Evaluation of the effective threshold energy of the Interband impact ionization in a deep-submicron silicon n-channel MOS transistor

Ground state and the spin precession of the Dirac electron in counterpropagating plane electromagnetic waves

“…where We intensively use indexing of various mathematical objects by points n = (n 1 , n 2 , n 3 , n 4 ) of the integer lattice L. The sequential numbering of these points, based on the use of g 4d (n), drastically simplifies both numerical implementation of the presented techniques and analysis of solutions, because it takes into account the specific Fourier spectra of the electromagnetic lattice and the wave function, as well the structure of the finite models described below and in more detail in [12]. It is well known, e.g., see Ref.…”

“…In appendix, we present a specific numeration of these basis matrices Γ k , k = 0, ..., 15, which makes it possible, in particular, to reconstruct any matrix Γ k from its number k, see Ref. [11]. Any 4 × 4 matrix V = 15 k=0 V k Γ k is uniquely defined by the set of its components D s (V ) = {V k } in the Dirac basis [Dirac set of matrix V , briefly, Dset of V ].…”

“…(6), see Ref. [11][12][13]. D-sets of matrices V (n, s) are presented in [11] as functions of the dimensionless parameters…”

“…The fundamental solution S, i.e., the operator of projection onto the solution subspace of the multispinor space V C , and the projection operator P of the infinite system of equations (15) are defined as follows [11] …”

“…III. Additional information on the numerical implementation of the presented approach and some results of its computer simulation can be found in [11][12][13].…”