Theory of surface and interface transverse elastic waves inN-layer superlattices

Abstract

N-layer superlattices are formed out of a periodic repetition of N different slabs. Such materials for Nϭ3 and 4 are now easily grown by molecular-beam epitaxy. We present closed form expressions for localized and resonant transverse elastic waves associated with the surface of a semi-infinite N-layer superlattice, with or without a cap layer, and in contact either with vacuum or with a homogeneous substrate. We also calculate the corresponding Green's-function and densities of states. These general results are illustrated by a few applications to four-layer superlattices made of Nb-Fe-Nb-Cu and Nb-Cu-Nb-Cu. We show that increasing the number of the layers in each unit cell of the superlattice increases, in general, the number of the minigaps and surface modes. We generalize some of the results obtained previously in the case of two-layer superlattices, namely, ͑i͒ the creation from an infinite superlattice of a free surface gives rise to ␦ peaks of weight ͑Ϫ 1 4 ͒ in the density of states, at the edges of the superlattice bulk bands; ͑ii͒ by considering together the two complementary semiinfinite superlattices obtained by cleavage of an infinite superlattice along a plane parallel to the interfaces, one always has as many localized surface modes as minigaps, for any value of the wave vector k ʈ ͑parallel to the interfaces͒. Finally, we investigate the localized and resonant modes associated with the presence of a cap layer on top of the superlattice. ͓S0163-1829͑96͒00944-7͔

Surface and interface shear horizontal acoustic waves in piezoelectric superlattices

“…L i are ␦ peaks associated with surface localized modes, whereas B i and T i refer to ␦ peaks of weight ͑Ϫ1/2͒ situated at the bottom and the top of the minibands [18][19][20][21][22] given by k 1 Dϭ0 and k 1 Dϭ . One can see that the positions of the surface modes are almost the same in the first two cases.…”

“…23 The details of the analysis are the same as for shear horizontal waves in elastic superlattices, 18,19 but with more complicated calculation. 20 Let us emphasize that, in the geometry of the SL substrate structure, the elements of the Green's function take the form g ␣␤ ( 2 ,k ʈ ͉n,i,x 1 ;nЈ,iЈ,…”

“…We have previously applied such a formalism to the case of purely shear horizontal elastic waves associated with a semi-infinite SL or to its interface with a substrate. 18,19 For these waves involving only one direction of vibration, the Green's function has been calculated analytically and the DOS has been obtained as a function of the frequency and the wave vector k ʈ ͑parallel to the interfaces͒. However, the direct calculation becomes very cumbersome once the shear horizontal waves couple to the electric potential, although it remains analytical.…”

Polar optical phonon states and their dispersive spectra of a wurtzite nitride superlattice with complex bases: Transfer-matrix method

“…In order to further improve the performance of microelectronics devices based on SL structures, people brought forward the complex-bases SL, i.e., n layers per period (n > 2) [8][9][10]. In contrast to the binary (two-layer per period) SLs, the complex bases SLs provide more degrees of freedom for the adjustable structural parameters, the miniband and minigap widths due to the introduction of additional layers in each SL period.…”

“…the propagating (PR) mode, the IO mode, the (2 n − 2) QC modes probably coexist in wurtzite complex-base SL structures. (ii) By using the transfer-matrix method [8,9], the polar optical phonon states and corresponding dispersive equations for the 2 n types of polar phonon modes in wurtzite nitride complex-bases SLs are given within the framework of the DCM and the Loudon's uniaxial lattice model. (iii) As an example, a wurtzite GaN/Al x Ga 1−x N/Al y Ga 1−y N SL with three-layer complex bases is chosen to check the theoretical scheme described in the present paper.…”