Topological insulator
A topological insulator is a material that behaves as an insulator in its interior or bulk while permitting the movement of charges (metallic) on its surface.

In the bulk of a topological insulator the electronic band structure
Electronic band structure
In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...

 resembles an ordinary band insulator, with the Fermi level
Fermi level
The Fermi level is a hypothetical level of potential energy for an electron inside a crystalline solid. Occupying such a level would give an electron a potential energy \epsilon equal to its chemical potential \mu as they both appear in the Fermi-Dirac distribution function,which...

 falling between the conduction and valence bands. On the surface of a topological insulator there are special states that fall within the bulk energy gap and allow surface metallic conduction. Carriers in these surface states have their spin
Spin quantum number
In atomic physics, the spin quantum number is a quantum number that parameterizes the intrinsic angular momentum of a given particle...

 locked at a right-angle to their momentum (spin-momentum locking or topological order). At a given energy the only other available electronic states have different spin, so the "U"-turn scattering is strongly suppressed and conduction on the surface is highly metallic. These states are characterized by an index (known as Z2 topological invariants) similar to the genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

 in topology, and are an example of topologically ordered
Topological order
In physics, topological order is a new kind of order in a quantum state that is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations...


Topologically protected edge states were predicted to occur in quantum well
Quantum well
A quantum well is a potential well with only discrete energy values.One technology to create quantization is to confine particles, which were originally free to move in three dimensions, to two dimensions, forcing them to occupy a planar region...

s (very thin layers) of mercury telluride sandwiched between cadmium telluride
Cadmium telluride
Cadmium telluride is a crystalline compound formed from cadmium and tellurium. It is used as an infrared optical window and a solar cell material. It is usually sandwiched with cadmium sulfide to form a p-n junction photovoltaic solar cell...

 (band inversion in Hg(Cd)Te was first reported in 1986 by Pankratov and collaborators), and were observed in 2007. They were predicted to occur in three dimensional bulk solids of binary compounds involving bismuth
Bismuth is a chemical element with symbol Bi and atomic number 83. Bismuth, a trivalent poor metal, chemically resembles arsenic and antimony. Elemental bismuth may occur naturally uncombined, although its sulfide and oxide form important commercial ores. The free element is 86% as dense as lead...

. A 3D "strong topological insulator" exists which cannot be reduced to multiple copies of the quantum spin Hall state.

The first experimentally realized 3D topological insulator state (topological surface states) was discovered in bismuth antimony. Shortly thereafter topologically protected surface states were also observed in pure antimony
Antimony is a toxic chemical element with the symbol Sb and an atomic number of 51. A lustrous grey metalloid, it is found in nature mainly as the sulfide mineral stibnite...

, bismuth selenide
Bismuth selenide
Bismuth selenide is a gray powder that is a compound of bismuth and selenium also known as bismuth selenide. It is a semiconductor and a thermoelectric material. While perfect stoichiometric bismuth selenide should be a semiconductor naturally occurring selenium vacancies act as electron donors...

, bismuth telluride
Bismuth telluride
Bismuth telluride is a gray powder that is a compound of bismuth and tellurium also known as bismuth telluride. It is a semiconductor which, when alloyed with antimony or selenium is an efficient thermoelectric material for refrigeration or portable power generation...

 and antimony telluride
Antimony telluride
Antimony telluride is an inorganic compound with the chemical formula Sb2Te3. It is a grey, crystalline solid, although its melting point, density and colour may depend on the crystalline form it adopts.-Synthesis:...

 using ARPES
Angle-resolved photoemission spectroscopy , also known as ARUPS , is a direct experimental technique to observe the distribution of the electrons in the reciprocal space of solids...

. Several other material systems are now believed to exhibit topological surface states. In some of these materials the Fermi level actually falls in either the conduction or valence bands due to naturally occurring defects, and must be pushed into the bulk gap by doping or gating. Topological order is encoded in the existence of a gas of helical Dirac fermions. Helical Dirac fermion
Helical Dirac fermion
A Helical Dirac fermion is a charge carrier that behaves as massless relativistic particle with its intrinsic spin locked to its translational momentum....

, which behaves like a massless relativistic particle, has been observed in a 3D topological insulator.

The Z2 topological invariants cannot be measured using traditional transport method and transport is not quantized by the Z2 invariants. An experimental method to measure Z2 topological invariants was demonstrated which provide a measure of the Z2 topological order.

The surface states of a 3D Topological insulator is thus a new type of 2DEG (two dimensional electron gas) where electron's spin is locked to its linear momentum. The topological surface states differ from Graphene due to the locking of spin and momentum. Spin momentum locking or topological order allows topological surface states to host Majorana
Majorana may refer to:* Majorana equation, a relativistic wave equation* Majorana fermion, a concept in particle physics* Majorana spinor, a concept in quantum field theory* Origanum majorana, a somewhat cold-sensitive perennial herb...

 particles if superconductivity is induced on the surface of 3D topological insulators via proximity effects.

The surface states in Z2 topological insulators can be destroyed by local perturbations that break the time reversal symmetry. As a result, the gapless edge/surface states of topological insulators are also not topologically protected in the strictest sense. They can be gapped/localized by local perturbations that break the time reversal symmetry. However, true topological states (such as fractional quantum Hall
Fractional quantum Hall effect
The fractional quantum Hall effect is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of e^2/h. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations...

states) do exist, which
are stable against any local perturbations that can break any symmetries.
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