Abstract : An energy band in a solid contains an infinite number of states which transform linearly as a space group representation induced from a finite dimensional representation of the isotropy group of a point in space. A band representation is elementary if it cannot be decomposed as a direct sum of band representations; it describes a single band. We give a complete classification of the inequivalent elementary band representations.]]>

Symmetry and classification of energy bands in crystals

SOLIDES

CLASSIFICATION

ENERGIE

GROUPES SPATIAUX

To appear in Proceedings XVth International Colloquium on Group Theretical Methods in Physics (Varna, June 1987)

Abstract : An energy band in a solid contains an infinite number of states which transform linearly as a space group representation induced from a finite dimensional representation of the isotropy group of a point in space. A band representation is elementary if it cannot be decomposed as a direct sum of band representations; it describes a single band. We give a complete classification of the inequivalent elementary band representations.

Abstract : An energy band in a solid contains an infinite number of states which transform linearly as a space group representation induced from a finite dimensional representation of the isotropy group of a point in space. A band representation is elementary if it cannot be decomposed as a direct sum of band representations; it describes a single band. We give a complete classification of the inequivalent elementary band representations.

MICHEL

BACRY

ZAK

P/87/42

IHES

10/1987

A4

11 f.

EN

TEXTE

PREPUBLICATION

P_87_42.pdf

1987

IHES

IHES

MICHEL

BACRY

ZAK

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1964-1989/P_87_42/P_87_42.pdf

Oui

Bures-sur-Yvette

EUCLIDE

CLOIZEAUX

ZAK

EVARESTOV

SMIRNOV

BLOCH

WIGNER

SEITZ

BRILLOUIN

HILBERT

HERMITE

FROBENIUS

MACKEY

BORN

KARMAN

ABEL

WYCKOFF

FOURIER

Structure and classification of band representations

CRISTALLOGRAPHIE

SOLIDES

ASTRONOMIE

ORBITE

ETOILES

MODELISATION

ENERGIE

ATOMES

Abstract : Band representaitons in solids are investigated in the general framework of induced representations by using the concepts of orbits (stars) and strata (Wyckoff positions) in their construction and classification. The connection between band representations and irreducible representations of space groups is established by reducing the former in the basis of quasi-Bloch functions wich are eigenfunctions of translations but ar not, in general, eigenfunctions of the Hamiltonian. While irreducible representations of space group ar fnite-dimensional and are induced from infinite-order little groups Gk for vectors K in the Brillouin zone, band representations are infinite-dimensional adn are induced from finite-order little groups Gr for vectors r in the Wigner-Seitz cell. This connection between irreductible representations and band representations of space groups shedd new light on the duality properties of the Brillouin zone and the Wigner-Seitz cell. As an introduction to band representations the induced representations of point groups wich id applied to the investigation of th equivalency of band representations. Based on this connection and on the properties of the crystallographic point groups a necessary condition id established for the inequivalency of band representations induced from maximal isotropy groups. For using this condition there is need fro the induced representaitons of oint groups and a full list of them is given in the paper. One is especially interested in irreducible-band representations which form the elementary building bricks for band representations. From the point of view of the physics, irreducible-band representations correspond to energy bands with minimal numbers of branches. A method id developped for finding all the inequivalent irreducible-band representations of space groups by using the induction from maximal isotropy groups. As a rule the latter leads to inequivalent irreducible-band representations. There are, however, few exceptions to this rule. A full list of such exceptions is tabulated in the paper. With this list at hand one can construct all the different irreducible-band representations of 2-dimensional space groups. For them we list the continuity chords of all their irreducible-band representations.

MICHEL

BACRY

ZAK

P/86/35

IHES

06/1986

A4

46 f.

EN

TEXTE

PREPUBLICATION

P_86_35.pdf

1986

IHES

IHES

MICHEL

BACRY

ZAK

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1964-1989/P_86_35/P_86_35.pdf

Oui

Bures-sur-Yvette

WYCKOFF

BLOCH

HAMILTON

BRILLOUIN

WIGNER

SEITZ

HILBERT

WEYL

HEISENBERG

EUCLIDE

CLOIZEAUX

FROBENIUS

MACKEY

POINCARE

LORENTZ

MINKOWSKI

DIRICHLET

FEDOROV

VORONOY

DELONE

BRAVAIS

ABEL

SYLOW