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arXiv:cond-mat/0411445 · Physical Review B

Electronic Raman response in anisotropic metals

Dietrich Einzel, Dirk Manske

Published Nov 18, 2004• 5 citations
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Abstract

Using a generalized response theory we derive the electronic Raman response function for metals with anisotropic relaxation rates. The calculations account for the long-range Coulomb interaction and treat the collision operator within a charge conserving relaxation time approximation. We extend earlier treatments to finite wave numbers $(\ensuremath{\mid}\mathbf{q}\ensuremath{\mid}⪡{k}_{\mathrm{F}})$ and incorporate inelastic electron-electron scattering besides elastic impurity scattering. Moreover we generalize the Lindhard density response function to the Raman case. Numerical results for the quasiparticle scattering rate and the Raman response function for cuprate superconductors are presented.

DOI
10.1103/physrevb.70.172507

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Using a generalized response theory we derive the electronic Raman response function for metals with anisotropic relaxation rates. The calculations account for the long-range Coulomb interaction and treat the collision operator within a charge conserving relaxation time approximation. We extend earlier treatments to finite wave numbers (\ensuremath∣q\ensuremath∣⪡kF)(\ensuremath{\mid}\mathbf{q}\ensuremath{\mid}⪡{k}_{\mathrm{F}})(\ensuremath∣q\ensuremath∣⪡kF​) and incorporate inelastic electron-electron scattering besides elastic impurity scattering. Moreover we generalize the Lindhard density response function to the Raman case. Numerical results for the quasiparticle scattering rate and the Raman response function for cuprate superconductors are presented.
We have carried out a comprehensive investigation of the quasiparticle properties of a two-dimensional electron gas, interacting via the long-range Coulomb interaction in the presence of bare mass anisotropy (i.e., with an elliptic noninteracting Fermi surface) by calculating the self-energy, the spectral function, the scattering rate, and the effective mass within the leading-order dynamical self-energy approximation. Our theory is exact in the high-density limit. We find anisotropic features of quasiparticle properties that are not captured by the commonly used isotropic approximation where the anisotropic effective mass is replaced by the isotropic averaged density-of-states mass. Some of these interesting results are as follows: (1) The many-body renormalization of the quasiparticle spectrum becomes highly anisotropic as the quasiparticle energy increases away from the Fermi energy; (2) the interaction-induced inelastic-scattering rate features a strong anisotropy, exhibiting an abrupt jump at different injected energies depending on the momentum direction of the injected electron; (3) the effective-mass enhancement is larger (smaller) for the light (heavy) mass, showing that the anisotropy is reduced by interactions. Our results and analysis show that the unjustified neglect of the mass anisotropy can lead to an incorrect description of quasiparticle properties of the anisotropic system and inaccurate estimates of physical quantities of interest although the use of an equivalent isotropic approximation using the density-of-states effective mass as is commonly and uncritically performed in the literature, works as a reasonable approximation in many situations. In addition to the complete random phase approximation theory for the anisotropic quasiparticles, we also provide a theory using the simpler plasmon-pole approximation, commenting on its validity for anisotropic self-energy calculations. We comment also on the interaction effect on the Fermi-surface topology, finding that the elliptic shape of the bare Fermi surface is preserved with suppressed ellipticity in the interacting system to a high degree of accuracy except in the very strongly interacting limit (and for very high bare mass anisotropy). Our theory provides a complete generalization of the existing isotropic many-body theory of interacting electrons to the corresponding anisotropic systems.
Linearizing the appropriate kinetic equation we derive general response functions including self-consistent mean fields or density functionals and collisional dissipative contributions. The latter ones are considered in relaxation time approximation conserving successively different balance equations. The effect of collisions is represented by correlation functions which are possible to calculate with the help of the finite temperature Lindhard random-phase approximation expression. The presented results are applicable to the finite temperature response of interacting quantum systems if the quasiparticle or mean-field energy is parametrized within Skyrme-type functionals including density, current, and energy dependencies which can be considered alternatively as density functionals. In this way we allow to share correlations between density functional and collisional dissipative contributions appropriate for special treatment. We present results for collective modes such as the plasmon in plasma systems and the giant resonance in nuclei. The collisions lead in general to an enhanced damping of collective modes. If the collision frequency is close to the frequency of the collective mode, resonance occurs and the collective mode is enhanced showing a collisional narrowing.
In this thesis we investigate the physics of two strongly correlated electron systems. The part I of this document is dedicated to the study of the real-time dynamics of the Kondo effect produced by a quench protocol. For it, we consider one or two metallic chains of interacting electrons (Hubbard chains) coupled to a magnetic impurity of spin 1/2. The quench protocol consists of assuming that, for times τ < 0, the chains are decoupled from the impurity, at τ = 0 the coupling is suddenly turned on and, for τ > 0 , the system evolves unitarily in time. We determine an analytical expression for the relaxation of the impurity magnetization in the time regime τ ≪ τK , where τK is the Kondo time. This result was confirmed by numerical calculations performed in our group. Based on these numerical results, we observe that the increase in the interaction U between the electrons in the chains leads to a faster decay of the impurity magnetization, showing that the electron-electron interaction favors the formation of the Kondo effect. Further we studied the time-dependent transport through the impurity, from which we determine, in the interacting case, an analytical expression for the time scale equivalent to the Kondo time. According to the expression we obtained, we conclude that the Kondo scale is reduced when U increases, in agreement with the numerical results mentioned above. In part II of this thesis we investigate the physics of superconducting cuprates, motivated by electronic Raman spectroscopy measurements of the charge density wave (CDW) phase observed in these materials. Through a phenomenological approach, we describe a charge order in the CuO2 plane. Analyzing the spectral properties of the system, we establish that the CDW phase should be more easily detectable in the ω > 0 (above the Fermi level) region of the local density of states. Our results for the B2g Raman response show a general dip-hump behavior, as actually seen in the experiments, which is independent of the system band structure. In contrast, the B1g Raman response is strongly band structure dependent. In the second part of the work, a more general model was proposed, based on the Yang-Rice-Zhang ansatz, which also takes into account the pseudogap (PG) phase. We conclude that the PG phase drives the doping dependence of the CDW energy scale, in agreement with the experiments. This result is suggestive of a PG phase governing the different energy scales of the exotic phases that appear in the underdoped region of the cuprate phase diagram.
We solve for the electronic Raman scattering response functions on an infinite-dimensional hypercubic lattice employing dynamical mean-field theory. This contribution extends previous work on the nonresonant response to include the mixed and resonant contributions. We focus our attention on the infinite-dimensional spinless Falicov-Kimball model, where the problem can be solved exactly, and the system can be tuned to go through a Mott-Hubbard-like metal-insulator transition. Resonant effects vary in different scattering geometries, corresponding to the symmetries of the charge excitations scattered by the light. We do find that the Raman response is large near the double resonance, where the transfered frequency is close to the incident photon frequency. We also find a joint resonance of both the charge-transfer peak and the low-energy peak when the incident photon frequency is on the order of the interaction strength. In general, the resonance effects can create order of magnitude (or more) enhancements of features in the nonresonant response, especially when the incident photon frequency is somewhat larger than the frequency of the nonresonant feature. Finally, we find that the resonant effects also exhibit isosbestic behavior, even in the A1g{A}_{1\mathrm{g}}A1g​ and B2g{B}_{2\mathrm{g}}B2g​ sectors, and it is most prominent when the incident photon frequency is on the order of the interaction energy.
We calculate the nonresonant Raman response, the single particle spectra and the charge-spin configuration for the electron-doped $t\text{\penalty1000-\hskip0pt}{t}^{\ensuremath{'}}$ Hubbard model using unrestricted Hartree-Fock calculations. We discuss the similarities and differences in the response of homogeneous versus inhomogeneous structures. Metallic antiferromagnetism dominates in a large region of the $U\text{\penalty1000-\hskip0pt}n$ phase diagram but at high values of the on-site interaction and for intermediate doping values inhomogeneous configurations are found with lower energy. This result is in contrast with the case of hole-doped cuprates where inhomogeneities are found already at very low doping. The inhomogeneities found are in-phase stripes compatible with inelastic neutron scattering experiments. They give an incoherent background in the Raman response. The B2g{B}_{2g}B2g​ signal can show a quasiparticle component even when no Fermi surface is found in the nodal direction.
For years, theories for Raman scattering have been confined to either the insulating or fully metallic state.While much can be learned by focusing attention on the metal or insulator, recent experimental work on the cuprate systems points to the desirability of formulating a theory for Raman response which takes one through a quantum critical point -the metalinsulator transition.Using the Falicov-Kimball model as a canonical model of a MIT, we employ dynamical mean-field theory to construct an exact theory for non-resonant Raman scattering.In particular we examine the formation of charge transfer peaks and pseudogaps as well as the low-energy dynamics.The results are qualitatively compared to the experimental B 1g Raman spectra in the cuprates, which probes the hot quasiparticles along the Brillouin zone axes.The results shed important information on normal state electronic transport and the pseudo-gap in the cuprates.