Inelastic X-ray scattering: a probe of the electronic properties of cuprate superconductors?

Journal of Physics and Chemistry of Solids 61 (2000) 473–475 www.elsevier.nl/locate/jpcs

Inelastic X-ray scattering: a probe of the electronic properties of cuprate superconductors? M. Altarelli a,*, P. Johansson b a

b

European Synchrotron Radiation Facility, Boite Postale 220, F-38043 Grenoble Cedex, France Department of Theoretical Physics, University of Lund, Solvegatan 14 A, S-223 62 Lund, Sweden

Abstract The need for a genuine bulk probe of the electronic properties and charge excitations of high-Tc materials, capable of mapping Fermi surfaces and superconducting energy gaps in momentum space, is briefly reviewed. The possible use of high-resolution inelastic X-ray scattering is discussed theoretically. It is argued that in principle X-ray scattering has the potential to perform such a role, but that the presently available intensity of undulator sources has to increase by at least one order of magnitude before these experiments become feasible. q 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Inelastic X-ray scattering

A large amount of invaluable information about high-Tc superconductors has been obtained by angle-resolved photoemission (see e.g. Refs. [1,2] and references therein). This technique allows the mapping of Fermi surfaces, superconducting gaps, and even of the elusive “pseudogaps” encountered in the normal state of underdoped cuprates. However, photoemission is a surface-sensitive technique and requires suitable cleavage properties. This means that, among the cuprates, the technique has been almost exclusively applicable to BSCCO. Inelastic neutron scattering has also been very valuable in mapping the magnetic dynamics of the high-Tcs (see e.g. Refs. [3,4] and references therein). Although neutrons are not coupled to charge excitations, these experiments show the interest of a probe with a wavelength sufficiently short to explore a large region of reciprocal space, and with an energy resolution of the order of a few meV or less. This energy resolution is commonly available in Raman scattering experiments with visible light (for a review, see Ref. [5]), which also provide useful information, but restricted to the immediate neighborhood of the origin of reciprocal * Corresponding author. Present address: Sincrotrone Trieste, 34012 Trieste, Italy. Tel.: 139-040-375-8003; fax: 139-0409380903. E-mail address: [email protected] (M. Altarelli).

space (typically 10 23 times smaller than the zone boundary vector). In view of the achievement of energy resolutions in the meV range in inelastic X-ray scattering experiments [6,7], it is interesting to envisage the application of this technique to the high-Tc materials: the association of this energy resolution with wavelengths l , 0:1 nm and with probing depths of the order of a few mm, is a very promising combination to investigate the cuprates. In fact, in the case of a quasi 2D Fermi surface, i.e. one with the same cut on every plane perpendicular to the c-axis, Fig. 1 shows that the momentum conservation condition for transitions across the Fermi surface is satisfied only at a discrete set of symmetry-related ! points on the surface, and that therefore the possibility of k space mapping is ensured. Our analysis [8] shows however that the limitations of the scattering volume imposed by photoelectric absorption in the high-Tc materials, which invariably contain heavy elements, require an improvement of the brilliance of presently available sources by a factor of order 20–50 to produce an acceptable count rate. The basic ingredients and the results of this analysis are summarized in the following. Resolutions in the meV range for hard X-rays are only possible for selected incoming photon energies, satisfying the Bragg condition very close to backscattering on highorder Si [hhh]-type reflections [6,7]. Therefore, the

0022-3697/00/$ - see front matter q 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S0022-369 7(99)00339-X

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M. Altarelli, P. Johansson / Journal of Physics and Chemistry of Solids 61 (2000) 473–475

∆

ky

s

+ q2 q

3

q

1

kx

+

+

d

Fig. 1. Schematic illustration of the Fermi surface and the superconducting gap for the anisotropic s and dx2 2y2 cases, in a quasi 2D superconductor. d-wave superconductivity may be caused by repulsive interactions which scatter electrons mainly between the corners (surrounded by boxes) of the Fermi surface. The arrows indicate excitations involving large momentum transfers but only small energy transfers. Such excitations can be created by means of inelastic X-ray scattering.

incoming photon energy cannot be tuned near an absorption edge, and the relevant cross section for inelastic X-ray scattering is the non-resonant one [9,10]: d2 s v ˆ r02 f …! 1 pf ·! 1i †2 S…! q ; v† dvf dVf vi

…1†

where the labels i, f refer to the initial and final photons, ! q and v denote the transferred wave vector and energy, r0 is the Thomson radius and the dynamic structure factor S…! q ; v† is defined by: S…! q ; v† ˆ …2† ! ! !0 ! 1 Z1 ∞ ! !0 d r dr dt e2ivt e2i q ·… r 2 r † kr…! r ; 0†r…r 0 ; t†l; 2p 2 ∞ In this equation, r…! r ; t† denotes the density of electrons in the interaction representation. In Ref. [8], for lack of any consensus on a better description of the superconducting state in the cuprates, the correlation function in Eq. (2) was evaluated using BCS-type wave functions of the appropriate symmetry …s or dx2 2y2 †: The effects of screening and of final-state interactions on the correlation function were considered. Although the momentum transfer of order 2kF is sufficient to reduce screening effects considerably in threedimensional metals, this is not necessarily the case in quasi 2D systems, where an RPA estimate shows an effective dielectric constant of order 2–3, for the ! q and v of interest. Screening effects within each plane must therefore be taken into account, whereas interplane screening is not effective due to the large interplane separation combined with large momentum transfers. Final-state interactions describe the effect of the interaction responsible for pairing on the probability of creating two quasi-particles by breaking a pair. In the case of s-wave pairing, the interaction is assumed to be a uniform nonretarded attractive interaction, whereas for d-wave pairing

Fig. 2. Inelastic scattering probability in the normal state and with an isotropic s-wave gap for different wave vectors ! q ˆ …X; 2X†p=a (the value of X is shown near each curve). Panel (a) corresponds to EF ˆ 0:65 eV and D ˆ 0 (normal state); (b) EF ˆ 0:65 eV and D ˆ 30 meV, including final-state interactions; (c) EF ˆ 0:74 eV; D ˆ 30 meV with (solid curves) and without (dotted curves) final-state interactions.

it is assumed to be a repulsive interaction scattering electrons between the different “corners” of the Fermi surface, enclosed by boxes in Fig. 1. The analysis performed in Ref. [8] shows that final-state interactions produce appreciable

M. Altarelli, P. Johansson / Journal of Physics and Chemistry of Solids 61 (2000) 473–475

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D ˆ 30 meV. Results are presented for two values of Fermi energy, EF ˆ 0:65 and 0:74 eV (the latter being very close to the “half-filling” condition, with a nearly square Fermi surface) and for wave vectors in the [1,1] direction, in which the d-wave gap function has nodes. Figs. 2 and 3 show that it should be possible to gain a lot of information about Fermi surfaces and gap functions. However, an estimate of the count rate for realistic experimental conditions shows that these experiments may have to wait for the next generation of synchrotron sources. An order of magnitude of the scattering rate may be obtained by inserting into Eq. (1) an estimate of S…! q ; v† based on the f-sum rule [9,11] and on the surface density of conduction electrons in each superconductor layer. In agreement with the calculated results presented in Figs. 2 and 3 this gives a scattering probability for incoming photons , 10212 eV21 sr21 per layer. Since absorption restricts the sampling depth to , 104 layers, and energy and momentum resolution should not be worse than 10 meV and 10 22 sr, respectively, for the presently available fluxes on the sample of the order of 10 10 photons per second in a few meV band [6,7], it is hard to do better than 10 22 counts per second, a signal one to two orders of magnitude too low to emerge from the background.

References Fig. 3. Inelastic scattering probabilities for the case of a dx2 2y2 gap. The calculations include screening, but neglect final-state interactions. The wave vectors are indicated as in Fig. 2. The Fermi energy is (a) EF ˆ 0:65 eV; (b) EF ˆ 0:74 eV:

effects in the s-wave case, but are negligible in the dx2 2y2 case. Results of the calculations are shown in Fig. 2 for the swave case and in Fig. 3 for the dx2 2y2 case, for a cuprate with a simple tight-binding band structure:

1! ˆ 22t‰cos…kx a† 1 cos…ky a†Š; k

…3†

In the calculations reported here, we chose the parameter t to give a bandwidth 8t ˆ 1:5 eV; the lattice parameter a ˆ  a background dielectric constant of 4, temperature 3:87 A; T ˆ 0; and a (maximum) superconducting gap of

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