Memory Function Formalism Research Articles

Dynamical thermoelectric transport properties of graphene: MF approach

In the present work, Seebeck coefficient and Figure of merit in terms of thermoelectric, electric, and thermal scattering rates employing powerful technique of memory function formalism in graphene system are investigated. This study leads to the dynamical thermoelectric transport behavior of graphene subjected to electron–phonon interactions using linear response theory. Within premises, explicit expression of generalized Drude scattering rate or imaginary part of memory function associated with thermoelectric/electric/thermal functions is studied. In zero frequency (DC regime) and low-temperature limit, Seebeck coefficient shows the linear-T dependence and, is found to be good in agreement with the experimental and theoretical results. Further, several frequency and temperature dependent transport coefficient results for electron–phonon scattering in graphene are performed. We found that S(ω,T) shows saturation at ω≫ωBG and raises with temperature below the Bloch–Grüneisen (BG) frequency regime (ω≪ωBG). We numerically investigated thermoelectric Figure of merit (ZT) with frequency/temperature domains and discussed.

From Green-Kubo to the full Boltzmann kinetic approach to heat transport in crystals and glasses

We show that vertex corrections to the quasi-harmonic Green-Kubo theory of heat transport in insulators naturally lead to a generalisation of the expression for the conductivity that could be derived from the linearized Boltzmann equation, when the effects of the full scattering matrix are accounted for. Our results, which are obtained from the Mori-Zwanzig memory-function formalism, provide a fully ab initio derivation of the linearized Boltzmann transport equation and establish a connection between two recently proposed unified approaches to heat transport in insulating crystals and glasses.

The Drude-Smith Equation and Related Equations for the Frequency-Dependent Electrical Conductivity of Materials: Insight from a Memory Function Formalism.

The front cover artwork is provided by Wei-Chen Chen and Rudolph A. Marcus. The image shows an electron undergoing backscattering in a disordered solid, a phenomenon commonly described using the Drude-Smith equation represented by the damping curve. Read the full text of the Article at 10.1002/cphc.202100299.

Front Cover: The Drude‐Smith Equation and Related Equations for the Frequency‐Dependent Electrical Conductivity of Materials: Insight from a Memory Function Formalism (ChemPhysChem 16/2021)

The Front Cover depicts an electron undergoing backscattering in a disordered solid, a phenomenon commonly described using the Drude-Smith equation. The equation, graphically represented by the damping curve, is shown to emerge as the “critically damped” case when treated with a more general formulation using a memory function. More information can be found in the Article by Rudolph A. Marcus and Wei-Chen Chen.

The Drude‐Smith Equation and Related Equations for the Frequency‐Dependent Electrical Conductivity of Materials: Insight from a Memory Function Formalism

The Drude‐Smith equation is widely used for treating the frequency‐dependent electrical conductivity of materials in the terahertz region. An attractive feature is its sparsity of adjustable parameters. A significant improvement over Drude theory for these materials, the theory includes backscattering of the charge carriers. It has nevertheless been criticized, including by Smith himself, because of the arbitrariness of a step in the derivation. We recall a somewhat similar behavior of back scattering in fluids observed in molecular dynamics computations and discussed in terms of memory functions. We show how theories such as Drude‐Smith and Cocker et al. are examples of a broader class of theories by showing how they also arise as particular cases of a memory function formalism that divides the interactions into short and long range.

Non-Drude Behaviour of Optical Conductivity in Kondo-Lattice Systems

The optical conductivity in a Kondo lattice system is presented in terms of the memory function formalism. I use Kondo-lattice Hamiltonian for explicit calculations. I compute the frequency dependent imaginary part of the memory function ($M^{\p\p}(\om)$), and the real part of the memory function $M^{\p}(\om)$ by using the Kramers-Kronig transformation. Optical conductivity is computed using the generalized Drude formula. I find that high frequency tail of the optical conductivity scales as $\sigma(\om) \sim \frac{1}{\om}$ instead of the Drude $\frac{1}{\om^2}$ law. Such a behaviour is seen in strange metals. My work points out that it may be the magnetic scattering mechanisms that are important for the anomalous behaviour of strange metals.

Analysis of Interictal EEG Signal Correlations for Diagnostics of Epilepsy

Principles are described for diagnosing epilepsy on the basis of the Memory Function Formalism. Auto- and cross-correlations of electroencephalograms from healthy subjects and patients with epilepsy recorded during the interictal period are studied. Characteristic types of the memory function power spectra and substantial changes in statistical memory effects in pathology are found. Disorders are revealed in the dynamic interconnection in areas of the cerebral cortex in epilepsy.

Strange semimetal dynamics in SrIrO3

The interplay of electronic correlations, multi-orbital excitations, and spin-orbit coupling is a fertile ground for new states of matter in quantum materials. Here, we report on a polarized Raman scattering study of semimetallic SrIrO3. The momentum-space selectivity of Raman scattering allows to circumvent the challenge to resolve the dynamics of charges with very different mobilities. The Raman responses of both holes and electrons display an electronic continuum extending far beyond the energies allowed in a regular Fermi liquid. Analyzing this response within a memory function formalism, we extract their frequency dependent scattering rate and mass enhancement, from which we determine their DC-mobilities and electrical resistivities that agree well with transport measurement. We demonstrate that its charge dynamics is well described by a marginal Fermi liquid phenomenology, with a scattering rate close to the Planckian limit. This demonstrates the potential of this approach to investigate the charge dynamics in multi-band systems.

A theory of resistivity in Kondo lattice materials: memory function approach

We have theoretically analysed DC resistivity (ρ) in the Kondo-lattice materials using the powerful memory function formalism. The complete temperature evolution of ρ is investigated using the Wölfle–Götze expansion of the memory function. The resistivity in this model originates from spin–flip magnetic scattering of conduction s-electron off the quasi-localized d or f electron spins. We find the famous resistivity upturn in lower temperature regime (kBT ≪ μd), where μd is the effective chemical potential of d-electrons. In the high temperature regime (μd ≪ kBT) we discover that resistivity scales as cube root of T (). Our results are in reasonable agreement with the experimental results reported in the literature.

Recent advances in the physics of polymer systems

Recent advances in the physics of polymer systems

Nonequilibrium electron relaxation in graphene

We apply memory function formalism to investigate nonequilibrium electron relaxation in graphene. Within the premises of two-temperature model (TTM), explicit expressions of the imaginary part of the memory function or generalized Drude scattering rate (1/[Formula: see text]) are obtained. In the DC limit and in equilibrium case where electron temperature (Te) is equal to phonon temperature (T), we reproduce the known results (i.e., 1/[Formula: see text][Formula: see text]T4 when T[Formula: see text][Formula: see text] and 1/[Formula: see text][Formula: see text]T when T[Formula: see text][Formula: see text], where [Formula: see text] is the Bloch–Grüneisen temperature). We report several new results for 1/[Formula: see text] where T[Formula: see text][Formula: see text][Formula: see text]Te relevant in pump–probe spectroscopic experiments. In the finite-frequency regime we find that 1/[Formula: see text] when [Formula: see text], and for [Formula: see text] it is [Formula: see text]-independent. These results can be verified in a typical pump–probe experimental setting for graphene.

Direct evaluation of the desired correlations: Verification on real data

In this paper, a new method for evaluation of the desired correlations is proposed. It allows to evaluate the “content” of the external factors ( l=1, 2,…,L) setting in the form of data arrays ym(l)(x) (m=1, 2,…,M) inside the given Ym(x) function that is supposed to be subjected by the influence of these factors. As contrasted to the conventional correlation analysis, the proposed method allows finding the “influence” functions bl(x) (l=1, 2,…,L) and evaluating the “remnant” array Gm(x) that is remained as a “quasi-independent” part from the influence of the factors ym(l)(x). The general expression works as a specific “balance” and reproduces the well-known cases, when bl(x) =Cl (it is reduced to the linear least square method with Gm(x) ≈ 0) and coincides with the remnant function Ym(x) ≈Gm(x), when the influence functions becomes negligible (bl(x) ≈ 0). The available data show that the proposed method allows to extract a small signal S(x) from the “pattern” background and it conserves its stability/robustness in the presence of a random fluctuations/noise. The method is rather flexible and allows to consider the cases of strong correlations, when the external factors act successively, forming the cause-and-effect chains. It can be generalized for expressions containing the bonds expressed in the form of memory functions. The proposed method adds new quantitative ties expressed in the form of the desired functions to the conventional correlation relationships expressed in the form of the correlation coefficients forming, in turn, the correlation matrices. New relationships allow to understand deeper the existing correlations and make them more informative, especially in detection of the desired deterministic and stable bonds/laws that can be hidden inside.

A comparative study of finite frequency scattering rate from Allen, Mitrović–Fiorucci, Shulga–Dolgov–Maksimov, Sharapov–Carbotte and memory function formalisms

We report a comparative study of scattering rates which are calculated using different formalisms such as [P. B. Allen, Phys. Rev. B 3, 305 (1971); S. V. Shulga, D. V. Dolgov and E. G. Maksimov, Physica C 178, 266 (1991); B. Mitrović and M. A. Fiorucci, Phys. Rev. B 31, 2694 (1985); S. G. Sharapov and J. P. Carbotte, Phys. Rev. B 72, 134506 (2005)] and memory function formalism (MemF) [H. Mori, Prog. Theor. Phys. 33, 423 (1965)]. An advantage of this study is that it sheds light on the physical assumptions used in these formalisms. In this context, we consider a case of electron–phonon interaction and discuss frequency- and temperature-dependent behavior of scattering rates. Further, a comparative study of scattering rate proceeds for distinct phonon density of states (PDOS) and electron density of states (EDOS). From the detailed analysis, we observe that the MemF is the most general one and others are based on restrictive assumptions as discussed in this work.

Polymer chain diffusion in polymer blends: A theoretical interpretation based on a memory function formalism

ABSTRACTThe diffusion of polymer chains in miscible polymer blends with large dynamic asymmetry—those where the two blend components display very different segmental mobility—is not well understood yet. In the extreme case of the blend system of poly(ethylene oxide) (PEO) and poly(methyl methacrylate)(PMMA), the diffusion coefficient of PEO chains in the blend can change by more than five orders of magnitude while the segmental time scale hardly changes with respect to that of pure PEO. This behavior is not observed in blend systems with small or moderate dynamic asymmetry as, for instance, polyisoprene/poly(vinyl ethylene) blends. These two very different behaviors can be understood and quantitatively explained in a unified way in the framework of a memory function formalism, which takes into account the effect of the collective dynamics on the chain dynamics of a tagged chain. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019, 57, 1239–1245

Thermalization of dipole oscillations in confined systems by rare collisions

We study the relaxation of the center-of-mass, or dipole oscillations in the system of interacting fermions confined spatially. With the confinement frequency $\omega_{\perp}$ fixed the particles were considered to freely move along one (quasi-1D) or two (quasi-2D) spatial dimensions. We have focused on the regime of rare collisions, such that the inelastic collision rate, $1/\tau_{in} \ll \omega_{\perp}$. The dipole oscillations relaxation rate, $1/\tau_{\perp}$ is obtained at three different levels: by direct perturbation theory, solving the integral Bethe-Salpeter equation and applying the memory function formalism. As long as anharmonicity is weak, $1/\tau_{\perp} \ll 1/ \tau_{in}$ the three methods are shown to give identical results. In quasi-2D case $1/\tau_{\perp} \neq 0$ at zero temperature. In quasi-1D system $1/\tau_{\perp} \propto T^3$ if the Fermi energy, $E_F$ lies below the critical value, $E_F < 3 \omega_{\perp}/4$. Otherwise, unless the system is close to integrability, the rate $1/\tau_{\perp}$ has the temperature dependence similar to that in quasi-2D. In all cases the relaxation results from the excitation of particle-hole pairs propagating along unconfined directions resulting in the relationship $1/\tau_{\perp} \propto 1/\tau_{in}$, with the inelastic rate $1/\tau_{in} \neq 0$ as the phase-space opens up at finite energy of excitation, $\hbar \omega_{\perp}$. While $1/\tau_{\perp} \propto \tau_{in}$ in the hydrodynamic regime, $\omega_{\perp} \ll 1/\tau_{in}$, in the regime of rare collisions, $\omega_{\perp} \gg 1/\tau_{in}$, we obtain the opposite trend $1/\tau_{\perp} \propto 1/\tau_{in}$.

Collective effects in human EEGs at cognitive activity

In this paper we use the Memory function formalism to study the collective phenomena in EEG signals from subjects with musical abilities (“musicians”) and from subjects with a low level of these abilities (“non-musicians”) at solving the three cognitive tasks. We have revealed the spectral features in the EEG collective dynamics of these groups, as well as sensor pairs, which have the a different interaction of signals.

Dynamical electrical conductivity of graphene

For graphene (a Dirac material) it has been theoretically predicted and experimentally observed that DC resistivity is proportional to T4 when the temperature is much less than Bloch–Grüneisen temperature () and T-linear in the opposite case (). Going beyond this case, we investigate the dynamical electrical conductivity in graphene using the powerful method of the memory function formalism. In the zero frequency regime, we obtain the above mentioned behavior which was previously obtained using the Bloch–Boltzmann kinetic equation. In the finite frequency regime, we obtain several new results: (1) the generalized Drude scattering rate, in the zero temperature limit, shows behavior at low frequencies () and saturates at higher frequencies. We also observed the Holstein mechanism, however, with different power laws from that in the case of metals; (2) at higher frequencies, , and higher temperatures , we observed that the generalized Drude scattering rate is linear in temperature. In addition, several other results are also obtained. With the experimental advancement of this field, these results should be experimentally tested.

Finite frequency Seebeck coefficient of metals: A memory function approach

We study the dynamical thermoelectric transport in metals subjected to the electron-impurity and the electron-phonon interactions using the memory function formalism. We introduce a generalized Drude form for the Seebeck coefficient in terms of thermoelectric memory function and calculate the latter in various temperature and frequency limits. In the zero frequency and high temperature limit, we find that our results are consistent with the experimental findings and with the traditional Boltzmann equation approach. In the low temperature limit, we find that the Seebeck coefficient is quadratic in temperature. In the finite frequency regime, we report new results: In the electron-phonon interaction case, we find that the Seebeck coefficient shows frequency independent behavior both in the high frequency regime (ω⪢ωD, where ωD is the Debye frequency) and in the low frequency regime (ω⪡ωD), whereas in the intermediate frequencies, it is a monotonically increasing function of frequency. In the case of the electron-impurity interaction, first it decays and then after passing through a minimum it increases with the increase in frequency and saturates at high frequencies.

All-coupling polaron optical response: Analytic approaches beyond the adiabatic approximation

In the present work, the problem of an all-coupling analytic description for the optical conductivity of the Froehlich polaron is treated, with the goal being to bridge the gap in validity range that exists between two complementary methods: on the one hand the memory function formalism and on the other hand the strong-coupling expansion based on the Franck-Condon picture for the polaron response. At intermediate coupling, both methods were found to fail as they do not reproduce Diagrammatic Quantum Monte Carlo results. To resolve this, we modify the memory function formalism with respect to the Feynman-Hellwarth-Iddings-Platzman (FHIP) approach, in order to take into account a non-quadratic interaction in a model system for the polaron. The strong-coupling expansion is extended beyond the adiabatic approximation, by including into the treatment non-adiabatic transitions between excited polaron states. The polaron optical conductivity that we obtain by combining the two extended methods agree well, both qualitatively and quantitatively, with the Diagrammatic Quantum Monte Carlo results in the whole available range of the electron-phonon coupling strength.

Memory function approach to correlated electron transport: A comprehensive review

Memory function formalism or projection operator technique is an extremely useful method to study the transport and optical properties of various condensed matter systems. A recent revival of its uses in various correlated electronic systems is being observed. It is being used and discussed in various contexts, ranging from nonequilibrium dynamics to the optical properties of various strongly correlated systems such as high temperature superconductors. However, a detailed discussion on this method, starting from its origin to its present day applications at one place is lacking. In this paper, we attempt a comprehensive review of the memory function approach focusing on its uses in studying the dynamics and the transport properties of correlated electronic systems.