Hierarchical Modeling of Thermal Transport from Nano-to-Macroscales

Understanding and predicting thermal transport at extremely short time and length scales is essential to further advance a variety of emerging technologies. The trend towards miniaturization of electronic devices has lead to device features in the sub-micron and nanometer range. In fact, transistors with gate lengths on the order of 65 nm are already in production, and silicon-on-insulator (SOI) transistors are predicted to reach a gate length of 45 nm by the year 2005 [1]. The thermal conductivity of silicon at these scales is smaller than that of bulk silicon due to the scattering of the energy carriers with the boundaries of the device [2, 3]. Heat dissipation becomes an important issue, since overheating can negatively affect the reliability of the transistor. In addition to applications in microelectronics, micro/nanostructures such as quantum wells, nanowires, nanotubes and superlattices (periodic arrays of thin films), are appearing in optoelectronic devices [4], semiconducting lasers [5] and thermoelectric applications [6]. The thermal properties of nanowires [7-9], superlattices [10-12] and nanotubes [13] have recently attracted attention because of their special characteristics [8, 9, 14-16]. Carbon nanotubes, for example, have the potential for very high thermal conductivity [13, 17], while the thermal properties of superlattices can be controlled by adjusting the thickness and composition of its components. Theoretical modeling can be an important tool to further the understanding of thermal transport. Different approaches are needed to study the thermal properties of devices whose components span length scales from the nanoscale to the macroscopic [18]. Phonons, which are quantized lattice vibrations [19, 20], are the predominant energy carriers in dielectrics and undoped semiconductors. Fourier diffusion yields erroneous predictions under the following two conditions: (a) when the mean free path of the energy carriers ( ) becomes comparable to or larger than the characteristic length scale of the device/system under consideration (L); and/or (b) when the time scale of the processes under consideration becomes comparable to or smaller than the relaxation time of the energy carriers [15, 21, 22]. Under these conditions, if the wave nature of the energy carriers can be neglected [23], the semi-classical Boltzmann transport equation (BTE) can be employed to solve for the distribution function of the energy carriers. Molecular dynamics (MD) involves solving the atomistic equations of motion, from which phonon and thermal properties can be obtained. Being an entirely classical method, molecular dynamics is strictly applicable to solids only at temperatures above Debye’s temperature (for silicon, D ~ 625K [19]), where the system behaves classically. Several corrections have been proposed in the literature to partly account for quantum effects in molecular dynamics simulations of thermal transport when these effects become relevant, such as when the temperature of the system is smaller than its Debye temperature, as reviewed in [24]. However, when the temperature and dimensions of the system are such that the characteristic