Phonon Green Function Research Articles

Phonon Thermal Transport in UO_{2} via Self-Consistent Perturbation Theory.

Computing thermal transport from first-principles in UO_{2} is complicated due to the challenges associated with Mott physics. Here, we use irreducible derivative approaches to compute the cubic and quartic phonon interactions in UO_{2} from first principles, and we perform enhanced thermal transport computations by evaluating the phonon Green's function via self-consistent diagrammatic perturbation theory. Our predicted phonon lifetimes at T=600 K agree well with our inelastic neutron scattering measurements across the entire Brillouin zone, and our thermal conductivity predictions agree well with previous measurements. Both the changes due to thermal expansion and self-consistent contributions are nontrivial at high temperatures, though the effects tend to cancel, and interband transitions yield a substantial contribution.

Anharmonic phonon behavior via irreducible derivatives: Self-consistent perturbation theory and molecular dynamics

Cubic phonon interactions are now regularly computed from first principles, and the quartic interactions have begun to receive more attention. Given this realistic anharmonic vibrational Hamiltonian, the classical phonon Green's function can be precisely measured using molecular dynamics, which can then be used to rigorously assess the range of validity for self-consistent diagrammatic approaches in the classical limit. Here we use the bundled irreducible derivative approach to efficiently and precisely compute the cubic and quartic phonon interactions of CaF$_2$, systematically obtaining the vibrational Hamiltonian purely in terms of irreducible derivatives. non frequency shifts and linewidths, We demonstrate that the 4-phonon sunset diagram has an important contribution to the optical phonon linewidths beyond $T=500$ K. Reasonable results are obtained even at $T=900$ K when performing self-consistency using the 4-phonon loop diagram and evaluating the 3-phonon bubble and 4-phonon sunset diagrams post self-consistency. Further improvements are obtained by performing quasiparticle perturbation theory, where both the 4-phonon loop and the real part of the 3-phonon bubble are employed during self-consistency. Our irreducible derivative approach to self-consistent perturbation theory is a robust tool for studying anharmonic phonons in both the quantum and classical regimes.

Perturbation theory and thermal transport in mass-disordered alloys: Insights from Green's function methods

Lowest-order quantum perturbation theory (Fermi's golden rule) for phonon-disorder scattering has been used to predict thermal conductivities in several semiconducting alloys with surprising success given its underlying hypothesis of weak and dilute disorder. In this paper, we explain how this is possible by focusing on the case of maximally mass-disordered ${\mathrm{Mg}}_{2}{\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Sn}}_{x}$. We use a Chebyshev polynomials Green's function method, which allows a full treatment of disorder on very large systems (tens of millions of atoms) to probe individual phonon linewidths and frequency-resolved thermal transport. We demonstrate that the success of perturbation theory originates from the specific form of mass disorder terms in the phonon Green's function and from the interplay between anharmonic and disorder scattering.

Phonon transport in disordered alloys: A Multiple-scattering approach

I present a formalism to calculate the configuration-averaged lattice thermal conductivity for substitutional random alloys. The method is based on multiple-scattering approach which can capture the effect of disorder-induced configuration fluctuations on single-particle and two-particle phonon Green functions in random alloys. The randomness of the system is dealt within the augmented space theorem. This is combined with a generalized Feynman diagrammatic technique to extract various useful results in the form of mathematical expressions. I show the structure of all possible scattering diagrams up to the fourth order and subsequently illustrate how to obtain Dyson's equation from a resummation of the diagrammatic series. I also study how disorder scattering affects two-particle Green functions associated with thermal response. It was shown explicitly how the disorder scattering renormalizes both the phonon propagators as well as the heat currents. I derive the relation between these renormalized heat currents and the self-energy of the propagators. I have also studied a different class of scattering diagrams which are not related to the self-energy but rather to the vertex corrections. The configuration-averaging scheme is straightforward to apply to other relevant quantities such as joint density of states, thermal diffusivity, etc., for random alloys. The developed formalism is applied to a realistic ${\mathrm{Au}}_{1\ensuremath{-}x}{\mathrm{Fe}}_{x}$ binary alloy. The effect of disorder-induced corrections (as compared to the simple virtual crystal approximation) turns out to be appreciable in this alloy. The configuration-averaged lattice thermal conductivity for ${\mathrm{Au}}_{50}{\mathrm{Fe}}_{50}$ shows a quadratic behavior in low-temperature regime ($T\ensuremath{\le}30$ K), which increases smoothly to a $T$-independent saturated value at high $T$. Simulated thermal diffusivity $D(\ensuremath{\nu})$ helps to numerically estimate the mobility $\mathrm{edge}\phantom{\rule{0.28em}{0ex}}({\ensuremath{\nu}}_{c})$ which in turn evaluates the fraction of localized states. $D(\ensuremath{\nu})$ is found to decrease smoothly (almost linearly) in the high-$\ensuremath{\nu}$ range, which when fitted to ${({\ensuremath{\nu}}_{c}\ensuremath{-}\ensuremath{\nu})}^{\ensuremath{\alpha}}$ gives the critical $\mathrm{exponent}\phantom{\rule{0.28em}{0ex}}(\ensuremath{\alpha})$ to be 1.018 for ${\mathrm{Au}}_{50}{\mathrm{Fe}}_{50}$ alloy. This agrees fairly well with the scaling and other theories of Andersen localization.

Towards ultrafast cooling through transient phonon currents: A closed-form solution

We develop a closed-form formula to calculate the transient thermal currents flowing through an arbitrary nanoscale phonon device in response to a sudden thermal switch. Our theory provides a solution to the problem in the far-from-equilibrium nonlinear response regime beyond the wide-band-like approximation and Drude regularization. We present calculations in a one-dimensional monatomic chain with Lorentzian-like thermal baths and show that the transient phonon currents are significantly larger than the long-time-limit steady-state phonon current. From the formula's clear mathematical structure, we also show that the transient oscillation periodicity and relaxation time are determined by the poles of the retarded phonon Green's function. In addition, higher temperatures of the thermal baths and stronger coupling between the baths and the central monatomic chain result in higher transient thermal currents. Our results suggest that ultrafast cooling of nanodevices through transient phonon currents is a promising route.

Ultrafast asymmetric Rosen-Zener-like coherent phonon responses observed in silicon

We investigate the spectral profiles of time signals attributed to coherent phonon generation in an undoped Si crystal. Here, the retarded longitudinal-optical (LO) phonon Green function relevant to the temporal variance of induced charge density of ionic cores is calculated by employing the polaronic quasiparticle model developed by the authors [Y. Watanabe et al., Phys. Rev. B 95, 014301 (2017); ibid., 96, 125204 (2017)]. The spectral asymmetry is revealed in the frequency domain of the signals under the condition that an LO phonon mode stays almost energetically resonant with a plasmon mode in the early time region; this lasts for approximately 100 fs immediately after the irradiation of an ultrashort pump-laser pulse. It is understood that based on the adiabatic picture in time, this asymmetry is caused by the Rosen-Zener coupling between both modes. The associated experimental results are obtained by measuring time-dependent electro-optic reflectivity signals, and it is proved that these are in harmony with the calculated ones. The spectra become more symmetric, as the photoexcited carrier density further changes from that meeting the above condition to higher and lower sides of carrier densities. Moreover, the effect of optical nutation of carrier density on the CP signals is addressed, and the present results are compared with the asymmetry caused by transient Fano resonance, and the spectral profiles observed in a GaAs crystal in the text.

Nonequilibrium electron dynamics in pump-probe spectroscopy: Role of excited phonon populations

We study the role of excited phonon populations in the relaxation rates of nonequilibrium electrons using a nonequilibrium Green's function formalism. The transient modifications in the phononic properties are accounted for by self-consistently solving the Dyson equation for the electron and phonon Green's functions. The pump induced changes manifest in both the electronic and phononic spectral functions. We find that the excited phonon populations suppress the decay rates of nonequilibrium electrons due to enhanced phonon absorption. The increased phonon occupation also sets the nonequilibrium decay rates and the equilibrium scattering rates apart. The decay rates are found to be time-dependent, and this is illustrated in the experimentally observed population decay of photoexcited $\mathrm{Bi}_{1.5}\mathrm{Sb}_{0.5} \mathrm{Te}_{1.7}\mathrm{Se}_{1.3}$.

Debye-Waller Factor in an Isotopically Disordered Semiconductor Crystal

The diagonal and non-diagonal parts for the Debye–Waller factor have been established using equation of motion technique of quantum dynamics and the Dyson equation approach. The double time temperature dependent phonon Green function has been taken to find the phonon linewidth and phonon shift. Renormalized mode frequency has been investigated in terms of electron–phonon coupling constant and temperature. The effect of electron–phonon interaction on the Debye–Waller factor has been studied in low temperature limit in low impurity concentration in semiconductor crystals.

Cumulant expansion for phonon contributions to the electron spectral function

We describe an approach for calculations of phonon contributions to the electron spectral function, including both quasiparticle properties and satellites. The method is based on a cumulant expansion for the retarded one-electron Green's function and a many-pole model for the electron self-energy. The electron-phonon couplings are calculated from the Eliashberg functions, and the phonon density of states is obtained from a Lanczos representation of the phonon Green's function. Our calculations incorporate ab initio dynamical matrices and electron-phonon couplings from the density functional theory code ABINIT. Illustrative results are presented for several elemental metals and for Einstein and Debye models with a range of coupling constants. These are compared with experiment and other theoretical models. Estimates of corrections to Migdal's theorem are obtained by comparing with leading order contributions to the self-energy, and are found to be significant only for large electron-phonon couplings at low temperatures.

Electric dipolar susceptibility of the Anderson-Holstein model

The temperature dependence of electric dipolar susceptibility \chi_P is discussed on the basis of the Anderson-Holstein model with the use of a numerical renormalization group (NRG) technique. Note that P is related with phonon Green's function D. In order to obtain correct temperature dependence of P at low temperatures, we propose a method to evaluate P through the Dyson equation from charge susceptibility \chi_c calculated by the NRG, in contrast to the direct NRG calculation of D. We find that the irreducible charge susceptibility estimated from \chi_c agree with the perturbation calculation, suggesting that our method works well.

Landauer formula for phonon heat conduction: relation between energy transmittance and transmission coefficient

The heat current across a quantum harmonic system connected to reservoirs at different temperatures is given by the Landauer formula, in terms of an integral over phonon frequencies \omega, of the energy transmittance T(\omega). There are several different ways to derive this formula, for example using the Keldysh approach or the Langevin equation approach. The energy transmittance T({\omega}) is usually expressed in terms of nonequilibrium phonon Green's function and it is expected that it is related to the transmission coefficient {\tau}({\omega}) of plane waves across the system. In this paper, for a one-dimensional set-up of a finite harmonic chain connected to reservoirs which are also semi-infinite harmonic chains, we present a simple and direct demonstration of the relation between T({\omega}) and {\tau}({\omega}). Our approach is easily extendable to the case where both system and reservoirs are in higher dimensions and have arbitrary geometries, in which case the meaning of {\tau} and its relation to T are more non-trivial.

Lattice Thermal Conductivity of GaAs

A new approach to evaluate the relaxation times of various collision events responsible for thermal transport has been reported through which various de ciencies of earlier models of lattice thermal conductivity have been resolved. These investigations involve the evaluation of the phonon Green functions via a non-perturbative approach. The new expressions of relaxation times expressions for scattering of phonons by boundaries, atomic impurities, phonon phonon scattering, electron phonon scattering are the new features of the theory. The lattice thermal conductivity of three samples of GaAs has been analyzed on the basis of modi ed Callaway model and fairly good agreement between theory and experimental observations has been reported.

Time autocorrelation function and Green-Kubo formula: Study on a disordered harmonic chain

We have considered heat conduction in a one-dimensional mass-disordered harmonic chain of N particles connected to two Langevin type reservoirs at different temperatures. An exact expression for the boundary heat current-current autocorrelation function in the nonequilibrium steady state (NESS) is obtained in terms of nonequilibrium phonon Green's functions. The time integral of the correlation function gives expected result, both in nonequilibrium as well as equilibrium cases. Using the form of this correlation function we show that asymptotic system size dependence of current fluctuation in NESS for a mass-disordered harmonic chain is N(-α) for different boundary conditions. For free and fixed boundary conditions we get α=1/2 and 3/2, respectively, while for pinned case the fluctuation decays exponentially with system size.

Vibrational properties of grain boundaries in nanocrystalline Ni using second moment potentials

The high frequency phonon properties of a computer-generated nanocrystalline (nc) Ni are investigated by directly calculating the onsite phonon Green function using a recursion technique based on a continued fraction representation. Past work found that a high frequency enhancement of the vibrational density of states exists and arises from localised vibrational modes forming within the nc grain boundary regions at atoms of reduced coordination [Phys. Rev. Lett. 92 (2004) p.035505]. In the present work, the positron wave function for such nanocrystalline grain boundaries was calculated and used to identify regions of localised free volume that, in turn, were found to correlate well with the localised vibrational modes. The work also demonstrates that, irrespective of the form of the central-force inter-atomic potential used, the observed effect can be more generally understood with respect to local variations in hydrostatic pressure within the grain boundary.

Electron–phonon coupling and longitudinal sound velocity of La0.5Ca0.5MnO3

Abstract In this paper, the microscopic theory of the relative change in velocity of sound with temperature of La0.5Ca0.5MnO3 is reported. The phonon Green function is calculated using the Green function technique of Zubarev in the limit of zero wave vector and low temperature. The lattice model electronic Hamiltonian in the presence of the phonon interaction with hybridization between the conduction electrons and the l-electrons is used. The relative change in velocity of sound at various temperatures is studied for different model parameters, namely the position of the l-level, the effective phonon coupling strength and hybridization strength. The phonon anomalies observed experimentally at different temperatures are explained theoretically. An abrupt change in velocity at Neel temperature (TN) is observed clearly. It is observed that different parameters influence the velocity of sound.

Microscopy model of the velocity of sound in La0.5Ca0.5MnO3

In this paper, the microscopic theory of the relative change in the velocity of sound with temperature of La0.5Ca0.5MnO3 is reported. The phonon Green function is calculated using the Green function technique of Zubarev (1960 Sov. Phys.—Usp.3 320) in the limit of zero wave vector and low temperature. The electronic Hamiltonian of the lattice model in the presence of the phonon interaction with the hybridization between the conduction electrons and l-electrons is used. The relative change in the velocity of sound at various temperatures is studied for different model parameters, namely the position of the l-level, the effective phonon coupling strength, the Coulomb interaction and the hybridization strength. The phonon anomalies observed experimentally at different temperatures are explained theoretically. An abrupt change in velocity at the Neel temperature (TN) is clearly observed. It is observed that different parameters influence the velocity of sound and the Neel temperature increases with increase in the Coulomb interaction.

Generalized Langevin equation for solids. I. Rigorous derivation and main properties

We demonstrate explicitly that the derivation by Adelman and Doll (AD) [J. Chem. Phys. 64, 2375 (1976)] of the generalized Langevin equation (GLE) to describe dynamics of an extended solid system by considering its finite subsystem is inconsistent because it relies on performing statistical averages over the entire system when establishing properties of the random force. This results in the random force representing a nonstationary process opposite to one of the main assumptions made in AD that the random force corresponds to a stationary stochastic process. This invalidates the derivation of the Brownian (or Langevin) form of the GLE in AD. Here we present a different and more general approach in deriving the GLE. Our method generalizes that of AD in two main aspects: (i) the structure of the finite region can be arbitrary (e.g., anharmonic), and (ii) ways are indicated in which the method can be implemented exactly if the phonon Green's function of the harmonic environment region surrounding the anharmonic region is known, which is, e.g., the case when the environment region represents a part of a periodic solid (the bulk or a surface). We also show that in general after the local perturbation has ceased, the system returns to thermodynamic equilibrium with the distribution function for region 1 being canonical with respect to an effective interaction between atoms, which includes instantaneous response of the surrounding region. Note that our method does not rely on the assumption made in AD that the stochastic force correlation function depends on the times difference only (i.e., the random force corresponds to a stationary random process). In fact, we demonstrate explicitly that generally this is not the case. Still, the correct GLE can be obtained, which satisfies exactly the fluctuation-dissipation theorem.

Heat transport in low-dimensional systems

Recent results on theoretical studies of heat conduction in low-dimensional systems are presented. These studies are on simple, yet nontrivial, models. Most of these are classical systems, but some quantum-mechanical work is also reported. Much of the work has been on lattice models corresponding to phononic systems, and some on hard particle and hard disc systems. A recently developed approach, using generalized Langevin equations and phonon Green's functions, is explained and several applications to harmonic systems are given. For interacting systems, various analytic approaches based on the Green-Kubo formula are described, and their predictions are compared with the latest results from simulation. These results indicate that for momentum-conserving systems, transport is anomalous in one and two dimensions, and the thermal conductivity kappa, diverges with system size L, as kappa ~ L^alpha. For one dimensional interacting systems there is strong numerical evidence for a universal exponent alpha =1/3, but there is no exact proof for this so far. A brief discussion of some of the experiments on heat conduction in nanowires and nanotubes is also given.

Theoretical study of NMR relaxation due to rattling phonons

We calculate the NMR relaxation rate due to quadrupolar coupling of the nucleus to a local, strongly anharmonic phonon mode. As a model potential for a “rattling” motion we consider a square-well potential. We calculate the free phonon Green's function analytically and derive the low and high temperature limits of the NMR relaxation rate. It is shown that the temperature dependence of the NMR relaxation rate possesses a peak in contrast to harmonic phonons but in qualitative agreement with a recent NMR study on KOs 2O 6. We discuss the influence of phonon renormalization due to electron–phonon interaction.

Sound velocity in La 0.5Ca 0.5MnO 3

In this Letter the microscopic theory of the relative change in velocity of sound with temperature of La 0.5Ca 0.5MnO 3 is reported. The phonon Green function is calculated using the Green function technique of Zubarev in the limit of zero wave vector and low temperature. The lattice model electronic Hamiltonian in the presence of the phonon interaction with hybridization between the conduction electrons and the l-electrons is used. The relative change in velocity of sound at various temperatures is studied for different model parameters namely the position of the l-level, the effective phonon coupling strength and hybridization strength. The phonon anomalies observed experimentally at different temperatures are explained theoretically. An abrupt change in velocity at Neel temperature ( T N ) is observed clearly. It is observed that different parameters influence the velocity of sound.