We calculate for La_{2}CuO_{4} the phonon-induced redistribution of the electronic charge density in the insulating, the underdoped pseudogap, and the more conventional metallic state as obtained for optimal and overdoping, respectively. The investigation is performed for the anomalous high-frequency oxygen bond-stretching modes, which experimentally are known to display a strong softening of the frequencies upon doping in the cuprates. This most likely generic anomalous behaviour of these modes is shown to be due to a strong nonlocal electron-phonon interaction mediated by charge fluctuations on the ions. We demonstrate that the softening of the modes is caused by nonlocally induced dynamic charge inhomogeneities in form of charge stripes along the CuO bonds with different orbital character. The dynamic charge inhomogeneities may in turn be considered as precursors of static charge stripe order as recently observed in

In recent work [

In the conventional metallic state, the electronic partial density of states (PDOS) at the Fermi level

Approaching the delocalization-localization transition from the conventional metallic region representative for optimal and overdoping, when

Our modeling of the pseudogap state in terms of compressible, metallic

Furthermore, our modeling of the pseudogap state corresponds with a smaller Fermi volume as compared with the conventional metallic Fermi-liquid state because the Fermi volume is determined by the density of the compressible metallic charge carriers (holes) of dominantly

Because particle density fluctuations scale with the compressibility of the system, the sum rules introduce varying and orbital dependent fluctuations of the particle number in the different metallic states of the HTSC according to our modeling. The tuning parameter for these fluctuations of the particle number is doping, and the pseudogap state may be considered as a state with reduced particle density fluctuations as compared with the optimal and overdoped state. Moreover, the partial incompressibility of the

Qualitatively, the incompressibility of the

In our orbital-based local picture, the pseudogap state of the cuprates looks like a “two-component” electronic structure where a real space organisation of the low lying charge excitations is achieved in form of a metallic charge response by mobile holes on the oxygen network in the CuO plane that is blocked at the incompressible insulator-like Cu sites. Inspection of the sum rule in (

In

Our modeling of the density response is strongly supported by corresponding calculations of phonon dynamics [

(a) Experimental results for the highest

Experimental Results

Calculated Results

It seems appropriate to explain at least the ideas of our modeling of phonon dynamics and charge response of the cuprates. A quantitative treatment of the theory and the modeling would be too extensive and can be found in [

The rigid part of the electronic charge response and the electron-phonon interaction (EPI) is approximated by an ab initio rigid-ion model (RIM), taking into account covalent ion softening in terms of (static) effective ionic charges calculated from a tight-binding analysis. In addition, scaling of the short-range part of certain pair potentials between the ions is performed to simulate further covalence effects in the calculation in such a way that the energy-minimized structure is as close as possible to the experimental one. The RIM with the corrections just mentioned then serves as an unbiased reference system for the description of the cuprate superconductors and can be considered as a first approximation for the insulating state of these compounds because of the strong ionic nature of bonding in the cuprates. Starting with such a rigid reference system nonlocal, electronic polarisation processes are introduced in the form of more or less localized electronic charge fluctuations (CF’s) at the outer shells of the ions. In the metallic state of the cuprates, especially the latter dominate the nonlocal contribution of the charge response and the EPI and are particularly important in the CuO planes. In addition, anisotropic dipole fluctuations are admitted in our approach. They prove to be specifically of interest for the ions in the ionic layers.

Comparing with Figure

Finally, it should be remarked that a constructive interplay of electron-ion interaction with short-ranged antiferromagnetic order is important for the Cu sites in the underdoped and insulating state, because electrons are transferred from the Cu ion where the Cu bond is compressed to the Cu ion where the bond is stretched, see for example, Figure

(a) Contour plot in the CuO plane of the displacement-induced charge density redistribution

Modelling of the electronic polarizability

In Figure

From Figure

In case of the full-breathing mode

The self-consistent changes of the potential felt by an electron due to nonlocal EPI also have been calculated for the OBSMs and yield a strong coupling of the order 100 meV [

In Figures

(a) Same as Figure

To the best of our knowledge in overdoped samples static stripe order has not been observed so far. This would mean that the pseudogap in the underdoped state may be required to nucleate dynamic stripes. In our approach this implies that for the emergence of static stripe order the incompressibility of the

Comparing the stripe patterns of the insulating and underdoped state with that of the more conventional metallic optimal and overdoped state (not shown) with a large FS as the locus of the crucial low energy excitations, we find a relatively small contribution of the

In the overdoped state, a growing importance of the

From our calculations it becomes obvious that a multiorbital approach is needed that includes besides Cu

Translational symmetry-breaking models of the electronic structure of the CuO plane generated by charge density wave order of a certain wavevector are discussed in the literature and used to characterize Fermi surface reconstructions of a large FS into small pockets, for recent work see [

Same as Figure

We again obtain a stripe-like charge pattern with translational symmetry in the

In summary, we have applied our modeling of the doping-dependent electronic states of the cuprates to the calculation of the phonon-induced charge inhomogeneities nonlocally induced by the

The interplay between the reduced particle density fluctuations related to the insulator-like behaviour of the localized atomic-like

_{c}superconductor LaCuO: phonon dynamics and charge response

_{2}CuO

_{4}in the self-interaction-corrected density-functional formalism

_{2}CuO

_{4+δ}

_{2}Sr

_{1.6}La

_{0.4}Cu

_{2}O

_{6+δ}superconductors

_{1.85}Sr

_{0.15}CuO

_{4}

_{2}Cu

_{3}O

_{7}studied by inelastic neutron scattering

_{2−x}Sr

_{x}CuO

_{4}(0 ≤

_{1.7}Sr

_{0.3}Cu

_{3}O

_{4}

_{1.86}Ce

_{0.14}CuO

_{4+δ}determined by inelastic X-ray scattering

_{1.85}Ce

_{0.15}CuO

_{4}

_{2}CuO

_{4}

_{2−x}Ba

_{x}CuO

_{4}(0.095 ≤

_{2}Cu

_{3}O

_{y}

_{2}Cu

_{3}O

_{y}from high-field Hall effect measurements