The high-Tc cuprate superconductors are close to antiferromagnetic order. Recent measurements of magnetic excitations have reported an intriguing similarity to the spin waves—magnons—of the antiferromagnetic insulating parent compounds, suggesting that magnons may survive in damped, broadened form throughout the phase diagram. Here we show by resonant inelastic X-ray scattering on Bi2Sr2CaCu2O8+δ (Bi-2212) that the analogy with spin waves is only partial. The magnon-like features collapse along the nodal direction in momentum space and exhibit a photon energy dependence markedly different from the Mott-insulating case. These observations can be naturally described by the continuum of charge and spin excitations of correlated electrons. The persistence of damped magnons could favour scenarios for superconductivity built from quasiparticles coupled to spin fluctuations. However, excitation spectra composed of particle–hole excitations suggest that superconductivity emerges from a coherent treatment of electronic spin and charge in the form of quasiparticles with very strong magnetic correlations.
One of the most distinctive and surprising features of the cuprate superconductors (SC) is their proximity to an antiferromagnetic (AFM) phase. Understanding what survives of magnetic correlations when the AFM insulator is doped is an outstanding issue. It is especially important for theoretical scenarios where magnetic fluctuations provide the pairing interaction at the origin of high-temperature superconductivity1. Theoretical studies of spin dynamics in layered cuprates can be broadly separated into two complementary approaches that work well in opposite limits. The insulating parent compounds are well described in terms of local spins and the Heisenberg hamiltonian2. The evolution of spin excitations on introducing a finite amount of carriers is treated by assuming a mixture of local spins described by the superexchange interaction J and itinerant carriers dispersing with hopping integral t, such as the t–J model3,4. Another class of theories build on an itinerant description, where the system’s behaviour is primarily governed by interactions at energies smaller than the fermionic bandwidth, while contributions from higher energies only lead to renormalizations. In this itinerant approach, correlations are usually introduced through the random-phase approximation (RPA)5,6,7, where the spin response is a continuum of electron–hole excitations, albeit this approach remains under debate8,9. These two approaches can be seen as approximations of the Hubbard model, which probably contains the relevant physics, including superconductivity10, and could form the basis of a comprehensive theory that bridges the gap between these two limits. Indeed, numerical calculations on clusters of finite size have shown to be consistent with experimentally measured electronic and magnetic excitation spectra11,12. However, a generally valid solution to the Hubbard model remains out of reach. Therefore, new experiments probing both spin and charge dynamics in the crucial intermediate doping range are especially valuable. A key question is whether the excitations of the many-body system can be separated into weakly hybridizing spin and charge excitations, or whether the spin and charge signals are two response functions to the same underlying dynamics.
Neutrons are a standard probe of magnetic excitations, and have provided the full magnon dispersion in selected insulating parent compounds13,14,15. In doped compounds, neutron spectroscopy has provided great insight on the low-energy part of the magnetic excitation spectrum, including an hourglass anomaly, the opening of a spin gap and build-up of a resonance in the superconducting state, which seems to be a generic feature in the cuprates and in some iron-based SC16,17,18. This anomaly may be at the origin of characteristic features in the electron spectral function measured by angle-resolved photoemission19. Dispersive excitations have been reported up to 150 meV (refs 20, 21), above which neutron spectroscopy becomes increasingly challenging.
An independent picture of magnetic and charge excitations has now been gained by resonant inelastic X-ray scattering (RIXS) experiments at the Cu K edge22 and Cu L3 edge23,24,25,26,27,28 in the cuprates and at the Fe L3 edge in the iron pnictides29. RIXS offers some interesting advantages, such as the possibility of measuring small-volume samples, additional flexibility from polarization control and complementary information from the Cu (single magnons) and O (two magnons) edges30. RIXS has enabled a complete mapping of the spin waves in the insulating cuprates. Namely, it revealed dispersive excitations along the magnetic zone boundary, in contrast to the predictions of the simple two-dimensional (2D) Heisenberg model, and therefore established the importance of longer-range interactions31. A significant breakthrough was the observation by RIXS in several families of SC cuprates of dispersions closely resembling the spin waves of the parent compounds, but progressively broadened with increasing doping32,33,34,35. In all these cases, the dispersion was studied only along the anti-nodal (100) direction, parallel to the Cu–O bonds in the CuO2 planes, for which the d−wave superconducting gap is the largest. Here we present high-resolution Cu L-edge RIXS data for optimally doped, underdoped and undoped Bi2Sr2CaCu2O8+δ (Bi-2212) samples, for both the nodal (110) and the anti-nodal (100) directions. We find well-defined dispersive features similar to the magnons of the insulating parent compound along the anti-nodal direction. However, along the nodal direction, these excitations collapse, with spectral weight extending down to zero-energy transfer. While the dispersive anti-nodal excitations could be interpreted as originating from correlated paramagnetic (PM) local moments, we show that both the nodal and the anti-nodal directions are qualitatively described by the continuum of spin excitations calculated for correlated itinerant electrons.
The doping and momentum dependence of the RIXS spectra
Figure 1 presents RIXS data for Bi1.6Pb0.4Sr2YCu2O8+δ (BSYCO; insulator), for underdoped SC Bi1.6Pb0.4Sr2Ca0.95Y0.05Cu2O8+δ (UD-2212; Tc=70 K) and for optimally doped Bi2Sr2CaCu2O8+δ (OPT-2212; Tc=92 K). All spectra shown in the figure were collected at 20 K with σ-polarized light, and the incident photon energy (930.5 eV) coincides with the peak of the Cu L3 X-ray absorption line shape (Fig. 1a). The scattering geometry is illustrated in Fig. 1b. The intensity of the scattered photons is plotted as a function of the energy loss, defined as the difference between the incident and emitted photon energies, for selected values of q, the transferred momentum projected on the ab plane. The spectra have been normalized to the same integrated intensity in a window encompassing the manifold of dd excitations around 2 eV.
Figure 1c,d shows spectra of insulating BSYCO, measured along the (100) and (110) high-symmetry directions. The line shapes are similar to those of other insulating cuprates, namely, La2CuO4 (LCO)24 and Sr2CuO2Cl2 (SCOC)25. One can identify the elastic peak, broadened by non-resolved phonon sidebands, and a sharp magnon peak, dispersing from zero energy at q=0 to ~0.3 eV at the boundary of the magnetic Brillouin zone (BZ). The magnon energy is ~60 meV larger at (π, 0) than at (π/2,π/2), due to interactions beyond nearest-neighbour spins, as already observed for LCO by neutrons13,14 and discussed elsewhere31. The high-energy tail of the magnon peak belongs to the multi-magnon continuum.
Data for the UD-2212 sample are illustrated in Fig. 1e,f. A dispersive inelastic feature is visible along the anti-nodal (100) direction (Fig. 1e). It is broader and relatively less intense than in BSYCO, but the dispersion is similar. This is consistent with the results on Bi-2212 in ref. 33. More generally, the momentum-dependent RIXS line shape is qualitatively similar to that of other SC cuprates along the anti-nodal direction32. The data in Fig. 1g,h for OPT-2212 show that the RIXS spectral function is fairly insensitive to the doping level, as previously reported from experiments performed with both σ- and π-polarized light32,33,34,35. Along the nodal (110) direction, the line shape is asymmetric and ~4 times broader than the resolution-limited elastic peak, revealing an underlying inelastic component. At variance with the (100) direction, it is not possible to identify a dispersive feature that could be interpreted as a broadened version of the magnon peak of the insulator. This is reinforced by a detailed analysis of the spectral line shape. Figure 2 illustrates representative spectra of UD-2212, for two values of momentum transfer along the (100) and (110) directions, together with fits performed along the lines in ref. 32. Each spectrum is decomposed into the sum of a paramagnon line shape, an elastic peak and a sloping background representing the tail of the higher-lying dd manifold. Satisfactory fits are possible for both spectra along (100). By contrast, a good agreement can never be achieved for the two representative spectra along the nodal direction. Alternatively, the spectra along the (100) direction could be reasonably well described by a suitably broadened magnon peak plus a multi-magnon tail, as in ref. 25, but such an approach would also fail along the nodal direction. Such discrepancy could not be appreciated from the previously published RIXS data. The anisotropic response is illustrated by the energy–momentum intensity map for UD-2212 (Fig. 3) where, for clarity, a resolution-limited elastic peak has been subtracted. The magnon dispersion for BSYCO, superimposed on the map, was extracted from the experimental spectra in Fig. 1 using the same analysis in ref. 25. The RIXS intensity follows the spin-wave dispersion of the insulator along the (100) direction. By contrast, along the (110) direction, the peak of the spectral weight distribution for the superconducting sample is consistently found below the magnon energy.
The evolution of the RIXS spectra with incident photon energy
The data in Figs 1 and 3 suggest that the spin-wave picture built on AFM-correlated local spins, which well describes the insulating cuprates, may not be the most appropriate starting point to describe the metallic (superconducting) phase. A dramatic distinction between the two cases is revealed by the photon energy-dependent RIXS data in Fig. 4. The schematic diagrams of Fig. 4a,b show a RIXS transition from the ground state G to an excited final state F in two different situations. In Fig. 4a, the coherent second-order optical transition goes through a sharp intermediate level I. This illustrates Cu L3 RIXS in an insulating cuprate, where the intermediate state I is the single 2p53d10 level and the final state F is the ground state plus a magnon. In this case, both at and above threshold, the photon energy loss is constant and equal to the energy separation between G and F. In Fig. 4b, by contrast, the intermediate state is a continuum, as in a metal with a well-defined Fermi surface. Above threshold, this opens the possibility of a real transition, followed by an incoherent radiative de-excitation (fluorescence) from a lower-energy state. The energy difference between the incident and emitted photons—the energy loss—now increases with the incident energy. RIXS measurements as a function of the incident photon energy above threshold can therefore distinguish the two situations.
Figure 4c shows spectra of insulating SCOC and UD-2212 measured near (π, 0) at various incident energies at and above the L3 threshold. In SCOC, the magnetic part of the response exhibit an intensity modulation and minor modifications of the line shape as a function of incident energy, but the corresponding energy loss remains constant, as expected. The case of UD-2212 is quite different. Here we observe a linear energy shift of both the dd manifold and the low-energy features (arrows) away from the elastic peak. A few hundred meV’s above threshold, the magnon peak is still visible in SCOC, but it has disappeared in the SC. This clearly demonstrates that, even in the anti-nodal direction, for a description of the low-energy excitations in RIXS it is imperative to consider the continuum of e–h excitations across the Fermi surface.
The previously reported similarity between RIXS spectra in the parent and doped compounds suggests that the doped spectra can be interpreted as damped magnons32,33,34,35. Calculations of the spin dynamical structure factor S(q,ω) performed in the framework of the Hubbard11,12 and t–J models32,36 have been able to reproduce several aspects of the experimental spectra, namely, the nearly doping-independent dispersion and progressive energy broadening. The results in Fig. 4, on the other hand, indicate that the inelastic spectrum in doped Bi-2212 belongs to the e–h continuum. From a different perspective, it has been argued that the RIXS data could be explained by the band structure alone, without taking into account collective spin fluctuations37.
Here we consider the possibility that in the doped compounds the RIXS signal can be described in terms of collective spin excitations from correlated itinerant electrons. This can be simply described in terms of the RPA spin susceptibility38, χs(q,ω)=χ0(q,ω)/[1–Uχ0(q,ω)], shown in Fig. 5a for the undoped AFM-ordered case and in Fig. 5b for the PM doped case (for further details, see Methods). An interesting feature of this expression is that for collective spin excitations the actual modification of the Lindhard particle–hole response function due to the Coulomb interaction U in the RPA denominator occurs only near (π,π), around the AFM wave vector itself. The dispersive behaviour of χs along the anti-nodal (0, 0)→(π, 0) direction is mainly due to the structure of the Lindhardt function. In Fig. 5b, white circles mark the peak energy in the elastic line-corrected RIXS spectra for UD-2212, extracted from Fig. 3. The calculated RPA χs qualitatively reproduces the salient experimental features, namely: (i) the persistence of a dispersive feature along the (0, 0)→(π, 0) direction; (ii) the collapse of the excitations to lower energy along the (0, 0)→(π,π) direction; and (iii) being a particle–hole continuum, it naturally explains the incident energy dependence reported in Fig. 4. The dispersive features in χs away from (π,π) are already present in the original e–h response function, but are enhanced by the RPA denominator. For smaller doping values, these excitations become the true spin-wave excitations of the AFM ground state present in the entire BZ. In this regard, the interpretation of the experimental data is close to that in ref. 37, with the difference that we assume as a basic starting point that RIXS measures the direct spin excitations.
The calculated dispersion of the spin excitations in Fig. 5b shows additional intense features that are seemingly absent in the experiment. In particular, in the calculations, we find low-energy excitations around q1~(π/3,0) and a broad spectral weight distribution below ω~0.1 eV between (π/3,π/3) and (π,π). Within RPA, these excitations arise from regions near the Fermi surface with opposite Fermi velocities, and are usually referred to as 2kF instabilities. Namely, the 1D wave vector q1 corresponds to the electron–hole scattering between two, nearly parallel, anti-nodal regions on the Fermi surface near the (π, 0) point of the BZ. The second type of scattering originates from the so-called nodal 2kF instability along the diagonal of the BZ. Due to finite doping, this wave vector is incommensurate and smaller than (π,π). In principle, these excitations should be visible in RIXS. However, recent calculations37, based on the same tight-binding parametrization, show that their intensities are strongly influenced by the RIXS matrix elements. They should become more visible in the strongly overdoped regime where both the proximity to the Mott insulator, which influences the Fermi surface, and the quasiparticle lifetime become less important.
For completeness, we also present in Fig. 5c the charge-response function. As mentioned above, within an itinerant approach, the collective charge and spin excitations arise initially from the particle–hole scattering of the electronic states located near the Fermi level. The only difference within the RPA treatment of the single-band Hubbard model is that summing corresponding diagrams yields in the PM-state different denominators for the charge and the PM spin susceptibilities. While the spin response is enhanced within RPA, the charge response is suppressed as χch(q,ω)=χ0(q,ω)/[1+Uχ0(q,ω)]. Nevertheless, the gross features of both functions look rather similar, except for the difference in the intensity and the downward shift in energy of the spin excitation. In particular, low-energy features around q1~(π/3,0) and (π/3,π/3) in the charge response are very similar to, but weaker than, the corresponding spin excitations. Therefore, in the doped compounds, both spin-flip and non-spin-flip processes contribute often at the same energies, thus making it difficult to disentangle them.
Interestingly, the strong spin fluctuations in doped cuprates also influence the excitations in the charge-response function. It has been argued39 that the remarkable correspondence between the position of the spin resonance observed in the inelastic neutron-scattering experiments and the pair-breaking electronic Raman response in the A1g symmetry could be a results of the collective spin fluctuations, transferred to the charge response via renormalization of the vertex function. Given the similarity between Raman scattering and RIXS, one would expect the same type of diagrams to contribute also to the RIXS intensity. In particular, the spin fluctuation contribution to the charge excitations can be written as39:
where we set the RIXS matrix elements to unity for the sake of clarity. Then, the total charge response is the one discussed in ref. 37 plus equation (1). We note that collective spin excitations must be well pronounced to be visible in the charge-response function. For example, in Fig. 5d,e) we compare in the parent AFM phase, where the excitations are the true magnons, and in the itinerant limit. One sees that the structure of the spin excitations is clearly visible only in the case of spin waves, while the itinerant PM response gives a featureless contribution to . This indicates that spin fluctuations strongly influence the charge RIXS response in the undoped and underdoped cuprates but will become less visible in the overdoped cuprates.
The anisotropy and energy dependence of the RIXS response reported here reveal new aspects of the magnetic excitations in the superconducting cuprates that need to be incorporated in a theory of the electronic structure of these materials. One can anticipate further progress from a more systematic survey of these features across the phase diagram. Specifically, studying how the incident energy dependence evolves from Raman-type response in the insulating parent compound to fluorescence-type response in the superconductor—for example, as a cross-over or as an abrupt transition—could provide a new type of insight to the entanglement of localization and strongly correlated quasiparticles. In addition, experiments implementing polarization analysis of both the incident and scattered beams would be especially valuable, as they would enable unambiguous separation of the magnetic signal from other spectral contributions12. Such experiments pose severe technical challenges, but will soon become possible in a new generation of RIXS instruments.
Cuprate single-crystal growth
Single crystals of optimally doped Bi2Sr2CaCu2O8+δ were grown by the travelling-solvent floating-zone method in a commercial two-mirror furnace (Cyberstar), under a 93% Ar–7% O2 atmosphere. The pre-reacted precursor rod (feed) and the crystallization seed were counter rotating at ≈20 r.p.m. and travelled through the hot zone at 0.2 mm per hour. The growth conditions at the flat liquid–solid interface were kept stable for several days. Plate-like crystals 2–3 mm long and 0.1–0.2 mm thick were cleaved (perpendicularly to the c axis) from the as-grown rod and annealed at 500 °C in p(O2)=10−3 bar for ≈100 h to fix and homogenize the optimal oxygen content. The critical temperature extracted for the magnetic transition was 92 K, with a transition width narrower than 2 K. Single crystals of insulating BSYCO and SCOC and of SC-underdoped Bi1.6Pb0.4Sr2Ca0.95Y0.05Cu2O8+δ (UD-2212; Tc=70 K) were grown using a self-flux method in the form of large platelets perpendicular to the c axis. All samples were characterized by X-ray diffraction and resistivity measurements.
The RIXS experiment
RIXS is a second-order optical process that probes charge-neutral single-particle and collective excitations, with charge or spin character30. RIXS measurements at the Cu L3 absorption edge (2p3/2→3d; Fig. 1a) were performed with the SAXES spectrometer40 of the ADRESS beam line41 at the Swiss Light Source, Paul Scherrer Institut (Villigen, Switzerland). The soft X-ray beam from the undulator source was linearly polarized either within (π polarization) or perpendicular to (σ polarization) the horizontal scattering plane. The scattering geometry is schematically illustrated in Fig. 1b. The fixed scattering angle was 130°, corresponding to a momentum transfer Q=0.85 Å−1. Samples were mounted with the c axis and either the (100) or the (110) direction in the scattering plane. The projection Q||=q of Q in the ab plane could be varied between ±0.73 Å−1 by rotating the crystal around a vertical axis. The energy calibration and the experimental resolution ΔE=130 meV were determined from repeated measurements of the elastic peak of a polycrystalline carbon sample. The incidence (emission) angle, measured from the sample normal, varied between 25°(−25°) at q=0 and 85°(35°). Self-absorption of the scattered beam is negligible over the whole range of transferred momenta. The magnetic RIXS cross-section exhibits a strong polarization dependence near grazing emission. By contrast, in the near-normal to grazing incidence geometry adopted here, the cross-sections for σ and π polarizations are similar42. Spectra measured with σ polarization exhibit a stronger elastic peak, which provides a useful internal energy calibration.
The RPA calculation
We used an RPA formalism within a single-band Hubbard model,
to describe the evolution of the spin excitations from the parent AFM state to the SC state at optimal doping. Here, creates (annihilates) an electron with spin σ and momentum k with the normal state tight-binding energy dispersion εk=−2t1 (cos kx+cos ky)−4t2 cos kx cos ky−2t3 (cos 2kx+cos 2ky)−2t4 (cos 2kx cos ky+cos kx cos 2ky)−μ with (t1, t2, t3, t4)=(126, −36, 15, 1.5) (all in meV). The chemical potential μ controls the doping, chemical doping x, defined as n=1+x.
First, we analyze the excitations in the parent AFM state. The Hubbard interaction term in the Hamiltonian equation (S1) leads in the Hartree–Fock factorization to the emergence of the commensurate AFM order, with corresponding spin-wave-type excitations in the transverse spin susceptibility43. After decoupling the Hubbard interaction via a mean-field approximation, and diagonalizing the resulting mean-field Hamiltonian via a conventional spin density wave Bogoliubov transformation, one obtains two electronic bands (labelled α and β) in the reduced BZ with dispersions , where is the spin density wave order parameter that is determined self-consistently for a given U, for which we took 3.2t1, and . For completeness, the self-consistent determination of the chemical potential must also be included.
The dynamical spin susceptibility for the transverse, +−, components is then defined as , where we took into account that the AFM ordering at Q=(π,π) doubles the unit cell43,44. Since commensurate AFM order breaks the translational symmetry and doubles the unit cell, the total susceptibility in the transverse channel is a 2 × 2 matrix:
Within RPA, the susceptibility is obtained via solving a Dyson equation, and the result can be expressed43 as:
where indicates the bare components.
This weak-coupling approach gives an adequate description of the spin excitations in the AFM state of the parent cuprate compounds in quantitative agreement with experiment45, as seen in Fig. 5a. An additional advantage of this approach is that it provides a simultaneous information on not only the collective spin excitations of the AFM state, but also the e–h continuum. Furthermore, this approach is naturally extended to the PM state.
At the next stage, we consider hole doping and analyse the spin excitations in the PM state. In this case, the expression for the transverse part of the spin susceptibility is reduced to the standard one χs(q,ω)=χ0(q,ω)/[1–Uχ0(q,ω)] where χ0(q,ω) is the Lindhard response function. We further use the value of U=1.5t1 to keep the system in the PM state and not in immediate vicinity to the AFM instability, so that the RPA denominator is far from divergence. Figure 5b illustrates the imaginary part of χs(q,ω), showing the e–h continuum along the two high-symmetry directions for optimal doping.
How to cite this article: Guarise, M. et al. Anisotropic softening of magnetic excitations along the nodal direction in superconducting cuprates. Nat. Commun. 5:5760 doi: 10.1038/ncomms6760 (2014).
The work at Lausanne is supported by the Swiss NSF. The work of I.E. is partially supported by the Ministry of Education and Science of the Russian Federation in the framework of the Program to Increase Competitiveness of NUST ‘MISiS’ (No. K2-2014-015). We acknowledge stimulating discussions with E. Demler and D. Benjamin and thank R. Gaal for the resistivity measurements. The SAXES instrument was jointly built by the Paul Scherrer Institut (Villigen, Switzerland), Politecnico di Milano (Italy) and École Polytechnique Fédérale de Lausanne (EPFL, Switzerland).
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