51.1 Symmetries in Solid-State Physics andPhotonics
Next to the optimization of numerical calculations, group theory can be ap-
plied to classify promising systems for further investigations, like in the case of the
search for multiferroic materials [19, 20]. In general, four primary ferroic prop-
erties are known: ferroelectricity, ferromagnetism, ferrotoroidicity, and ferroelas-
ticity. The magnetoelectric coupling, of special interest in applications, is a sec-
ondary ferroic effect. The occurrence of multiple ferroic properties in one phase
is connected to specific symmetry conditions a material has to accomplish.
Defects in solids and at solid surfaces play a continuously increasing role in ba-
sic research and applications (diluted magnetic semiconductors, p-magnetism in
oxides). For example, group theory allows to get useful information in a general
and efficient way (cf. [21, 22]) treating defect states in the framework of perturba-
tion theory.
More recently, a close connection between high-energy physics and condensed
matter physics has been established, where effective elementary excitations with-
in a crystal behave as particles that were formally described in elementary particle
physics. A promising class of materials are Dirac materials like graphene, where
the elementary electronic excitations behave as relativistic massless Dirac fermi-
ons [23, 24]. Degeneracies and crossings of energy bands within the electronic
band structure together with the dispersion relation in the neighborhood of the
crossing point are closely related to the crystalline symmetry [25, 26].
In Figure 1.2b, a scanning electron microscope (SEM) image of macroporous
silicon is shown. The special etching technique provides a periodically structured
dielectric material that is referred to as a photonic crystal. The propagation of
electromagnetic waves in such structures can be calculated starting from M-
’s equations [27, 28]. The resulting eigenmodes of the electromagnetic field
are closely connected to the symmetry of the structured dielectric. Group theory
can be applied in various cases within the field of photonics. Subsequently, a few
examples are mentioned. The photonic bands of two-dimensional photonic crys-
tals can be classified with respect to the symmetry of the lattice. The symmetry
properties of the eigenmodes, found by means of group theory, decide whether
this mode can be excited by an external plane wave [29]. Metamaterials are com-
posite materials that have peculiar electromagnetic properties that are different
from the properties of their constituents. Group theory can be used for design
and optimization of such materials [30]. Group theoretical arguments also help
to discuss the dispersion in photonic crystal waveguides in advance. Clearly, this
approach represents a more sophisticated strategy in comparison to relying on a
trial and error approach [31, 32]. If a magneto-optical material is used for a pho-
tonic crystal, time-reversal symmetry is broken due to the intrinsic magnetic field.
In this case, the theory of magnetic groups can be used to study the properties of
such systems [33].
The goal of this book is to discuss the variety of possible applications of com-
putational group theory as a powerful tool for actual research in photonics and
electronic structure theory. Specific examples usingthe Mathematica package
GT-
Pack
will be provided.