Green function, quasi-classical Langevin and Kubo–Greenwood methods in quantum thermal transport

Abstract

With the advances in fabrication of materials with feature sizes at the order of nanometers, it has been possible to alter their thermal transport properties dramatically. Miniaturization of device size increases the power density in general, hence faster electronics require better thermal transport, whereas better thermoelectric applications require the opposite. Such diverse needs bring new challenges for materials design. Shrinkage of length scales has also changed the experimental and theoretical methods to study thermal transport. Not surprisingly, novel approaches has emerged to control phonon flow. Besides, ever increasing computational power is another driving force for developing new computational methods. In this review, we discuss three methods developed for computing vibrational thermal transport properties of nano-structured systems, namely Green function, quasi-classical Langevin, and Kubo-Green methods. The Green function methods are explained using both nonequilibrium expressions and the Landauer-type formula. The partitioning scheme, decimation techniques and surface Green functions are reviewed, also a simple model for reservoir Green functions is shown. The expressions for the Kubo-Greenwood method are derived, and Lanczos tridiagonalization, continued fraction and Chebyshev polynomial expansion methods are discussed. Additionally, the quasi-classical Langevin approach, which is useful for incorporating phonon-phonon and other scatterings is summarized.

Ballistic thermoelectric properties of monolayer semiconducting transition metal dichalcogenides and oxides

“…Using L n , one can express the electrical conductance (G), Seebeck coefficient (S), and the electrical part of the thermal conductance (κ el ) as G = e 2 L 0 , S = (L 1 /L 0 )/eT , and κ el = (L 2 − L 2 1 /L 0 )/T , respectively. Phonon thermal conductance is calculated using Landauer formalism [69,70],…”

Temperature-dependent thermal transport of single molecular junctions from semi-classical Langevin molecular dynamics

“…Previously, both the NEGF and classical MD (CMD) have been used to study thermal transport in SMJs [15,16,18,[52][53][54]. However, both methods have their shortcomings.…”

An Atomistic Study of the Thermoelectric Signatures of CNT Peapods

“…Unlike the L m functions, where the effective range of integration is largely limited to a relatively narrow region around the chemical potential (due to the derivative of the Fermi distribution), the calculation of the thermal conductance involves the whole phonon spectrum. To compute both transmission coefficients τ el and τ ph , we use the Green’s function formalism and the common partitioning scheme in which the whole system is split into three regions, namely, left and right reservoirs (electronic and thermal) and a scattering region (see Figure a). ,− Since the system is periodic, both the reservoirs and the central region have the same atomistic structure so that there is no scattering at the reservoir–central region interface. τ el is obtained as where G r is the retarded Green’s function of the central region computed as G r = ( EI – H – Σ L r – Σ R r ) −1 , with H as the Hamiltonian matrix, E the energy of the electrons, and I the unit matrix.…”