Phonon Band Structure and Thermal Transport Correlation in a Two-atom Unit Cell

The phononic band structure and thermal conductivity of a family of a two-atom unit cell Lennard-Jones crystals are predicted using molecular dynamics simulations. The structure consists of alternating layers of atoms with different masses, leading to anisotropic thermal properties. An increase in the mass ratio results in an increase in the width of the band gap and a decrease in the value of its central frequency. The thermal conductivity decreases with an increase in the mass ratio, and in all cases is lower than that for a monatomic unit cell. The thermal conductivity increases with an increase in the ratio of the central gap frequency to its width. The rate of this increase decreases and becomes less temperature-dependent at lower mass ratios. The results could be utilized towards the development of guidelines for the design of materials with extreme thermal transport properties. NOMENCLATURE a lattice constant A constant B constant E energy F force vector ∗Address all correspondence to this author. k thermal conductivity kB Boltzmann constant K spring constant m mass m∗ mass ratio N number of atoms r, r particle position, particle separation, particle separation vector S, S heat current, heat current vector t time T temperature v particle velocity vector V volume Greek ∆ band gap εLJ Lennard-Jones energy scale κ wave number σLJ Lennard-Jones length scale τ time constant φ potential energy ω angular frequency 1 Copyright c © 2004 by ASME Subscripts ac acoustic i summation index, particle label j summation index, particle label lg long-range L longitudinal LJ Lennard-Jones op optical sh short-range 1 mass label 2 mass label Superscripts ∗ dimensionless INTRODUCTION The phonon dispersion relation of a crystal describes its frequency space characteristics. Along with phonon scattering, the dispersion plays a significant role in controlling thermal transport in dielectric materials. Limited work has been done to quantitatively relate dispersion characteristics to thermal transport properties such as the thermal conductivity. An example is the use of band diagram features and/or quantities (e.g., phase and group velocities) in the single mode relaxation time formulation of the Boltzmann transport equation [1, 2]. In this case, the dispersion is usually greatly simplified. The importance of accurately modeling the dispersion in such a formulation has recently been investigated [3, 4]. The dispersion is generally treated at a qualitative level. For example, optical phonon dispersion branches are often flat, which results in low phonon group velocities. Based on this fact their contribution to the thermal conductivity is generally assumed negligible. Dong et al. [5] have addressed the connection between dispersion and thermal transport in the prediction of the thermal conductivity of a family of materials made from germanium using molecular dynamics (MD) simulations. In Fig. 1 of their paper, they show the dispersion curves for a diamond structure, a clathrate cage, and for the same cage structure filled with strontium atoms. While the range of frequencies accessed by the vibrational modes in these structures is comparable, the dispersion characteristics are quite different. The large unit cell of the clathrate cage significantly reduces the frequency range of the acoustic phonons, the carriers generally assumed to be most responsible for thermal transport. This results in an order of magnitude reduction in the thermal conductivity. In the filled cage, the guest atoms have a natural frequency that cuts directly through the middle of what would be the acoustic phonon branch, and the thermal conductivity is reduced by a further factor of ten. This work clearly demonstrates how changes in the dispersion characteristics can have a direct impact on the thermal transport. Numerous experimental studies have found similar results [i.e., that guest (rattler) atoms can reduce the conductivity] (e.g., [6, 7]). Apart from these efforts, there have been very few investigations that have attempted to rigorously establish the connection between dispersion characteristics, which include the size and location of band gaps, as well as velocity quantities, and thermal transport properties. The present work explores the three-way relationship in a crystal between: (i) the unit cell structure and properties, (ii) the associated dispersion characteristics, and (iii) the bulk thermal transport behavior. As a starting point, attention is narrowed to a two-atom unit cell. By modeling simple systems, phenomena can be observed that might not be discernable in more complex structures. A Lennard-Jones (LJ) system containing two different masses is considered. Insights from the present study could lead to the development of a systematic technique for the atomiclevel design of materials with desired thermal transport properties. This capability could facilitate the introduction of novel, yet realizable, materials with extremely high, or low, thermal conductivities. Examples of applications include electronic devices enjoying enhanced cooling characteristics and efficient thermal insulators for chemical processing. In the context of the continuum description of periodic materials, the idea of tailoring the frequency band structure has previously been applied in the areas of electromagnetics [8] and elastodynamics [9-11]. We begin by introducing the basic formulations for the MD simulations and thermal conductivity prediction method [the Green-Kubo (GK) approach]. The zero-temperature LJ phonon dispersion curves are then computed for different mass ratios. This is followed by the prediction of thermal conductivity for the different mass ratios over a wide range of temperatures. Discussion is presented with regards to the magnitude of the thermal conductivity, its directional dependence, and relation to the unit cell. The relationship between the thermal conductivity and the associated band structure is then explored. We conclude with a discussion of future research directions. MOLECULAR DYNAMICS SIMULATIONS In an MD simulation, the classical position and momentum space trajectories of a system of particles are determined using interatomic forces (which are calculated from an appropriate interatomic potential) and the Newton laws of motion. Here, systems described by the LJ potential, φi j(ri j) = 4εLJ [( σLJ ri j )12 − ( σLJ ri j )6] , (1) are considered. In Eq. (1), φi j is the potential energy associated with a pair of particles (i and j) separated by a distance 2 Copyright c © 2004 by ASME