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Electronic Structure and Topology of Novel Materials

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Topological insulator

A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material. A topological insulator is an insulator for the same reason a "trivial" (ordinary) insulator is: there exists an energy gap between the valence and conduction bands of the material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator.

Topological order

In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian geometric phases of degenerate ground states. Microscopically, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.

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 forbidden bands). Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules.

Periodic table

The periodic table, also known as the periodic table of the elements, arranges the chemical elements into rows ("periods") and columns ("groups"). It is an organizing icon of chemistry and is widely used in physics and other sciences. It is a depiction of the periodic law, which says that when the elements are arranged in order of their atomic numbers an approximate recurrence of their properties is evident. The table is divided into four roughly rectangular areas called blocks.

Block (periodic table)

A block of the periodic table is a set of elements unified by the atomic orbitals their valence electrons or vacancies lie in. The term appears to have been first used by Charles Janet. Each block is named after its characteristic orbital: s-block, p-block, d-block, f-block and g-block. The block names (s, p, d, and f) are derived from the spectroscopic notation for the value of an electron's azimuthal quantum number: sharp (0), principal (1), diffuse (2), or fundamental (3).

Topological space

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness.

Group (periodic table)

In chemistry, a group (also known as a family) is a column of elements in the periodic table of the chemical elements. There are 18 numbered groups in the periodic table; the 14 f-block columns, between groups 2 and 3, are not numbered. The elements in a group have similar physical or chemical characteristics of the outermost electron shells of their atoms (i.e., the same core charge), because most chemical properties are dominated by the orbital location of the outermost electron.

Topological quantum number

In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations.

Period (periodic table)

A period on the periodic table is a row of chemical elements. All elements in a row have the same number of electron shells. Each next element in a period has one more proton and is less metallic than its predecessor. Arranged this way, elements in the same group (column) have similar chemical and physical properties, reflecting the periodic law. For example, the halogens lie in the second-to-last group (group 17) and share similar properties, such as high reactivity and the tendency to gain one electron to arrive at a noble-gas electronic configuration.

Wannier function

The Wannier functions are a complete set of orthogonal functions used in solid-state physics. They were introduced by Gregory Wannier in 1937. Wannier functions are the localized molecular orbitals of crystalline systems. The Wannier functions for different lattice sites in a crystal are orthogonal, allowing a convenient basis for the expansion of electron states in certain regimes. Wannier functions have found widespread use, for example, in the analysis of binding forces acting on electrons; the existence of exponentially localized Wannier functions in insulators was proved in 2006.