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# lecture18 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 18 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 18 In last lecture, we deal with the Boltzmann transport equation s f dd f ff td t d t t ∂∂ ⎛⎞ +• ∇= ⎜⎟ ⎝⎠ rp where the subscripts ( r and p ) in the gradient operators represent the variables of the gradient. The scattering term o s f f f t τ =− was discussed based on two particle interactions. Here () , τω k is the relaxation time. For equilibrium distribution f 0 , we have 2 0 3/2 - 2 1 Bose-Einstein distribution (phonon) 1 1 f Fermi-Dirac distribution (electron) 1 Displaced Maxwell velocity distribution (molecules) 2 B B B kT E m B e e m ne ω µ π = + vu = Note: The relaxation time is due to combined factors and can be evaluated numerically by adding all possible influence together. This idea can be used in calculating the band structure by the Boltzmann transport equation. For two-particle interactions (above figure (a)), we have 12 3 ωω += == = (energy conservation), 31 + 2 G+k =k k (momentum conservation), where zero G corresponds to normal process, otherwise it is umklapp scattering. Generally speaking, electrons will collide with electrons, phonons, and impurities in the crystals. The electron-phonon scattering causes the electrical resistance. k 1 , ν 1 k 2 , ν 2 ν 3 = ν 1 + ν 2 k 3 ν 1 , k 1 ν 2 , k 2 ν 3 , k 3 (a) (b)

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2.57 Fall 2004 – Lecture 18 2 Note: At low temperatures, phonon has low energy and the electron-phonon scattering is negligible. Therefore, the impurity-electron scattering is the main cause of electrical resistance. For gas molecules, in Lecture 2 we have derived the collision obeys 2 1 2 nD π Λ= , v τ . Note: (1) To simplify, most time we view ( ) , τω k as ( ) . (2) The time () , k is only applicable to elastic scattering (see chapter 8), such as electron-electron scattering (energy conserved). It is not accurate for electron-phonon scattering. In this situation, we have o ep s ff f gT T t
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lecture18 - 2.57 Nano-to-Macro Transport Processes Fall...

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