Band Representations Of Space Groups Research Articles

Electron state symmetries and optical transitions in semiconductor superlattices: II. grown along the [110] and [111] directions

Using the method of induced band representations of space groups, we have performed a complete group-theory analysis of electron state symmetries in superlattices grown along the [110] and [111] directions. The spin - orbit interaction has been taken into account. The formulae giving, for arbitrary numbers of monolayers m and n, the arrangements of atoms over the Wyckoff positions have been obtained. The selection rules for both direct and phonon-assisted optical transitions have been derived. Using results of the group-theory analysis, we predict some variations in the optical transitions when m and/or n are varied. We also propose polarized-light optical experiments to establish the detailed electron-band structures of the superlattices.

Electron state symmetries and optical transitions in semiconductor superlattices: I. grown along the [001] direction

Using the method of induced band representations of space groups, we have performed a complete group-theory analysis of electron state symmetries in superlattices grown along the [001] direction. The spin - orbit interaction has been taken into account. The selection rules for both direct and phonon-assisted optical transitions have been derived. Using both the results of our group-theory analysis and data on the various electronic-structure calculations which have appeared in the literature, we predict some variations in the optical transitions when m and/or n are varied. We also propose optical experiments to decide among the numerous and sometimes contradictory results of those band calculations.

Continuity of phonon spectra for diamond and hexagonal close packed structures.

Band representations of space groups are used for showing that crystals with diamond or hexagonal close packed structures have phonon spectra that extend from zero to some maximal frequency without a gap in between. This gapless feature follows from the vectorial nature of lattice vibrations, from the continuity that is introduced by band representations, and from the symmetry of Bloch functions for nonsymmorphic space groups. Some other space groups having two identical atoms in a unit cell and leading to the same continuity feature for their phonon spectrum are listed.

Normal modes of BiSrCaCuO high-temperature superconductors: layer-by-layer approach

The non-superconducting Bi 1−x( Sr 1−y Ca y) x O 1.5− x 2 compounds and the Bi 2Sr 2 Ca n−1 Cu n O 2 n+4 ( n=1.2) superconductors have been synthesized and investigated. Using the method of band representations of space groups the complete group symmetry analysis of the normal vibrational modes has been made. Comparing the Raman spectra of the row of bismuth compounds and using the results of the group theory analysis all the strong RSS-lines have been attributed.

Site symmetry approach to lattice dynamics of semiconductor superlattices

We present here the site symmetry approach to lattice dynamics of superlattices which permits us to connect by symmetry the local atomic displacements and normal vibrational modes over the entire Brillouin zone. The atomic arrangements over the Wyckoff positions for (GaAs) m(AlAs) n and (Si) m(Ge) n superlattices oriented along [001] for different sets of m and n are determined. We obtained the phonon symmetry for k ≠ 0 using the developed theory of the band representations of space groups and derived the selection rules for the first and second-order infrared and Raman spectra. Raman spectra obtained for the (GaAs) 7(AlAs) 18 and (Si) 3(Ge) 3 superlattices are interpreted in terms of the presented theory.

Berry's phase for energy bands in solids.

Berry's phase is defined for the dynamics of electrons in periodic solids and an explicit formula is derived for it. Because of the special torus topology of the Brillouin zone a nonzero Berry phase is shown to exist in a one-dimensional parameter space. Symmetry of the Bloch functions in the Brillouin zone leads to the quantization of Berry's phase. A connection is established between the latter and the Wyckoff positions in the crystal in the framework of band representations of space groups. Berry's phase can therefore be used for labeling energy bands in solids.

Tensor fields in crystals and band representations of space groups

AbstractThe connection between tensor field and band representations of space groups is established. A tensor field representation is a direct sum of several simple band representations. Its irreducible components are easily found from the tables of simple band representations. The vibrational tensor field representations for some high‐Tc superconductors are considered in this context.

Double‐Valued Band Representations of Monoclinic and Orthorhombic Space Groups

AbstractThe simple double‐valued band representations of the space groups C, C, C, C, D, D, D‐D are generated. These space groups describe the symmetry of the most widely spread low‐symmetry crystal structures. The dependence of the band representations on the choice of origin and setting in the space group description is discussed.

Symmetry of Localized Crystalline Orbitals and Double‐Valued Band Representation

AbstractThe double‐valued band representations of space groups are considered. The simple double‐valued band representations of the space groups Oh5, Oh7, and Td2 are generated for the first time. The tables of the induced double‐valued representations of the crystallographic point groups are given. These tables are necessary for some applications of the simple band representations of space groups in physical problems. The symmetry of localized orbitals in Ge crystals is analyzed basing on the results of the relativistic band structure calculation.

Application of Band Representations of Space Groups in the Theory of Phase Transitions and Point Defects in Crystals

AbstractIt is shown that simple band representations tables can be used to solve the following two problems: 1) to find the irreducible components of the space group representations induced by the irreducible representations of their site symmetry subgroups; 2) to find the irreducible components of the site symmetry group representations subduced by irreducible representations of a space group. The solution of both the problems requires also the induced representation table for the crystallographic point grousp which is given here. The space group D is considered as an example. The simple band representations table for this group is also given. The application of the band representations of space groups in the theory of second‐order phase transitions and in the theory of point defects in crystals is discussed.

Application of the Band Representations of Space Groups in the Theory of Electronic States of Crystalline Solids. II. Band Representations for Some Space Groups

AbstractThe simple band representations of the space groups O, T, and D are tabulated. The results of the band structure calculations for some crystals with the symmetry of these space groups are analysed from the point of view of the band representation theory. The possible applications of band representations in the electronic state theory of local defects in solids are discussed.

Application of the Band Representations of Space Groups in the Theory of Electronic States of Crystalline Solids. I. General Consideration of the Band Representations

AbstractBand representations are considered in the framework of the ordinary representation theory of space groups. The band representation characterizes the symmetry of all the electronic states of an energy band. Any band representation coincides with a so‐called simple band representation or is a direct sum of the latter. The whole number of simple band representations is finite for any space group. The type of a band representation determines the symmetry of the localized functions related with a given energy band.

Band representations of space groups

All the irreducible band representations of a space group are shown to be induced from a set of inequivalent relevant symmetry centers in the Wigner-Seitz cell. A connection is established between representations and band representations of space groups by using the Born---von K\'arm\'an boundary conditions. Continuity chords are used for proving the equivalency theorem which enables one to distinguish between equivalent and inequivalent band representations. As examples we consider a one-dimensional crystal and the ${D}_{6h}^{4}$ space group for a hexagonal close-packed structure.

Continuity chords of bands in solids: The diamond structure

The concept of a continuity chord is defined for denoting all those Bloch states at different symmetry points in the Brillouin zone that by symmetry and continuity can, in principle, belong to one band of a solid. The continuity chords for different bands are derived on the basis of band representations of space groups. A specific example of the diamond group ${{O}_{h}}^{7}$ is considered in detail and the continuity chords are calculated for all the possible symmetry types of bands for this group.

Band representations and symmetry types of bands in solids

Symmetry types of bands in solids are specified by means of band representations of space groups. This is a new kind of representation that corresponds to bands of energies rather than to single energies as in the case of usual representations. It is shown that each band representation defines a symmetry type of a band by specifying the symmetry of localized orbitals with respect to a whole lattice of point group centers. In this symmetry specification the quasicoordinate $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ in the Wigner-Seitz cell plays a similar role to what is played by the quasimomentum $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ in the symmetry specification of Bloch states in the Brillouin zone.