International Journal of Thermal Sciences 48 (2009) 1467–1476 Contents lists available at ScienceDirect International Journal of Thermal Sciences www.elsevier.com/locate/ijts Prediction of the thermal conductivity anisotropy of Si nanofilms. Results of several numerical methods Damian Terris a , Karl Joulain a , Denis Lemonnier a , David Lacroix b,∗ , Patrice Chantrenne c a LET, UMR 6608 CNRS, ENSMA, Université de Poitiers, B.P. 40109, 86960 Futuroscope Cedex, France b LEMTA, Nancy Université, CNRS, B.P. 70239, 54506 Vandœuvre Cedex, France c CETHIL, UMR 5008 CNRS, INSA-Lyon, Université Lyon 1, 69621 Villeurbanne Cedex, France article info abstract Article history: Received 28 July 2008 Received in revised form 9 January 2009 Accepted 9 January 2009 Available online 3 February 2009 PACS: 63.20.-e 65.80.+n 02.60.Cb Keywords: Nanofilm Silicon Thermal conductivity Numerical simulation The purpose of this work is to predict the in-plane and cross-plane thermal properties of crystalline silicon films. Several thicknesses from 20 nm to 6 μm and mean temperatures between 20 and 500 K have been investigated. Heat transport properties in silicon films have been studied through three different techniques: a semi-analytical method based upon the Kinetic Theory, a deterministic solution of the Boltzmann Transfer Equation (BTE) through the Discrete Ordinate Method and a statistical handling of the BTE by means of Monte Carlo Method. Each technique requires a model for the bulk material dispersion curves and the collision times of the different scattering processes. The three techniques have been validated through their correct prediction of silicon bulk thermal conductivity. Comparisons with in-plane thermal conductivity calculations and measurements have been also discussed. Thus, the cross-plane thermal conduction properties have been predicted. The expected temperature and thickness variations of the thermal conduction properties have been observed: the cross-plane thermal conduction appears to be less efficient than the in-plane thermal conduction, which proves that a significant anisotropy exists.  2009 Elsevier Masson SAS. All rights reserved. 1. Introduction Nanotechnologies development is one of the major current is- sues in electronics. Transistors with nanometric sizes are now commonly accepted and already achieved in laboratories [1]. For such devices heat dissipation is a challenging problem that has to be solved [2,3]. Moreover, nanostructured technology improvement reaches limits for which the classical heat transfer laws are not valid [4]. Consequently, thermal properties of these nanostructured materials need to be known as their typical size goes down. Cross-plane and/or in-plane thermal conductivity measure- ments have already been performed for superlattices [5–8] and single nanofilms [9–13]. Thermal conductivity exhibits a strong anisotropy in superlattices which is also expected for nanofilms. Nevertheless, these measurements are not numerous. Despite, thermal properties are key parameters for multilayered processor design. As a consequence, development of numerical tools aimed at the assessment of thermal properties in nanostructures has en- countered a wide interest for semiconductors of various shapes (nanofilms, nanotubes, nanowires, ...). * Corresponding author. E-mail addresses: Karl.Joulain@ensma.fr (K. Joulain), David.Lacroix@lemta.uhp-nancy.fr (D. Lacroix), patrice.chantrenne@insa-lyon.fr (P. Chantrenne). In semiconductor crystals, solid state physics teaches us that heat flows through the lattice vibrations and heat carriers can be seen as quasi-particles called phonons. Historically, thermal con- ductivity in crystalline materials has been derived from the phonon mean free path [14], using the gas Kinetic Theory. Precursor studies by Klemens [15,16], Callaway [17] and Holland [18] have put into practice this method assuming the single relaxation time approx- imation to model the collision term occurring in the Boltzmann transport equation. With this formalism, boundary and impurity scattering as well as three phonon processes (normal and umk- lapp) contribute to a single relaxation time. The resulting models have successfully predicted the bulk thermal conductivity for vari- ous semiconductors (Si, Ge, with several dopant concentration) at low and high temperatures. One of the three methods used here is based on this approach and is semi-analytical [19]. This method is the fastest to implement. It easily gives values of the thermal conductivity. The other two methods are based upon the resolu- tion of a transport equation (BTE). Majumdar [20] has developed such a phonon radiative transfer equation (PRTE) which is derived from the BTE introducing a phonon specific intensity through the multiplying factor v¯ hωD(ω) (where v is the phonon velocity in a given direction and D(ω) the density of states). The PRTE can be solved by deterministic means such as those widely developed for neutron and photon propagation. Radiation numerical modeling tools [21,22] like the Discrete Ordinate Method (DOM), used in this 1290-0729/$ – see front matter  2009 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2009.01.005