Statistical Algorithms for Simulation of Electron Quantum Kinetics in Semiconductors - Part II

The applied electrical field destroys the spherical symmetry of the field less Barker-Ferry equation. The dimensionality of the task increases and furthermore no general integration domain can be specified due to the correlation of the phase space and time coordinates. In this part of the work we propose and integral formulation which decouples these coordinates. The equation is solved by the randomized iterative Monte Carlo algorithm introduced in Part I. An analysis of the quantum effects demonstrated by the solutions is presented. 1 Integral Form of the Barker-Ferry Equation The quantum-kinetic equation, explored in Part I, has been obtained in a framework of a physical model which describes the relaxation of semiconductor electrons initially excited by a laser pulse [1]. The equation appears as a simplified Barker-Ferry (B-F) equation [2] written for the case of zero electric field. The original formulation of the B-F equation accounts for the effect of the electric field on the process of collision the intra collisional field effect. It is argued that this effect plays a negligible role in the stationary solution of the quantum-kinetic equation [3]. Here we investigate the transient problem, i.e. electron phonon relaxation of initially excited electrons in the presence of an applied electric field E. The B-F equation has the following integro-differential form: ∂f(k, t) ∂t + F · ∇kf(k, t) = (1) ∫ t 0 dt′ ∫ dk′ {S(k′,k, t, t′)f(k′(t′), t′)− S(k,k′, t, t′)f(k(t′), t′)} S(k′,k, t, t′) = 2V (2π)3 2 |gq| exp(−Γ (t− t′))× [ (nq + 1) cos (∫ t t′ dτΩ(k(τ),k′(τ)) ) + nq cos (∫ t t′ dτΩ(k′(τ),k(τ)) )] , S. Margenov, J. Wasniewski, and P. Yalamov (Eds.): ICLSSC 2001, LNCS 2179, pp. 183–190, 2001. c © Springer-Verlag Berlin Heidelberg 2001 184 M. Nedjalkov et al. where F = eE/ , nq is the Bose function, ωq generally depends on q = k′ − k, k(t′) = k− F(t− t′); Ω(k(τ),k′(τ)) = (k(τ))− (k ′(τ)) + ωq . The damping factor Γ is considered independent of the electron states k and k′. This is reasonable since Γ weakly depends on k and k′ for states in the energy region above the phonon threshold, where the majority of the electrons reside due to the action of the electric field. An application of the method of characteristics leads to the following integral form of (1): f(k, t) = φ(k(0)) + ∫ t