Notes9 - ECE215B/Materials206B Fundamentals of Solids for...

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ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 1 Transport Theory #4 Boltzmann Formalism We have seen how ionized-impurity scattering of charged particles is associated with a strong dependence of scattering time τ on the particle kinetic energy U* K . In addition we expect that τ and perhaps other aspects of the transport process, such as the number of particles involved, will depend on particle identity, i.e., classical (Maxwell-Boltzmann), Fermionic, etc. This leads us to the Boltzmann transport formalism, arguably the most important advancement of transport theory ever made because of its inherent ability to handle multiple physical effects automatically and self-consistently. 1 At the foundation of Boltzmann are a set of three axioms each far more plausible than those of kinetic theory. (1) First, each particle in the transport process can be described by a point in mechanical “phase space” spanned by position and momentum, p r G G & . This phase space is six dimensional and is shared by all N particles of the population. In other words, at any moment in time, there are N points in this space, each designated by & rp ii G G where i is the particle index. (2) The evolution of each particle point in time is given by the equations of motion which, in turn, depend on the external forces through the appropriate mechanical laws of motion. When Boltzmann developed the formalism, the only known laws where those of Newton, for which i i dr pm dt = G G ( 1 ) i i dp F dt = G G (2) where F is the external force and m is the particle mass. 1 while we apply the Boltzmann transport only to solids, it is equally well suited to fluid transport, plasma transport, and even radiative (i.e., photonic) transport.
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ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 2 (3) The entire population of particles is described by a probability distribution function f that is the mean number of particles at each point in phase space at any given time. Mathematically we can write, f = [( ) , ( ) ] frt pt ii GG . So by the chain rule of elementary calculus, ballistic dr dp df f f dt r dt p dt ∂∂ =− G G (3) The first term is called the diffusion term since it is proportional to the gradient of the particle number in real space (i.e., with respect to r). The second term is called the drift term since from (2) it is proportional to the force on the particle. The subscript “ballistic” reminds us that there is nothing in (3) to account for the collisions (or scattering) of the particles necessary to maintain an equilibrium state in the solid. The negative signs in (3) are important here and represent the fact that the particle transport always tends to go in the opposite direction to the gradient vs r G or p G . To see this, we examine a specific example of (3), shown diagrammatically in Fig. 1, of one-dim diffusive (along x axis) transport in r G space with v x uniform (independent of x) and positive, and no applied forces. This implies 0; v df df dr dx x dr dx dt dt ⇒=> =≡= constant.
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This note was uploaded on 01/17/2012 for the course ECE 215B taught by Professor Brown during the Spring '08 term at UCSB.

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Notes9 - ECE215B/Materials206B Fundamentals of Solids for...

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