Linearized Boltzmann Transport Equation Research Articles

Mobility and hall factor calculations for a superlattice

Low field mobility and Hall factors, as a function of the superlattice period, are calculated for a non-degenerate electron gas in a miniband of a GaAs/GaAlAs superlattice with a band offset of 250 meV at 300 K. Interface roughness and polar optical phonon scattering are considered. The effect of the superlattice structure on the phonon modes is taken into account by using the dielectric continuum model, which has been shown to be a good approximation. Mobilities parallel and perpendicular to the interfaces are calculated when the magnetic field B = 0, from iterative numerical solutions of the linearized Boltzmann transport equation. It is shown that these solutions may be used to calculate the Hall factors when B is small. Interface roughness scattering dominates the mobility behavior when the superlattice period d < 90 A ̊ . We find two independent Hall factors, both of which remain close to 1 for all values of the superlattice period.

NEUMANN SERIES SOLUTION FOR THE INTEGRAL BOLTZMANN TRANSPORT EQUATION

In this paper, a method based on Neumann series is proposed for solving stationary linear integral Boltzmann transport equations with variable energy for the case of both distributed and isotropic deltalike sources, by working in Lp(1⩽p⩽∞). The convergence of the Neumann series and convergent rate are established. In addition, a relation between stationary transport equation and time-dependent one is given.

Numerical methods for the linear Boltzmann transport equation in slab geometry

AbstractAn iterative method for computing numerical solutions of a finite‐difference system corresponding to the linear Boltzmann equation in slab geometry is presented. This iterative scheme gives a straightforward marching process starting from the given boundary and initial conditions. It is shown that with a suitable initial iteration the sequence of iterations converges monotonically to a unique solution of the finite‐difference system. This monotone convergence leads to improved upper and lower bounds of the solution in each iteration, and to the well‐posedness of the discrete system in the sense of Hadamard. It also leads to the convergence of the discrete system to the continuous system as the mesh size of the space–velocity–time variables approaches to zero. Under a mild restriction on the time‐increment the discrete system is numerically stable, independent of the mesh‐size of the space and velocity. An error estimate for the computed solution due to simultaneous initial and iteration error is obtained. Also given are some numerical results for the time‐dependent and the steady‐state solutions.

Discrete Symmetries of the Linear Boltzmann equation

Abstract Group theoretical considerations originating from Wigner are applied to the few group diffusion equation (DE) and to the linear Boltzmann transport equation (TE). The irreducible subspaces invariant under the symmetries of the TE or DE determine specific incoming current patterns. If these patterns are used in the response matrix formulation, the response matrix will be diagonal. A method for obtaining a solution that belongs to a given irreducible subspace is provided. These considerations are utilized in a coarse-mesh program. It is shown that the solution to a boundary condition problem is determined by the irreducible decomposition of the boundary condition and by the solutions belonging to the irreducible boundary conditions. Finally, the solution of the TE in an infinite lattice is investigated and the results are utilized to solve the TE in a finite lattice.

Theoretical study of the linear dispersion relation and the stationary energy-transport velocity of a coupled polar LO phonon-plasmon system

A semiclassical investigation of the linear dispersion relation and the stationary energy-transport velocity associated with coupled LO phonon-plasmon waves in polar semiconductors is made. The influence of a free-carrier drift velocity on the frequency dispersion of the modes is considered. The complex wave vector of the coupled mode is obtained from the roots of an effective dielectric function which has been calculated on the basis of the appropriate linearized Boltzmann transport equation, the Maxwell equations, the long-wavelength equation of motion of the polar-optical lattice vibrations, and the Born and Huang expression for the lattice polarization. The minor contribution arising from the collision-drag effect associated with impurity scattering is included in the general framework of the theory. With main emphasis on small or even vanishing electronic drift velocities, inverse dispersion relations valid for low-density Maxwellian plasmas are obtained and discussed. The limiting cases, outside the quantum region, where local or extremely nonlocal approaches can be adopted, are examined. By means of a simple nonlinear Boltzmann equation the energy-balance equation associated with the LO phonon-plasmon mode is derived. A general expression for the stationary energy-transport velocity associated with damped (or amplified) coupled modes is established. Finally, the basic concepts of velocity of energy propagation are applied to classical plasmas.

Space dependent electron transport by an integral approach

The aim of the present paper is to study the diffusion of a rarefied electron gas in a random distribution of field particles, and in presence of an impressed external electric field. Electrons are emitted by a plane source, and undergo scattering or ionization collisions against field particles. The analysis is carried out resorting to the linear Boltzmann transport equation for the electron distribution function, and following a recently developed integral formulation. Several results of physical interest are presented and briefly discussed.

Asymptotic solution to the Boltzmann transport equation for degenerate polar semiconductors

A new method for solving the linearized Boltzmann transport equation, originally developed for metals, is applied to degenerate polar semiconductors assuming that the electrons are scattered primarily by phonons of the optical mode. The solution is represented as an asymptotic power series in a non-dimensional temperature variable, whose coefficients are polynomials in a non-dimensional energy variable. The approximate analytic form of the nth late term in the series is found to be strikingly similar to the corresponding term in the previous application of the method to metals. The asymptotic series may be summed by the method of terminants, and the electrical conductivity is obtained to a high accuracy, even at low temperatures.

A few analytical solutions to the linear Boltzmann transport equation with an application to channeling

Some exact solutions to the linear Boltzmann transport equation in a one-dimensional space are presented. These results may be applied to demonstrate the trend towards statistical equilibrium of fast channeled particles in the transverse plane. Statistical mechanics of channeled particle motion is then discussed in terms of the random string approximation and further considerations.

General calculation of the collision integral for the linearized Boltzmann transport equation

We develop and describe a general method for evaluating collision integrals in the linearized Boltzmann transport equation which eliminates the necessity to repeat similar integration steps for each force law. Integrations not dependent on scattering cross-section variables have been carried out once and for all. The two mathematical innovations which facilitate these general integrations are (i) the development of an expansion of the Burnett functionXNML(x+y) into products of Burnett functions of argument x with other functions; and (ii) the use of representations of the full rotation group to transform from space-fixed axes to axes aligned with the relative velocity vector of colliding atoms. The relations so derived allow rapid evaluation of the collision integral from a knowledge of the scattering cross section.

Sound propagation and relaxation times for a fermi liquid

The techniques for solving exactly the linearized Boltzmann transport equation of a Fermi liquid are applied to sound propagation. By separating out the collision integral into spherical harmonics, we show that different circular polarizations propagate independently; the present derivation is believed to be more general than previous ones. As an aid to calculation, a set of relaxation times is defined, there being one time for each spherical harmonic of the disturbance. The relation between these relaxation times and the single relaxation time introduced by Khalatnikov and Abrikosov is discussed. Using the relaxation times, we set up equations by which the dispersion of sound (longitudinal waves) can be calculated without approximating the collision integral. In the hydrodynamic limit, these equations have a simple solution, and we show that the use of a single relaxation time is rigorously valid in this limit. A relation is derived between this relaxation time and the viscosity, and the attenuation of sound is calculated without appeal to hydrodynamics. A similar analysis for transverse waves shows that in the hydrodynamic limit a single relaxation time is again valid, and this time is shown to be equal to the one that describes sound.

Rarefied gas flow between parallel plates

AbstractThe flow of a rarefied gas between parallel plates has been studied via the linearized Boltzmann transport equation. Using a general boundary condition, which includes an arbitrary ratio of specular to diffuse reflection from the wall, we have derived an integral equation for the mass flow velocity. The integral equation is solved by using a replication property of the kernel and application of the method of Muskelishvili.The total volumetric flow rate is obtained and a slip boundary condition is deduced for use with the hydrodynamic equations.Certain aspects of the eigenvalue spectrum associated with the Boltzmann equation are discussed.

Biorthogonal Angular Polynomial Expansion of the Two-Dimensional Transport Equation

A new system of biorthogonal polynomials is developed for the angular expansion of the directional flux in the linear Boltzmann transport equation. It is shown in systems infinite in one space dimension that the angular integral in the Boltzmann equation can be reduced to a weighted integral over the unit circle. The corresponding system of orthogonal functions is found to be a system of two sets of polynomials in two variables. Recursion relations and an addition theorem are derived for these polynomials. The angular dependence of the particle flux is expanded in each set of these polynomials. Systems of partial differential equations are derived for the expansion coefficients, that is, for angular moments of the particle flux. One of these systems is shown to be a specific linear combination of the equations obtained when the directional flux is expanded in spherical harmonics functions specialized for the geometry considered. It is shown that this same system, in (x, y) geometry, reduces simply to the spherical harmonics equations in one-dimensional plane geometry.

Numerical solution of the Boltzmann Transport Equation

A numerical model for solution of the linear Boltzmann Transport Equation is formulated. By applying the same techniques used in the derivation of the analytic equation, a discrete analog of the Boltzmann equation is derived for a finite cell in phase space. Initially undetermined coefficients in the analog are determined by requiring the numerical formulation to include properties (e.g., particle conservation) of the analytic equation. Terms occurring in the finite-cell analog are defined, and two treatments of the angular dependence are illustrated. A discrete ordinates representation is derived based on a connected straight-line-angular representation. This formulation maintains optical reciprocity and may be generalized. The second portion of the paper describes the systematic derivation of difference relations necessary to complete solution of the numerical formulation. Both representation schemes, based on assumed forms of particle fluxes in the cell, and characteristic schemes are examined. Difficulties encountered in extrapolations made by the use of representation schemes are clarified, and insight is gained for methods that can be used to prevent flux oscillations in numerical calculations.

Magnetic field dependence of electromagnetic fields in a semi-infinite metal: General formalism

There have been several recent measurements of the highfrequency surface impedance of very pure metals at very low temperatures in variously oriented magnetic fields ranging in strength from just a few gauss to many thousands of gauss. In order to understand these we have been led to examine the explicit effect of the surface of the metal on the electronic motion and the resulting electric field distribution. The linearized Boltzmann transport equation has been solved so as to include all discontinuities in the distribution function which arise from diffuse scattering of electrons at the metal-vacuum interface for an arbitrary Fermi surface and static magnetic field orientation. We are led to a partial differential equation which must be satisfied by the parameter characterizing the earliest time that an electron can scatter onto a given trajectory. Expressing the bulk scattering of electrons as a relaxation into the local equilibrium distribution assures us that the equation of continuity connecting the current and the charge density is satisfied. Combining the current calculated from our distribution function with Maxwell's equations, we find a system of singular (or an alternative system of nonsingular) integral equations for the selfconsistent electric field transform. In order to discover the form of the singularities in the electric field transform we examine the singular form of the integral equations and introduce a many dimensional Hilbert problem, the dimensionality being determined by the particular Fermi surface and magnetic field orientation. Even though in general the many dimensional Hilbert problem has not been solved, in the regime where our methods are necessary (nonlocal behavior, anomalous-skin-effect conditions) we can obtain a good approximate factorization into a set of one-dimensional Hilbert problems which can be readily solved. Moreover, the general form of our solution for the Fourier transformed electric field allows us to exhibit explicitly all of its singularities. Besides the infinite-medium inverse dielectric tensor, the singularities are also determined by another factor which is related to the discontinuous transport of current into the bulk of the metal; the

The General Solution of the Boltzmann Transport Equation and Boundary-Value Problems

The general solution of the linearized Boltzmann transport equation is derived for an infinite medium, a semi-infinite medium and a medium bounded by two parallel planes. The particular solution required by the nature of the boundary scattering of the electrons is also obtained for these bounded media, subject to the following assumptions: a fraction p of the electrons arriving at the boundary of the medium is scattered specularly, while the rest are scattered diffusely; p is independent of the direction of motion of the electrons. The general solution of the linearized Boltzmann transport equation agrees completely with Chambers' expression, and the formulation presented by Mattis and Dresslhaus, Rodriguez, and Suzuki et al. , all of whom have used the specular reflection boundary condition for the semi-infinite medium, is shown to be essentially correct.

Sound Absorption in Liquid Helium II,T&lt;0.5°K

The expression $\ensuremath{\Gamma}=\frac{A\ensuremath{\Omega}{T}^{4}}{\ensuremath{\rho}{{c}_{0}}^{3}}$ for the absorption of sound in liquid ${\mathrm{He}}^{4}$ II for $T&lt;0.5\ifmmode^\circ\else\textdegree\fi{}$K, where $T$ is the absolute temperature, is shown to be a simple consequence of the principles of the excitation theory of a Bose liquid and a few other reasonable postulates. ($\ensuremath{\Gamma}$ is the absorption, $\frac{\ensuremath{\Omega}}{2\ensuremath{\pi}}$ is the frequency and ${c}_{0}$ the velocity of the sound, and $\ensuremath{\rho}$ is the density.) As in the work of Woodruff and Ehrenreich on sound absorption in insulating crystals, the analysis is based on the linearized Boltzmann transport equation and the Blount formula for the energy loss from a system of interacting excitations driven by a sound wave. It is valid for frequencies such that $\ensuremath{\Omega}\ensuremath{\tau}\ensuremath{\gg}1$ and $\ensuremath{\Omega}\ensuremath{\ll}\frac{\mathrm{KT}}{\ensuremath{\hbar}}$, where $\ensuremath{\tau}$ is the relaxation time for the thermal phonons and $K$ is Boltzmann's constant. The coefficient $A$ is related to the rate of change of the sound velocity with density. An attempt is made to determine the exact magnitude of $A$ within the framework of the present considerations by postulating a relative motion of the normal and superfluid components, but this approach leads to new difficulties.

Solution of the linearized Boltzmann transport equation for the slab geometry

Solution of the linearized Boltzmann transport equation for the slab geometry