Brillouin Zone Folding Research Articles

Electronic Band Structure of SiC Polytypes: A Discussion of Theory and Experiment

After a brief discussion of the origin of polytypes and a few general remarks on the relationship between polytypism and properties of SiC, we focus on a comparative study of their electronic band structures. We first explain how the different band structures can be put on the same footing by examining Brillouin zone folding effects. Then we discuss the dependency of some of the important eigenvalues on hexagonality. Next, we discuss some of the available spectroscopic information on the band structures. Finally, we examine some of the details near the gaps in further detail such as the location of the conduction-band minima, the effective masses and the crystal field splittings and masses of the upper valence bands. Open questions and areas where experimental verification is needed are pointed out.

Acoustic SH phonons in a superlattice with (111) interfaces

Transmission rates and dispersion relations of SH phonons in a (111) superlattice are calculated. Unlike the case of a (100) superlattice, phonon stop bands appear inside the folded Brillouin zone in general rather than at the center and edges. A superlattice structure factor is introduced appropriately for phonon reflection or Bragg conditions in the superlattice. More importantly, the transfer matrix contains information not only about the dispersion relation, but also about the condition for frequency bands or stop bands in the dispersion relation, and the extinction rule for the Bragg reflection. The locations of the stop bands inside of the folded zone are defined as a function of frequency and incidence angle of the phonons.

A Perspective on Next-Generation Silicon Devices

The minimum size of metal oxide semiconductor field-effect transistors assessed in terms of their circuit performance, the possibility of quantum-mechanical engineering of the electronic properties of silicon through the modification of the energy band with mechanical stress, Brillouin zone folding, and artificially built-in nanometer-scale potential as well as through the utilization of low dimensional electrons, and single electronics as a quantum limit of nano-power electronics are reviewed. Furthermore, effects of the statistical fluctuation of device edge definition in fabrication and dopant distribution on the minimum size of silicon devices are discussed and the importance of dopant location control is emphasized.

Magnons dans les super-réseaux ferromagnétiques à 3 - couches

A brief study of bulk and surface magnons in 3-layer Heisenberg ferromagnetic superlatices is presented. Each layer is modelled by a simple-cubic lattice of spins coupled via nearest-neighbor exchange interactions. The superlattice has a larger periodicity in the direction perpendicular to the slabs and therefore many magnon branches in the folded Brillouin zone. In the gaps existing between these magnon branches appear the surface-localized magnons. We give closed-form expressions for bulk and surface magnons in 3-layer superlattices, together with the corresponding dispersion curves for one specific case. An application is also done for a Fe/Co superlattice

Ultrathin SimGen strained layer superlattices-a step towards Si optoelectronics

Ultrathin SimGen (m monolayers (ML) Si, n ML Ge) strained layer superlattices (SLS) have been grown by molecular beam epitaxy. The optical properties of these structures depend on the concept of band-structure engineering by Brillouin zone folding and strain adjustment of the SLS by a Si1-ybGeyb alloy buffer layer. The energies and the oscillator strengths of the bandgap and intersubband transitions have been studied theoretically for SimGen SLS with a variety of period lengths, particularly those of m+n=10. Various characterization tools such as X-ray diffraction, transmission electron microscopy, Raman spectroscopy, photoluminescence (PL) and photocapacitance measurements have been used to analyse growth quality, interface sharpness, morphology, strain distribution and optical properties of the superlattice experimentally. The PL data indicative of the quasidirect energy gap of the 10 ML strain-symmetrized SLS in the near-infrared spectral regime (h nu approximately 0.8 eV) are presented and discussed as well as complementary photocapacitance measurements on a p-n doped Si4Ge4 SLS diode. The fabrication of test mesa diodes from Si/Ge SLS structures is described. Finally, device applications offering the possibility of monolithic integration of superlattice devices with complex silicon-based electronic circuits are outlined.

Electroluminescence and photoluminescence from Si1−xGex alloys grown on (100) silicon by molecular beam epitaxy

Abstract Photoluminescence has been observed from Si 1− x Ge x alloy layers, superlattices and non-periodic multilayers where x was varied from 0 to 0.6 A p-type Si 0.82 Ge 0.18 alloy layer 200 nm thick, grown by molecular beam epitaxy (MBE), has been fabricated into a mesa diode which operated as a light-emitting diode, emitting at 1.4 μm at temperatures up to 80 K. This diode was selected from a series of heterostructures which exhibited intense photoluminescence with internal quantum efficiencies in the range 1%–10% at low temperatures. Photoluminescence in the wavelength range 1.2–1.7 μm has been observed from thick (100–200 nm) Si 1− x Ge x alloys and Si 1− x −Si strained layer superlattices with a range of dimensions which was large compared with the unit cell; i.e. where Brillouin zone folding effects were negligible. The intense Si 1− x Ge x alloy photoluminescence peak had a halfwidth of about 80 meV and the peak energy was found to shift consistently and predictably with the germanium fraction. Photoluminescence peak energies at 4.2 K varied from 620 to 990 meV for germanium fractions where 0.53 > x > 0.06. The photoluminescence and electroluminescence peaks were consistently about 120 meV below the established bandgap for tetragonally strained Si 1− x Ge x . In general, Si 1− x Ge x strained layers grown at low temperatures typical of MBE (below 500°C) exhibited low photoluminescence intensity. However, post-growth annealing in the 500–700°C temperature range enhanced luminescence efficiency by up to two orders of magnitude. A comparison is also made of the broad intense photoluminescence observed from MBE material with the weak near-band-edge spectra obtained from Si 1− x Ge x -Si grown by low temperature chemical vapour deposition.

Diamond polytypes and their vibrational spectra

A series of diamond polytype structures are described and their IR and Raman vibrational modes predicted. The diamond polytypes are analogous to the well-known silicon carbide polytypes. The intermediate 6H diamond polytype was recently identified by single crystal electron diffraction of vapor precipitated diamond powder. In addition, end member polytypes of 3C (cubic diamond) and 2H diamond (hexagonal lonsdaleite) have been previously established, and polytypes such as 4H, 8H, 15R, and 21R diamond are predicted, but may be difficult to isolate and identify. The various diamond polytype structures differ only in the stacking sequences of identical puckered hexagonal carbon layers. These identical carbon layers lie parallel to the cubic 3C {111} and the hexagonal 2H {001} planes. A new method for uniquely labeling the structural layers in the polytype stacking sequences is presented. Factor group analysis was used to determine the IR and Raman selection rules for five diamond polytypes with structures intermediate between those of end members diamond and lonsdaleite. Brillouin zone folding techniques were used to determine band positions, in analogy with analyses of SiC polytypes discussed in the literature. The results predict that (i) all diamond polytypes are Raman active, (ii) limiting polytypes 3C and 2H are not IR active, and (iii) polytypes 4H, 6H, 8H, 15R, and 21R have IR active modes.

SiGe strained layer superlattices

High quality SiGe strained layer superlattices can be achieved by low temperature molecular beam epitaxy. Strain distribution and Brillouin zone folding effects have a large influence on the band structure. Photoluminescence from strain-symmetrized Si 6Ge 4 superlattices shows the first evidence for the achievement of a quasi-direct fundamental energy gap. Raman spectroscopy is used as a versatile tool to characterize short-period SiGe superlattices.

Pulsed growth and electronic structure of layered CuFe films on Cu(100)

The authors have attempted the preparation of CuFe superlattice structures by pulsed metal deposition in an atomic layer-by-layer fashion upon a Cu(100) substrate at 300 K. The electronic structure of a single crystal 6*(1 ML Fe:1 ML Cu)(100)CuFe film has been probed by angle-resolved ultraviolet photoemission spectroscopy. Experimental photoemission results have been critically compared with both fully self-consistent linear muffin-tin orbital (LMTO) predictions of the ground state electronic structure and with photocurrent calculations using time reversed LEED formalism. Theoretical predictions indicate the expected large rearrangements in the valence electron states which are shown to be large enough in certain E(k) points to provide a definite 'fingerprint' of superlattice formation. Despite LEED/AES measurements which indicate a high quality p(1*1) single crystal film exhibiting strong compositional modulation they show by comparing experimental and theoretical angle-resolved photoemission spectra that the attempted Brillouin zone folding in the growth direction has not been imposed. Their result suggests that formation of metallic superlattice with short repeat periods (2 ML) is a nontrivial task despite the apparent ideality of the CuFe system for this purpose. Results presented here are supportive of nonideality in the growth mechanism of Fe/Cu(100).

Localized vibrational modes in superlattices.

The effect of an impurity cell on the vibrational properties of superlattices is studied theoretically. The continuous equation governing the elastic-wave motion in a superlattice is transformed exactly to a discrete form which is analogous to the equation for the displacement in one-dimensional discrete lattices. With the use of this equation isolated frequencies associated with the vibrations localized near the impurity cell are predicted in the band gaps at the center and boundary of the folded Brillouin zone of the host superlattice. Numerical calculation reveals further the existence of impurity states in the intrazone gaps due to intermode Bragg reflection. The calculated phonon transmission rate shows local enhancements due to these localized states, suggesting their observability by phonon spectroscopic experiment with a quasimonochromatic phonon detector.

Folded, confined, interface, surface, and slab vibrational modes in semiconductor superlattices

Vibrational modes of semiconductor superlattices are related to those of their bulk constituents. Reduction in the translational symmetry enables observation by optical spectroscopy of symmetry forbidden modes of the bulk materials. Interesting but subtle concepts, such as Brillouin zone folding and mode confinement, arise. Moreover, electrostatic interface phonons exist in ideal infinite superlattices. In practical superlattices with a finite number of periods, slab and surface vibrational modes also occur. Most of the past experiments have been performed by means of Raman spectroscopy on GaAs Al xGa 1−xAs superlattices; recent work on Ge Si systems reveals new features. The properties of these modes, their symmetries and relationship to the modes of the bulk components are discussed.

Acoustic phonons in multiconstituent superlattices.

Acoustic phonons propagating parallel to the growth direction of a periodic superlattice composed of more than two different constituent layers are studied theoretically. We give general formulas for the phonon dispersion relation and the width \ensuremath{\Delta}\ensuremath{\nu} of frequency gaps existing at the center and boundary of the folded Brillouin zone. We also calculate the structure factor for phonons and find that \ensuremath{\delta}-function peaks corresponding to Bragg reflection have prefactors proportional to \ensuremath{\Delta}\ensuremath{\nu}. This relation can be used to explain the relative magnitude of transmission dips found at frequencies satisfying the Bragg condition.

Bulk and surface phonons in fcc-fcc superlattices using the matching procedure.

The general theory developed in the preceding paper [Phys. Rev. B 38, 5931 (1988)] is applied to an fcc-fcc superlattice with (100) interfaces and nearest-neighbor interactions. Equations of motion have been derived for ${\mathrm{q}}_{\mathrm{\ensuremath{\parallel}}}$ along the [011] direction. Examples are reported for both even and odd motions, including for the latter an interpretation in terms of Brillouin-zone folding. The equations of the odd surface states of a semi-infinite superlattice with a surface parallel to its interfaces are also derived and solved.

Acoustic-phonon propagation in superlattices

Bragg reflection of acoustic phonons propagating through a periodic superlattice is explored in detail. We examine Bragg reflection for a single vibrational mode and relate it to gaps, or stop bands, in the phonon dispersion relation at the center and boundary of the folded Brillouin zone. Bragg reflection due to the coupling of different polarization modes is similarly linked to intrazone stop bands. We also calculate phonon transmission rates, and explore in depth the spatial distribution of the transmitted phonon intensity. These results are compared with experimental phonon images that provide a two-dimensional map of the acoustic-phonon transmission as a function of propagation direction.

Calculation of surface phonon dispersion on Ni(100) and Ni(100)+c(2×2) along the (010) direction by means of the matching procedure. II

For pt.I see ibid., vol.20, p.4725 (1987). A detailed application of the matching procedure is presented in order to calculate the surface phonon dispersion on Ni(100) and Ni(100)+c(2*2)-O in the (010) direction. The phonons even and odd with respect to the (010) plane are analysed with special regard to the effect of adsorbate-induced Brillouin zone (BZ) folding. Studying each phonon symmetry separately enables one to identify the respective contributions from the BZ centre and edge in the electron energy-loss spectra. The equations of motion for Ni(100)+c(2*2)-O are worked out from those for the Ni(100) case. Information on the dispersion of bulk phonons projected onto the (100) surface and the associated Van Hove singularities is obtained by solving a secular equation. The surface phonon and resonance dispersion curves are found by matching the solutions of the secular equation to the surface conditions, which amounts to studying the scattering of incoming bulk waves by the surface. The eigenvectors obtained on the (100) surface permit one to perform a perturbation treatment of the adsorbate-substrate coupling. Finally, a comparative discussion with Green-function-based techniques is given.

Effects due to spin ordering in layered MPX3compounds revealed by inelastic light scattering

Light scattering experiments on MPX3 compounds with M identical to Fe, Ni or Zn and X identical to S or Se and their MxM1-x'PX3 alloys at temperature ranging from 5 to 300K are presented. The peaks observed at room temperature are attributed to Raman-active normal modes of vibrations. Below the Neel temperatures of the iron and nickel antiferromagnetic compounds, three classes of effects are observed: (i) effects due to the appearance of a magnetic superstructure, which is different from the crystallographic cell and which induces Brillouin zone folding; (ii) effects due to spin-spin coupling evidenced by the temperature dependence of the integrated phonon Raman intensity which follows the reduced nearest-neighbour spin correlation function; (iii) effects due to the interference between single-phonon states and spin-split-off electronic bands. These effects are observed in the NiPS3 low-temperature spectrum which shows a broad Raman background on which are superimposed the phonon peaks undergoing lineshape deformations and antiresonance.

Magnetic ordering induced Raman scattering in FePS3 and NiPS3 layered compounds

Abstract

Band theory of superlattices.

We present a first-principles theory of the band structure of lattice-matched superlattices. We formulate the one-electron problem of superlattices in the Bloch representation of one of the constituent materials. In this formulation, the symmetry breaking along the superlattice axis leads to partitioning and folding of the original homogeneous crystal Brillouin zone onto a thin region around the center of the zone. A single homogeneous crystal Bloch function becomes a multicomponent wave function. Each component represents a superlattice subband. We use Loewdin's theorem to derive series expressions for the subband energies and wave functions, and discuss the relation between these solutions and the eigenvalues and eigenfunctions corresponding to quantum-well models. We apply the general results to a superlattice composed of two different types of simple two-band model materials.

Surface phonon dispersion of clean and oxygen covered Pt(111)

The dispersion of the vibrational modes of the Pt(111) surface in the absence of impurities as well as in the presence of oxygen has been measured by inelastic helium scattering along the \\ ̄ gG $ ̄ M azimuth. In the case of the ordered p(2×2)O superstructure, the additional dispersion branch appearing as a result of the Brillouin zone folding was also measured. The experimental results show the first direct evidence for the appearance of a surface Rayleigh phonon gap at the new zone boundary. The gap was determined to be ~ 0.85 meV at the M̄ 0 point. Lattice-dynamical calculations show that this relatively large gap is due to bond stretching interactions between the adsorbate and substrate atoms. The dispersion curve in the presence of disordered oxygen, obtained by annealing (1200 K) the Pt surface in an oxygen atmosphere and assumed to be located subsurface on the basis of ion scattering data has been measured. The frequency of the Rayleigh phonon at the M̄ point is 12% below the clean surface value. This is at variance with the case of disordered oxygen on metal surfaces (O/Ni(100)) which has been shown to have a negligible influence on the Rayleigh dispersion curve. According to the lattice dynamical calculations this 12% frequency lowering corresponds to a reduction of the interlayer force constant k 12 by about 30% and thus supports the subsurface location of the oxygen atoms.

Theory of bulk and surface magnons in Heisenberg ferromagnetic superlattices.

We present a theoretical study of the bulk and surface magnons of a semi-infinite stack of two different ferromagnetic films. Each film is modeled by a simple-cubic lattice of spins coupled via nearest-neighbor exchange (Heisenberg ferromagnet). The superlattice has a larger periodicity in the direction perpendicular to the slabs and therefore many magnon branches in the folded Brillouin zone. In the gaps exisiting between these magnon branches appear the surface-localized magnons. The simplicity of our model allows one to obtain in closed form the bulk and (001) surface Green's functions for this magnetic superlattice. The analytic knowledge of these functions enables us to study easily all the bulk and and surface magnetic properties of a ferromagnetic superlattice. We give here the analytic expression we obtained for the folded bulk magnons and also the expression that gives the surface-localized modes, which may appear within the extra gaps which exist between the folded bulk bands. One figure for a specific case illustrates these results.