lecture3 - 2.57 Nano-to-Macro Transport Processes Fall 2004...

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2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 3 8. Micro & Nanoscale Phenomena 8.1 Classical size effects In section 7, the characteristic length of the box is much longer than the mean free path Λ . Therefore, the collisions between molecules and the wall are neglected in our derivation and thermal conductivity is regarded as the bulk property of the gas. However, there are many applications in which Λ becomes comparable or larger than the size of the system. The classical size effects occur in such situations. Example 1: Λ > d for a disk drive Example 2: Λ ~ d or Λ > d for thin films d=10~20 nm Λ =100 nm Thin film k Disk drive In example 2, we can further reduce the film thickness to enhance the size effects. With measured data for k and specific heat c , the mean free path in silicon can be estimated by cv Λ k = 3 , where v is sound velocity. The approximated mean free path Λ is around 40 nm, while the actual value is around 300 nm. The size effects occur for silicon films with thickness less than Λ . Note: in some thermal insulation applications, we also use porous materials whose pore sizes are comparable to or less than Λ . The thermal conductivity of the air trapped in the pores will be significantly reduced. 8.2 Quantum size effects According to quantum mechanics, electrons and phonons are also material waves; the finite size of the system can influence the energy transport by altering the wave characteristics, such as forming standing waves and creating new modes that do not exist in bulk materials. For example, electrons in a thin film can be approximated as standing waves sitting inside a potential well of infinite height. The condition for the formation of such standing waves is that the wavelength, λ , satisfies the following relation D = λ n / 2 ( n =1,2,…), where D is the width of the potential well. According to the de Broglie relation, the wavelength is / λ = h p , 2.57 Fall 2004 – Lecture 3 1
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where h is Planck constant ( h =6.6x10 -34 Js), p is momentum. The energy of the electron n is thus E = p 2 /2 m or E = p 2 = h 2 ( ) 2 . 2 m 8 m D n=1 n=2 D 1) For a free electron, D =1 mm, we have 2 34 2 E = n ( 6.6 × 10 34 ) 2 ~10 n << k T = 4.14 × 10 21 J at room temperature. × 10 3 B 8 × 9.1 10 31 2) For D =1e-8 m, we calculate E k T . Further reducing D results in more observable E. > B 8.3 Fast transport For many materials, we have τ = 10 12 10 11 s . Laser pulse can be as short as a few femtosecond, we cannot use diffusion theory when the time scale is shorter than the relaxation time.
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