Abstract
We consider the problem of heat transport by vibrational modes between Langevin thermostats connected by a central device. The latter is anharmonic and can be subject to large temperature difference and thus be out of equilibrium. We develop a classical formalism based on the equation of motion method, the fluctuation–dissipation theorem and the Novikov theorem to describe heat flow in a multi-terminal geometry. We show that it is imperative to include a quartic term in the potential energy to insure stability and to properly describe thermal expansion. The latter also contributes to leading order in the thermal resistance, while the usually adopted cubic term appears in the second order. This formalism paves the way for accurate modeling of thermal transport across interfaces in highly non-equilibrium situations beyond perturbation theory.
Highlights
Transport theories of non-interacting quantum systems based on the Keldysh formalism, which treats non-equilibrium flow of charge or heat carriers in a one-dimensional (1D) geometry have been developed in the past
Following the seminal work of Caroli et al, many other groups worked on similar formalisms and proved a formula for the transmission through the system, widely used for both non-interacting electrons and phonons
Using a classical method to derive an expression for the heat current, we argue that to leading order, it is necessary to include quartic terms in the Hamiltonian, in order to properly describe both the thermal expansion and the dominant temperature dependence effect in the heat current
Summary
Transport theories of non-interacting quantum systems based on the Keldysh formalism, which treats non-equilibrium flow of charge or heat carriers in a one-dimensional (1D) geometry have been developed in the past. The equilibrium version of it, namely T = Tr[GΓLG†ΓR] , where G and Γ are, respectively, the retarded Green’s function and the escape rates to the leads, was established by Meir and Wingreen [2] and in a similar form by Pastawski [3] in 1991 This formula holds for a non-interacting (or harmonic, in the case of phonons) system near equilibrium, meaning the chemical potential or temperature gradients are to be infinitesimally small. The non-equilibrium anharmonic phonon problem has been addressed in the past by Mingo [4] and separately by Wang [5] in 2006 They used the many-body perturbation approach of non-equilibrium quantum systems based on the Keldysh formalism, ( called the Non-Equilbrium Green’s Function or NEGF method) and derived a lowestorder approximation for the transmission function. Using a classical method to derive an expression for the heat current, we argue that to leading order, it is necessary to include quartic terms in the Hamiltonian, in order to properly describe both the thermal expansion and the dominant temperature dependence effect in the heat current
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