Quadratic Electron-phonon Coupling Research Articles

Renormalized phonon spectrum in the Su–Schrieffer–Heeger model

Motivated to understand phonon spectrum renormalization in the ground state of the half-filled Su–Schrieffer–Heeger model, we use the Born–Oppenheimer approximation together with the harmonic approximation to evaluate semi-analytically the all-to-all real-space ionic force constants generated through both linear and quadratic electron-phonon coupling. We then compute the renormalized phonon spectrum and the corresponding lattice zero-point energy (ZPE) as a function of the lattice dimerization. Crucially, the latter is included in the system’s total energy, and thus has a direct effect on the equilibrium dimerization. We find that inclusion of a small quadratic coupling leads to very significant changes in the predicted equilibrium dimerization, calling into question the use of the linear approximation for this model. We also argue that inclusion of the ZPE is key for systems with comparable lattice and electronic energies, and/or for finite size chains. Our method can be straightforwardly generalized to study similar problems in higher dimensions.

Temperature dependence of nitrogen-vacancy optical center in diamond

Diamond, a wide band gap semiconductor material, has been attracting interest in several fields from electrics and optics to biomedicine and quantum computing due to its outstanding properties. These properties of diamond are related to its unique lattice and optically active defect centers. In this paper, the dependence of nitrogen-vacancy (NV) center on measurement temperature is studied by using the low-temperature photoluminescence (PL) spectroscopy in a temperature range of 80–200 K. The results show that with the increase of the measurement temperature, the zero phonon lines of NV defects are red-shifted, its intensity decreases and its full width at half maximum increases. These results are attributed to the synergetic process of the lattice expansion and quadratic electron-phonon coupling. The NV<sup>—</sup> and NV<sup>0</sup> centers have similar values in the quenching activation energy and the thermal softening coefficient, resulting from their similar structures. The small differences may be associated with the electron-phonon coupling. The broadening mechanism of the NV centers is carefully distinguished by <inline-formula><tex-math id="Z-20200615115616-1">\begin{document}$T^3,\; T^5,\; T^7$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200395_Z-20200615115616-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200395_Z-20200615115616-1.png"/></alternatives></inline-formula> Voigt function fitting with the relation. These results show that the full width at half maximum of the Gaussian component of NV<sup>—</sup> and NV<sup>0</sup> centers are randomly distributed near 0.1 meV and 2.1 meV, respectively, while the full width at half maximum of the Lorentz component of NV<sup>—</sup> and NV<sup>0</sup> centers increase with measurement temperature increasing. The full width at half maximum of Lorentz of NV<sup>—</sup> and NV<sup>0</sup> centers conform to the <inline-formula><tex-math id="Z-20200615115631-1">\begin{document}$ T^3 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200395_Z-20200615115631-1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20200395_Z-20200615115631-1.png"/></alternatives></inline-formula> relationship. It can be proved that under the action of the fluctuating field, the zero phonon lines of the NV defects exhibit an obvious homogeneous widening mechanism.

Influence of the growth temperature on the Si-V photoluminescence in diamond thin films

The influence of growth temperature (350 ÷ 1100 °C) on the intensity of Si-V colour centres photoluminescence was studied in diamond thin films. The films were grown by a microwave plasma enhanced chemical vapour deposition system. The film quality and surface morphology were characterised by Raman spectroscopy and scanning electron microscopy, respectively. For selected samples, the temperature behaviour of steady-state photoluminescence emission spectra was studied within the range 11 ÷ 300 K as well. The photoluminescence properties are related to the film growth temperature. We found that 800 °C is the optimal growth temperature, at which the highest intensity of the Si-V centre photoluminescence was observed. For all the samples, the blue shift in the position of the Si-V centre photoluminescence zero-phonon line is observed with decreasing temperature, which is attributed to the effects of lattice contraction and quadratic electron–phonon coupling. The zero-phonon line narrowing is discussed regarding vibrations of the perturbed lattice.

Determination of temperature dependent parameters of zero-phonon line in photo-luminescence spectrum of silicon-vacancy centre in CVD diamond thin films

Abstract In this work we present a methodological approach to the temperature dependence of photoluminescence (PL) emission spectra of the silicon-vacancy centre in diamond thin films prepared by chemical vapour deposition. The PL spectra were measured in the temperature range of 11 - 300 K and used to determine the temperature dependence of the zero-phononline full-width at half-maximum and of the peak position. Experimental data were fitted by models of lattice contraction, quadratic electron-phonon coupling, homogeneous and inhomogeneous broadening. We found that the shift of peak position and peak broadening reflect polynomial dependence on temperature. Moreover, a proper setting of monochromator slits width is discussed with respect to line profile broadening.

Going beyond the linear approximation in describing electron-phonon coupling: Relevance for the Holstein model

Using the momentum average approximation we study the importance of adding higher-than-linear terms in the electron-phonon coupling on the properties of single polarons described by a generalized Holstein model. For medium and strong linear coupling, even small quadratic electron-phonon coupling terms are found to lead to very significant quantitative changes in the properties of the polaron, which cannot be captured by a linear Holstein Hamiltonian with renormalized parameters. We argue that the bi-polaron phase diagram is equally sensitive to addition of quadratic coupling terms if the linear coupling is large. These results suggest that the linear approximation is likely to be inappropriate to model systems with strong electron-phonon coupling, at least for low carrier concentrations.

Two-phonon absorption operator in spectroscopy

While the raising operator aˆ+ does not have eigenvectors, coherent states are eigenvectors of the lowering operator aˆ. Therefore, operating with the operator exp[kaˆ+2] on coherent states poses a challenge, where k is an arbitrary constant. The result of this operation on coherent states has not yet been reported in a closed form. A closed-form expression for this problem is derived. The applicability and correctness of the result are tested by calculating the electronic dipole moment time correlation function and the corresponding Franck–Condon factors for systems with distorted Hamiltonian which arises in case of quadratic electron–phonon coupling which is significant in quantifying electronic dephasing.

Temperature dependence of the R1 linewidth in Al 2O 3:Mn 4+: A spectral hole-burning and FLN study

The temperature dependence of the R 1 linewidth in Al 2O 3:Mn 4+ is reported. Transient holes broaden rapidly from 2.5 to 40 K due to the one-phonon process between the E ¯ and 2 A ¯ levels of the 2 E excited state. The linewidth varies from 3 MHz at 2.5 K to 2640 GHz at 294 K and the line shifts by −1440 GHz. Above 50 K the two-phonon Raman process becomes dominant. The quadratic electron–phonon coupling constant is larger for Al 2O 3:Mn 4+ compared to ruby, resulting in a stronger phonon sideband in Al 2O 3:Mn 4+. The best description of the linewidth data are yielded by a model using pseudo-local modes.

Zero-phonon line: Effect of quadratic electron–phonon coupling

The recently proposed theory of vibronic transitions in impurity centres of solids [10] is applied to zero-phonon lines in optical spectra. New equations for the relative intensity, position and width of the line are derived, which take into account both the linear and the quadratic electron–phonon couplings. The joint effect of these couplings on the line intensity is found to be different for absorption and luminescence.

Bipolaron in thet−JModel Coupled to Longitudinal and Transverse Quantum Lattice Vibrations

We explore the influence of two different polarizations of quantum oxygen vibrations on the spacial symmetry of the bound magnetic bipolaron in the context of the t-J model by using exact diagonalization within a limited functional space. Linear as well as quadratic electron-phonon coupling to transverse polarization stabilize d-wave symmetry. The existence of a magnetic background is essential for the formation of a d-wave bipolaron state. With increasing linear electron-phonon coupling to longitudinal polarization the symmetry of a d-wave bipolaron state changes to a p wave. Bipolaron develops a large anisotropic effective mass.

Frequency dependence of the quadratic electron-phonon coupling constant in a polymer glass: Direct measurement by single-molecule spectroscopy

The frequency dependence of the quadratic coupling constant between the electron system of the chromophores and the quasilocal vibrational modes of the matrix, $B({\ensuremath{\omega}}_{0})$, and its distribution was measured in a disordered system via single-molecule spectroscopy. A correlation between $B$ and ${\ensuremath{\omega}}_{0}$ was found which does not follow a power law. The analysis of this result allows us to suggest that, for a correct description of the broadening of optical impurity spectra in disordered solids, the applicability of the weak-coupling limit and the long-wavelength approximation should be considered separately for each single molecule interacting with its local environment. A simple formulation of the electron-phonon coupling theory for the dye-matrix system as a whole is, in general, not appropriate.

Novel theoretical approach in photoemission spectroscopy: application to isotope effect and boron-doped diamond

A new path-integral theory is developed to calculate the photoemission spectra (PES) of correlated many-electron systems. The application to the study on Bi2Sr2CaCu2O8 (Bi2212) and boron-doped diamond (BDD) is discussed in details. It is found that the isotopic shift in the angle-resolved photoemission spectra of Bi2212 is due to the off-diagonal quadratic electron-phonon (e-ph) coupling, whereas the presence of electron-electron repulsion partially suppresses this effect. For the BDD, a semiconductor-metal phase transition, which is induced by increasing the e-ph coupling and dopant concentration, is reproduced by our theory. Additionally, the presence of Fermi edge and phonon step-like structure in PES is found to be due to a co-existence of itinerant and localized electronic states in BDD.

Pigment Spectra and Intermolecular Interaction Potentials in Glasses and Proteins

A model is proposed for chromophore optical spectra in solids over a wide range of temperatures and pressures. Inhomogeneous band shapes and their pressure dependence, as well as baric shift coefficients of spectral lines, selected by the frequency, were derived using Lennard-Jones potentials of the ground and excited states. Quadratic electron-phonon coupling constants, describing the thermal shift and broadening of zero-phonon lines, were also calculated. Experimentally, thermal shift and broadening of spectral holes were studied between 5 and 40 K for a synthetic pigment, chlorin, embedded in polymer hosts. The baric effects on holes were determined by applying hydrostatic He gas pressure up to 200 bar, at 6 K. Absorption spectra of pheophytin a, chlorophyll a, and β-carotene in polymers and plant photosystem II CP47 complex were measured between 5 (or 77) and 300 K, and subject to Voigtian deconvolution. A narrowing of inhomogeneous bandwidth with increasing temperature, predicted on the basis of hole behavior, was observed as the shrinking of Gaussian spectral component. The Lorentzian broadening was ascribed to optical dephasing up to 300 K in transitions with weak to moderate linear electron-phonon coupling strength. The thermal broadening is purely Gaussian in multiphonon transitions (S 2 band of β-carotene, Soret bands of tetrapyrrolic pigments), and the Lorentz process appears to be suppressed, indicating a lack of exponential dephasing. Density, polarity, polarizability, compressibility, and other local parameters of the pigment binding sites in biologically relevant systems can be deduced from spectroscopic data, provided that sufficient background information is available.

Isotopic shift in angle-resolved photoemission spectra of Bi2Sr2CaCu2O8 due to quadratic electron–phonon coupling

Abstract In connection with the experiment on oxygen isotope effect of Bi 2 Sr 2 CaCu 2 O 8 with the angle-resolved photoemission spectroscopy (ARPES), we theoretically study the isotope-induced band shift in ARPES by the Hartree–Fork and quantum Monte Carlo methods. We find that this band shift can be clarified based on a quadratically coupled electron–phonon ( e –ph) model. The large ratio of band shift versus phonon energy change is connected with the softening effect of phonon, and the positive–negative sign change is due to the momentum dependence of the e –ph coupling.

A didactic model of solvent shifts, inhomogeneous broadening, and linear and quadratic electron–phonon coupling

A simplified model of optical spectra of chromophores embedded in disordered solids is proposed. Using the repulsive–dispersive potentials of the ground state and the excited state in Lennard–Jones form the expressions are obtained for inhomogeneous site energy distribution function, and the linear and quadratic coupling strength of electronic transition to local phonons. Experimentally, the shift of spectral holes was studied for chlorin in polymer hosts at temperatures between 5 and 40 K and by applying hydrostatic He gas pressure up to 200 bar. The opposite frequency dependencies of line shifts induced by temperature and pressure are in agreement with the model, if the potential minimum of the excited state is shifted to a shorter distance.

Influence of Temperature and Pressure on Shape and Shift of Impurity Optical Bands in Polymer Glasses

The shape, broadening, and shift of optical absorption spectra of molecular impurity centers in polymer glasses are considered in terms of inhomogeneous energy distributions and coupling of electronic transitions to vibrations. Persistent spectral hole burning was applied for frequency-selective probing of zero-phonon lines. The shift and broadening of spectral holes were studied between 5 and 50 K and by applying a hydrostatic He gas pressure up to 200 bar. Broadband absorption spectra were recorded between 5 and 300 K in poly(methyl methacrylate) and polyethylene. In addition to "normal" thermal broadening, due to the first- and second-order electron phonon coupling, several narrowing components were predicted on the basis of frequency dependent hole behavior. Thermal expansion of the matrix and the relaxation of local strains, previously accumulated on cooling below the glass temperature can lead to shrinking of the inhomogeneous width. A Voigt treatment of absorption band shapes reveals that the Gaussian component can indeed suffer remarkable narrowing. Inhomogeneous band shapes and the frequency-dependent thermal and baric line shifts were rationalized with the aid of a pair of two-body Lennard-Jones potentials. The shift of potential well minima is a crucial factor influencing solvent shifts, inhomogeneous band shapes, pressure shift coefficients, and quadratic electron phonon coupling constants.

Lennard-Jones approach to optical zero-phonon spectra of impurities in glasses

Frequency dependent properties of zero-phonon lines are considered on the basis of persistent hole burning in the inhomogeneous optical bands of dye-doped polymer glasses. The energy spread of optical transitions was expressed as a difference of Lennard-Jones (L-J) potentials of the ground state and the excited state. Pressure shift coefficients of spectral holes depend on the difference of the first derivatives of L-J potentials. From the second derivatives the quadratic electron–phonon coupling constants were calculated and compared to temperature induced shift and broadening of spectral holes. A softening of local phonons in glasses or defect crystals containing empty space can be associated with the vanishing of force constants at intermolecular distances exceeding the equilibrium value by a mere 11%.

Coupling of Electrons to the Multiphonon Complexes and its Relevance to Superconductivity

We discuss the phonon mechanism of superconductivity accounting for the anharmonic oscillations and quadratic electron-phonon coupling. We show that these interactions give rise to the multiphonon complexes which couple to the charge carriers. We prove that electron pair coupled to a quartaphonon forms a multiphonon polaron with binding energy exceeding 300 K.

Nickel–nitrogen complexes in synthetic diamond: the 1.660 eV luminescence system

In this paper, we report the effects of applied uniaxial stress and temperature dependence of the 1.660eV zero phonon line (ZPL) observed in high-pressure high-temperature synthetic diamonds, grown using an Ni solvent catalyst and annealed at 1600°C. It is shown that stress dependence of the 1.660eV line can be understood in terms of an electric dipole transition between A and B states at a centre of monoclinic I symmetry, with the electric dipole lying in the plane normal to the main symmetry axis at an angle of ∼75° to the 〈1−10〉 crystal axis.The shape of the vibronic sideband is explained using a simple electron–lattice coupling model. The temperature effects on the ZPL are discussed in the framework of linear and quadratic electron–phonon coupling.

Thermal effects on zero-phonon holes in the optical spectra of molecular probes in polymer glasses

Purely electronic zero-phonon transitions in impurity molecules are of interest as highly sensitive local probes of solids. Spectral holes were created at different positions in the inhomogeneous optical-absorption bands of tetrapyrrolic chromophores in polymer glasses, and the thermal shift and broadening were investigated between 5 and 50 K. In addition, hole shifts were determined under He gas pressure up to 200 bars with the aim to separate the phonon-induced line shift from the ``trivial'' component caused by thermal expansion of the matrix. A prominent frequency dependence of a pure thermal, phonon-induced shift was discovered. The shift becomes progressively more negative (bathochromic) with increasing transition energy in the inhomogeneous ensemble of chromophores. In theory the corresponding negative quadratic electron-phonon coupling constant W reflects phonon mode softening in the excited state. In several cases the holes burned on the low-frequency edge show practically no thermal shift. The line broadening that depends on the square of W remains almost constant over the inhomogeneous contour. The role of intermolecular repulsive and attractive potentials on local phonon frequencies is elucidated.

Thermal hole shifts in the optical spectra of a dye-doped amorphous polymer

Temperature and pressure effects were investigated on spectral holes burned in the inhomogeneous S 1 absorption band of octaethylporphine in poly(methyl methacrylate) glass. The temperature-induced hole shift and broadening were recorded between 7 and 30 K. The dynamic, phonon-induced shift depends strongly on burning position. According to theory, negative quadratic electron–phonon coupling (QEPC) constant corresponds to a decrease of phonon frequencies in the excited state. This mode-softening behaviour prevails on the short-wavelength part of absorption, where the thermal shift is large and bathochromic. The absence of shift on the red edge cannot be explained with negligible absolute values of QEPC constants, since thermal hole broadening (that depends on the square of QEPC constant) is large and essentially independent on burning position.