSCO and YBCO could be the joint type. We believe that there is much to be learned from studying such materials, which offer relatively simple crystalstructure (single C u - O layers with minimal site disorder or unintentional oxygen non-stiochiometry). For obvious reasons the charge carriers (holes or electrons) will be called 'electrons' throughout the paper. We would consider here that coupled 2D electronic-plasmon and ionic-plasmon modes exist. The low-frequency ionic plasmon mode and not the other is assumed to play an important role in bringing about superconductivity in addition to the normal quantized lattice interactions (e-ph interactions). We then derive a Tc equation applicable to prototype layered SC. The study will then be extended to still higher HTCSCs, e.g. YBCO, in which the isotope effect is small. The possibilities of contribution from the 1D chain and also of interlayer hopping as a secondary role are discussed briefly. The consequences, such as substantial suppression of isotope effects and the importance for various normal state properties of inclusion of nonadiabatic effects in interactions beyond Migdal's theorem will also be discussed. 2. MODEL AND FORMALISM
The experimental evidence concerning the symmetry of the order parameter in cuprates has shifted over the past few years. Nevertheless, substantial evidence favouring s-wave pairing continues [10]. We neglect, for simplicity, any anisotropy in the c-axis, since the carriers involved are on the CuO2 plane. As far as the in-plane gap of the material is concerned, the overall anisotropy is believed to be not so large as to alter the s-wave pairing description [39]. The vertex correction here is relatively small compared to the case of fullerene compounds where the ratio of pairing energy to Fermi energy is ,,~2-3 times larger. Following Yu et al. [25] we do not consider the vertex terms (a discussion will be presented later on) at this moment in order to maintain the clarity and manageability of the formalism. In a layered 2D system the plasmon has a wide excitation band which starts from k = 0. This encourages all the electrons within the Fermi surface to feel an attractive e-e interaction through plasmon exchange. Thus the contribution of both phonons and plasmons (providing an additional mechanism leading to an increase of To) will be studied here using the
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Temperature Green's function method, and the generalized Eliashberg equation. We write the energy and momentum dependent order parameter A as follows [32, 39, 44]:
A(~.,k)=(2~2Z)-'T~ f dk' [j=~rb,plFJ(Wn - ~m, k - k ') l F+ (Wm, k ')
(1)
Here k is the 2D electron wave vector, Z the renormalization function. If ( refers to electron energy referred to the Fermi level, the anomalous Green's function which describes the pairing is
F+(wm,k ,) = A(wm,k,)[w2 + (2 + A2(Wm, T)]-I (2) The vertex ['ph describes e-ph interaction with two electron ends and one phonon end. Similarly for the corresponding case for Fpl. Since the pairing is caused by conventional phonon exchange, Fph can be written in terms of the phonon Green's function Dph and the corresponding e-ph coupling c o n s t a n t Aph, which can then be expressed in terms of the characteristic phonon frequency ~'~ph as Fph = AphDph(0dn -- 0Ym,k - k') 2 2 + (Wn -- 03m)2]-1 = Aph[~ph[["~ph
(3)
We now consider electron-plasmon (e-pl) interaction. Here we assume coupled 2D ionic and electronic plasmon modes whose existence is the key consequence of a theory developed by Gersten [45]. One of the coupled modes, w+, is basically e-pl like and is not of much interest. The low frequency mode, w_, however, plays an important role in bringing about superconductivity. Some of the experimental consequences posed by the existence of this low frequency mode w_ have already been discussed [45]. The characteristic frequency f~pl, in this case, can be expressed as apl :
(2~rN~ikF/M%)l/2Qe
(4)
where kF is the Fermi wave vector, M the mass of the unit cell. N o is the reciprocal of the unit cell projected perpendicular to the c-axis. The ionic charge Q is based on the standard valence for the ions in the CuO2 plane(s). It determines how much ionic charge lies in the conducting plane(s). Finally % is the background dielectric constant. Now Fpl consists of the Coulomb repulsion u¢ (in 2D case, bare Coulomb), and its product with the plasmon Green's function Dpl [46]: Fpl = Uc + ucDpl = Uc + Ucfl21(k)[ca2 - f~2pl(k)]-I (5)
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where an effective D function [33] has been used. It follows then that effective attraction corresponds to the region w < f~pl. The renormalization function is given by [32]
Z(iwn) = 1 + 7rT/wn E
A(iWm - iwn)
m ~m
[W2m+ A2(iWm)],/2
(6)
Here iwn = iTrT(2n-l) is the nth Matsubara frequency and n --- 0, +1, 4-2, ..., ..., with T being the temperature. The term, A(iWm- iw,,) in eqn (6) is related to the electron-phonon (or other boson) spectral density a2F(w) (w = frequency of the exchanged boson) through the relation
A(iw,,, - iw,,) = 2
f
f~a2F(f~)df~
[~2 + (~n -
~)2]
-- A(m - n)
(7)
This can be reduced to the much simpler form if a two square-weU model (see Ref. [31]) is taken as an approximation. In this model A(iw,n - iwn) = A; I~nl, tWml < we = O; otherwise
(8)
where we is the cut-off used in order that the sum over m in eqn (I) converges. Thus the effective renorrnalization parameter is given by A(rn = n) = A(0) = A = f 2a2F(f~)df~/f~
(9)
If in the equation for Z we neglect the gap we get
Z(iwn) = 1 + 7rTIwn E
A(iWm - iwn)sgn(wm)
m
= 1+ A
(10)
In the combined phonon-plasmon mechanism the gap equation (with the help of model gap of eqns (8) and (9)) reduces to the approximate form at T = T¢: A(iwn) = . "~ph 7rT¢' ~ A(iWm)
I +AT
~
IWm[
+ ~pl ~r~7,a(i~) l+~r
c~
I~l
(11)
In terms of the digarnma function ~(N), the above equation reduces to l =
APh
1 -t- Ar [~b(fZPh/2~rT¢+ 1/2) - 'I~(1/2)1 +
(!b(f]pl/ETrTc + 1/2) - ~(1/2)]
Applying the appropriate approximation for # ( N ) this yields the required transition temperature Tc
2e6 AP Ap [- ~ ) Te = --~- (Qph) Ph(~pl) p'e h~h+h"
(12)
Here 6 is the usual Euler's constant and Ap = Aj/(Aph + Apl), w i t h j for 'ph' or 'pl'. The eqn (12) looks similar to that derived by Kresin [33] but is, in fact, quite different in both Qpl and in the exponential term. Kresin [33] derived his equation with the assumption of strong Aph(Aph>> Apl). But no such assumption as regards the strength of coupling has been utilized in this paper. The formula through eqn (4) contains the magnitude of the electron charge, thus it makes no difference whether the mobile carriers are electrons or holes. As can be seen, the eqn (12) does not contain the Coulombic pseudopotential term #c. It is taken to be zero, keeping in mind that a non-zero #¢ reproduces the same phenomenology (as long as A is not too small), except for slightly higher values of the coupling, which depends on the value of #c. As a matter of fact the repulsion is reduced due to vertex correction [47]. The behaviour of any an/sotropic energy gap and Tc in the presence of the 'anticipated' anisotropic Fermi surface and two coupling mechanisms is no doubt a complex problem, which needs more careful consideration than has been given here. But still the above equation can be considered to be not too far from a good approximation. The above Tc equation may also be written in a form more suitable for our purpose as
2e6 ~ -- - -7r (Uph To--
,~P 1"2 ~P I-'+~h+~'] ) ph(asQ/M / ) p,e ~Ph+xP'
(13)
where a = e¢27rN~ikF/%, which is roughly constant at constant pressure for the 2D cuprate materials considered here. s is the number of CuO2 planes in the unit cell.
2.1. Non-adiabatic corrections For an exact solution of the problem of HTC superconductivity one needs to look beyond the Migdal approximation [20]. Non-adiabatic effects and the e - p h interaction have been studied recently by several workers [47-51]. The generalized gap equation, including the non-adiabatic effects, has been shown in Fig. 1 in Ref. [50], the double electronic lines indicate the inclusion of all the self-energy effects. These are the two vertex corrections and cross phonon scattering. N/col et al. [48] neglected the momentum dependence of the self-energy but used local approximation in their study of vertex corrections. The k-averaging approximation by Krishnamurthy et al. [49] yielded a modest reduction in Tc corresponding to the van Hove singularity in the density of states (DOS). Correction due to the cross phonon scattering, in addition to the vertex correction has been
High-To in 2D cuprate superconductors done by Grimaldi et al. [50]. Cappelluti et al. [51] also performed similar calculations. Under the scheme of Grimaldi et al. [50] non-adiabatic effects show a complex structure as a function of exchanged frequency and momenta. In order to see in a simple way how the correction is brought about we consider the structure of the vertex and the renormalization factor. The structure of the vertex is A(wn, Wm, q, wo, EF) = 1 + Aev(wn, Wm, q,~o, EF) where the symbols are as defined in Ref. [50]. Evidently the correction term multiplies the bare vertex. Let us ignore the momentum dependence. In this case the renormalization function in the lowest order turns out to be [47] Z(iw o : O) = 1 + A[Zwo/Ev + 1]-1
which shows roughly the correction introduced (i.e. a reduced Z factor) for ~o/EF ~ O.
3. NUMERICAL RESULTS 3.1. L S C O We take LSCO as a test case. Here we have a single layer in the unit cell. The interlayer interaction is neglected because of the small out of plane coherence length and large interlayer distance ,-,6.6 A compared to ~3.1 A typical for other multilayer structured materials. The characteristic plasmon frequency is determined to be ~'~pl = 8.53 x 1012 Hz (see Table 1). Oo = 390 K is taken from Ref. [52]. We consider that at present Oo datum yields a much more reasonable estimate of f~ph. This may be compared with an estimation of Allen et aL [43] giving f~ph ~ 200K, with an uncertainty of almost 100%. We first evaluate A t ( = Aph + Apl) for LSCO. The values of the Sommerfeld constant -y reported in Marsiglio et at. [54] and Ramirez (see Ref. [55]) are taken into consideration. The average of these two values turns out to be 8.5 (mJ/mole.Cu.K2). The average band structure DOS, N(0) is 0.97 (states/eV. f.u. spin) [43]. Utilizing these, we obtain Ar = 0.85 + 0.05. The details are given elsewhere [53]. At this point it is worth mentioning that Marsiglio et al. [54] have examined the combined phonon-exciton mechanism for LSCO based on the study of the Table 1. Parameters used in calculating 9tp] (after Ref. [53]) Parameters
PCS 58: 7-E
Values
kF
4.70 x 107cm -1
~0
4.0 7.04 x 1014cm-2
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isotope effect. They have taken ~')ex = 500meV and varied )~ex" They fixed Te = 36K for LSCO to get AT ",~ 0.9, and Aex "~ Aph = 0.45. Evidently Marsiglio et al. [54] have taken the electronic excitons into consideration, manifested in the extremely high energy of the exciton (500 meV is nearly five times the Fermi energy of LSCO (see Ref. [53]). Also the procedure adopted by these authors violates the Migdal theorem [20]. Thus there is considerable doubt in this approach. Our analysis yields, )kph = 0.69 and/~pl ~--- 0.16 from which one arrives at Tc "-~ 36 K. As regards isotope effects we find that eqn (13) predicts a shift in the Tc of 0.8 K when all the 160 is substituted by 180. This is to be compared with experimental value of ~0.5 K [40, 41]. Now we consider the pressure dependence of Tc. It follows from eqn (4) that as pressure increases Ni increases and Tc is raised as a result. According to band structure calculations (see Ref. [57]) the DOS is flat near the Fermi level in La2CuO 4. So N(0) should not depend strongly on pressure. Utilizing this we observe that, if N 0 changes by ,,,2.8 % due to a change of 1 kbar of pressure, A T c ~ 1.7 K, due to P = 6 kbar. This is in agreement with the experimental value [56]. The linear behaviour of resistivity as a function of temperature is a puzzling feature of HTCSCs. Nearlinear temperature dependence is observed in LSCO with x ,,~ 0.15-0.18 [57], where measurements have been reported from T = 50-1000 K. As T is lowered towards zero the resistivity appears to extrapolate linearly to zero until Tc is reached-thereafter it falls suddenly to zero. Linear behaviour of p(T) occurs in many metals when T > OD, i.e., the thermal energy must exceed the characteristic energy. Since the characteristic temperature of the low frequency mode in our case if ,,~65 K, the near-linear behaviour is somewhat contained in our formalism at least up to 28 K above Tc for LSCO. The joint mechanism model presented here is more consistent with present band-structure calculations for the values of DOS and EF. The question is whether band-structure calculation can be applied directly in such correlated systems. However, fairly convincing evidence is accumulating in favour of band theory [32].
3.2. Implications o f the f o r m a l i s m to other H T S C s
Several HTCSCs have 2D structural features with several layers. Some of them have 1D C u - O chain. YBa2Cu307 possesses two active CuO2 conducting layers and 1D CuO chain in the unit cell. The distance between the nearest planes is --,3.1 .~ and that between the chain and the plane is ,,~4.3 A. Very recently Maly et aL [13] in their theoretical calculations find two gaps appropriate to each of the multiple bands. But
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A . K . M . A . ISLAM and S. H. NAQIB
they also observe that van Hove singularities stabilize some order parameters over others and elongate the gap functions along to four van Hove points, thereby leading to a substantial region of gaplessness. First we assume that the chains nearly provide room for plane-chain scattering without being superconducting by themselves, that is, without interlayer interaction on CuO chains [25]. Then we assume that all C u - O layers in a unit cell are strongly correlated and can be taken as one with effective Q (which is the net bound valence charge of ions of the layers). This in turn increases the characteristic plasmon-frequency. The Fermi momentum in YBCO is comparable to that of LSCO, and Ni remains nearly the same for both the SCs. The mass of the unit cell, M ~ 1.66 times of that of LSCO. We take the value of the Sommerfeld constant 7(0) = 11.7 mJ/mole K 2. Widely scattered values of 7 are found in the literature (see Ref. [55]). An average of these unusually scattered values turns out to be ,-all. Instead we have chosen the average of three consistent sets of values of 12, 12, 11 reported in three published papers (see Ref. [55]). The DOS N(EF) at Fermi level is taken to be ,~ 0.93 (state/eV f.u. spin) [43], which then yields Ar '~ 1.66. This may be compared with the value obtained in self-consistent localdensity approximation band structure calculation of )~ph ~ 1-1.5 (see Ref. [17]). OD is taken to be 465K (see Ref. [28]). Proceeding as before we get Tc ,,~ 80 K for Ar = 1.66 with /~pl//~ph ~- 0.23, the same ratio is found for LSCO. The estimated value is reasonable considering the uncertainty of the input data used. It should be noted that a further enhancement of this value could be obtained through vertex correction. A similar approach can be adopted for layered SCs with still higher T~ values. Prospects of pairing enhancement of HTSCs containing planes with 1D chains has been discussed by Kresin and Morawitz [31]. They find additional enhancement of Tc qualitatively due to the coupling between carriers in the planes and quasi- 1D plasmons. We do not consider here any charge transport along the c-axis for multi-layered SCs as the planes are somewhat isolated. Moreover the carrier concentrations in the planes are low. But it is interesting to note that pairing enhancement due to an interlayer Cooper-pair tunnelling has been considered recently [59, 60]. Byczuk and Spalek [60] considered both the intrinsic inplane pairing and the two interplanar Josephson tunneling process, which they claim is valid for both Fermi- and non-Fermi liquids. If s is the number of tightly spaced identical CuO2 planes, and separated by a large distance, but in contact with each other through non-superconducting region, there is a weaker Cooper-pair tunnelling. Such inter-
layer hopping could enhance Tc for multi-layered materials including YBCO, which is given by [60]
Aro :~ cos(~-~+0. 4. DISCUSSION We have developed here a general approach to the pairing mechanism of superconductivity with 2D systems. The Tc equation yields reasonable values of transition temperature for both LSCO and YBCO. In deriving the Te formula we have used some approximation, the implication of which has been analysed by us. It is found that for LSCO, eqn (12), due to some approximations, slightly overestimates the predicted Tc. But because of uncertainties in some of the input parameter values, the approximation should not be of much significance here. The vertex corrections to the electron self-energy are reduced by the order of Migdal's parameter m = ~ph/Er. F o r AxC6o(x = 3, A = K, Cs) compounds, m ~ 0.5-1. Lower values for HTC cuprate SCs occur (~0.3 for LSCO for phonon interaction). Although smaller compared to fullerenes, these are still not small enough to neglect the correction for phonon vertex. It has been observed [50] that a predominance of small momentum scattering leads to an enhancement of Tc with respect to the adiabatic theory with the same coupling [50-51]. The behaviour of the correction to Tc has been evaluated and displayed in Fig. 5 in Ref. [50]. Vertex corrections are expected to have implications on both superconducting and normal state properties (like transport, photoemission, life times effects [49]). For example, the isotope effect becomes negligibly small depending on the value of Migdal's parameter. Using the formalism developed in Refs [47, 48] we have roughly estimated that flox(YBCO) may be as low as 0.1. The analysis is far from complete. For example we have not incorporated vertex corrections and the Coulombic pseudopotential term in the formula derived. Instead we have discussed some possible consequences using the formalism of other works [47-51]. A more realistic description of the various contributing factors is to be undertaken in a future work to learn more about the systems.
Acknowledgements--The authors acknowledge the help received from the University of Rajshahi under an Annual Research Programme. The work was performed in part, at Jawaharlal Neheru Centre for Advanced Scientific Research (Indian Institute of Science Campus), Bangalore, where it was supported by JNCASR-IISC-ICTP AssociateshipProgramme. REFERENCES
1. Bednorz, J. G. and Muller, K. A., Z. Phys. B64, 189 (1986).
High-Te in 2D cuprate superconductors 2. Uchida, S. H., Takagi, T., Kitazawa, K. and Tanaka, S., Jpn. J. AppL Phys. 26, L1 (1987). 3. Sherman, A. and Schreiber, M., Physica C 253, 23 (1995). 4. Rietschel, H., Pintschovius, L. and Reichardt, W., Physica B 186-188, 813 (1993). 5. Pickett, W. E., Krakauer, H., Cohen, R. E. and Singh, D. J., Science 255, 46 (1992). 6. Levi, B. G., Physics Today, p. 19 (1996) for a survey and references therein. 7. Cox, D. L., Maple, M. B., Physics Today, p. 32 (1995). 8. Sun, A. G., Gajewski, D. A., Maple, M. B. and Dynes, R. C., Phys. Rev. Lett. 72, 2267 (1994). 9. Chaudhury, P. and Lin, S. Y. Phys. Rev. Lett. 72, 1084 (1994). 10. Schrieffer, J. R., SolidState Commun. 92, 129 (1994). 11. Scalapino, D. J., Physics Reports 250 329 (1995) and references therein. Scalapino, D. J. Superconductivity (Edited by R. D. Parks), Vol. 1. Dekker, New York (1969). 12. Dynes, R. C., Solid State Commun. 92, 53 (1996). 13. Maly, J., Liu, D. Z., Levin, K., Phys. Rev, B 53, 6786 (1996). 14. Wu, D. H., Mao, J., Mao, S. N., Peng, J. L., Xi, X. X., Venkatesan, T., Greene, R. I. and Anlarge, S. M., Phys. Rev. Lett. 70, 85 (1993). 15. Openov, L. A., Elesin, V. F. and Krasheninnikov, A. V., Physica C 257, 53 (1996). 16. O'Donovan, C. and Carbotte, J. P., Solid State Commun. 97, 799 (1996). 17. Zeyher, R. and Kulic, M. L., Phys. Rev. B 53, 2850 (1996I). 18. Varelogiannis, G., Phys. Rev. B 51, 1381 (1995-11). 19. Eliashberg, G. M., Soy. Phys. JETP 11, 696 (1960); Nambu, Y. Phys. Rev. 117, 648 (1960). 20. Migdal, A. B., Soy. Phys. JETP 7, 996 (1958). 21. Holcombe, M. J., Perry C. L., Collman J. P. and Little, W. A., Phys. Rev. B 53, 6734 (1996). 22. Temmerman, W. M., Szotek, Z., Gyorffy, B. L., Anderson, O. K and Jepsen, O., Phys. Rev. Lett. 76, 307 (1996). 23. Varelogiannis, G., Phys. Lett. A 192, 125 (1994). 24. Benedetti, P., Grimaldi, C., Pietronero, L. and Varelogiannis, G., Europhys. Lett. 28, 351 (1994). 25. Yu, B.D., Kim, H. and Ihm, J., Phys. Rev. B 45, 8007 (1992). 26. Allen, P. B., Comments Cond. Mat. Phys. 15, 327 (1992). 27. Coffey, D. and Coffey, L., Phys. Rev. Lett. 70, 1529 (1993). 28. de Wette, F. W., Comments Cond. Mat. Phys. 15, 225 (1991). 29. Kresin, V. Z., Phys. Lett. A 122, 434, (1987). 30. Ruvalds, J., Phys. Rev. B 35, 8869 (1987). 31. Kresin, V. Z. and Morawitz, H., Phys. Rev. B 37, 7854 (1988). 32. Carbotte, J. P., Rev. Mod. Phys. 62, 1027 (1990). 33. Kresin, V. Z., Phys. Rev. B 35, 8716 (1987); J. Mater. Res. 2, 793 (1987).
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34. Kresin, V. Z. and Wolf, S. A., Phys. Rev. B 41, 4278 (1990). 35. Takada, M., J. Phys. Soc. Jpn. 45, 786 (1978) and the references therein. 36. Malozovski, Y. M., Bose, S. M,, Longe, P. and Fan, J. D., Phys. Rev. B 48, 10504 (1993). 37. Varelogiannis, G., Phys. Rev. B 51, 3941 (1995). 38. Weber, W., Phys. Rev. Lett. 58, 2154 (1987). 39. Shiina, Y. and Nakamura, Y. O., Solid State Commun, 76, 1189 (1990). 40. Batlogg, B., Kourouklis, G., Weber, W., Cava, R. J., Jayaraman, A., White, A. E., Short, K. T., Rupp, L. W. and Rietman, E. A., Phys. Rev. Lett. 59, 912 (1987). 41. Franck, J. P., Harker, S. and Brewer, J. H., Phys. Rev. Lett. 71, 283 (1993). 42. Nickel, J. H., Morris, D. E. and Ager, J. W., Phys. Rev. Lett. 70, 81 (1993). 43. Allen, P. B., Pickett, W. E. and Krakauer, H., Phys. Rev. B 37, 7482 (1988); Freeman, A. J., Yu, J. and Fu, C. L., Phys. Rev. B 36, 7111 (1987). 44. Kresin, V. Z., Gutfreund, H. and Little, W. A., Solid State Commun. 51, 339 (1984). 45. Gersten, J. I., Phys. Rev. B 37, 1616 (1988). 46. Pashitskii, E., Soy. Phys. JETP 28, 1267 (1969). 47. Pietronero, L. and Strassler, S., Europhys. Lett. 18, 627 (1992). 48. Nicol, E. J. and Freericks, J. K., Physica C 235-240, 2379 (1994). 49. Krishnamurthy, H. R., Newns, D. M., Pattnaik, P. C., Tsuei, C. C. and Chi, C. C., Phys. Rev. B 49, 3520 (1994). 50. Grimaldi, C., Pietronero, L. and Str~issler,S., Phys. Rev. Lett. 75, 1158 (1995). 51. Cappelutti, E. and Pietronero, L., Phys. Rev. 53, 932 (1996). 52. Physical Properties of High Temperature Superconductors H (Edited by D. M. Ginsberg), p. 30. World Scientific, Singapore (1990). 53. Islam, A. K. M. A. and Naqib, S. H., Paper submitted to the 19th Annual BAAS Conference, Jahangirnagar University, Savar, Dhaka, Bangladesh (1996). 54. Marsiglio, F., Akis, R. and Carbotte, J. P., Physica C 153-155, 227 (1988). 55. Poole, C. P. Jr., Datta, T. and Farach, H. A., Copper Oxide Superconductors, p. 192. John Wiley, New York (1988). 56. Schirber, J. E., Venturini, E. L., Kwak, J. F., Ginley, D. S. and Morosin, B., J. Mater. Res. 2, 421 (1987). 57. Batlogg, B., Hwang, H. Y., Takagi, H., Kao, H. L., Kwo, J. and Cava, R. J., J. Low Temp. Phys. 95, 23 (1994). 58. Erskine, D., Hess, E., Yu, P. Y. and Stacy, A. M., J. Mater. Res. 2, 783 (1987). 59. Chakravarty, S., Sudbo, A., Anderson, P. W. and Strong, S., Science 261, 337 (1993). 60. Byczuk, K. and Spalek, J., Phys. Rev. B 53, R518 (1996-II).
PII: S0022-3697(96)00145-X
J. Phys. Chem SolidsVol 58, No. 7, pp. 1153-1159, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/97 $17.00 + 0.00
POSSIBLE EXPLANATION OF HIGH-Tc IN SOME 2D CUPRATE SUPERCONDUCTORS A. K. M. A. I S L A M * a n d S. H. N A Q I B f Jawaharlal Neheru Centre for Advanced Scientific Research, Indian Institute of Science Campus, Bangalore, India, ~'Department of Physics, Rajshahi University, Rajshahi-6205, Bangladesh
(Received 12 June 1996) Abstract--The present paper deals with the superconducting state of a prototype of high transition temperature superconductor (SC) Lal.s5Sr.lsCUO4 with strong 2D features and relatively low carrier concentration. The study is carried out within the Eliashberg framework by considering a two-component total vertex consisting of electron pairing mediated by quantized lattice waves as well as plasmons. For the latter part the low-frequency ionic-plasmon mode has been assumed to play an important role in preference to electronic-plasmon in bringing about superconductivity. We then derive an expression for the transition temperature Tc applicable to layered SCs. A comparison of the calculated Te and other properties with experiments provides support to this type of generalised joint phonon-plasmon mechanism in LSCO. An extension of this formalism to still higher Tc-materials, e.g. in YBCO, is encouraging. The possibility of interlayer hopping and the consequences of inclusion of non-adiabatic effects in interactions beyond Migdal's theorem are also discussed. © 1997 Elsevier Science Ltd. All rights reserved.
Keywords: A. 2D cuprate SCs, D. high Te mechanism, D. non-adiabatic effects, D. pressure effect D. resistivity behaviour.
1. INTRODUCTION Nearly a decade after the discovery of the first cuprate superconductors (Ses) [1, 2] the origin of superconductivity in these and other recently discovered high-Tc (HTC) compounds remains elusive and no general concensus has been reached [3-27]. The larger isotope effect in highly doped La2_xSrxCu04 and a small but still non-zero effect in other perovskites still possess problems. Together with a small isotope effect (especially in cuprates with the higher To) the observation led to the idea of non-phonon mechanism of high Tc [3]. On the other hand there are a number of experimental facts including the considerable phonon softening at the superconducting transition which indicate a pronounced electron-phonon (e-ph) interaction in some of these cuprate superconductors [4]. A variety of pairing mechanisms and superconducting states can be proposed in view of a phase diagram as depicted in Ref. [11]. These involve e - p h or chargetransfer models in which the orbital pairing has s-wave or anisotropic s-wave symmetry, models based upon the antiferromagnetic (AF) nature of the undoped system which involve dx2y: pairing or other possibilities, strongly correlated semion gauge models and so on (see Ref. [11] for citations). As results from
*Permanent address: Department of Physics, Rajshahi University, Rajshahi-6205, Bangladesh.
phase-sensitive experiments accumulate (see Refs [6, 7] for a summary), the nature of the wavefunction in HTCSC, is somewhat clearer, but the pieces do not yet all fit. Some of the recent experiments are consistent with a 'd-wave picture' in which the electrons are paired in a state with l = 2. There are also significant measurements which say that it is controversial or give evidence for s-wave pairing [8, 9]. The tunnelling experiments done by Sun and Dynes [8] tend to rule out d-wave gap anisotropy. The order parameter symmetry in layered cuprate SCs is also one of the most important issues currently under debate [10-12]. Despite favouring evidence for dx2_y~state in BSCCO and others there are still experimental uncertainties in YBCO [12, 13]. There is clear evidence for the conventional BCS-like pairing in edoped HTC cuprates [14], but experimental results on hole-doped materials still remain largely controversial (see Ref. [15]). Moreover the gap anisotropy even in hole-doped materials is low and not as high as commonly believed [15]. The relevance of Eliashberg theory for the description of HTCSC has been considered by many authors (see [18]). Varelogiannis [18] claims that exotic features can be reproduced and understood in the context of s-wave, conventional, strong coupling M i g d a l E l i a s h b e r g - N a m b u (MEN) theory of superconductivity [10]. Characteristic anomalies in the tunnelling and photoemission data of Bi2Sr2CaCu202 like
1153
1154
A.K.M.A. ISLAM and S. H. NAQIB
a diplike structure and a second peak structure can be obtained within Eliashberg theory [18]. The gap ratio values and the disagreement of the gap measurement by infrared spectroscopy with the other gap measurements in YBaECU307 has also been shown to be understood in the context of conventional strongcoupling theory [18]. The author further remarks that his approach using s-wave pairing is successful and he clearly disagrees with the conclusion of Ref. [26] that these structures are evidence of d-wave. The behaviour of the low energy dynamics of Y BaECU307 can be taken as evidence of conventional superconductivity [23]. The experimental implications of thermal-difference reflections spectroscopy in Tl-, Bibased compounds in addition to YBCO has been discussed thoroughly by Holcombe et al. [21]. They observed that both the temperature and energy dependence of the structure of the R s / R N spectra may be adequately described within Eliashberg theory with an e-boson coupling function with low and high energy components. Thus the situation in HTCSC, unlike heavy fermion SCs (as UPt3 or UBel3 where l = 2 pairing is evident) has been naggingly controversial, both experimentally and from the point of view of theory [10]. In spite of the progress on the material aspects of HTSCs, there remain widely divergent views regarding the nature of both the normal and the SC states as well as the origin of the pairing mechanism responsible for the superconductivity (see Refs [11, 12]). Numerous pieces of circumstantial evidence have been combined with heuristic theoretical arguments to make a compelling case that pairs are bound with non-phonon 'glue' (see Ref. [7]). In addition to the conventional e-ph interaction serious calculations explaining the whole spectrum of interactions are being explained using excitons, spin-fluctuation, holes, spin bags, valence fluctuations, acoustic plasmons, marginal Fermi Liquid, RVB and so on [12]. The proposed models for nonconventional SCs are actually motivated by anomalies in normal state properties. According to Benedetti et al. [24] the extension of these concepts to the superconducting state is not obvious and the conventional Eliashberg theory remains the only framework for the analysis of the superconducting properties. But some important questions that have been raised (e.g. the symmetry of the pairing, or the scattering rate behaviour near To) still remain open. Such questions can only be clarified when contradictions in the experimental results disappear [22]. To summarize, a fully consistent picture of the experimental data is still lacking at present and fresh efforts are needed to reduce the possibilities further and to understand the pairing mechanism and the Fermi or non-Fermi liquid nature of the normal state.
Hence despite intensive studies of the possible involvement of alternate microscopic mechanism, the e-ph interaction remains the only proven underlying cause of superconductivity, and it may play an important role or even crucial role in the basic mechanism of HTCSCs (see Ref. [12]). In the background of all these controversies one continues to discuss HTCSCs using the conventional or generalized electron-boson mechanisms. Several researchers [29-38] made a number of theoretical studies, some immediately after the discovery of the layered HTCSCs. Kresin [29] derived a formula for T¢ (appficable in the strong coupling regime) for LSCO. The analysis made by Shiina and Nakamura [39] shows that for materials with T¢ _> 30 K use of McMillan or Allen-Dynes formulae leads to an incorrect result. Weber [38] calculated the e-ph spectral function ct2F(w) for LSCO. The analysis yields a value of coupling strength A = 2.5 and a single-spin electronic density of states (DOS) which is at least half of the band structure value. This points towards a smaller A value in LSCO, which could be achieved through hardening of the e-ph spectrum with more coupling to the high frequency modes, or it could mean that we have a joint phonon-excitonic mechanism. For LSCO the e-ph mechanism is present-but the question is to what extent? The scenario is more puzzling when we consider the well-known material YBazCu307 where a pure theoretical consideration gives A ,-~ 8.6 with Coulomb pseudopotential #¢ = 0.13 (see Ref. [32]). Here the oxygen isotope effect range from 0.03 to 0.17 with a probable average of 13ox= 0.05 [40-42]. Measurements of isotope effect on Cu and Ba sites show they are very small and often below resolution. The reasons for the apparent absence, or at least smallness of the isotope effect in HTCSC, and what this means for the e-ph interaction are still debated. There may be several factors which can suppress isotope effects. Both a calculation of Allen et al. [43] and isotopic measurements show that LDF theory overestimates Drude frequency and suggest the need for additional pairing mechanisms besides e-ph, particularly for YBCO. Franck et al. [41] investigated Cu- and O-isotopic effects in LSCO. Nickel et al. [42] studied parital 180 substitution in YBCO. The results show that large parts of the phonon spectrum influence the transition temperature, de Wette and co-workers (see Ref. [28]) have published lattice dynamical calculations of the phonon spectrum in LSCO and YBa2Cu3OT_~(6=0-1). The spectra for SC and non-SCs (6 = 1) show marked shifting and difference (in addition to difference in structure) in an expected way. All these results show that large parts of the phonon spectra influence the transition temperature--an effect of strong e-ph interaction.
High-Tc in 2D cupratc superconductors
In this paper we discuss afresh the role of a joint phonon and plasmon mechanism in the 2D layered La2_xSrxCuO 4 ( x = 0 . 1 5 ) material. This type of mechanism could give a fairly consistent picture for all SCs--both conventional and HTCSC. The conventional ones fall in the pure phonon limit, while LSCO and YBCO could be the joint type. We believe that there is much to be learned from studying such materials, which offer relatively simple crystalstructure (single C u - O layers with minimal site disorder or unintentional oxygen non-stiochiometry). For obvious reasons the charge carriers (holes or electrons) will be called 'electrons' throughout the paper. We would consider here that coupled 2D electronic-plasmon and ionic-plasmon modes exist. The low-frequency ionic plasmon mode and not the other is assumed to play an important role in bringing about superconductivity in addition to the normal quantized lattice interactions (e-ph interactions). We then derive a Tc equation applicable to prototype layered SC. The study will then be extended to still higher HTCSCs, e.g. YBCO, in which the isotope effect is small. The possibilities of contribution from the 1D chain and also of interlayer hopping as a secondary role are discussed briefly. The consequences, such as substantial suppression of isotope effects and the importance for various normal state properties of inclusion of nonadiabatic effects in interactions beyond Migdal's theorem will also be discussed. 2. MODEL AND FORMALISM
The experimental evidence concerning the symmetry of the order parameter in cuprates has shifted over the past few years. Nevertheless, substantial evidence favouring s-wave pairing continues [10]. We neglect, for simplicity, any anisotropy in the c-axis, since the carriers involved are on the CuO2 plane. As far as the in-plane gap of the material is concerned, the overall anisotropy is believed to be not so large as to alter the s-wave pairing description [39]. The vertex correction here is relatively small compared to the case of fullerene compounds where the ratio of pairing energy to Fermi energy is ,,~2-3 times larger. Following Yu et al. [25] we do not consider the vertex terms (a discussion will be presented later on) at this moment in order to maintain the clarity and manageability of the formalism. In a layered 2D system the plasmon has a wide excitation band which starts from k = 0. This encourages all the electrons within the Fermi surface to feel an attractive e-e interaction through plasmon exchange. Thus the contribution of both phonons and plasmons (providing an additional mechanism leading to an increase of To) will be studied here using the
1155
Temperature Green's function method, and the generalized Eliashberg equation. We write the energy and momentum dependent order parameter A as follows [32, 39, 44]:
A(~.,k)=(2~2Z)-'T~ f dk' [j=~rb,plFJ(Wn - ~m, k - k ') l F+ (Wm, k ')
(1)
Here k is the 2D electron wave vector, Z the renormalization function. If ( refers to electron energy referred to the Fermi level, the anomalous Green's function which describes the pairing is
F+(wm,k ,) = A(wm,k,)[w2 + (2 + A2(Wm, T)]-I (2) The vertex ['ph describes e-ph interaction with two electron ends and one phonon end. Similarly for the corresponding case for Fpl. Since the pairing is caused by conventional phonon exchange, Fph can be written in terms of the phonon Green's function Dph and the corresponding e-ph coupling c o n s t a n t Aph, which can then be expressed in terms of the characteristic phonon frequency ~'~ph as Fph = AphDph(0dn -- 0Ym,k - k') 2 2 + (Wn -- 03m)2]-1 = Aph[~ph[["~ph
(3)
We now consider electron-plasmon (e-pl) interaction. Here we assume coupled 2D ionic and electronic plasmon modes whose existence is the key consequence of a theory developed by Gersten [45]. One of the coupled modes, w+, is basically e-pl like and is not of much interest. The low frequency mode, w_, however, plays an important role in bringing about superconductivity. Some of the experimental consequences posed by the existence of this low frequency mode w_ have already been discussed [45]. The characteristic frequency f~pl, in this case, can be expressed as apl :
(2~rN~ikF/M%)l/2Qe
(4)
where kF is the Fermi wave vector, M the mass of the unit cell. N o is the reciprocal of the unit cell projected perpendicular to the c-axis. The ionic charge Q is based on the standard valence for the ions in the CuO2 plane(s). It determines how much ionic charge lies in the conducting plane(s). Finally % is the background dielectric constant. Now Fpl consists of the Coulomb repulsion u¢ (in 2D case, bare Coulomb), and its product with the plasmon Green's function Dpl [46]: Fpl = Uc + ucDpl = Uc + Ucfl21(k)[ca2 - f~2pl(k)]-I (5)
1156
A . K . M . A . ISLAM and S. H. NAQIB
where an effective D function [33] has been used. It follows then that effective attraction corresponds to the region w < f~pl. The renormalization function is given by [32]
Z(iwn) = 1 + 7rT/wn E
A(iWm - iwn)
m ~m
[W2m+ A2(iWm)],/2
(6)
Here iwn = iTrT(2n-l) is the nth Matsubara frequency and n --- 0, +1, 4-2, ..., ..., with T being the temperature. The term, A(iWm- iw,,) in eqn (6) is related to the electron-phonon (or other boson) spectral density a2F(w) (w = frequency of the exchanged boson) through the relation
A(iw,,, - iw,,) = 2
f
f~a2F(f~)df~
[~2 + (~n -
~)2]
-- A(m - n)
(7)
This can be reduced to the much simpler form if a two square-weU model (see Ref. [31]) is taken as an approximation. In this model A(iw,n - iwn) = A; I~nl, tWml < we = O; otherwise
(8)
where we is the cut-off used in order that the sum over m in eqn (I) converges. Thus the effective renorrnalization parameter is given by A(rn = n) = A(0) = A = f 2a2F(f~)df~/f~
(9)
If in the equation for Z we neglect the gap we get
Z(iwn) = 1 + 7rTIwn E
A(iWm - iwn)sgn(wm)
m
= 1+ A
(10)
In the combined phonon-plasmon mechanism the gap equation (with the help of model gap of eqns (8) and (9)) reduces to the approximate form at T = T¢: A(iwn) = . "~ph 7rT¢' ~ A(iWm)
I +AT
~
IWm[
+ ~pl ~r~7,a(i~) l+~r
c~
I~l
(11)
In terms of the digarnma function ~(N), the above equation reduces to l =
APh
1 -t- Ar [~b(fZPh/2~rT¢+ 1/2) - 'I~(1/2)1 +
(!b(f]pl/ETrTc + 1/2) - ~(1/2)]
Applying the appropriate approximation for # ( N ) this yields the required transition temperature Tc
2e6 AP Ap [- ~ ) Te = --~- (Qph) Ph(~pl) p'e h~h+h"
(12)
Here 6 is the usual Euler's constant and Ap = Aj/(Aph + Apl), w i t h j for 'ph' or 'pl'. The eqn (12) looks similar to that derived by Kresin [33] but is, in fact, quite different in both Qpl and in the exponential term. Kresin [33] derived his equation with the assumption of strong Aph(Aph>> Apl). But no such assumption as regards the strength of coupling has been utilized in this paper. The formula through eqn (4) contains the magnitude of the electron charge, thus it makes no difference whether the mobile carriers are electrons or holes. As can be seen, the eqn (12) does not contain the Coulombic pseudopotential term #c. It is taken to be zero, keeping in mind that a non-zero #¢ reproduces the same phenomenology (as long as A is not too small), except for slightly higher values of the coupling, which depends on the value of #c. As a matter of fact the repulsion is reduced due to vertex correction [47]. The behaviour of any an/sotropic energy gap and Tc in the presence of the 'anticipated' anisotropic Fermi surface and two coupling mechanisms is no doubt a complex problem, which needs more careful consideration than has been given here. But still the above equation can be considered to be not too far from a good approximation. The above Tc equation may also be written in a form more suitable for our purpose as
2e6 ~ -- - -7r (Uph To--
,~P 1"2 ~P I-'+~h+~'] ) ph(asQ/M / ) p,e ~Ph+xP'
(13)
where a = e¢27rN~ikF/%, which is roughly constant at constant pressure for the 2D cuprate materials considered here. s is the number of CuO2 planes in the unit cell.
2.1. Non-adiabatic corrections For an exact solution of the problem of HTC superconductivity one needs to look beyond the Migdal approximation [20]. Non-adiabatic effects and the e - p h interaction have been studied recently by several workers [47-51]. The generalized gap equation, including the non-adiabatic effects, has been shown in Fig. 1 in Ref. [50], the double electronic lines indicate the inclusion of all the self-energy effects. These are the two vertex corrections and cross phonon scattering. N/col et al. [48] neglected the momentum dependence of the self-energy but used local approximation in their study of vertex corrections. The k-averaging approximation by Krishnamurthy et al. [49] yielded a modest reduction in Tc corresponding to the van Hove singularity in the density of states (DOS). Correction due to the cross phonon scattering, in addition to the vertex correction has been
High-To in 2D cuprate superconductors done by Grimaldi et al. [50]. Cappelluti et al. [51] also performed similar calculations. Under the scheme of Grimaldi et al. [50] non-adiabatic effects show a complex structure as a function of exchanged frequency and momenta. In order to see in a simple way how the correction is brought about we consider the structure of the vertex and the renormalization factor. The structure of the vertex is A(wn, Wm, q, wo, EF) = 1 + Aev(wn, Wm, q,~o, EF) where the symbols are as defined in Ref. [50]. Evidently the correction term multiplies the bare vertex. Let us ignore the momentum dependence. In this case the renormalization function in the lowest order turns out to be [47] Z(iw o : O) = 1 + A[Zwo/Ev + 1]-1
which shows roughly the correction introduced (i.e. a reduced Z factor) for ~o/EF ~ O.
3. NUMERICAL RESULTS 3.1. L S C O We take LSCO as a test case. Here we have a single layer in the unit cell. The interlayer interaction is neglected because of the small out of plane coherence length and large interlayer distance ,-,6.6 A compared to ~3.1 A typical for other multilayer structured materials. The characteristic plasmon frequency is determined to be ~'~pl = 8.53 x 1012 Hz (see Table 1). Oo = 390 K is taken from Ref. [52]. We consider that at present Oo datum yields a much more reasonable estimate of f~ph. This may be compared with an estimation of Allen et aL [43] giving f~ph ~ 200K, with an uncertainty of almost 100%. We first evaluate A t ( = Aph + Apl) for LSCO. The values of the Sommerfeld constant -y reported in Marsiglio et at. [54] and Ramirez (see Ref. [55]) are taken into consideration. The average of these two values turns out to be 8.5 (mJ/mole.Cu.K2). The average band structure DOS, N(0) is 0.97 (states/eV. f.u. spin) [43]. Utilizing these, we obtain Ar = 0.85 + 0.05. The details are given elsewhere [53]. At this point it is worth mentioning that Marsiglio et al. [54] have examined the combined phonon-exciton mechanism for LSCO based on the study of the Table 1. Parameters used in calculating 9tp] (after Ref. [53]) Parameters
PCS 58: 7-E
Values
kF
4.70 x 107cm -1
~0
4.0 7.04 x 1014cm-2
1157
isotope effect. They have taken ~')ex = 500meV and varied )~ex" They fixed Te = 36K for LSCO to get AT ",~ 0.9, and Aex "~ Aph = 0.45. Evidently Marsiglio et al. [54] have taken the electronic excitons into consideration, manifested in the extremely high energy of the exciton (500 meV is nearly five times the Fermi energy of LSCO (see Ref. [53]). Also the procedure adopted by these authors violates the Migdal theorem [20]. Thus there is considerable doubt in this approach. Our analysis yields, )kph = 0.69 and/~pl ~--- 0.16 from which one arrives at Tc "-~ 36 K. As regards isotope effects we find that eqn (13) predicts a shift in the Tc of 0.8 K when all the 160 is substituted by 180. This is to be compared with experimental value of ~0.5 K [40, 41]. Now we consider the pressure dependence of Tc. It follows from eqn (4) that as pressure increases Ni increases and Tc is raised as a result. According to band structure calculations (see Ref. [57]) the DOS is flat near the Fermi level in La2CuO 4. So N(0) should not depend strongly on pressure. Utilizing this we observe that, if N 0 changes by ,,,2.8 % due to a change of 1 kbar of pressure, A T c ~ 1.7 K, due to P = 6 kbar. This is in agreement with the experimental value [56]. The linear behaviour of resistivity as a function of temperature is a puzzling feature of HTCSCs. Nearlinear temperature dependence is observed in LSCO with x ,,~ 0.15-0.18 [57], where measurements have been reported from T = 50-1000 K. As T is lowered towards zero the resistivity appears to extrapolate linearly to zero until Tc is reached-thereafter it falls suddenly to zero. Linear behaviour of p(T) occurs in many metals when T > OD, i.e., the thermal energy must exceed the characteristic energy. Since the characteristic temperature of the low frequency mode in our case if ,,~65 K, the near-linear behaviour is somewhat contained in our formalism at least up to 28 K above Tc for LSCO. The joint mechanism model presented here is more consistent with present band-structure calculations for the values of DOS and EF. The question is whether band-structure calculation can be applied directly in such correlated systems. However, fairly convincing evidence is accumulating in favour of band theory [32].
3.2. Implications o f the f o r m a l i s m to other H T S C s
Several HTCSCs have 2D structural features with several layers. Some of them have 1D C u - O chain. YBa2Cu307 possesses two active CuO2 conducting layers and 1D CuO chain in the unit cell. The distance between the nearest planes is --,3.1 .~ and that between the chain and the plane is ,,~4.3 A. Very recently Maly et aL [13] in their theoretical calculations find two gaps appropriate to each of the multiple bands. But
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A . K . M . A . ISLAM and S. H. NAQIB
they also observe that van Hove singularities stabilize some order parameters over others and elongate the gap functions along to four van Hove points, thereby leading to a substantial region of gaplessness. First we assume that the chains nearly provide room for plane-chain scattering without being superconducting by themselves, that is, without interlayer interaction on CuO chains [25]. Then we assume that all C u - O layers in a unit cell are strongly correlated and can be taken as one with effective Q (which is the net bound valence charge of ions of the layers). This in turn increases the characteristic plasmon-frequency. The Fermi momentum in YBCO is comparable to that of LSCO, and Ni remains nearly the same for both the SCs. The mass of the unit cell, M ~ 1.66 times of that of LSCO. We take the value of the Sommerfeld constant 7(0) = 11.7 mJ/mole K 2. Widely scattered values of 7 are found in the literature (see Ref. [55]). An average of these unusually scattered values turns out to be ,-all. Instead we have chosen the average of three consistent sets of values of 12, 12, 11 reported in three published papers (see Ref. [55]). The DOS N(EF) at Fermi level is taken to be ,~ 0.93 (state/eV f.u. spin) [43], which then yields Ar '~ 1.66. This may be compared with the value obtained in self-consistent localdensity approximation band structure calculation of )~ph ~ 1-1.5 (see Ref. [17]). OD is taken to be 465K (see Ref. [28]). Proceeding as before we get Tc ,,~ 80 K for Ar = 1.66 with /~pl//~ph ~- 0.23, the same ratio is found for LSCO. The estimated value is reasonable considering the uncertainty of the input data used. It should be noted that a further enhancement of this value could be obtained through vertex correction. A similar approach can be adopted for layered SCs with still higher T~ values. Prospects of pairing enhancement of HTSCs containing planes with 1D chains has been discussed by Kresin and Morawitz [31]. They find additional enhancement of Tc qualitatively due to the coupling between carriers in the planes and quasi- 1D plasmons. We do not consider here any charge transport along the c-axis for multi-layered SCs as the planes are somewhat isolated. Moreover the carrier concentrations in the planes are low. But it is interesting to note that pairing enhancement due to an interlayer Cooper-pair tunnelling has been considered recently [59, 60]. Byczuk and Spalek [60] considered both the intrinsic inplane pairing and the two interplanar Josephson tunneling process, which they claim is valid for both Fermi- and non-Fermi liquids. If s is the number of tightly spaced identical CuO2 planes, and separated by a large distance, but in contact with each other through non-superconducting region, there is a weaker Cooper-pair tunnelling. Such inter-
layer hopping could enhance Tc for multi-layered materials including YBCO, which is given by [60]
Aro :~ cos(~-~+0. 4. DISCUSSION We have developed here a general approach to the pairing mechanism of superconductivity with 2D systems. The Tc equation yields reasonable values of transition temperature for both LSCO and YBCO. In deriving the Te formula we have used some approximation, the implication of which has been analysed by us. It is found that for LSCO, eqn (12), due to some approximations, slightly overestimates the predicted Tc. But because of uncertainties in some of the input parameter values, the approximation should not be of much significance here. The vertex corrections to the electron self-energy are reduced by the order of Migdal's parameter m = ~ph/Er. F o r AxC6o(x = 3, A = K, Cs) compounds, m ~ 0.5-1. Lower values for HTC cuprate SCs occur (~0.3 for LSCO for phonon interaction). Although smaller compared to fullerenes, these are still not small enough to neglect the correction for phonon vertex. It has been observed [50] that a predominance of small momentum scattering leads to an enhancement of Tc with respect to the adiabatic theory with the same coupling [50-51]. The behaviour of the correction to Tc has been evaluated and displayed in Fig. 5 in Ref. [50]. Vertex corrections are expected to have implications on both superconducting and normal state properties (like transport, photoemission, life times effects [49]). For example, the isotope effect becomes negligibly small depending on the value of Migdal's parameter. Using the formalism developed in Refs [47, 48] we have roughly estimated that flox(YBCO) may be as low as 0.1. The analysis is far from complete. For example we have not incorporated vertex corrections and the Coulombic pseudopotential term in the formula derived. Instead we have discussed some possible consequences using the formalism of other works [47-51]. A more realistic description of the various contributing factors is to be undertaken in a future work to learn more about the systems.
Acknowledgements--The authors acknowledge the help received from the University of Rajshahi under an Annual Research Programme. The work was performed in part, at Jawaharlal Neheru Centre for Advanced Scientific Research (Indian Institute of Science Campus), Bangalore, where it was supported by JNCASR-IISC-ICTP AssociateshipProgramme. REFERENCES
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