We consider processes determining the steady-state population of longitudinal optical (LO) phonons being intensively produced in the quantum well (-10041 width) of a GaAs/GaAlA s h t e erostructure by electrons accelerated by a strong external electric field. The emitted LO phonons are confined in the well layer [ 1,2] and transfer their energy to acoustic phonon system through anharmonic decay into longitudinal acoustic (LA) phonons. The latter, in the first approximation, emerge almost monoenergetically and distributed homogeneously in-an infinitely thin spherical layer of radius klo M w,/2vl M 0.6 x lo8 cm-' in the reciprocal space, here w, is the frequency of LO phonon, and y is the group velocity of LA phonon [3].
To demonstrate stimulated feedback of LA phonons on the LO decay processes, we have to trace the stages of decay and propagation of these resulting LA phonons, to obtain the LA phonons density in the near vicinity of the well, and to find the occupation numbers of individual phonon a-states.
Using simple analytical functions to model LA and TA dispersion laws, wl(k) = vlk and wt(k) = wt tanh (k/b) -where wt = 1.48 x 1013 set-' is the edge frequency of the TA phonons, and b = 4.43 x lo7 cm-' is fitting parameter -we have calculated the rates of the main inelastic processes involving in high-frequency LA phonon propagation, LA-, LA+TAand LA -+ TA + TA decays (see Fig. 1, a andb). The pronounced hierarchy of these processes allows us to neglect the first process altogether and to simplify significantly the general kinetic equation for the LA phonon distribution function nl(z, w,0). The only elastic scattering process which matters in the relevant frequency range, LA + LA, has the rate n(w) = 47rG,03G4, with coefficient G, = 5.59 x 106sec-', LY = 0.59 being transverse-to-longitudinal-velocity ratio. We assume the operating temperature to be zero and use the natural boundary conditions of absolute absorption at zero-temperature sink, nl( L, w, 6) = 0 for cosfl < 0. On the other hand, we take into account the phonon generation in the quantum well at x = 0 and the specular reflection of the phonons impinging on this plane from the bulk by imposing the other boundary condition, cos e(nl(O, w, @) -nr(O, w, z -0)) = (2go/vlwt)6(w -w,/2), where go is the LA phonon generation rate (in the further numerical estimates we will take go = 2 x 1O25 cm-2gec-1).