Size-dependent Model for Thin Film Thermal Conductivity

We present a closed-form classical model for the size dependence of thin film thermal conductivity. The model predictions are compared to Stillinger-Weber silicon thin film thermal conductivities (in-plane and cross-plane directions) calculated using phonon properties obtained from lattice dynamics calculations. By including the frequency dependence of the phonon-phonon relaxation times, the model is able to capture the approach to the bulk thermal conductivity better than models based on a single relaxation time. INTRODUCTION Thin films are ubiquitous in electronic and optoelectronic applications. By confining materials to sub-micron dimensions, it is possible to control the transport of electrons and phonons. For example, the quantum wells in the active region of a lightemitting diode (LED) allow for the recombination of electrons and holes to produce photons [1, 2]. The closely-spaced interfaces in a semiconductor superlattice scatter phonons while allowing electrons to pass, qualities necessary for a good thermoelectric material [3–5]. ∗Address all correspondence to this author. The thermal conductivity of a thin film in both the in-plane and cross-plane directions is less than that of the corresponding bulk material [6–9]. The reduced thermal conductivity makes it difficult to remove excess heat (e.g., generated by non-radiative recombination in an LED). This reduction is due to (i) changes in the phonon density of states (sometimes referred to as phonon confinement), and (ii) phonon-boundary scattering. The density of states effect is only expected to be important in very thin films {e.g., thinner than about 20 nm for a Stillinger Weber (SW) silicon film with free surfaces [8] and thinner than about 2 nm for SW silicon thin films bounded by large extents of SW germanium [10]} and is not discussed here. Our focus is on the boundary scattering effect in films with bulk-like density of states. Modeling work at different levels of sophistication has attempted to predict the thermal conductivity reduction in thin films. Flik and Tien [11] and Majumdar [12] proposed simple models based on a single phonon group velocity and mean free path (i.e., under the gray approximation). Sellan et al. used lattice dynamics calculations to predict phonon properties for the full Brillouin zone of bulk Stillinger-Weber (SW) silicon [9]. From these, they included boundary scattering using the Matthiessen rule and predicted thin film thermal conductivity in the cross-plane direction using a solution to the Boltzmann trans1 Copyright c ⃝ 2011 by ASME port equation (BTE). They found that their thermal conductivities approached the bulk value at a slower rate than the Majumdar model. Turney et al. [8] found a similar result for the in-plane direction using a boundary scattering model that does not require the Matthiessen rule. While accurate and allowing for consideration of all the phonon modes, the required lattice dynamics calculations are computationally demanding and do not allow for quick calculations for a range of materials. In a recent paper, Sellan et al. proposed a closed-from classical model for the cross-plane thermal conductivity of a thin film and used it to assess size effects in molecular dynamics thermal conductivity predictions [13]. Here, we compare the predictions of their model, along with an extension to the in-plane direction, to lattice-dynamics based predictions of thin film thermal conductivity. We find that the Sellan model better captures the trends in the lattice dynamics thermal conductivity predictions compared to the Flik and Majumdar models, particularly for the in-plane direction. MODEL DERIVATION We first review the Sellan thermal conductivity model [13]. The film has thickness L and is shown in Fig. 1. The in-plane and cross-plane directions correspond to the x and z directions. The derivation begins with an expression for the thermal conductivity in the i direction that is obtained by solving the BTE under the relaxation time approximation and using the Fourier law [14,15]: ki(L) =∑ ν ∑ κ cphvg,i(κ ,ν)τ(κ ,ν ,L). (1) The summation is over all phonon modes with wave vector κ and dispersion branch ν , the volumetric specific heat of each mode, cph, is kB/V for the classical systems considered here, and vg,i(κ ,ν) is the i component of the group velocity vector, vg(κ ,ν). The phonon transport is described using a set of modeand system size-dependent relaxation times, τ(κ ,ν ,L), defined as the average time between successive scattering events. The mode-dependent phonon mean free path, Λ(κ ,ν ,L) is defined as |vg(κ ,ν)|τ(κ ,ν ,L). We begin by converting the summation over all phonon modes in Eq. (1) to an integral over the first Brillouin zone, resulting in ki(L) = V 8π3 ∑ν ∫ cphvg,zτdκ . (2) Considering the cross-plane (z) direction, assuming that optical phonons do not contribute to thermal conductivity, and transz, Cross-Plane