Thermal rectification in quantum graded mass systems

Abstract

We show the existence of thermal rectification in the graded mass quantum chain of harmonic oscillators with self-consistent reservoirs. Our analytical study allows us to identify the ingredients leading to the effect. The presence of rectification in this effective, simple model (representing graded mass materials, systems that may be constructed in practice) indicates that rectification in graded mass quantum systems may be an ubiquitous phenomenon. Moreover, as the classical version of this model does not present rectification, our results show that, here, rectification is a direct result of the quantum statistics. A fundamental challenge in statistical physics is the derivation of macroscopic phenomenological laws of thermodynamic transport from the underlying microscopic Hamiltonian systems. However, after decades, a first-principle derivation of the Fourier's law of heat conduction, for instance, is still a puzzle [1]. Many works have been devoted to the theme [2], most of them by means of computer simulations. But, in these problems where the central question involves the convergence or divergence of thermal conductivity, sometimes there is a significant difficulty to arrive at precise conclusions from numerical results [3]. Thus, the necessity of analytical studies together with the huge complexity of the associated nonlinear dynamical systems led to several works considering approximative schemes or simplified models [4] -since the pioneering work of Debye, the microscopic models to describe heat conduction are mainly given by systems of anharmonic oscillators, leading to problems without analytical solutions. Many works involve, e.g., the use of approximations such as Boltzmann equations, master equations for effective models, the analysis of Green-Kubo formula, etc. An example of simple (effective) model that can be analytically studied is the classical or quantum harmonic chain of oscillators with self-consistent reservoirs. It has been proposed a while ago [5,6], but it is always revisited [7,8]. In such model, each site of the chain is coupled to a reservoir; the first and last sites are coupled to "real" thermal baths, while the inner reservoirs only mimic the absent anharmonic interactions. The self-consistent condition means that there is no heat flow between an inner reservoir and its site in the steady state: the inner baths act only as a phonon scattering mechanism, such as 1

From Thermal Rectifiers to Thermoelectric Devices

“…The advantage of such ap-proach lies in its simplicity and independence from microscopic details of inelastic processes. Probe terminals have been widely used in the literature and proved to be useful to unveil nontrivial aspects of phase-breaking processes [49], heat transport and rectification [22,23,118,52,24,112,13,120], and thermoelectric transport [73,57,56,75,76,124,126,135,136,122,67,15,11,27,25]. The approach can be generalized to any number n p of probe reservoirs.…”

Energy rectification in quantum graded spin chains: Analysis of the XXZ model

“…Consequently, this scarcity of quantum results, the present ambient of device miniaturization, together with the possibility of effects of quantum nature, makes the detailed study of energy transport in genuine quantum models a program of great importance. As an example of change in the transport properties due to the quantum/classical nature of the model, we recall that thermal rectification has been observed in the quantum graded harmonic chain of oscillators with inner baths [10,11], but it is absent in the classical version of the same model [12]. A natural candidate for a quantum model describing these transport phenomena would be the quantum version of the chain of anharmonic oscillators, but even its classical version is already a problem of extreme difficulty [13].…”

Effective approach for anharmonic chains of oscillators: Analytical description of negative differential thermal resistance

“…For the quantum version, due to quantum distributions associated to the baths, the thermal conductivity depends on temperature and thermal rectification appears in the graded chain [14,16]. In the high temperature regime, however, the properties of the classical and quantum models become similar (as expected).…”