Boltzmann transport mean free path

The same k p scheme has been extended to the study of transport properties of CNTs. The conductivity calculated in the Boltzmann transport theory has shown a large positive magnetoresistance [18], This positive magnetoresistance has been confirmed by full quantum mechanical calculations in the case that the mean free path is much larger than the circumference length [19]. When the mean free path is short, the transport is reduced to that in a 2D graphite, which has also interesting characteristic features [20]. 

The early theories for the transport coefficients were based on the concept of the mean free path. Excellent summaries of these older theories and their later modifications are to be found in standard text books on kinetic theory (J2, K2). The mean-free-path theories, while still very useful from a pedagogical standpoint, have to a large extent been supplanted by the rigorous mathematical theory of nonuniform gases, which is based on the solution of the Boltzmann equation. This theory is... 

Up to now we have discussed two extreme limits, the band picture on the one hand, and strong localization associated with interruptions in the metallic chains on the other. In fact, from work on thin metallic films and metallic glasses it is known that there is an intermediate region, that of weak localization. This occurs when the mean free path for elastic scattering (Lel) is only somewhat larger than, or comparable with, that for inelastic processes (Lin). In the first approximation there are corrections to the Boltzmann transport formula which depend on the ratio Lin/Lel in different ways for one-, two-, and three-dimensional materials. Weak localization... 

The lattice constant of the x = 3 face-centered cubic unit cell is 14.28 A (4). Accordingly, the carrier density is 4.1 x 10 cm , with four C o molecules and twelve donated electrons p>er unit cell. This charge density corresponds to a Fermi wave vector kf = 0.50 A which, when substituted into a Boltzmann equation description of the minimum resistivity gives = 2.3 A for the electronic mean free path. This unphysically small implies that, even at X = 3, the Boltzmann equation is inadequate for describing a system where intergranular transport may still be limiting the conductivity. 

That is, instead of determining the transport properties from the rather theoretical Enskog solution of the Boltzmann equation, for practical applications we may often resort to the much simpler but still fairly accurate mean free path approach (e.g., [12], section 5.1 [87], chap. 20 [34], section 9.6). Actually, the form of the relations resulting from the mean free path concept are about the same as those obtained from the much more complex theories, and even the values of the prefactors are considered sufficiently accurate for many reactor modeling applications. 

To avoid the account of the edge effects let us consider rather long structures (L > 50 nm), i.e. we will consider the armchair single-wall carbon nanotubes with the length greater than electron mean free path [2-6]. To describe the electron-phonon transport in nanotubes like that the semiclassical approach and the kinetic Boltzmann equation for one-dimensional electron-phonon gas can be used [4,6]. In this connection the purpose of the present study is to develop a model of electron transport based on a numerical solution of the Boltzmann transport equation. 

Maxwell [95] was able to obtain these fairly accurate expressions for the transport coefficients which describe their primary dependence of upon temperature, pressure, mass and size of the molecules in the gas based on rather crude arguments. Historically, the mean free path theory given by Maxwell [95] predates the more accurate theory based on the Boltzmann equation by about half a century. 

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