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Bosonic mode interpretation of novel STM and related experimental results, within boson-fermion modelling of HTSC

This paper seeks to synthesize much recent work on the HTSC materials around the latest energy resolved scanning tunnelling microscopy (STM) results from Davis and coworkers. The conductance diffuse scattering results in particular are employed as point of entry to discuss bosonic modes, both of condensed and uncondensed form. The bosonic mode picture is essential to understanding an ever growing range of observations within the HTSC field. The work is expounded within the context of the negative-U, boson-fermion modelling long advocated by the author. This general approach is presently seeing much theoretical development, into which I have looked to couple many of the experimental advances. While this formal theory is not yet sufficiently detailed to cover adequately all the experimental complexities presented by the real cuprate systems, it is clear it affords very appreciable support to the line taken. An attempt is made throughout to clarify why and how it is that these novel phenomena are tied so very closely to this particular set of materials.


§1. Introduction to the energy resolved STM work of Hoffman et al.
Recently Hoffman et al [1] have uncovered in an energy resolved 4K STM study of local tunnelling conductance into Bi-2212 that over the energy range 5 to 30 meV additional novel scattering information is forthcoming. Through a Fourier transformation of their real space local conductance maps they show that structured incommensurate elastic scattering of the incoming quasiparticles is being incurred. This scattering is quite distinct from that previously reported due to (i) low energy impurity effects, (ii) magnetic effects, (iii) charge stripe effects, (iv) vortex effects, and (v) the Bi-O layer superlattice modulation. The newly found modulations in the tunnelling conductance signal are much more diffuse than those from (v), and the action at the wavevectors involved is seen besides to be strongly energy dependent. The dispersion characteristics manifest in the new scattering are moreover not those of the antiferromagnetic magnons of low doping, nor likewise of the lattice phonons. The resolved wavevectors (q A and q B ) are found to relate directly to the basal Fermi surface geometry, k F , and the scattering is coupled furthermore with tip binding energies roughly equal in value to the superconducting gap. As a result Hoffman et al have tentatively presented their 4K STM conductance data in terms of the introduced electrons being scattered from the Bogoliubov quasiparticles. The density of states peak for the latter then should of course map out the energy below k F of the bottom of the superconducting band gap as structured by the d x2-y2 symmetry HTSC order parameter ∆(k F ). However upon closer examination of the STM results we will show that while at each level of sample hole doping reported the locus from the full set of conductance modulation wavevectors indeed creates a shape in k-space very close to the ARPES-determined normal state Fermi surface, the associated binding energies E q (θ) are consistently significantly greater than the gap ∆(θ) but less than 2∆(θ) − prior at least to the 'hot spot' being reached (see below). It will be argued that this observation is in accord with the existence of a dispersed uncondensed boson mode, as figures in the negative-U two-subsystem treatment of HTSC from the current author [2]. Formal development in modelling of the mixed boson-fermion and/or negative-U form recently has been much expanded by Micnas et al [3], by Domanski et al [4], and by Casas, de Llano, Solis et al [5], closer examination being made of questions relating in particular to the electronic specific heat, entropy and condensation energy data from Loram et al [6] and to the Uemura type rendering of the µSR data [7].
A matter that we will return to in due course but one which at this point must be registered is the misleading reading of the true value of the superconducting gap from the ARPES spectra adopted by Hoffman et al. In the latter spectra the leading peak has to be viewed -unlike in earlier type readings of such data, and taken up now in [1] -as much more closely defining 2∆ than it does ∆. ∆ in fact is better monitored by the inflexion point of the leading edge. At optimal doping in YBCO-123 and BSCCO-2212 it is the maximal gap values 2∆ o which fall just short of 40 meV.
Thermally activated experimentation such as specific heat and nmr, along with phonon line-width and electronic Raman analyses [2e ( §E16) and 2f (ref 36)] all concur on such a 2∆ ο value (≅ 320 cm -1 ). This yields then the well-known strong-coupling 2∆ o /kT c ratio of approximately 5.5. A proper reading of 2∆ directs interpretation of what Hoffman et al's new results mean along significantly different lines from those followed in [1]. The proper interpretation of the ARPES 3 spectrum from Bi-2212 has been a long and complex task − one involving the joint effects of the local boson mode and bilayer splitting, in addition to matters of quasiparticle delocalization.
What is most striking about the new STM results, as indicated above, is that for any given doping level the maximal (*) scattering signal identified in the conductance occurs at some energy E q * appreciably less than 2∆(k*,p',T = 0), for coupled q A */q B *. The latter wavevectors specify a set of equivalent points within k-space which fall well removed from the (π,0) saddles in the band structure, and likewise from the renowned 'hot spots' sited on lines (0,π) to (π,0), etc. where those lines intersect the Fermi surface. In the spin fluctuation interpretation of HTSC the 'hot spots' are to be associated with the highly characteristic (π,π) scattering physics [8], preeminently in evidence in the striking 'resonance mode' of inelastic neutron scattering, where the scattering is normally treated in standard nesting fashion [9]. By contrast our own negative-U interpretation of the π,π scattering involves not true magnetic spin excitation but a singlet-triplet spin flip pair breaking of local pairs [2]. The maximal scattering in the new STM data, besides arising in k-space as close to the d x2-y2 nodes as to the hot points, does so at a binding energy E q * which as noted above is only ≈ ½.2∆ ο (k*,p'). It would appear that the STM conductance signal becomes maximized where/when the dispersed boson mode of the postulated negative-U model stands not too severely perturbed either by pair formation or pair dissolution, these in the model coming in the vicinity respectively of the hot spots and of the gap nodes. Hoffman et al observe that the new diffuse peaks appearing in the (r:k) Fourier analysis of the STM conductance data clearly seems to reflect the geometry of the Fermi surface. In fact the two wavevectors q detailing the scattering events in k-space seem in elastic fashion to span at selected energy E below the Fermi surface between equivalent degenerate points within the zone, to specify |q A | and |q B |, and how secondly the latter two peak scattering positions shift progressively in k space as the binding energy of the STM tip is incrementally augmented. By combining this information one is able, upon accepting the above dispositions for q A and q B , to mirror the Fermi surface. The outcome was not actually displayed in [1] and is shown now in fig.   1. Included in the figure is a circle about the zone corner that would for an idealized, homogeneous, band-like situation in the 2D limit correspond to F.S.for a hole doping p h of (1)+ 0.16, the band filling to yield maximum T c in all cuprate HTSC systems.

{figure 1}
The HTSC compounds of course are not ideal metals but very highly correlated, are not homogeneous but chemically mixed-valent and dynamically charge-and spin-striped [2e,d], and are not in the 2D limit but hold measurable 3D character. In the Bi-2212 case the latter automatically will introduce direct intra-bilayer interaction and for a standard material this is going to lead to a two-sheeted d x2-y2 symmetry Cu-O pdσ* band complex. Here the fuller component (re electrons, and the one supplying the outermost electron F.S. sheet) will be the one for which the intra-bilayer interaction entails c-axis 'bonding' as opposed to 'antibonding' phasing between the pair of pdσ* wavefunctions introduced by the two Cu-O chessboard arrays per unit cell. Following much initial searching for this 3D splitting several more recent ARPES works employing better resolution are able now to detect both components of the pdσ* band [10][11][12], at least for optimally and overdoped material. Because the same groups however when working with Bi-2223 observe only two dominant spectral features, not three [13], and more tellingly yet with Bi-2201 continue to report two clear features, not just one [14], this justifies our continued adherence to the position reexpressed recently by Campuzano, Norman and Randeria [15] in their very comprehensive review of the ARPES work -namely that the universal peak and hump circumstance apparent in all these spectra is primarily the consequence of strong correlation and scattering, and arises quite independently of any multisheet coupling. The consensus to emerge from these more refined 5 ARPES determinations now as regards the overall shape of the Fermi surface is that the latter departs appreciably from the circular form of the idealized surface present in figure 1 [17], and in all probability this will not be attainable either with very highly overdoped material, witnessing the parallel null result obtained for the 3D s-electron mixed-valent superconducting system (Ba/K)BiO 3 [18]. All HTSC materials are characterized by chronic scattering and this is not just because they are poorly formed or intrinsically nonstoichiometric. Indeed with PrBa 2 Cu 4 O 8 , upon compromising greatly the potential there within the Cu-O basal planes for metallic conductivity (and indeed superconductivity), this chain-bearing material in fact becomes rendered a notably improved metal, more coherent and Fermi-liquid-like, even in its c-axis direction [19].
(Recall in the case of Pr-124 that despite it macroscopically being crystallographically perfect, one finds the basal plane electric and magnetic behaviour to be totally transformed under the strong Pr4f-Cu3d hybridization. The latter f-d mixing with its spin coupling inhibits production of the RVB spin-singlet condition so essential to permitting HTSC to arise.) Within the author's negative-U two-subsystem perception of HTSC  [22]. The above extremely strong e-e scattering actually persists to well out beyond p h = 0.3 [23], i.e. to well beyond where the novel superconductivity can any longer be sustained.

{figure 2}
In figure  reading it as being much more closely related to 2∆ than to ∆ as originally taken in [24]. Both the E q (θ) and the 2∆(θ) values are somewhat difficult to quantify exactly and call for a certain amount of smoothing of the raw data. Nonetheless several features stand out from the plots given in  Norman and coworkers have for some time advocated that the peculiar form of the ARPES spectrum below T c , with its peak, dip, hump structure, points to the interaction of the quasiparticle states with some bosonic mode [25]. The natural implication of the presence of such a feature is that it relates critically to the superconductivity if not actually causing it. Not only does the above spectral activity figure at the gap antinodes, but at the gap nodes too there is evidence of a perturbation of simple behaviour, a clear kink being evident in the quasiparticle dispersion curves, this growing more marked below T c . When Johnson et al first reported the latter behaviour they attributed it to spin fluctuations [26]. Subsequently it was suggested by Lanzara and coworkers [27], from analogy with what is found in a standard strong-coupling superconductor like Mo [28], that phonons were responsible. The energy is now however too large (~ 60 meV) to be appropriate for phonons in general, and the situation would have to be more akin to some specific optic phonon coupling, such as is present in a CDW/PSD system like 2H-TaSe 2 [29]. Although this action might relate in the cuprates to incipient stripe formation, it is not evident why then the kinking would grow sharply at T c to show a component growing in magnitude as the superconducting order parameter. Despite spin-singlet pseudogapping being strongly in evidence [9], Norman and colleagues would look to have settled for a magnetic attribution to the mode's origin [30,8a], in view largely of its similarity in energy to the much discussed 'magnetic resonance' excited at Q = (π,π) in neutron inelastic spin scattering − for optimally-doped Bi-2212 a quite sharp feature positioned around 43 meV [9]. The alternative perspective already advocated in [2a,b] is that the mode being sensed in the ARPES measurements is associated with local bosonic pairs - This interpretation subsequently received further support in similar, more wide-ranging optical work by Little et al [33], by Kabanov, Demsar et al [34], and above all by Li et al [35], results examined at length in [2a]. The situation throughout has been perceived as a fragmented two-subsystem one (see fig 4 in [2h]), at low doping often being driven dynamically towards stripes [2e,d]. This highly perturbed geometrical condition (not entirely unlike that in a quasicrystal), when taken in conjunction with a negative-U pair resonance stationed in close degeneracy with the Fermi energy, assures that the level of particle-particle scattering will be extreme. A high flux of interchange between fermionic and bosonic states, as well as between boson states which are condensed and those that remain outside the Bose condensate, is inevitable.
Calculations of a negative-U Hubbard type consistently have indicated that when, as here, the overall effective negative-U value (per pair) emerges as being of the order of the bandwidth, the greatest T c values will at that stage be met with [36]. In the present case |U eff | is indeed, as As listed in [3,4,5], theoretical (homogeneous) negative-U boson-fermion equilibrium models recently have been much extended beyond the original work of Micnas, Ranninger and Robaszkiewicz [37] and of Friedberg and Lee [38]. One key matter specifically to find themselves back now near the hot spots on the two saddles from which originally they were abstracted in the original negative-U pair production process. The above pair elimination necessitates a net momentum transfer to its electrons from each scattering neutron of (π,π), with (π − ε,ε) going to the one emerging on the x-axis saddle and the balance of (ε,π − ε) to the one coming onto the y-axis saddle (see [2b, fig 3]. For the case of bilayered YBCO and BSCCO what particularly is noteworthy in the neutron scattering process is that it predominantly occurs associated with a large incommensurate z-axis momentum transfer which corresponds in direct space to the Cu-O bilayer spacing. Eschrig and Norman [30b] following the lead of Fong, Bourges et al [9] interpret the latter detail as evidence of a dominant magnetic aspect to conditions in the material, and view the bonding and antibonding channels evident in these bilayer systems as being associated respectively with 'even' and 'odd' parity magnetic spin coupling. They pursue this direction despite the sizeable spin gap, with no evidence in optimally or overdoped material of any genuinely magnetic correlation length divergence towards low T [40]. By contrast the negative-U point of view would see the z-axis momentum transfer as being required by charge symmetry, as a local pair is undone and its components separated to nearest-neighbour cells. The two electrons, being left spin-parallel by the action of the neutron, are no longer permitted to occupy the same Cu-O coordination unit (as d 10 p 6 ). The process is illustrated in detail in figure 3.
It should be pointed out here that the above z-axis 'complication' is in no way essential to the viability of the HTSC mechanism, it relating solely to pair break up. It is simply a symmetry requirement imposed by the crystal structure. When, as with Tl-2201, one has a single layer structure the neutron spin-flip scattering peak centres on the k z = 0 plane [9b]. It remains to be found how this structure will simplify details also of the ARPES spectrum for Tl-2201.

{figure 3}
The It has at this point to be emphasized that the above dispersed mode is distinct from the bosonic mode concentrated upon by Eschrig and Norman in their ARPES-related work [8a,30].
The latter mode runs virtually undispersed throughout the outer part of the zone. In the present work that mode will be associated directly with the condensate itself of local pair bosons and lies to higher binding immediately below the dispersed mode of uncondensed pairs. Extrapolation of the line relating to the dispersed mode in figure 2 down into the (π,0) saddle-point region serves to place the ground state 'bosonic resonance' (labelled Ω res by Eschrig  ionic) the system is, the farther below the Fermi level the local pair negative-U state resides. We will return shortly to this second mode and to its relation to the neutron scattering resonance peak.
With somewhat underdoped 2212-BSCCO it has been observed that 60 meV is in fact an energy echoed in the self-energy driven 'kink' feature in the ARPES-determined quasiparticle dispersion curves readily visible once clear of the saddle direction [26]. There, under the prevailing crystalline and condensate symmetries, the superconductive and bilayer gapping effects are lessened and the state structure simpler. The persistence of this observed band kinking to well above T c marks a continued presence there of minority negative-U local pairs, while the loss at T c of the Meissner effect, etc. registers the extinction at that point of the majority Cooper pair population, and along with this a relinquishing of global phase coherence. Note in the present view the spin pseudogap existing to a hundred degrees and more above T c is to be ascribed not so much to superconductive inter-pair phase fluctuation as to dynamic RVB spin gapping. As is stated in [2] this RVB gapping is perceived as the means towards preparing the system for the production of local-pair spin-singlets, as well as it being key in their preservation. The absence of any anomalous peaking in the c-axis IR conductance just above the superconducting energy gap within underdoped yet 'well-formed' Y-124 corroborates that the fluctuational behaviour to be seen far beyond T c is not primarily superconductive in character [42]. Full participation of the bulk of fermions in the seeded Cooper pairing is required before the systems can take up an overall condition brought somewhat more standard in its transport properties. Once below T c the dramatic fall-off in the chronic saddle-point scattering of local pair creation and excitation brings about very rapid growth in the residual electronic mean free path, as disclosed for example in the thermal Hall data [43] and in the very steep decline in nmr relaxation rate [44].
Just what the nature of the low dispersion Ω res mode is, and its relationship to the neutron scattering peak, has been most hotly debated [45]. The uniqueness of both matters within the superconducting field suggests a very close link with the local pair condensate itself. This similarly is true as regards the observed strong and very specific coupling into the problem of the basalplane Cu-O bond-stretching longitudinal optic phonons. Over the outer parts of the zone these particular phonons find themselves severely depressed in energy [41,46], and in a manner quite dissimilar to that for some regular Fermi surface dictated soft-mode CDW/PLD circumstance [47] (see §6). The fact that this coupling, as with the kink in the quasiparticle dispersion above, is observed too in the k,k,0 direction rules out any connection with stripe phase formation. The virtually dispersionless form of the ARPES-revealed mode as contrasted with the mode revealed in the new STM work is striking. It brings to mind the phenomenon of second sound in liquid 4 He.
Upon looking into the relevant literature it soon is uncovered that the appropriate excitations have long been postulated for strong-coupling superconductors. What is more, their study actually was advanced some years ago to 2D geometry by Belkhir and Randeria [48] in the wake of developments with the cuprates − and was indeed performed within a negative-U context.
What was explored in reference [48] is how the collective mode spectrum of a In probing the dielectric/optical response of a material it is well known that a plasma excitation is best monitored through the electron energy loss (EELS) function Im{−1/(1+ε(ω,q))}. It is this same function that one looks to also for information regarding longitudinal optic phonons.
HREELS work on a metal demands high surface quality, it in effect monitoring the surface resistivity. The low temperature cleavage of BSCCO affords one excellent opportunity to secure reliable plasma excitation results, a project in fact carried through back in 1992 by Li, Huang and Lieber [49]. Sure enough what was reported was a single strong spectral peak centred at 60 meV, this displaying furthermore a thermal behaviour which clearly associates it with the superconductivity. The above workers in fact related the signal to pair breaking. Notwithstanding this they reported the 60 meV peak to show a T dependence that despite looking to collapse sharply at T c would if treated as a mean-field BCS feature actually extrapolate to an 'onset' temperature of 150 K. Accordingly it would seem that our boson mode elicits appreciable energy loss only as it becomes heavily damped upon the engagement below T c of the majority Cooper pair population.
The presence of a multi-sourced pseudogapping up to temperatures of 150 K and beyond is endorsed in a great many types of experiment, not least optical, but the cleanest indicator to date that this temperature range retains some 'superconductive' content is provided by the very recent Nernst effect measurements from Wang et al [50]. The Nernst effect involves the production of an electric cross-potential (y-axis) signal as the passage of charge down a (x-axis) thermal gradient is subjected to a large and mutually perpendicular (z-axis) magnetic field − the thermal equivalent of the Hall effect. The signal issues from normal carriers in addition to entities associated with the superconductivity, but whenever of course the latter exist they are dominant.
The above paper actually focusses on the signal that can come from the flux vortices once these enter their 'liquid state', depinned from the lattice and able to drift in the thermal gradient.
However because the signal Wang et al record clearly extends up 20 or more degrees above T c it is evident that a local pair boson attribution becomes there most appropriate. §5. Expressions of the mixed-valent inhomogeneity.
What additionally is so instructive about these Nernst data [50] is that they supply a direct means of assessing H c2 (at which applied field the strong Nernst signal component naturally is taken to zero). Although with most HTSC samples H c2 lies well beyond the 30 teslas experimentally available in [50], it is discovered, as a result of there emerging simple scaling rules for both reduced fields and temperatures, that the zero temperature critical field can be extracted In view of the above it is rather strange Loram, Tallon and Liang in a recent preprint [51] have chosen to play down the fragmented circumstances prevailing in the cuprate systems, especially when underdoped. They declare that the rather sharp aspect to what occurs at critical 13 doping would imply a more uniform state, and they advance their view by reference to sharp signals seen in nmr work specifically for 81 Y and 17 O. However the type of disorder being addressed in [51] is restricted to that associated solely with static doping disorder and frozen charge segregation, as the recent STM results could at first sight appear to flag [52]. anomaly [54]. Within the fluctuational mêlée the local effective doping level has proved in the LSCO case to be such that it can stand far different from the mean value [53], to such a degree that magnetic moments can in fact emerge at select sites within even mildly underdoped material.
This is what it has been argued in [2d] leads to the weak spin-flip effects recorded in µSR [55] and neutron diffraction [56].
Indeed it is the Swiss cheese view of the situation, supported at one time by Loram, Tallon and coworkers [57], although now given a dynamic time scale − more like cheese fondue. The percolation threshold is a sharp threshold and clearly one of considerable import for global superconductivity within these strongly coupled systems. The µSR results early on revealed how the averaged n s count achieved under hole doping p climbs linearly, and with n s likewise T c , as marked in transport related properties − in the case of µSR via the screening controlled penetration depth [7]. Under thorough analysis the specific heat results [6] disclose how, nonetheless, a fully proportionate condensation energy is not attained in advance of critical doping; only with the latter is proper superconductive coherence between the different entities and structural fragments within the system fully attained. It is in this way that HTSC systems come to acquire glassy characteristics below T c , especially when underdoped [58].
A striking close up affirmation of the local conditions prevailing in the superconducting state of HTSC materials is to be found in the low temperature, high spatial and energy resolution STM mappings of BSCCO samples published in a fairly recent paper from Davis, Lang and coworkers [59]. Let us inspect this remarkable new data from the perspective of the current paper. Spatially one finds a condition highly fragmented as regards energy gap magnitude, and this within a

Just published this month by Davis, McElroy and coworkers is yet another STS paper with
novel results to bear strongly upon the above [63]. Following on from the Hoffman paper [1], they include now a reconstruction of the F.S. such as was given earlier in figure 1, but secured this time by identifying and simultaneously fitting all eight symmetry equivalent elastic scattering vectors at energy E within the Brillouin zone, q 1 to q 8 , apposite to the Fermi surface geometry, not just the shortest two detected previously (then labelled q A and q B ). This new more detailed work still asserts though an identity between the quasiparticle Bogoliubovon states and the STM conductance registered states. By contrast the latter we shall continue to claim actually lie at appreciably different binding energy to the former (assessed via ARPES). We feel the STM detected states in reality do not display the dispersion and intensity characteristics to match Davis and colleagues' interpretation of events. The experimental mode dispersion curve extracted in [63] remains much like that given in figure 2 above, if anything presenting a slightly countersigmoid form, and certainly does not give one any confidence at all that it could swing round to zero binding by θ = 45 o . Perhaps the best argument to support the Bogoliubovon interpretation profferred by McElroy et al in [63] is that quite similar results now are revealed to show up at positive bias also. We shall return to this latter point in due course.
Following on from the discussion of previous sections it is best however to deal first with the observations appended to the new paper [63] in its closing paragraph. The authors put on record there, along with figure 4, that the direct space conductance mapping signal, g(r,ω), exhibits under specific local conditions a novel fine-grained 'tweed' structure. The period of this tweed is just one unit cell, and it materializes within the map only when and wherever the tip tunnelling energy matches the locally relevant superconducting gap maximum. Both italicized observations serve to indicate that the tweed structure is associated with the strong scattering activity proceeding at the zone edge. Scattering under a reciprocal lattice vector G signals the occurrence of a coupled umklapp phonon process, and one which arises seemingly regardless of whatever the local gap size maximum at the saddles might actually be. In the present model such lattice involvement already has been mentioned in regard to satisfying momentum conservation as local pairs (i) are shifted between the (00) or (π,π) negative-U related states and the saddles, or secondly (ii) convert at the saddles into +k/-k Cooper pairs. In both cases the net momentum per pair demanded for these changes is close to (1,0) or (0,1).
Just which phonons might be involved here is to be gleaned from inspection of the phonon dispersion curves obtained via inelastic neutron scattering. As noted already, there has been discovered to occur appreciable phonon softening (vis-à-vis the corresponding Mott insulating material) of certain basal LO modes towards the basal zone edge [41]. Clear intensity changes furthermore are found to develop between branches and also within a branch as a function of temperature. More striking still, there at first sight would seem to be present 'extra' features in the spectrum beyond those branches anticipated from a standard normal mode phonon analysis for the given crystal structure. Additional extensive neutron scattering data clarifying these matters have just been published by Chung et al [64], and they deserve close examination now with regard to phonon coupling, this due not only to the changes above but also to the local pair boson modes discussed earlier. resonance excitation itself, which as was discussed at some length in [65] disperses to numerically smaller energies upon a reduction from π,π in the crystal momentum input being suffered there.) That comparable effects to the above occur too in the phonon dispersion behaviour of LSCO [41] signifies that in YBCO 7 they do not enter solely as a consequence of the crystallographic presence of the chains. The standard shell model does not indicate any such a strong divergence as that observed to develop between and within the relevant Σ and ∆ modes.

Chung et al's work [64] deals in fact with
Why the LO Σ 1 phonon mode with its displacement vectors perpendicular to the chains should couple more strongly with the plasmon mode than does LO ∆ 1 , with its chain-paralleling displacements, would seem largely to be a matter of compatible symmetries. The former mode involves, note, the bumping together of the chains, and this should effect carrier passage between the charge stripes, which in YBCO are known to follow the chain direction [66]. More significant is the reason as to why the really strong coupling sets in only as k increases beyond ~ ½(π/a o ), i.e.
as the wavelength falls below 4a o . The latter distance, recall, is expressly equal to the pair coherence length ξ for optimally doped HTSC cuprates, and as pointed out in [48] it only is for wavelengths inside this limit that the local pair plasmon mode is able properly to form and so to couple to the lattice.
Very short wavelength charge pair plasma oscillations are not confined to the above basal plane states. In the bi-, tri-and higher order layer HTSC compounds there is below T c at least partially coherent superconductivity in the c-axis direction, with pair transfer between adjoining ). Munzar and Cardona demonstrate, in addition, that the action of these interplane plasma oscillations is to produce a Raman scattering efficiency of such magnitude as to clearly be source to the anomalous form of the A 1g electronic Raman feature that so long has plagued satisfactory interpretation of this latter type of Raman work [68]. Because the pair plasma mode is interacting so directly with the lattice vibrations, and strongly once below T c , it is not surprising that oxygen isotope effects have extensively been recorded within HTSC systems [71]. The latter effects are however quite different in form from those associated with the standard, phonon mediated, retarded interaction of the BCS mechanism. Indeed it is observed under p change within any given HTSC system that the isotopic shift in regard to T c itself is pointedly dropping to zero at the very composition to yield the highest T c − or, more probably, the highest condensation energy per pair and shortest coherence length, the critical doping p ≈ 0.19. Very significantly, however, there is in operation at precisely this juncture a strong isotopic mass exponent below T c with regard to the penetration depth λ (∝ n s /m* b ). Because n s is not appreciably affected by isotopic substitution this must mean an isotopic shift is operative deriving from the bosonic effective mass m* b . The electronic masses are responding here, via the plasma mode coupling discussed above, to the eigen-energy changes being isotopically imposed upon the phonon modes. On cooling from 300 to 4 K, m* e itself has been evaluated to mount steadily from 2 to 4 m e [69].
Since the kinking in the quasiparticle dispersion curves and likewise the development of the peak, dip, hump structuring to the ARPES energy spectrum, each become more pronounced with an increase in layering number, n, across a sequence such as Bi-2201, Bi-2212, Bi-2223 [72], one would now much like to find out what happens to the above penetration depth isotope effect.
The most probable scenario is that these layering effects actually issue from the known reduction with n in the basal plane lattice parameter, the latter rendering the negative-U, closed-shell state more stabilized, but without any rise here in pair breaking as would arise through underdoping. It correspondingly is registered that uniaxial pressure applied within the basal plane produces a steep growth in T c (p) max [73], and so potentially directly in n s .
As is to be expected from the current modelling, the mass enhancements arising in the saddle regions should be greater than those encountered within the nodal regions of the Fermi surface. This plus a sharp growth in renormalization of the imaginary part of the self energy below T c under the action of the boson mode is very clearly to be found in the new ARPES data published very recently by Kim et al [74] (scanning from π,0 across k F towards π,π). Apparent too in this new data is that no fundamental difference of response exists between the bonding and antibonding interlayer coupled sub-bands. The kinking is more evident in the former simply because it is the fuller. Above T c the antinodal coupling constant is found to become rapidly much decreased (although still around 1), whilst the mass (but not the scattering rate) is rendered more or less isotropic. Note the quasiparticle mass renormalization resulting from the self-consistent action of the electron pair mode manifests quite different thermal characteristics to what would be expected if the mode were primarily phononic in nature or one of spin fluctuations. §7. More on the nature of the HTSC mechanism.
In the present form of modelling what in large measure dictates the sensitivity of response to parameter changes like pressure and doping is the precise location of the double-loading negative-U fluctuational state in relation to the Fermi energy − or once below T c to the chemical potential. Remember the small energy ~ -50 meV by which the state is to sit below E F is but a tiny fraction of |U eff |, which in the present case is ≅ 3 eV (per pair) [2a,c]. The latter in turn is less than half the actual state adjustment energy accompanying double-loading shell closure, since around 4 eV is negated in the requirement to 'back off' the Coulomb repulsion energy. Accordingly a high sensitivity of T c to all parameter changes is to be anticipated if these remarkable effects indeed do establish themselves as one reaches close degeneracy between the negative-U state and E F , and a ready interconversion between charge fermions and bosons.

20
This degenerate condition very recently has been referred to by Domanski [75] as a 'Feshbach resonance'. It is demonstrated formally in [75] that when the boson and fermion subsystems are disentangled (via application of a continuous canonical transformation) the residual fermion-fermion quasiparticle interaction itself presents highly resonant structure above T c , which for T < T c becomes reduced to that of the superconducting gap. A further key result for our purposes obtained in [75] is that the overall boson-fermion system foregoes the particle-hole symmetry which characterizes the standard Bogoliubov analysis of the simpler BCS mechanism.
Domanski is not the only one to investigate this symmetry breakage between the hole and electron Bogoliubov-related pair states in the superconducting phase, the matter being extensively  [27]. In that case, though, there would then be no compelling reason ever to depart from the Bogoliubov e-h symmetry of the traditional mechanism.
Batle et al [5a] cloud the situation somewhat further by noting, after Hirsch [76], that the great majority of known superconductors have p-type Hall coefficients, as against most nonsuperconducting metals being n-type in their transport properties. This carrier electron-hole asymmetry of course is far from synonymous though with the BCS related usage of those terms.
The present author already has commented upon how the observed preponderance of p-type materials possessing elevated T c values appears ascribable to chemical bonding effects, even within the simplest of systems. Where such effects become particularly evident is with the 21 superconducting, high pressure, semimetallic, homopolar-bonded forms of elements like B, Si, P, S and I [77], and seen even more recently in Li [78]. One might include here cluster structures like the C 60 [79] and Ga 84 [80] derivatives and the graphite-sulphur composites [81]. Nonetheless the general level of difference found between the ordinary ±V pair tunnelling results is encouraging. It would be extremely interesting now to attempt to follow the STM scattering work to higher temperatures, if manageable without surface contamination.
One thing very apparent through all this work is that the reason the HTSC problem has been with us for so long is that it is a truly complex problem. The previously simplest of experiments repeatedly debouche into a remarkable pan of intricacy which it may or may not be profitable to pursue. The nmr results were an early marker of this [53,90] and the ARPES data more recently have witnessed a similar transition from being treated as beguilingly straightforward to being recognized as highly complicated in nature [91]. What can be said however, as with the evolution of this paper, is that, wherever one probes, the boson-fermion, two-subsystem, negative-U approach appears able to cope with more problems than it generates. It is sufficiently complicated of itself to hold the required degree of flexibility to pursue all the various twists and turns that the experimental work exposes. It is self-evident neither the phonon scenario of HTSC nor the spin-fluctuation one possess the necessary degree of built-in complexity to do this. I would refer those still thinking along the latter lines to examine the 'inversion' of the photoemission spectra recently presented by Verga, Knigavko and Marsiglio [92], as they seek to account for the observed kinking in the quasiparticle dispersion curves. The spectral form for α 2 .F(ω) of the mode coupling extracted there within an Eliashberg type treatment (see figure 6) is quite unlike what is expected for spin fluctuations, and it in addition contains far too much weight at high energies to be relevant to phonons alone.
One of the chief reasons for seeing the spin-fluctuation scenario actively supported for so long is its perceived compliance with neutron scattering data. Wherever the latter though has been dominated by data coming from LSCO (in consequence of the large crystals available) that is unfortunate since LSCO is the most ionic of all HTSC systems, consequent upon its particular  [95]. They deemed that out-of-plane 'impurities' had to be responsible -presumably the dopant ions. However our view of events is even more 'intrinsic' yet than this. Besides the above sizeable elastic (and strongly angular dependent) term, there exists too of course the very substantial inelastic scattering activity between carriers and between carriers and the lattice. These effects go forward to yield those strong and very characteristic resistivity terms, linear in both T and ω [96], that Varma early on referred to in his Marginal Fermi Liquid formulation of HTSC behaviour as issuing from a 'magic polarizability' [97]. This carrier scattering is at least an order of magnitude more intense than would occur if the transport properties were simply as covered by a standard Boltzmann treatment and the given LDA band structure. It is not just the unusual T and ω dependencies to the observed transport properties which are in question, but, above all, the sourcing of their strikingly large prefactors. ) ∝ T 2 and ∆ρ B ∝ T -4 . However, as they now acknowledge, this is not the right way to reach these relations [99]. It was stressed in [2e] that the latter are not in toto Fermiology results; they constitute indeed a clear expression of that intense fermion-boson activity which precipitates HTSC within the present resonant negative-U scenario. Clearly the two-particle, two-subsystem, umklapp and DOS pseudogap aspects to the problem all will require to be incorporated explicitly.

24
Let us now take stock of the present theoretical situation. The magic polarizability looked to within the MFL theory, with its rather momentum independent and structureless form both energetically and thermally, is in accord with a resonant interconversion/quantum critical fluctuation as comes in the negative-U scenario. The action we have seen extends to the lattice, with the accommodation of momentum conservation as quasiparticles transfer between the saddles, the zone corners and the zone centre. The appropriate zone edge phonons are observed to couple into the various charge processes. The local pair bosons still are present to some degree above T c and they form dispersed propagating modes within the pseudogap. Below The 40 meV 'resonance peak' recorded in the neutron scattering experiments [9] is not a bosonic mode capable of supplying the 'glue' for HTSC. Indeed it marks the disruptive excitation of the S = 0 superconducting pairs, involving S = 1 spin flipping, as a pair is taken back from K = (0,0) via (π,π) to its component fermions near (π,0) and (0,π). The observed z-axis component to this excitation in YBCO-123 and BSCCO-2212 is to do not with magnetic interlayer spin coupling but with the appropriate accommodation of the two dissociating fermionic spins within the bilayer unit cell. The paper by Abanov, Chubukov and colleagues [45b], in preprint form entitled 'What the (π,π) resonance peak can do' [100], acknowledged the spin-exciton view of events endorsed 25 there to be quite distinct from that of antiferromagnetic magnon mediation. Nevertheless they continued still to look towards the action of Fermi sea spanning to build up the response function χ'(q,ω) around q = Q = (π,π) and ω(Q) = Ω res . This 'across the body of the zone' physics must be contrasted with our 'to the corner of the zone' physics. In order to acquire the large requisite 'spinfermion' coupling constant, g, and dimensionless fermion self-energy coupling constant, λ, of roughly ¾ eV and 2 respectively it was demonstrated necessary that the 'magnetic' coherence length be just 2a o . Accordingly the bosonic coupling mode looked to within [45b] is of very highly overdamped spin excitations, and in this note contrary to what popularly is often advocated − an antiferromagnetic spin fluctuation mode acting as bosonic mediator in BCS-related fashion.
Pointedly the above ξ value for the spin coherence length of just 2a o means it is identical to the range of the superconductive coupling in the local pair view [2b,101], a result that exposes the magnetic attribution of the coupling as not being unique. It is self-evident g and λ have to be quite large in order to yield HTSC, but the neutron peak has rather little spectral weight or role in events.
Clearly the 'hot spots' on the Fermi surface dominate scattering activity in the zone and dictate the novel transport behaviour above T c , besides the d-wave form to the superconductive gapping below, but this action patently is not due to Fermi surface nesting. The uncondensed boson modes dispersing upwards from the hot spots, which the energy resolved STM scattering experiments from Davis and coworkers [1,63] now look to bear witness to, reassert the local pair character to the novel events current in these materials − as above all, of course, does the extreme smallness of ξ o sc .
It would appear to the author essential now to try to draw together the many disparate theoretical approaches pursued in the past, including the spin fluctuation and MFL ones, around the negative-U and boson-fermion modelling developed in some detail by de Llano, Tolmachev and colleagues [5,102], Tchernyshyov and Ren [103], Letz and Gooding [104], Domanski, Ranninger, Romani, Tripodi and coworkers [4,89,105], and several others. In accomplishing this it will be decisive to embrace properly the inhomogeneous two-subsystem nature of the mixedvalent cuprates, as the STM results of [1,63] and related papers serve so clearly to illustrate.
Acknowledgments: I would like to dedicate this paper to Prof B L Gyorffy on the occasion of his 65th birthday and for his helping to make my time in Bristol a long, lively and productive one. At this moment of my own retirement I thank too my wife (P.A.W.) for her inestimable help over a still longer period in seeing this paper and its predecessors brought to fruition. Additionally thanks are due to Dr N E Hussey for his comments on the current manuscript. Table 1.
Characteristics of zone boundary phonons in YBa 2 Cu 3 O 7 propagating in basal plane perpendicular and parallel to the chains; i.e. along the a axis (irreducible representations Σ) and b axis (irreducible representations ∆) respectively. Information drawn from Chung et al [64], figs. 1 and 4 and text. (Note small errors exist there in figure 1, with 1e being labelled LO rather than TO, and superfluous oxygen atoms appearing in 1c, 1e and 1f at the chain level).
figure label [64] in plane propag n vector (re chain) Σ vs ∆ splitting is from presence of chains destroying tetragonal symmetry.
Σ1 and Σ4 relate to there being two basal planes per YBCO unit cell. (Likewise ∆2 and ∆3).
-of which only ungerade (antisymmetric) inter-layer combination generally seen for integer l.  behaviour is to be contrasted with that included in the figure of the first sharp peak in the ARPES data under comparable conditions, this constructed from the data of Mesot et al [24]. The latter curved plot (dashed) is taken to be a moderately close guide to the functional form of the superconducting gap 2∆(θ) − not ∆(θ) as stated in [24]. It is not justified, moreover, to take directly the non-sinusoidal form of this plot as representing faithfully any true deviation from a strictly d x2-y2 form for the superconducting order parameter.