Mott insulator

Encyclopedia

**Mott insulators**are a class of materials that should conduct electricity

Electricity

Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...

under conventional band theories

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...

, but are insulators when measured (particularly at low temperatures). This effect is due to electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

-electron interactions which are not considered in conventional band theory.

The bandgap in a Mott insulator exists between bands of like character, such as 3d character, while the bandgap in charge transfer insulators

Charge transfer insulators

Charge transfer insulators are a class of materials predicted to be conductors following conventional band theory, but which are in fact insulators due to a charge transfer process. Unlike Mott insulators, where the insulating properties arise from electrons hopping between unit cells, the...

exists between anion and cation states (see lecture slides ), such as between O 2p and Ni 3d bands in NiO.

## History

Although the band theory of solids had been very successful in describing various electrical properties of materials, in 1937 Jan Hendrik de BoerJan Hendrik de Boer

Jan Hendrik de Boer was a Dutch physicist and chemist.De Boer was born in Ruinen, now De Wolden, and died in 's-Gravenzande...

and Evert Johannes Willem Verwey pointed out that a variety of transition metal oxides predicted to be conductors by band theory (because they have an odd number of electrons per unit cell) are insulators. Nevill Mott and R. Peierls then (also in 1937) predicted that this anomaly can be explained by including interactions between electrons.

In 1949, in particular, Mott proposed a model for NiO

Nickel(II) oxide

Nickel oxide is the chemical compound with the formula NiO. It is notable as being the only well characterized oxide of nickel . The mineralogical form of NiO, bunsenite, is very rare. It is classified as a basic metal oxide...

as an insulator, where conduction is based on the formula

_{2}→ Ni

^{3+}O

^{2−}+ Ni

^{1+}O

^{2−}.

In this situation, the formation of an energy gap preventing conduction can be understood as the competition between the Coulomb potential

*U*between 3

*d*electrons and the transfer integral

*t*of 3

*d*electrons between neighboring atoms (the transfer integral is a part of the tight-binding

Tight binding (physics)

In solid-state physics, the tight binding model is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method used...

approximation). The total energy gap is then

*E*_{gap}=*U*− 2*zt*,

where

*z*is the number of nearest-neighbor atoms.

In general, Mott insulators occur when the repulsive Coulomb potential

*U*is large enough to create an energy gap. One of the simplest theories of Mott insulators is the 1963 Hubbard model

Hubbard model

The Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...

. The crossover from a metal to a Mott insulator as is increased can be predicted within the so-called Dynamical Mean Field Theory

Dynamical mean field theory

Dynamical Mean Field Theory is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in Density Functional Theory and usual band structure calculations, breaks down...

.

## Mottness

*Mottness*denotes the additional ingredient, aside from antiferromagnetic ordering, which is necessary to fully describe a Mott Insulator. In other words, we might write

*antiferromagnetic order + mottness = Mott insulator*

Thus, mottness accounts for all of the properties of Mott insulators that cannot be attributed simply to antiferromagnetism.

There are a number of properties of Mott insulators, derived from both experimental and theoretical observations, which cannot be attributed to antiferromagnetic ordering and thus constitute mottness. These properties include

- Spectral weight transfer on the Mott scale
- Vanishing of the single particle Green functionGreen's function (many-body theory)In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators....

along a connected surface in momentum space in the first brillouin zoneBrillouin zoneIn mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice... *Two*sign changes of the Hall coefficientHall effectThe Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...

as electron doping goes from to (band insulatorsElectronic band structureIn 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...

have only one sign change at )- The presence of a charge (with the charge of an electron) boson at low energies
- A pseudogap away from half-filling ()

## Applications

Mott insulators are of growing interest in advanced physicsPhysics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

research, and are not yet fully understood. They have applications in thin-film magnetic heterostructures and high-temperature superconductivity

High-temperature superconductivity

High-temperature superconductors are materials that have a superconducting transition temperature above . From 1960 to 1980, 30 K was thought to be the highest theoretically possible Tc...

, for example.

## See also

- Hubbard modelHubbard modelThe Hubbard model is an approximate model used, especially in solid state physics, to describe the transition between conducting and insulating systems...
- Tight-binding approximationTight binding (physics)In solid-state physics, the tight binding model is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon superposition of wave functions for isolated atoms located at each atomic site. The method is closely related to the LCAO method used...
- Electronic band structureElectronic band structureIn 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...
- Mott Criterion
- Dynamical Mean Field TheoryDynamical mean field theoryDynamical Mean Field Theory is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in Density Functional Theory and usual band structure calculations, breaks down...
- (Mott) Variable range hopping