Semi-infinite Superlattice Research Articles

Surface states in double-well superlattices: symmetric vs asymmetric interwell coupling

Electronic level structure and the density of states (DOS) of a semi-infinite superlattice (SL), whose period consists of two GaAs quantum-well layers and two AlxGa1−xAs quantum-barrier layers (the so-called double-well basis), are investigated within an envelope-function approximation. A special attention is paid to surface effects due to the SL termination by a substrate or a cladding layer. In particular, the existence and properties of surface-localized states, as well as a consequent modification of the extended states forming SL minibands, are studied for various terminating conditions, determined by the choice of substrate and the sequence of SL layers approaching the SL surface. The resulting surface electronic structure is shown to be critically dependent on the geometry of the SL basis: For symmetrically (asymmetrically) coupled wells, crossing (anticrossing) of SL minibands occurs, which dramatically influences the energy spectrum of SL surface states as well as the local DOS distributions at the SL end.

Density of states of a superlattice with a δ defect in the subsurface region

The effect of a spatially and strongly confined ( δ-like) defect layer on the electronic properties of a semi-infinite superlattice (SL) is investigated, with emphasis placed on local density-of-states (LDOS) modifications at the SL end. The SL energy spectrum and the LDOS distributions, calculated within a modified Kronig–Penney type of approach, are studied for variable δ-defect parameters (i.e. its strength and position with respect to the SL surface) and different SL terminating conditions (depending on the choice of substrate). The possibility of a dramatic redistribution of extended states forming the SL minibands as a result of a single δ-defect sheet insertion in the subsurface SL region is indicated and discussed in detail.

Localized and resonant guided elastic waves in an adsorbed layer on a semi-infinite superlattice

The pseudoguided acoustic modes associated with an adlayer deposited on a substrate may become very weak resonances, difficult to detect experimentally, due to their interaction with the substrate modes. In this paper, we show that the observation of these modes can be made possible, or at least facilitated, when the substrate is a superlattice instead of being a homogeneous medium. This is essentially due to the existence of minigaps, and also of two types of polarizations for the minibands, which may prohibit the propagation of the guided modes of the adsorbed layer into the superlattice and therefore increase the degree of localization of these guided modes. These features are especially relevant at high acoustic velocities (in particular around and above the longitudinal velocity of sound in the adlayer) or in the case of a hard layer where the resonances may become very weak if the substrate is made of a homogeneous material. The purpose of this paper is to demonstrate theoretically this new phenomenon with a few illustrations. The localized and resonant modes associated with the adlayer are obtained from a calculation of the total and local density-of-states by means of a Green's function method. The existence and behavior of the guided modes are discussed as a function of the nature and thickness of the adsorbate, the nature of the layer in the superlattice that is in contact with the adlayer, and the wave vector ${k}_{\ensuremath{\parallel}}$ (parallel to the layers).

Surface and interface shear horizontal acoustic waves in piezoelectric superlattices

The propagation of acoustic waves of shear horizontal polarization in infinite and semi-infinite superlattices made of two piezoelectric media is studied within a Green’s function method. Localized modes induced by a free surface of the superlattice or a superlattice/substrate interface are investigated theoretically. These modes appear as well-defined peaks of the total density of states inside the minigaps of the superlattice. The spatial localization of the different modes is studied by means of the local density of states. The surface of the superlattice and the superlattice/substrate interface are considered to be either metallized or nonmetallized. We show the possibility of the existence of interface modes, which are without analogue in the case of the interface between two homogeneous media (the so-called Maerfeld–Tournois modes). We also generalize to piezoelectric superlattices a rule about the existence and number of surface states, namely when one considers two semi-infinite superlattices together obtained by the cleavage of an infinite superlattice, one always has as many localized surface modes as minigaps, for any value of the wave vector k∥ (parallel to the interfaces). Specific applications of these results are given for CdS–ZnO superlattices with a free surface or in contact with a BeO substrate.

Surface electronic structure of periodically δ-doped superlattice

Surface electronic structure of a semi-infinite superlattice (SL) intentionally δ-doped in each well- or barrier-layer is studied using a terminated Kronig-Penney-type of model interspersed with a periodic array of δ-function potentials representing the deep-center defect sheets. It is shown that SL minibands can be tailored by varying the properties of inserted δ-defects, e.g., the lowest energetic miniband can be shifted below the bottom of SL quantum wells (as a result, the main SL energy gap becomes smaller than those of the host semiconductors) and miniband alignement can occur (as a result, the bandwith is enhanced by an order of magnitude). Special attention is paid to the effect of δ-doping on the properties of surface states (SS's), i.e., the states appearing within energetic minigaps and localized at the SL/substrate interface. It is found that by an appropriate choice of the δ-defect weight and location within the SL period, SS with a required energy position relative to minigap edges and a desired extension into the SL bulk can be achieved. Moreover, in contrast to standard (i.e., undoped) SL's, a possibility of SS existence for the substrate identical to SL barriers is shown in periodically δ-doped SL's.

Surface electronic structure of a step-well-basis superlattice

Surface electronic structure of a semi-infinite polytype superlattice (SL) with period consisting of three different layers (the so-called step-well basis) is investigated using a transfer-matrix approach within an effective-mass approximation. Explicit analytical formulas for the bulk dispersion relation as well as the energy expression and existence condition for surface states are given for arbitrary terminating medium and their simplification for particular SL terminations by a substrate identical to one of the SL constituents is discussed. Dependence of surface-state properties (i.e., its energy position and the degree of localization) on SL parameters (i.e., layer thicknesses and potential barrier heights) and different surface configurations (depending on a sequence of SL layers approaching the surface as well as on the choice of substrate) is studied for a triple-constituent ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$-based SL. Additionally, surface-state wave functions are plotted in order to examine the associated space-charge distributions.

The reflectivity of a film with a grating in contact with a semi-infinite bimetallic superlattice

We study the influence of a semi-infinite bimetallic superlattice on the `excitation' of surface polaritons appearing in a film with a grating surface. The transfer-matrix formalism and the Rayleigh-Fano approach are applied to study the optical response for p-polarized light. We analyse the reflectivity and the dispersion relation of the surface plasmon-polaritons, considering Al and Mg for the semi-infinite superlattice and Mg for the film. The `optical' dispersion relation of surface plasmons is constructed using the minima of the reflectivity. This relation shows a dependence on the corrugation strength such that, as this corrugation grows, the corresponding curve shifts to lower frequencies.

Electronic surface states and miniband structure of superlattices with multiple layers per period

The electronic structure of a semi-infinite complex-basis superlattice (SL) with $N$ layers per period is investigated, with emphasis placed on the effect of the SL surface (i.e., the SL/substrate interface). The bulk dispersion relation as well as the energy expression and existence condition for surface states are derived using the transfer-matrix method within an envelope-function approximation. Some common properties of a symmetric termination of the SL potential (i.e., when the substrate is identical to the last layer of the SL basis) are discussed and it is shown that---contrary to binary SL's---surface states can appear in complex-basis SL's also without perturbing the SL potential at the surface. These general results are illustrated by application to ${\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ SL's with four-layer (two-well and two-barrier) bases.

Theory of far-infrared attenuated total reflection from surface magnetic polaritons in rare-earth superlattices

We present a theory for the attenuated total reflection (ATR) spectra to probe surface magnetic polaritons that propagate in semi-infinite superlattices made up of a rare-earth material intercalated by a nonmagnetic layer. We consider both the spiral and the cone states of the rare-earth constituent, apart from its simple ferromagnetic (antiferromagnetic) alignment due to the nearest (next-nearest) exchange coupling. We use the framework of the effective medium theory to determine the magnetic polariton’s dispersion relation, since it is very suitable to explain experimental observations of excitations in the far-infrared. Numerical results are presented to illustrate the ATR spectra, with emphasis in the surface polariton’s nonreciprocal aspect shown in the cone state of the rare-earth material.

Electromagnetic surface modes of a dielectric superlattice: the supercell method

We present a study of the reliability of the supercell method in calculations of surface electromagnetic modes. For a truncated superlattice constituted of nonabsorbing dielectric layers, we demonstrate that the numerical solutions obtained by this method for transverse-electric waves agree with those based on the Bloch theory for the semi-infinite superlattice. A slab of superlattice with at least nine unit cells yields satisfactory convergence to an analytic dispersion relation for the surface modes. In addition, we apply the supercell method to study in detail the dependence of transverse-electric and transverse-magnetic surface waves on the cut-off position in the cell next to the surface. As a specific case, we choose a TiO2/SiO2 superlattice—layers with relatively high dielectric contrast in the visible spectrum. We find the surface modes strongly dependent on the position of the surface. In fact, they appear only for certain terminations. By plotting the field amplitudes, we show that there exist different possibilities for the guidance of surface waves. The variation of the penetration depth of these modes is also discussed.

Electronic surface states in superlattice with complex basis

The electronic structure of a semi-infinite superlattice (SL) with a complex basis consisting of two identical wells coupled via two different barriers is investigated. The possibility of the surface state existence for a periodic termination of the SL potential (i.e., without perturbing the SL potential at the surface) is shown for such a two-barrier basis, in contrast to the conventional one of single well and barrier. This main conclusion is based on analytical considerations, however, some numerical results for the GaAs/AlAs SL are also presented.

Theory of surface and interface transverse elastic waves in N-layer superlattices.

N-layer superlattices are formed out of a periodic repetition of N different slabs. Such materials for N=3 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, (i) the creation from an infinite superlattice of a free surface gives rise to \ensuremath{\delta} peaks of weight (-1/4) in the density of states, at the edges of the superlattice bulk bands; (ii) by considering together the two complementary semi-infinite 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 ${\mathbf{k}}_{\mathrm{\ensuremath{\parallel}}}$ (parallel to the interfaces). Finally, we investigate the localized and resonant modes associated with the presence of a cap layer on top of the superlattice. \textcopyright{} 1996 The American Physical Society.

Surface and interface optical waves in superlattices: transverse electric localized and resonant modes

We investigate the existence and behaviour of localized and resonant optical waves associated with the surface of a semi-infinite superlattice or its interface with a substrate, considering the case of transverse electric modes. In this paper, we present an analytic determination of the response function (Green function) for a semi-infinite superlattice with or without a cap layer, and for a superlattice in contact with a substrate. This calculation enables us to obtain both local and total densities of states as functions of the frequency and the wavevector (parallel to the interfaces). Besides the surface and interface localized modes which occur as delta peaks inside the gaps, one can also obtain resonant states as well defined peaks of the density of states inside the bulk bands of the superlattice. In particular, we emphasize the different types of mode induced by a cap layer. In addition, we show that the creation of a semi-infinite superlattice gives rise to delta peaks of weight -1/4 in the density of states at every edge of the superlattice bulk bands.

Control of quasi-bound states by electron Bragg mirrors in GaAs/Al0.3Ga0.7As quantum wells

Electron Bragg mirrors grown on both sides of identical quantum wells were used for confining above-barrier states at different energies. By varying the thickness of the electron mirrors, a confined above-barrier state (a quasi-bound state) changed by over 110 meV in a 40 Å wide GaAs/Al0.3Ga0.7As quantum well. Formation of quasi-bound states due to the electron Bragg mirrors was theoretically investigated using the envelope function approximation and semi-infinite superlattice limit of the electron mirror layers. Theoretical calculations for absorption spectra involving quasi-bound states are compared with experimental results.

Electronic surface states in a semi-infinite effective-mass superlattice

The electronic structure of a semi-infinite effective-mass superlattice (EMSL) being in contact with a substrate is investigated taking into account the effect of the EMSL surface (i.e. the EMSL/substrate boundary). In particular, the influence of the EMSL bulk and surface parameters on the existence, energy position within the minigap, and localization properties of surface states is studied for different EMSL terminating layers. Additionally, the effect of a (small) conduction-band edge discontinuity between the EMSL layers on the surface electronic structure is examined and the maintenance of the basic features of the surface-state energy curve is found. The usefulness of the obtained results for the possible device applications of the EMSL is pointed out.

Electronic states of a semi-infinite superlattice with an embedded quantum well

We study the electronic states at the Gamma point of a heterostructure, formed by a semiinfinite substrate (AlAs), a well material (GaAs) and a semi-infinite superlattice (AlAs/GaAs), grown along the (001) direction. We employ an empirical tight binding (ETB) Hamiltonian together with the surface Green function matching (SGFM) method. We study the effect of the well layer width and the relative thicknesses of the layers forming the superlattice on the energy values and spatial localization of the different states (near the main gap at the Gamma point) of this heterostructure.

Surface and interface s-polarized optical waves in superlattices

The existence of localized and resonant optical waves associated with the surface of a semi-infinite superlattice or its interface with a substrate is reported, considering the case of s-polarization (transverse electric modes). These modes appear as well defined peaks of the density of states, either inside the minigaps or inside the bulk bands of the superlattice. The density of states, function of the frequency ω, and the wave vector k ∥ (parallel to the interface), are obtained from an analytic determination of the response function for a semiinfinite superlattice with or without a cap layer.

Magnetostatic modes in a finite superlattice

We develop a direct method by which we exactly derive a general dispersion relation for magnetostatic modes in an N-cell finite superlattice composed of two different alternating magnetic layers. We also confirm mathematically the consistency condition for the existence of surface modes on a semi-infinite superlattice. We obtain some new analytical conclusions about surface modes.

Electrons and excitons in an imperfect superlattice in electric fields

We present variational calculations of excitonic states in a superlattice coupled with a wide quantum well in electric fields. The electronic states in the structure are analyzed by using both exact solutions of the one-dimensional Schrödinger equation and the simple tight-binding approximation. We demonstrate the latter method to be well applicable to calculating and designing complicated irregular superlattices. The electron spectrum can be conveniently interpreted as a result of field-induced mixing and anticrossing of electron quantized states in the enlarged quantum well with non-equidistant Stark-ladder states in the semi-infinite ideal superlattice. The electron-hole Coulomb attraction results in a relative redistribution between the extended and the localized states in the exciton. The allowance for this redistribution has a particularly strong influence upon the exciton oscillator strength and radiative lifetime.

Electromagnetic wave propagation in magnetic multilayers— Non-reciprocal reflections

A theory of EM wave propagation through magnetic multilayers and superlattices is presented based on the propagation matrix of a magnetic film. By using theP matrix, the transmission and reflection coefficients of layered magnetic media, including: (1)semi-infinite magnetic surfaces, (2) magnetic multilayers, (3) semi-infinite magnetic superlattices are obtained. The numerical results show that the EM modes of a magnetic layer system is excited and manifested as the sharp dips in theS-polarized reflection and the dispersion curves of the magnetic polaritons can be measured by a method similar to the attenuated total reflection (ATR) technique.