Investigation of Transport Coefficients in Semimetals by the Variational Method I. General Solution of the Problem

Abstract

A variational procedure for the solution of the Boltzmann equation with a n arbitrary number of variational parameters is developed. In the frame of the method it is possible to carry out t h e optimization of trial function, as well as to check the convergence of the solution. It is shown that. in semimetals the probability matrix of charge carrier scattering on acoustic phonons entering the expression for the collision operator has a non-diagonal form in the coordinate system of the effective mass tensor. General expressions are obtained for the transport coefficients allowing to determine their numerical values.PaapaBo~ntra BapHaqnoman npouenypa pelrremn KHtIeTwIec~oro ypani~eiiuc! Gontu-

Transport properties of bismuth in non-quantizing magnetic fields: pseudo-parabolic variational calculation

“…The primary aim of the present work is to calculate the transport tensors of bismuth by usinga variational technique in which all the essentialfeaturesof the pseudo-parabolic model are retained. Thus from the analytical point of view the basic differences between the present treatment and that of Gitsu et a I [ 3 ] are in the form of the dispersion relation and in the energy dependence of…”

“…It is found that the method should converge whether the deformational potentials are taken to depend on energy according to the pseudo-parabolic model or to be energy-independent. It seems, therefore, that the energy dependence of may not be the main reason for the deterioration of convergence in Gitsu et ai [3], at least in the low-temperature range.…”

“…In the first two references a so-called pseudo-parabolic model, which depends mainly on the Lax non-parabolic dispersion relation and a relaxation time having a specific energy dependence, has been utilized. In Gitsu er ai [3] a variational technique has been employed to find the solution of the Boltzmann equation as a power series in energy. However, the convergence of their method deteriorated for non-parabolic dispersionand theirresultswere thusobtainedonlyfortheparaboliccase.…”

“…The dispersion relation is taken here to be nonparabolic and is taken to depend on energy according to the result of Heremans and Hansen [4]. The energy dependence ofthe deformational potentials, which are the quantities that appear in thecalculations, has beenretrievedfrom theenergydependence of These were taken to be independent of energy in Gitsu el al [3]. Also, as regards the numerical calculations, the temperature dependence of the deformational potentials was obtained in Gitsu et a1 [3] by treating them as fitting parameters, and the masses of electrons and holes were taken to be independent of temperature.…”

“…The energy dependence ofthe deformational potentials, which are the quantities that appear in thecalculations, has beenretrievedfrom theenergydependence of These were taken to be independent of energy in Gitsu el al [3]. Also, as regards the numerical calculations, the temperature dependence of the deformational potentials was obtained in Gitsu et a1 [3] by treating them as fitting parameters, and the masses of electrons and holes were taken to be independent of temperature. In the present calculations the whole set of input data of the pseudo-parabolic model (from Mikhail era1 [I] and Hansen and Mikhail [2]) will be used after the convergence of the variational method is achieved.…”