Phonon Boltzmann equation

If, by internal means, the system is near c2> only a small amount of energy is necessary to drive the system into the highly excited state. Furthermore, we have been able to show that oscillations on the hysteresis are possible for A detailed inspection of the transport equations (generalizeS nonlinear Peierls-Boltzmann equations for phonons) shows that nonlinear kinetics, dissipation and energy supply via transport are indispensable for such a behaviour to occur. 

If the exciton-phonon coupling is sufficiently weak, the solution of the equation for the correlation function (B (t)Bm(t)B, (0)Bm (0)) is equivalent to the solution of the Boltzmann equation (11). In this coherent limit, the exciton states of... 

The kinetic Boltzmann equation was solved to describe the non-stationary electron-phonon transport in armchair single-wall carbon nanotubes. The equation was solved numerically by using the finite difference approach. The current in the armchair singlewall carbon nanotubes was calculated. 

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. 

Briithng et al. have treated the conductivity of quasi-one-dimensional CDW systems in the framework of the Boltzmann equation (see e.g. [M2] or [M3]) with scattering from longitudinal acoustic phonon [23, 24]. For the temperature dependence of the conductivity u(T), they derived an integral equation which, aside from the temperature-dependent energy gap 2A(7), contains only a single materials parameter C ... 

Schnakenberg.J, Electron-Phonon Interaction and Boltzmann Equation in Narrow Band Semiconductors (Vol. 51)... 

If I ib phon neglected altogether the charge carriers and phonons are completely decoupled. The carrier wavefunction is described by a simple band with dispersion relation given by k) = —2tcos kL) (k is the wavevector and L the intermolecular distance). The carriers are completely delocalized and their motion is usually described semiclassically by the Boltzmann equation [58]. Carrier with wavevectors k are scattered to a new state k by impurities or phonons. In the simplest possible case, the average time between two collisions can be taken as a constant h independent from the initial state k. It can be further assumed that the distribution of final states k after a collision is simply the equilibrium distribution. The mobility under all these assumptions can be written as... 

The Bloch-Griineisen formula is a special case of a more general expression. Within a variational solution of the Boltzmann equation we can, to a good approximation, write the resistivity Pei-ph that is limited by the scattering of conduction electrons by phonons as... 

About Rep, it decreases as temperature decreases, due to the fact that the number of phonons decreases. A full treatment of the problem, however, can only be obtained by solving the Boltzmann transport equation, which has only been solved for the case of quasi-free electrons. Further information and approximate solutions can be found in ref. [7,106,107], The general result of these calculations shows that at low temperature T < 0D/1O), the thermal resistance Rep is of the form b- T2. 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

Ladd, A., B. Moran, and W.G. Hoover, Lattice Thermal Conductivity A Comparison of Molecular Dynamics andAnharmonic Lattice Dynamics. Physical Review B, 1986. 34 p. 5058-5064. McGaughey, A.J. and M. Kaviany, Quantitative Validation of the Boltzmann Transport Equation Phonon Thermal Conductivity Model Under the Single-Mode Relaxation Time Approximation. Physical Review B, 2004. 69(9) p. 094303(1)-094303(11). 

Raman lines corresponding to the optical phonons can be seen symmetrically shifted from the incident laser line. For ease of eomparison to IR speetroscopy, frequency shifts are expressed in wavenumbers (cm ). The Stokes shift is most commonly measured at room temperatures, as from simple thermodynamics using Boltzmann s equation, there are very few vibrations in most materials at room temperature which can contribute to anti-Stokes scattering. 

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