Phonon Self-energy Research Articles

Phonon self-energy in a d-wave superconductor with retardation, finite temperature, and impurities.

We give a strong-coupling real-frequency-axis formulation of the phonon-self-energy change on entering the superconducting state for a d-wave superconductor. The effect of impurity scattering in the Born (weak scattering) and unitary (T-matrix approach) limits is considered and numerical results presented for phonon shifts and changes in width.

Spin-phonon renormalization of the excitation energy in a dilute two-dimensional antiferromagnet.

The contribution of spin-lattice coupling to the phonon self-energy is calculated in a dilute two-dimensional antiferromagnet. The calculations are based on a bond percolation model of the CuO2 plane of high-T(c) copper oxides recently proposed by some of the authors. Numerical results are presented for the dampings of phonon excitations relating to three different situations. They are two magnons (Case A), one magnon and one fracton (Case B), and two fractons (Case C) involved in the phonon self-energy process. At low temperatures, the damping rate is much smaller than the frequency of excitations, leading to well-defined phonons for all the cases. The overall features of the GAMMA(q) approximately q curves possess similar characteristics: GAMMA(q) increases linearly with q in the small-q region, reaching a maximum at an intermediate q, and after that scales nearly as q-2 until q approaches the edge of the Brillouin zone. The temperature-dependent phonon linewidths arising from spin-lattice coupling are also presented. We point out the sensitivity of the interaction involving fractons on bond concentration of the network. Relevance of the calculated results to experimental data are discussed.

Midinfrared conductivity in orientationally disordered doped fullerides

The coupling between the intramolecular vibrational modes and the doped conduction electrons in ${\mathit{M}}_{3}$${\mathrm{C}}_{60}$ is studied by a calculation of the electronic contributions to the phonon self-energies. The calculations are carried out for an orientationally ordered reference solid with symmetry Fm3\ifmmode\bar\else\textasciimacron\fi{}m and for a model with quenched orientational disorder on the fullerene sites. In both cases, the dispersion and symmetry of the renormalized modes is governed by the electronic contributions. The current-current correlation functions and frequency-dependent conductivity through the midinfrared are calculated for both models. In the disordered structures, the renormalized modes derived from even-parity intramolecular phonons are resonant with the dipole excited single-particle spectrum, and modulate the predicted midinfrared conductivity. The spectra for this coupled system are calculated for several recently proposed microscopic models for the electron-phonon coupling, and a comparison is made with recent experimental data which demonstrate this effect.

Raman-active phonons and mode softening in superconducting HgBa2CuO4+ delta.

Polarized micro-Raman spectra of a single crystallite of ${\mathrm{HgBa}}_{2}$${\mathrm{CuO}}_{4+\mathrm{\ensuremath{\delta}}}$ have been measured and used to assign phonon modes of ${\mathit{A}}_{1\mathit{g}}$ and ${\mathit{E}}_{\mathit{g}}$ symmetry intrinsic to the superconducting phase. Predictions of the peak positions, linewidths, and Raman cross sections from compounds with similar bonding, i.e., HgO, Y-124, and Y-123, are closely matched and suggest assignments of the high-frequency ${\mathit{A}}_{1\mathit{g}}$ modes at 592 and 568 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ to apical oxygen stretch vibrations of the ideal structure and of a Cu or O defect (in the Hg plane). The low-frequency 168, 161, and 75 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ modes are assigned to O and Ba vibrations of ${\mathit{E}}_{\mathit{g}}$,${\mathit{A}}_{1\mathit{g}}$ and ${\mathit{E}}_{\mathit{g}}$ symmetry, respectively. An anomaly in the phonon frequency and linewidth of the 568-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ ${\mathit{A}}_{1\mathit{g}}$ mode is observed below ${\mathit{T}}_{\mathit{c}}$. If interpreted as a change in the phonon self-energy due to the onset of superconductivity, this behavior is consistent with a gap energy near 568 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, i.e., 8.7${\mathit{kT}}_{\mathit{c}}$ in ${\mathrm{HgBa}}_{2}$${\mathrm{CuO}}_{4+\mathrm{\ensuremath{\delta}}}$ and constitutes a first estimate of the superconducting gap in this material. The Raman tensor elements for the assigned ${\mathit{A}}_{1\mathit{g}}$ and ${\mathit{E}}_{\mathit{g}}$ modes are given in absolute units.

Raman scattering in a correlated superconductor II. A finite temperature study

The temperature dependence of the Raman spectrum of copper oxide superconductors is calculated assuming that the normal state of these systems is a correlated metal and that the superconductivity arises due to the correlation bag mechanism proposed earlier (Debanand Sa, S.N. Behera (1992), here after referred to as I). The correlated metallic state is characterized by the presence of a strong interaction between the doped charge carriers and the quanta of fluctuations of the pre-existing resonating valence bonds in the normal state. It is shown that the presence of such an interaction can give rise to the marginal Fermi liquid behaviour, which follows from the pair polarisability function. The model therefore, successfully explains the constant intensity background observed in the Raman spectrum due to the scattering by the charge carriers. In order to understand the observed temperature dependent shift and change in width of the phonons, their spectral density functions are calculated. Two different kinds of interaction between the charge carriers and phonons were derived in I, i.e. (i) the usual electron-phonon interaction where a phonon decays by exciting an electron-hole pair and (ii) the other where the decay process involves a quasi-particle pair. It was postulated that the localised phonons couple to the charge carrier by the later process whereas the propagating ones interact by the former. The temperature dependence of the phonon self-energies due to the two processes is calculated and the spectral density functions are computed. The calculated temperature dependent shift and change in width of the phonons show qualitative agreement with the observed result.

The moments method and damped systems

The moments method was shown to be very useful for determining the linear response of very large harmonic systems. In the presence of anharmonicity or electron-phonon coupling, the phonon self-energy appears in the one-phonon propagator. The phonon frequency is then renormalized and the system is damped. In this paper, we show that it is possible to extend the moments method to compute the response of these damped systems. The method involves introducing a two-dimensional density functional that can be calculated by the moments techniques and which allows the computation of the response function. We illustrate these results via applications to several models: systems with constant damping, systems of coupled relaxation modes and oscillators, and systems with electron-phonon coupling.

Temperature dependence of phonon intensities in tantalum below 4.2 K.

An unexpected temperature dependence of the phonon peak intensity in tantalum, as measured by neutron scattering, is observed between 1.2 and 6 K, that is, in a temperature range where the phonon self-energy is dominated by the electron-phonon interaction. A possible relation between the observed behavior of the phonon intensity and the results of a tunneling experiment is presented. Possible sources of the anomalous behavior presently observed are also considered.

Superconducting gap in Pr x Y1?x Ba2Cu4O8 and YBa2?y Sr y Cu4O8 probed by infrared phonon self-energies

We report a reflectivity study of thez-polarized TO-phonons of Pr x Y1−xBa2Cu4O8 and YBa2−ySr y Cu4O8 alloys in the temperature range 10–300 K. Anomalies of the frequency and linewidth of the plane-oxygen vibration atω∼300 cm−1 due to the opening of the superconducting gap are found to occur upon crossing the superconducting transition temperatureT c . Phonon self-energy effects are strongly dependent onT c , providing evidence for a relative shift of the gap with respect to the energy of phonon.

D-wave superconductivity and Raman shifts

The q → = 0 phonon self-energy is calculated and related to changes in Raman phonon widths and shifts upons entering the superconducting state. Results are presented for different order parameter symmetry: s-wave, planar d-wave and an anisotropic order parameter pertinent to a model of a layered superconductor. Comparison is made with results of Raman and infrared experiments on RBa 2Cu 3O 7 compounds.

Self-consistent calculation of the phonon self-energy and thermal conductivity for disordered solids.

We present a self-consistent calculation of the phonon self-energy as a function of frequency for disordered solids by using a Green's-function approach. The disordered solids consist of two species of atoms (the host lattice and defect atoms), which are arranged randomly in the three-dimensonal lattice sites. The present calculation contains the contributions to second order of the defect density. The previous calculation gives exactly the phonon self-energy to first order of the defect density. We show that for small defect density the present calculation only modifies the previous calculation slightly, but for large defect density the present calculation differs appreciably from the previous calculation. Our numerical calculations are done in the Debye model. We calculate the thermal conductivity as a function of temperature. We find that the present calculation is able to explain the observed ${\mathit{T}}^{2}$ behavior of the thermal conductivity for disordered solids at low temperatures.

Phonon-self-energy effects due to an electronic mechanism in high-Tcsuperconductors

The phonon self-energy at q=0 and zero temperature is calculated for a conventional electron-phonon superconductor, Pb, and for an equivalent high-${\mathit{T}}_{\mathit{c}}$ superconductor, assuming an electronic mechanism for the superconductivity (to be specific and as a convenient illustrative example, we use the marginal-Fermi-liquid phenomenology). Comparison between the results demonstrates that for an s-wave strong-coupling superconductor, phonon shifts and widths exhibit the same behavior regardless of the mechanism for superconductivity. In addition, the marginal-Fermi-liquid model shows no unusual behavior at zero temperature in the phonon self-energy, with the exception of a possible narrowing of the phonon width below 2${\mathrm{\ensuremath{\Delta}}}_{0}$ in the clean limit. As an aside, we also present renormalized BCS formulas that agree well with the full numerical strong-coupling calculations and could be easily implemented for use in model fits.

Anharmonic effects and the two-particle continuum in the Raman spectra of YBa2Cu3O6.9, TlBa2CaCu2O7, and Tl2Ba2CaCu2O8.

Anharmonic interactions of the apex O vibrations in ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{6.9}$, ${\mathrm{TlBa}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{7}$, and ${\mathrm{Tl}}_{2}$${\mathrm{Ba}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{8}$ single crystals are investigated by Raman spectroscopy. The temperature dependence of the O phonon frequency and linewidth is fit using the results of a perturbation calculation of Raman scattering from anharmonic vibrations with very good results, provided both third- and fourth-order terms in the phonon self-energy are included. Temperature-dependent asymmetric Fano line shapes of the apex O modes, which are observed especially well in ${\mathrm{TlBa}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{7}$, suggest significant coupling of these modes to the background continuum. From the temperature dependence of the background intensity we deduce that the Raman scattering continuum in the region 100--1000 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ contains significant two-particle scattering. The zz polarization of the observed scattering suggests the background is due to two-phonon rather than two-magnon scattering. The Fano line shape and the temperature dependence of the asymmetry can then be attributed to anharmonic interactions.

Effect of d-wave energy-gap symmetry on Raman shifts.

We have calculated the shift and change in width, due to superconductivity, for a q=0 Raman-active phonon in a two-dimensional, tight-binding band structure with a d-wave pairing interaction. We find that additional structure appears in the phonon self-energy due to the van Hove singularity, which occurs in two dimensions. A more general model of a superconductor with nodes in the gap function is also examined and related to both the d-wave model and another model for layered superconductors. In contrast to the usual isotropic s-wave case where a phonon with frequency less than twice the gap value is softened and narrowed, we find softening with broadening in these models. Comparison of the anisotropic models is made with experiment.

Enhancement of self-energy effects of phonons with finite wave vectors due to Fermi-surface nesting.

The changes in phonon self-energy due to superconductivity are investigated for all wave vectors in the Brillouin zone, for two types of gap-function symmetry (s or d wave). We also study the phonon self-energy as a function of frequency, and for various electron concentrations. We adopt a simple tight-binding model that includes the important feature of a nested Fermi surface at half-filling. The changes in phonon self-energy arise from electron-phonon coupling. However, we make no assumption regarding the mechanism giving rise to superconductivity. We present possible signatures of an s- or d-wave gap function observable through inelastic-neutron-scattering experiments

Green's-function self-energies in the two-dimensional Holstein model.

We examine the behavior of the electron and phonon self-energies obtained from quantum Monte Carlo simulations of the two-dimensional Holstein model as a function of Matsubara frequency and momentum. For the half-filled system, there is a finite-temperature phase transition to an insulating charge-density-wave state, and the electron self-energy develops a distinctive divergence. The renormalized phonon frequency softens strongly with temperature at the commensurate wave vector. Away from half-filling, the electron self-energy behaves in a manner consistent with that of a Fermi liquid.

Phonon-Raman scattering in a correlated superconductor

The phonon spectral density function is calculated to explain the observed anomalies in the Raman spectrum of the cuprate superconductors. These being strongly correlated systems, their ground state is taken to be the resonating valence bond (RVB) state. In this state, the phonons are shown to couple predominantly to the RVB pair excitations besides the usual interaction with the charge carrier density. It is proposed that the superconductivity in the system appears due to the correlation bag mechanism where the collective excitations of the RVB state mediate an attractive interaction between the charge carriers which leads to pairing. The dependence of the transition temperature on the RVB gap parameter as well as the dopant concentration is demonstrated. The phonon self-energies, which involve the pair correlation function and the density response function, are evaluated at zero temperature for the normal state as well as the superconducting states. It is shown that the pair correlation function strongly renormalises the phonon spectrum. Its spectral density shows a broad peak corresponding to the RVB-amplitude mode, which qualitatively resembles the observed Raman spectrum in the superconducting state.

Phonon self-energy to the fourth order: An application to an anharmonic diatomic linear lattice.

Various contributions responsible for phonon-frequency shifts and widths to the fourth order in the anharmonic perturbation are reformulated and expressed in a computationally suitable form. Numerical calculations of these expressions have also been made for the case of a diatomic linear lattice. Some useful results relating to the expressions for self-energy and to the diatomic linear lattice are presented. The numerical results indicate the relative importance of various contributions. The importance of decay of phonons for width calculations accounting for various decay mechanisms is also pointed out.

Optical and vibrational properties of doped zinc selenide epitaxial layers

In this review article, we present photoluminescence and/or Raman scattering properties of ZnSe epitaxial layers which were doped with either nitrogen, oxygen, indium, or lithium. In particular, we present photoluminescence data due to recombination at discrete donor-acceptor pairs in ZnSe epitaxial layers implanted with nitrogen with a dose of 10 13 cm -2. The experimental pair data is explained with the help of a theoretical computation of a type-II spectrum involving In donors substituting for Zn and N acceptors for Se. We also report two-hole transitions involving up to 6S 3 2 states of the nitrogen acceptor. The experimental values of the energy positions of these excited states are in good agreement with those obtained using an effective-mass calculation. A sharp line superimposed on the broad donor-acceptor pair band, whose peak position has a constant separation from the excitation energy, is also observed. This separation is ≈0.9 meV larger than the 1S-2S energy spacing for the nitrogen acceptor. We show that this line could be either due to resonant inelastic scattering of the exciting photons by the acceptor impurities, or due to selective excitation of the discrete donor-acceptor pairs. We also present photoluminescence and excitation data to show that oxygen substituting isoelectronically in ZnSe gives rise to a pair of transitions A + (2.7895 eV) and B (2.7877 eV) as a result of the exchange interaction between the trapped electron-hole pair. The former is attributed to total angular momentum J= 1, Г 4 representation and is electric-dipole allowed while the latter is assigned to J=2 belonging to Г 3+Г 5 representation and is electric-dipole forbidden. Based on this model, we explain several experimental observations including varying PL intensity of the B line from sample to sample, relatively rapid disappearance of the B line as function of increasing temperature and strong longitudinal-optical (LO) couplings of A + and B lines. Raman spectroscopy was used to investigate the coupling mechanisms between LO-phonons and electronic excitations in n-type ZnSe layers. The layers, grown by molecular-beam epitaxy, were intentionally doped below the Mott criterion for the insulator-metal transition. The nature of the electron-phonon interaction is determined by the degree of electron localization, which was effectively changed by temperature and donor concentration. The LO-phonons couple to plasmons when electrons are thermally excited into the conduction band or at donor sites. In both cases, unbound phonons are observed. From the renormalized phonon frequencies at high temperature, values of free electron concentration as a function of temperature were established. They are in excellent agreement with Hall-effect determinations. At low temperatures, the phonon Raman profiles are asymmetric and show Fano-type line shapes. The electronic continuum responsible for the phonon self-energies at low temperatures was identified as Raman scattering by bound electrons. In addition, Raman spectroscopy was used to establish the presence of a free hole gas and its coupling to the longitudinal optical phonons in Li-doped ZnSe epilayers. The phonon spectra shift to higher frequencies and broaden with increasing acceptor concentration and temperature, as was the case for indium-doped layers, in accordance with the expectation for coupled phonon-plasmon modes. Values for the concentration and mobility of the holes were obtained from an analysis of the spectral lineshapes. They agree with those determined by other methods. A linear relationships was found between the spectral broadening and the hole concentration. In addition, electronic Raman scattering (ERS) from holes bound to Li acceptors was also studied in ZnSe epilayers and correlated with the net acceptor concentration determined by capacitance versus voltage measurements. The ERS spectra reveal several transitions between the ground 1s and the excited states of the Li acceptors as well as transitions to a continuum of delocalized valence band states. Values of excitation energies for the bound hydrogenic states and the ionization energy of the acceptors were measured. The strength of the ERS signal normalized to the phonon scattering depends linearly on net acceptor concentration.

Mean field analysis of phonon softening in high Tc superconductors

Different experiments indicate softening of the phonons at or below the superconducting transition. We analyze in mean field the phonon self-energy in the superconducting state and found that phonons do not get softened significantly for an s-wave coupled superconductor.

The influence of Au and Pr on the superconductivity-related gap in RBa 2Cu 3O 7−δ

The superconductivity-related self-energies of phonons in the high- T c superconductor RBa 2Cu 3O 7−δ (R = rare earth) have recently made possible the determination of a value of the superconducting gap. 1 The addition of impurities has the apparent effect of shifting this gap value to higher energies. We interpret these results in terms of a symmetrization of the gap at the Fermi surface by the impurities.