7.3 Multigroup Equations to the 2D-BBP Equations

In this section, we introduce multigroup equations for investigating the transport of electrons and phonons at a heterojunction for the spatially homogeneous case, which means that f ^v and g do not depend on r . To begin with, we state the approximations used for describing the dispersion laws for the confined electrons and the 3D phonons. The electron energy E ^v( k ) in the in vth energy subband and the electron wave vector k are related by the spherical parabolic energy momentum rule (7.5). This implies that the modulus of the electron wave vector in the subband v is determined by

(7.64)

for the given energy E. Here, ? denotes the Heaviside step function. The energies of longitudinal optical (LO) phonons ? ? _LO are related to the phonon wave vector q via the usual Einstein approximations (cf. Sec. 2.3)

(7.65)

The electron system is described by a set of one-particle distributions functions f ^v( k , t), which gives the probability to find an electron in the infinitesimal volume d ² k around k at time t in the the vth subband. The evolution equation for f ^v( k , t) is the 2D BTE (7.59a). By expressing k in polar coordinates

(7.66)

with the polar angle ? between k