Electron-boson Spectral Density Research Articles

Planckian behavior of cuprates at the pseudogap critical point simulated via flat electron-boson spectral density

Planckian behavior has been recently observed in La1·76Sr0·24CuO4 at the pseudogap critical point. The Planckian behavior takes place in an intriguing quantum metallic state at a quantum critical point. Here, the Planckian behavior was simulated with an energy-independent (or flat) and weakly temperature-dependent electron-boson spectral density (EBSD) function by using a generalized Allen's (Shulga's) formula. We obtained various optical quantities from the flat EBSD function, such as the optical scattering rate, the optical effective mass, and the optical conductivity. These quantities are well fitted with the recently observed Planckian behavior. Fermi-liquid behavior was also simulated with an energy-linear and temperature-independent EBSD function. The EBSD functions agree well with the overall doping- and temperature-dependent trends of the EBSD function obtained from the optically measured spectra of cuprate systems, which can be crucial for understanding the microscopic electron-pairing mechanism for high-Tc superconductivity in cuprates.

Fermi liquid-like behaviour of cuprates in the pseudogap phase simulated via T-dependent electron-boson spectral density

We investigated the temperature- and frequency-dependent optical scattering rates in the pseudogap phase of cuprates using model pseudogap and electron-boson spectral density (EBSD) functions. We obtained the scattering rates at various temperatures below and above a given pseudogap temperature using a generalized Allen’s (or Sharapov’s) formula, which has been used to analyse the measured optical spectra of correlated electron systems with a non-constant density of states at finite temperatures. The pseudogap and EBSD functions should be temperature dependent to simulate the Fermi liquid-like behaviour of underdoped cuprate systems observed in optical studies. Therefore, the observed Fermi liquid-like behaviour can be understood by considering the combined contribution from the T-dependent EBSD function and the T-dependent pseudogap. We also obtained the optical conductivity spectra from the optical scattering rates and analyzed them to investigate intriguing electronic properties. We expect that our results will aid in understanding the Fermi liquid-like optical response in the pseudogap phase and in revealing the microscopic pairing mechanism for superconductivity in cuprates.

Doping-dependent superconducting physical quantities of K-doped BaFe_2As_2 obtained through infrared spectroscopy

We investigated four single crystals of K-doped BaFe_2As_2 (Ba-122), Ba_{1-x}K_xFe_2As_2 with x = 0.29, 0.36, 0.40, and 0.51, using infrared spectroscopy. We explored a wide variety of doping levels, from under- to overdoped. We obtained the superfluid plasma frequencies (Omega _{textrm{sp}}) and corresponding London penetration depths (uplambda _{textrm{L}}) from the measured optical conductivity spectra. We also extracted the electron-boson spectral density (EBSD) functions using a two-parallel charge transport channel approach in the superconducting (SC) state. From the extracted EBSD functions, the maximum SC transition temperatures (T_c^{textrm{Max}}) were determined using a generalized McMillan formula and the SC coherence lengths (xi _{textrm{SC}}) were calculated using the timescales encoded in the EBSD functions and reported Fermi velocities. We identified some similarities and differences in the doping-dependent SC quantities between the K-doped Ba-122 and the hole-doped cuprates. We expect that the various SC quantities obtained across the wide doping range will provide helpful information for establishing the microscopic pairing mechanism in Fe-pnictide superconductors.

Correlation effects obtained from optical spectra of Fe-pnictides using an extended Drude-Lorentz model analysis

We introduce an analysis model, an extended Drude–Lorentz model, and apply it to Fe-pnictide systems to extract their electron–boson spectral density functions (or correlation spectra). The extended Drude–Lorentz model consists of an extended Drude mode for describing correlated charge carriers and Lorentz modes for interband transitions. The extended Drude mode can be obtained by a reverse process starting from the electron–boson spectral density function and extending to the optical self-energy and, eventually, to the optical conductivity. Using the extended Drude–Lorentz model, we obtained the electron–boson spectral density functions of K-doped BaFe2As2 (Ba-122) at four different doping levels. We discuss the doping-dependent properties of the electron–boson spectral density function of K-doped Ba-122. We also can include pseudogap effects in the model using this approach. Therefore, this approach is very helpful for understanding and analyzing measured optical spectra of strongly correlated electron systems, including high-temperature superconductors (cuprates and Fe-pnictides).

Electron-boson spectral density functions of cuprates obtained from optical spectra via machine learning

The electron-boson spectral density (or glue) function can be obtained from measured optical scattering rate by solving a generalized Allen formula, which relates the two quantities with an integral equation and is an inversion problem. Thus far, numerical approaches, such as the maximum entropy method (MEM) and the least squares fitting method, have been applied for solving the generalized Allen formula. Here, we developed a new method to obtain the glue functions from the optical scattering rate using a machine learning approach (MLA). We found that the MLA is more robust against random noise compared with the MEM. We applied the new developed MLA to experimentally measured optical scattering rates and obtained reliable glue functions in terms of their shapes including the amplitudes. We expect that the MLA can be a useful and rapid method for solving other inversion problems, which may contain random noise.

Extended Drude Model Analysis of the Optical Spectra of Correlated Electron Systems in a d-Wave Superconducting State

Correlation information in strongly correlated electron systems can be obtained using an extended Drude model. An interesting method related to the extended Drude model analysis of superconducting optical data was proposed recently, and it has attracted attention from researchers. This method aims to extract the optical self-energy of quasiparticles (or residual unpaired electrons) from measured optical data in the superconducting state. However, this residual optical self-energy is a partial optical self-energy. The interpretation and significance of this partial optical self-energy is unclear. We investigate this method using a reverse process with simple electron-boson spectral density functions. With our obtained results, we conclude that the residual (or partial) optical self-energy is difficult to interpret because it contains unphysical features, in particular, a negative optical effective mass. The present study clarifies the extended Drude analysis method for superconducting optical data.

Analysis of optical data using extended Drude model and generalized Allen’s formulas

Extended Drude model formalism has been successfully utilized for analyzing optical spectra of strongly correlated electron systems including heavy-fermion systems and high-Tc superconducting iron pnictides and cuprates. Furthermore, generalized Allen’s formulas have been developed and applied to extract the electron-boson spectral density function from measured optical data of high temperature superconductors including cuprates in various material phases. Here we used a reverse process to obtain various optical quantities starting from two typical electron-boson spectral density model functions for three intriguing (normal, pseudogap, and d-wave superconducting) material phases in cuprates. We also assigned the calculated optical results to designated regions in the phase diagram of hole-doped cuprates and compared them with the corresponding measured optical spectra of Bi2Sr2CaCu2 (Bi-2212). This comparison suggested that this way of optical data analysis can be a convincing method to study correlated electrons in the copper oxide superconductors and other superconducting systems as well.

Optical investigation of BaFe2(As0.77P0.23)2 : Spin-fluctuation-mediated superconductivity under pressure

Temperature-dependent reflectivity measurements in the frequency range 75--8000 ${\mathrm{cm}}^{\ensuremath{-}1}$ were performed on ${\mathrm{BaFe}}_{2}{({\mathrm{As}}_{0.77}{\mathrm{P}}_{0.23})}_{2}$ single crystals under pressure up to $\ensuremath{\sim}5\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$. The obtained optical conductivity spectra have been analyzed to extract the electron-boson spectral density ${\ensuremath{\alpha}}^{2}F(\mathrm{\ensuremath{\Omega}})$. A sharp resonance peak was observed in ${\ensuremath{\alpha}}^{2}F(\mathrm{\ensuremath{\Omega}})$ upon the superconducting transition, persisting throughout the applied pressure range. The energy and temperature dependences of this peak are consistent with the superconducting gap opening. Furthermore, several similarities with other experimental probes such as inelastic neutron scattering (INS) [D. S. Inosov et al., Nat. Lett. 6, 178 (2010)] give evidence for the coupling to a bosonic mode, possibly due to spin fluctuations. Moreover, electronic correlations have been calculated via spectral weight analysis, which revealed that the system stays in the strongly correlated regime throughout the applied pressure range. However, a comparison to the parent compound showed that the electronic correlations are slightly decreased with P doping. The investigation of the phase diagram obtained by our optical study under pressure also revealed the coexistence of the spin-density wave and the superconducting regions, where the coexistence region shifts to the lower pressure range with increasing P content. Moreover, the optimum pressure range, where the highest superconducting transition temperature has been obtained, shows a nonlinear decrease with increasing P content.

Anomalous behavior of coupling constant near the magnetic phase transition of Ni-doped Ba-122 pnictides

We investigated Ni-doped Ba-122 pnictide single crystals at two different doping levels (BaFe2−xNixAs2; x = 0.05 (underdoped) and 0.10 (optimally doped)) using optical spectroscopy. We analyze measured optical data with an extended Drude model and a generalized Allen’s formalism. We extracted the electron-boson spectral density function using an approximate single-band approach from measured optical scattering rate. We observed that an anomalous behavior in the coupling constant obtained from the electron-boson spectral density function takes place near the magnetic transition temperature of the underdoped sample. This anomaly has not been observed in cuprate system. Nevertheless, overall temperature and doping dependent properties of the electron-boson spectral density of pnictides are similar to those of cuprates. This may indicate that these two different high-temperature superconducting systems share a same mechanism for their superconductivity.

Intrinsic temperature-dependent evolutions in the electron-boson spectral density obtained from optical data.

We investigate temperature smearing effects on the electron-boson spectral density function (I2χ(ω)) obtained from optical data using a maximum entropy inversion method. We start with two simple model input I2χ(ω), calculate the optical scattering rates at selected temperatures using the model input spectral density functions and a generalized Allen’s formula, then extract back I2χ(ω) at each temperature from the calculated optical scattering rate using the maximum entropy method (MEM) which has been used for analysis of optical data of high-temperature superconductors including cuprates, and finally compare the resulting I2χ(ω) with the input ones. From this approach we find that the inversion process can recover the input I2χ(ω) almost perfectly when the quality of fits is good enough and also temperature smearing (or thermal broadening) effects appear in the I2χ(ω) when the quality of fits is not good enough. We found that the coupling constant and the logarithmically averaged frequency are robust to the temperature smearing effects and/or the quality of fits. We use these robust properties of the two quantities as criterions to check whether experimental data have intrinsic temperature-dependent evolutions or not. We carefully apply the MEM to two material systems (one optimally doped and the other underdoped cuprates) and conclude that the I2χ(ω) extracted from the optical data contain intrinsic temperature-dependent evolutions.

Electron–boson spectral density function of correlated multiband systems obtained from optical data: Ba0.6K0.4Fe2As2 and LiFeAs

We introduce an approximate method which can be used to simulate the optical conductivity data of correlated multiband systems for normal and superconducting cases by taking advantage of a reversed process in comparison to a usual optical data analysis, which has been used to extract the electron–boson spectral density function from measured optical spectra of single-band systems, like cuprates. We applied this method to optical conductivity data of two multiband pnictide systems (Ba0.6K0.4Fe2As2 and LiFeAs) and obtained the electron–boson spectral density functions. The obtained electron–boson spectral density consists of a sharp mode and a broad background. The obtained spectral density functions of the multiband systems show similar properties as those of cuprates in several aspects. We expect that our method helps to reveal the nature of strong correlations in the multiband pnictide superconductors.

Extended Drude model analysis of n-doped cuprate, Pr0.85LaCe0.15CuO4

We investigated optical properties of an electron-doped copper oxide high temperature superconductor, Pr0.85LaCe0.15CuO4 (PLCCO) single crystal. We obtained the optical conductivity from measured reflectance at various temperatures. We found our data contained c-axis longitudinal optical (LO) phonon modes due to miscut and intrinsic lattice distortion. We applied an extended Drude model to study the correlations between charge carriers in the system. The LO phonons appear as strong sharp peaks in the optical scattering rate. We tried to remove the LO phonon modes by using the energy loss function, which also shows the LO phonons as peaks, and could not remove them completely. We extracted the electron-boson spectral density function using a generalized Allen's formula. We observed that the resulting electron-boson density show similar temperature dependence as hole-doped cuprates.

Reverse process of usual optical analysis of boson-exchange superconductors: impurity effects on s- and d-wave superconductors

We performed a reverse process of the usual optical data analysis of boson-exchange superconductors. We calculated the optical self-energy from two (MMP and MMP+peak) input model electron-boson spectral density functions using Allen's formula for one normal and two (s- and d-wave) superconducting cases. We obtained the optical constants including the optical conductivity and the dynamic dielectric function from the optical self-energy using an extended Drude model, and finally calculated the reflectance spectrum. Furthermore, to investigate impurity effects on optical quantities we added various levels of impurities (from the clean to the dirty limit) in the optical self-energy and performed the same reverse process to obtain the optical conductivity, the dielectric function, and reflectance. From these optical constants obtained from the reverse process we extracted the impurity-dependent superfluid densities for two superconducting cases using two independent methods (the Ferrel–Glover–Tinkham sum rule and the extrapolation to zero frequency of −ϵ1(ω)ω2); we found that a certain level of impurities is necessary to get a good agreement on results obtained by the two methods. We observed that impurities give similar effects on various optical constants of s- and d-wave superconductors; the greater the impurities the more distinct the gap feature and the lower the superfluid density. However, the s-wave superconductor gives the superconducting gap feature more clearly than the d-wave superconductor because in the d-wave superconductors the optical quantities are averaged over the anisotropic Fermi surface. Our results supply helpful information to see how characteristic features of the electron-boson spectral function and the s- and d-wave superconducting gaps appear in various optical constants including raw reflectance spectrum. Our study may help with a thorough understanding of the usual optical analysis process. Further systematic study of experimental data collected at various conditions using the optical analysis process will help to reveal the origin of the mediated boson in the boson-exchange superconductors.

Electron-boson spectral density of LiFeAs obtained from optical data

We analyze existing optical data in the superconducting state of LiFeAs at T = 4 K, to recover its electron-boson spectral density. A maximum entropy technique is employed to extract the spectral density I2χ(ω) from the optical scattering rate. Care is taken to properly account for elastic impurity scattering which can importantly affect the optics in an s-wave superconductor, but does not eliminate the boson structure. We find a robust peak in I2χ(ω) centered about ΩR ≅ 8.0 meV or 5.3 kBTc (with Tc = 17.6 K). Its position in energy agrees well with a similar structure seen in scanning tunneling spectroscopy (STS). There is also a peak in the inelastic neutron scattering (INS) data at this same energy. This peak is found to persist in the normal state at T = 23 K. There is evidence that the superconducting gap is anisotropic as was also found in low temperature angular resolved photoemission (ARPES) data.

High-energy fluctuation spectra in cuprates from infrared optical spectroscopy

Coupling of the charge carriers in the high-temperature superconducting oxides to bosonic modes has been widely reported using a variety of experimental probes. These include angular resolved photoemission (ARPES), scanning tunneling spectroscopy, Raman scattering, and infrared optical spectroscopy. The energy scale investigated has been mostly limited to a relatively small range up to 300 meV or so. Although some ARPES experiments report boson structure up to 800 meV in the dressed electron dispersion curves, the data are not analyzed to recover the spectral density of the fluctuation spectrum. We have extended to higher energies up to 2.2 eV the usual maximum entropy technique used to invert optical data so as to obtain an electron-boson spectral density. This has required that we include in our inversions, the calculated (local-density approximation) particle-hole symmetrized energy-dependent electronic density of states. Our analysis reveals that significant spectral weight remains in the fluctuation spectra up to 2.2 eV in the Bi-2212 family and 1.2 eV in Bi-2201 for all doping levels considered.

Evolution of electron–boson spectral density in the underdoped region of Bi2Sr2−xLaxCuO6

We use a maximum entropy technique to obtain the electron–boson spectral density from optical scattering rate data across the underdoped region of the Bi2Sr2−xLaxCuO6 (Bi-2201) phase diagram. Our method involves a generalization of previous work which explicitly includes finite temperature and the opening of a pseudogap which modifies the electronic structure. We find that the mass enhancement factor λ associated with the electron–boson spectral density increases monotonically with reduced doping and closer proximity to the Mott antiferromagnetic insulating state. This observation is consistent with increased coupling to the spin fluctuations. At the same time the system has reduced metallicity because of increased pseudogap effects which we model with a reduced effective density of states around the Fermi energy with the range of the modifications in energy set by the pseudogap scale.

Determination of boson spectrum from optical data in pseudogap phase of underdoped cuprates

Information on the nature of the dominant inelastic processes operative in correlated metallic systems can be obtained from an analysis of their AC optical response. An electron-boson spectral density can usefully be extracted. This density is closely related to the optical scattering rate. However, in the underdoped region of the high Tc cuprate phase diagram a new energy scale (the pseudogap) emerges, which alters the optical scattering and needs to be taken into account in any fit to data. This can influence the shape and strength of the recovered boson spectral function. Including a pseudogap in an extended maximum entropy inversion for optimally doped Bi-2212 is more consistent with existing data than when it is left out as done previously.

Electron-boson spectral density function of underdopedBi2Sr2CaCu2O8+δandYBa2Cu3O6.50

We investigate the electron-boson spectral density function, $I^2\chi(\omega,T)$, of CuO$_2$ plane in underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi-2212) and underdoped YBa$_2$Cu$_3$O$_{6.50}$ (Y-123) using the Eliashberg formalism. We apply a new (in-plane) pseudogap model to extract the electron-boson spectral function. For extracting the spectral function we assume that the spectral density function consists of two components: a sharp mode and the broad Millis-Monien-Pines (MMP) mode. We observe that both the resulting spectral density function and the intensity of the pseudogap show strong temperature dependences: the sharp mode takes most spectral weight of the function and the peak position of the sharp mode shifts to lower frequency and the depth of pseudogap, $1-\tilde{N}(0,T)$, is getting deeper as temperature decreases. We observe also that the total spectral weight of the electron-boson density and the mass enhancement coefficient increase as temperature decreases. We estimate fictitious (maximum) superconducting transition temperatures, $T_c(T)$, from the extracted spectral functions at various temperatures using a generalized McMillan formula. The estimated (maximum) $T_c$ also shows a strong temperature dependence; it is higher than the actual $T_c$ at all measured temperatures and decreases with temperature lowering. Since as lowering temperature the pseudogap is getting stronger and the maximum $T_c$ is getting lower we propose that the pseudogap may suppress the superconductivity in cuprates.

Eliashberg analysis of optical spectra reveals a strong coupling of charge carriers to spin fluctuations in doped iron-pnictideBaFe2As2superconductors

The temperature and frequency dependences of the optical conductivity of Co and Ni-doped BaFe2As2 are analyzed and the electron-boson spectral density a2F(w) extracted using Eliashberg's formalism. The characteristic energy of a large peak in the spectrum around 10 meV coincide with the resonance peak in the spin excitation spectra, giving compelling evidence that in iron-based superconductors spin fluctuations strongly couple to the charge carriers and mediate superconductivity. In addition the spectrum is found to evolve with temperature towards a less structured background at higher energies as in the spin susceptibility.

Magnetic resonance in the model high-temperature superconductorHgBa2CuO4+δ

Inelastic neutron-scattering measurements of single-${\text{CuO}}_{2}$-layer ${\text{HgBa}}_{2}{\text{CuO}}_{4+\ensuremath{\delta}}$ reveal an antiferromagnetic resonance with energy ${\ensuremath{\omega}}_{\text{r}}=56\text{ }\text{meV}$ $(\ensuremath{\approx}6.8{k}_{\text{B}}{T}_{\text{c}})$ below the superconducting transition temperature ${T}_{\text{c}}\ensuremath{\approx}96\text{ }\text{K}$. The resonance is energy-resolution limited and exhibits an intrinsic momentum width of about $0.2\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$, consistent with prior work on several other cuprates. The rather large value of ${\ensuremath{\omega}}_{\text{r}}$ is identical to the characteristic energy of the electron-boson spectral density obtained from recent optical conductivity work, consistent with the notion that the charge carriers are strongly coupled to magnetic fluctuations.