Size effect on thermal conductivity of thin films Guihua Tang, Yue Zhao, Guangxin Zhai, Zengyao Li, Wenquan Tao School of Energy & Power Engineering, Xi’an Jiaotong University, China
1 Background 2 Local mean free path method 3.1 3 3.2 4 4 Outline Results 1: Local thermal conductivity distribution Results 2:Overall thermal conductivity Conclusions
1. Background • Boundary or interface scattering becomes important when the characteristic length (film thickness, wire diameter) is comparable with the mean free path. • The thermal conductivity (as well as other transport coefficients, viscosity) becomes size dependent. • Numerous important applications of nanoscale thermal conduction (electronic devices cooling, thermal insulator, thermalelectric conversion, etc.)
Z y The Phonon Gas X is the lattice volumetric specific heat; is the average speed of phonons; is the phonon mean free path. • Specific heat of solid: Lattice vibration in solids. Harmonic oscillator model of an atom • Conduction in insulatorsis dominated by lattice waves or phonons. • Simple expression of thermal conductivity based on the kinetic theory
is bulk mean free path Apply the Matthiessen’s rule • Classical size effect based ongeometric consideration (1) • In the ballistic transport limit, L<<Lb, the MFP is L • L>>Lb, the MFP is the bulk mean free path Lb • Intermediate region:
L A thin film for paths originated from the boundary surface (m≈3) Simple interpolation between the two expressions • Classical size effect based ongeometric consideration (2) • When L<<Lb, assuming that all the energy carriers originate from the boundary surface • L>>Lb, the MFP is the bulk mean free path Lb
L (m≈3) A thin film for paths originated from the centre Interpolation • Classical size effect based ongeometric consideration (3) • The direction of transport was not considered and the anisotropic feature cannot be captured • Filk and Tien employed a weighted average of the mean free path components in the parallel and normal directions of a thin film
L Interpolation (m≈4/3) • Classical size effect based ongeometric consideration (4) A thin circular wire for paths originated from the centre
Thin Film Thin Wire p is the probability of specular scattering on the boundary • Classical size effect based onBoltzmann Transport Equation (BTE) • The relaxation time approximation was adopted. • The distribution function was assumed to be not too far away from equilibrium.
For a thin film: 2. Local mean free path method • For an unbounded phonon gas, the probability of a phonon gas can travel between two consecutive collisions with other phonons at location x and x+dx would be of the form: The probability of a phonon gas having a free path between x and x+dx • When the gas is bounded, a number of phonons will be terminated by the boundary, thus effective MFP < Lb
3. Results Local thermal conductivity distribution in a semi-infinitefilm
L Local thermal conductivity distribution in a thin film
4. Conclusions • An equation to calculate the size-dependent film thermal conductivity has been derived. No Matthiessen’s rule; No interpolation • Local thermal conductivity distribution in the thin film has been obtained. • The present solution seems to overpredicts reduction in thermal conductivity compared to the data in references when Knudsen number is larger than 1. • More cases are needed for further validation and extension to complicated geometric structures.
Thanks for your attention! 09/07/2010