Edge States
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In
solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state p ...
, edge states are the topologically protected electronic states that exist at the boundary of the
material A material is a matter, substance or mixture of substances that constitutes an Physical object, object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical property, physical ...
and cannot be removed without breaking the system's
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
.


Background

The
electronic band structure In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or '' ...
of materials is primarily studied based on the extent of the
band gap In solid-state physics and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to t ...
, the gap between highest occupied valence bands and lowest unoccupied conduction bands. The possible
energy level A quantum mechanics, quantum mechanical system or particle that is bound state, bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical mechanics, classical pa ...
of the material that provides the discrete energy values of all possible states in the energy profile diagram can be represented by solving the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
of the system. This solution provides the corresponding energy
eigenvalues and eigenvectors In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
. Based on the energy eigenvalues, conduction band are the high energy states (energy ) while valence bands are the low energy states (). In some materials, for example, in
graphene Graphene () is a carbon allotrope consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice, honeycomb planar nanostructure. The name "graphene" is derived from "graphite" and the suffix -ene, indicating ...
and zigzag
graphene quantum dot Graphene quantum dots (GQDs) are graphene nanoparticles with a size less than 100 nm. Due to their exceptional properties such as low toxicity, stable photoluminescence, chemical stability and pronounced quantum confinement effect, GQDs are co ...
, there exists the energy states having energy eigenvalues exactly equal to zero () besides the conduction and valence bands. These states are called edge states which modifies the electronic and optical properties of the materials significantly.


References

{{Reflist Physics