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Slow fracton modes in antiferromagnetic correlations of high-Tc superconductors

In cuprate superconductors superconductivity develops as a unique cross over between the two extremal phases characterizing these compounds. The extremal phases are known to be an antiferromagnetic (AF) Mott insulator one, on one side, and a non-superconducting itinerant metal on the other. Numerous inelastic neutron scattering studies on underdoped samples shows most directly that the static AF order of the parent phase does not disappear by doping but transforms in such a way that at least fluctuating and local AF order persists, coexisting with superconductivity. Structurally, the model of stripe-type correlations accounts for the coexistence while its genuine dynamics might hide the most interesting physics of cuprates, including the pairing mechanism. Indeed, there is an impressive accumulation of knowledge in favour of AF correlations underlying the pairing. Some crucial properties of these correlations have been recently pointed out: they should involve small velocity collective modes of the wave vector identical to that of static AF order while their energy scale (<50 meV) should be rather low compared with any other involved. In this work we show on phenomenological ground that the very concept of a disordered AF phase interpenetrated with non-magnetic boundaries comprises, irrespective of structural details of stripe order, the dynamical ingredients (fractons) which may be important for pairing. The model elaborated below introduces a new energy scale (10-50 meV) associated with slow (or almost localized) and well-defined AF excitations, interpreting simultaneously most of the neutron scattering results in a consistent and a natural way.

High-resolution studies of dissipation in weak link network of polycrystalline cuprates have recently identified a fractal dissipative regime inside which a self-organized current-carrying medium is an object of fractal geometry 8 . Taking into account that the Josephson-coupled disordered medium represents a good description of the problem of phase coherence not only between the CuO 2 planes but also inside them (due to stripes and/or other forms of charge separation and clustering, filamentary fragmentation, etc.) the presence of fractal geometry in self-organized structures of CuO 2 planes seems rather plausible as well. In this work we investigate elastical modes in such a hypothetical (but still physically justified) two dimensional (2D) fractal lattice. In particular, we study AF spin fluctuations inside a model which allows a fractal geometry of the relevant AF cluster; the latter, being sample-sized and homogeneous in the parent compound, gets progressively more ramified by hole doping. (The liquid-crystal-like stripe phases 2 represent of course a closely related visualization).
Generally, the dynamical features of such a problem are well-known: instead of extended modes (phonons or magnons) of elastic quasi-continuum the vibrational modes in fractal lattices are localized excitations, known as fractons 9,10 . The fractality of real lattices depends on wave vector of the considered excitation so the fracton modes start to dominate only above some critical wave vector, i.e., above the frequency of phonon-to-fracton crossover 10 . Also, there is a considerable accumulated knowledge about fractons specifically in classical diluted antiferromagnets both in their theoretical 12 and experimental 13,14 aspects. In these classical systems the magnetic component responsible for static AF order (e.g., Mn 2+ ion in MnF 2 or in RbMnF 3 ) is substituted at random by non-magnetic substituent (e.g., Zn 2+ or Mg 2+ , respectively). Inelastic neutron scattering studies on samples comprising substituent in concentrations approaching threshold concentration (above which there would be no magnetic order) revealed dynamical features consistent with predictions for AF fractons 13,14 . However, besides some doubts regarding proper assignment of specific experimental features 15 there are still some uncertainties about the proper analytical form of the dynamical structure factor to be applied in modeling fractal antiferromagnets.
Applying this knowledge to the problem of underdoped cuprates we first simplify our elaboration by limiting ourselves to the case of localized magnetic moments (spins associated to Cu 2+ ions) only. There is a consensus now that this is entirely appropriate for the spin wave phase of parent AF compounds but also equally inappropriate for the overdoped non-superconducting systems being compatible with itinerant electron description. The present work explores primarily the underdoped range. Localized description certainly makes sense for low hole concentration but becomes questionable close to optimal doping. There is however an increasing number of arguments 11 pointing out the localized nature of incommensurate magnetism even in optimally superconducting samples (oxygen doped La 2 CuO 4+y ) so one could expect that the model we elaborate here may be applied at least to a reasonably wide range of underdoped concentrations. Now we define our 'building block' which, playing the role of a non-magnetic ion in diluted antiferromagnets, are to be placed at random in the initially homogeneous and antiferromagnetic CuO 2 plane. The fluctuating stripe correlations are certainly the most convincing interpretation for the incommensurate magnetism in superconducting cuprates and the local stripe order seems to be an inevitable element of cuprate physics. Our building blocks are therefore the regions with stripe order, disordered in sizes and, possibly, in other features. By 'stripe order' we mean, in accordance with the localized magnetism of the present model, the original 1 pattern of 'the three spins wide AF stripes separated by non-magnetic antiphase domain boundary' 1 . The internal structure of the building block complicates somewhat the geometrical percolation scenario of the diluted antiferromagnets; thus, we do not attempt to define the threshold hole concentration here. However, one expects that the main qualitative elements of an crossover to fracton dynamics, in increasing hole concentration, remain valid.
Qualitatively, the scenario for underdoped cuprates would be the following: With no holes added the magnetic collective modes are high-velocity (hv = 650meVÅ, typically) regularly damped spin waves (Fig.1a) characterized by dominantly 2D dynamics. The latter follows from the very high inplane superexchange (100meV, typically) compared with the interplane one. Added holes introduce random obstacles for spin waves localizing them into fracton modes at short wave lengths (q > q c ), i.e., above the hole-concentration-dependent cross over energyhω c =hvq c . This takes place when ramified AF cluster becomes self-similar (fractal) on all spatial scales smaller than 1/q c ; subsequently, the crossover frequency progressively decreases as hole concentration (thus cluster fractality) increases 10 .
Quantitatively, all of these dependencies must obey specific power-laws involving several statical and dynamical exponents 9,10 (e.g., correlation length and dynamical one, fractal and AF fracton dimensions) which are, at least approximately, known 10 . The dynamical structure factor S(q, ω) has to include these power-laws and to assure known asymptotic behaviour characterizing the homogeneous (ω ≪ ω c ) and the fractal (ω ≫ ω c ) regime. The problem of dynamical structure factor for AF fractons has been treated by many approaches 10 most of which are founded on the effective-medium considerations 16,17 . Out of several probably equivalent results for S(q, ω) in this work we use the form of Polatsek and Entin-Wohlman 18 as it explicitly demonstrates the localization features of fracton modes (Fig.1b). Although the latter result suffers from general shortcomings of all effective medium approaches, why it cannot quantitatively be correct in all details 10 , its general Green's function background assures its basic usefulness. This form has been successfully used in studies of fractons in silica aerogels 19 , and, more recently, in fused silica 20 . Therefore, the form we propose to determine the scattering cross section of AF spin waves in cuprate CuO 2 planes of increasing fractality reads as where v(ω) and Γ(ω) are velocity of spin waves and damping (lifetime) function, respectively. The frequency dependence of these two quantities makes the present S(q, ω) different, together with its prefactor, from standard Lorentzian 'shape functions' generally used in scattering problems and, particularly, in the inelastic neutron studies on diluted antiferromagnets 21 . The asymptotic forms of v(ω) and Γ(ω) are precisely known 10 : At low ω (ω < ω c ), v = v 0 , the magnon velocity, while at high ω (ω > ω c ), in fractal regime, it must obey v(ω) ∝ ω z (z is dynamical exponent). In the same limits the damping Γ(ω) crosses over from the phonon (magnon)-like Rayleigh law ( Γ(ω) ∝ ω d+1 /ω d c , d is Euclidean dimension) to the localized Ioffe-Regel (Γ(ω) ∝ ω) regime 22 where both the dispersion and the wave vector description becomes ill-defined. For the specific frequency dependence of v and Γ we use the heuristic expressions of Courtens et.al 19 shown to work well in different fractal systems 20 . Taking into account a predominantly 2D nature of magnetism in CuO 2 planes the expressions are: where m characterizes the sharpness of the crossover. Most of the features of the present model have a weak dependence on the choice of m and the 'traditional' 19,20 value m = 4 has been used. Also, in calculation of z = d f /d af , where d f is the fractal andd af the AF fracton dimension, a well-accepted 10 values d f ≈ 1.9 andd af ≈ 0.9 have been used. We note however that our results depend very little, at least qualitatively, on the fine tuning of exponents. Fig.1 visualizes S(q, ω) of Eq.1 taking into account real physical parameters of YBCO superconductor. Now we compare S(q, ω) with the experimental results for imaginary part of generalized susceptibility χ " (q, ω) on cuprates which determines the scattering cross section. These results are summarized in, e.g., Refs. 23,24. Results at fixed q (ω-scan) or at fixed ω (q-scan) depend very much on convolution with instrumental resolution 25 involving the integration of intrinsic S(q, ω) inside usually a sizable resolution ellipse. Hence, a better defined experimental quantities are momentum-or energy-integrated χ " (q, ω). The local (Brillouin zone averaged) susceptibility χ " 2D (ω) ∝ χ " (q, ω)d 2 q is particularly well investigated 23,24 . In hole doped samples χ " 2D (ω) shows a characteristic maximum (Fig.2b,c) in the range of 20-50 meV while the parent compounds reveal approximately energy independent χ " 2D 23,24 , in agreement with S SW (q, ω) for spin waves. It is important to note that a sizable area of χ " 2D (ω) peaks is positioned, in graphs involving absolute calibration, above the spine wave level. The latter reflects a substantial redistribution of spectral weight in frequency domain as the holes are added to the planes. The present model for S(q, ω) readily accounts for the redistribution. The 2D integration of our model's S(q, ω), Eq.1, gives: where Q is Brillouin zone boundary and We note again that v and Γ are ω-dependent (Eq.2). The result for S 2D (ω), which makes a central result of this paper, is shown in Fig.2a for one particular choice of parameters together with the level which would characterize magnons propagating with velocity v 0 . Except behaviour just above the origin there is a full reproduction of experimental curves. The physical reason for the redistribution in our model is clear immediately from Fig.1: For very small energies the magnetic modes start to propagate as magnons but soon, by approaching ω c , slow down and accumulate in the region around ω c taking away the spectral weight that magnons would have in the hole-free lattice up to high energies. We discover therefore thathω c represents a natural new scale of energy characterizing the AF correlations in cuprates. S 2D (ω) of Eq.3 perfectly models most of the experimental results by choosinghω c from the interval 10 − 50meV . The present model attributes the experimental peaks in χ " 2D (and also in χ " (q, ω), see below) to dynamics of crossover quantified, in this work, by Eqs.2,3. One should note, however, that the main features are not determined by some special choice of spectral function S(q, ω), such as the one of Eq.1; instead, the features stem from universal principles of matching the two different regimes characterized by different geometrical constraints. In our model there is no principal difference between the narrow 'resonant' peak 26 , setting-in below T c , and the broad off-resonant 4,26 peaks as they both reflect the dynamics of crossover. The experimental differences may come primarily from the involved parameters, such as the damping scale Γ 0 . Indeed, it seems very plausible that the macroscopic superconductivity of the optimum-doped samples allows higher level of fractal self-organization accompanied with the increased value of Γ 0 . One easily verifies that the increased Γ 0 induces, in turn, a more pronounced peak in S 2D (ω). Now we analyze results for fixed fixed q (ω-scans) or at fixed ω (q-scans) in the light of our model, Fig.3. In literature, there are many data 26 for χ " with q fixed at AF wave vector, q = (0.5, 0.5) in reciprocal lattice units (rlu), measured on a variety of underdoped and optimally doped samples (no significant magnetic correlations has been observed in overdoped range). These results reveal a pronounced peak with the maximum positioned usually inside the interval 20-50 meV but with other details (e.g., temperature dependence, energy width) being dependent on the sample family and the extent of doping. In particular, the resonant peak remains as the only magnetic contribution to χ " in optimally doped samples (not yet observed in La 2−x Sr x CuO 4 family). In our model the region of Brillouin zone which belongs to the close vicinity of q = (0.5, 0.5) (say, for 0.5rlu < q < 0.6rlu, Fig.3) is subject to an abrupt transformation of modes from a purely dispersive (spine wave) to almost localized (fracton) ones.
A ω-scan taken nominally at q = (0.5, 0.5) actually integrates the neighbouring region in which this transformation takes place. In the integration, the dispersive mode gives an ω-independent background contribution while the modes accumulating close to ω c contribute to the net ω-dependence of χ " ((0.5, 0.5), ω). This is illustrated and further commented in Fig.3.
Finally, we discuss the q-scans. The neutron data have shown that there is a characteristic linear relationship 4,27 between the q-space width of χ " (q, ω) and T c at constant ω . The slope of this dependence, having the dimension of velocity 4 , has been used 4,7 as an evidence for the slow excitation, perhaps with a crucial contribution to formation of the superconducting state. In our model, a pronounced q-width of χ " (q, ω) is a natural consequence of progressive slowing down and localization of magnetic modes close to ω c . The peak q-width depends on the value of ω c determined, in turn, by the hole concentration. Fig.4 illustrates the increase of the q-width by decreasing ω c demonstrating that an approximately linear relationship is well-obeyed in a broad range of crossover frequencies.
We conclude therefore that there is a high level of agreement between the present model and the experimental results for imaginary susceptibility and that the slow fracton modes could provide a basis for interpretation of the neutron studies. We also claim that these modes could provide a basis for understanding pairing in high-T c superconductors. Generally, the fracton-mediated superconductivity has been treated theoretically in framework of quasi-crystals 28 by solving an appropriate Eliashberg equation. On other hand, the coupling of carriers to 40 meV 'resonant spin excitations' has recently been recognized 6 as strong enough to account for high values of T c in cuprates. The present work demonstrates that the fracton modes possibly underly these excitations and we believe that they could be a clue for 'uncovering the nature of slow moving charge objects' 7 , thus providing 'a crucial step for our understanding of high-T c superconductivity' 7 . The two asymptotic dynamics is shown: dispersive spin waves (shown in a)) and localized fractons (shown in b)). Lattice parameters of YBCO superconductor have been assumed. Known velocity of spin waves v0 has been used as a parameter. Some reduction of v0 by hole doping has been both observed and expected. The form of Eq.2 has been used for damping; however, in a) the damping was somewhat additionally suppressed in order to visualize the spin waves. The main physical parameter is the crossover frequency ωc. It decreases by hole doping and defines the energy scale of localizing fracton modes. The feature of localization is reflected by vanishing wave vector dependence for ω > ωc, where the mode extension becomes parallel to q-axis. Inset shows a set of ω-scans of S(q, ω), Eqs.1,2, for q fixed at q = 0.6 differing only in the designated choice for ωc. The amplitude of S(0.6, ω), taken athω = 25meV , has been chosen as a relative measure for the q-width. One notes a broad range of ωc characterized by approximately linear behaviour of q-width. Full line is a guide for the eye.