What is metal-to-insulator transition and origin of the problem?

What is metal-to-insulator transition? 

Why strong-static correlation is responsible for this transition?

To answer this question, we must first look at the band structure picture of solid state materials.

Our basic understanding on the band structure theory is as follows:

The electronic states form bands in materials and there is an energy level called Fermi energy which determines weather a material is metal or insulator. 

If the Fermi energy lies inside a band, this allows free movement of electrons and thus it is a metal. If the Fermi energy lies in forbidden states (in the gap) it acts as insulator. 

However, it was found that certain materials are insulators even though their Fermi level lies inside the bands. One of the examples is NiO.  It was explained that the strong interactions (electrostatic interactions by nature) make the movement of electrons extremely difficult. This interaction is termed as strong-correlation. Because of this strong-correlation, the a material actually act as an insulator (metal-to-insulator transition has occurred) even though it should act as a metal.

A clear note has been given in the general discussion of "Metal-insulator transitions" by Masatoshi Imada, Atsushi Fujimori and Yoshinori Tokura, Reviews of Modern Physics, Vol. 70, No. 4, October 1998 of Tokyo University, Japan. 

Here is an excerpt from this Masatoshi Imada et al, Reviews of Modern Physics, Vol. 70, No. 4, October 1998.

"The first successful theoretical description of metals, insulators, and transitions between them is based on noninteracting or weakly interacting electron systems. The theory makes a general distinction between metals and insulators at zero temperature based on the filling of the electronic bands: For insulators the highest filled band is completely filled; for metals, it is partially filled. In other words, the Fermi level lies in a band gap in insulators while the level is inside a band for metals. In the noninteracting electron theory, the formation of band structure is totally due to the periodic lattice structure of atoms in crystals. This basic distinction between metals and insulators was proposed and established in the early years of quantum mechanics (Bethe, 1928; Sommerfeld, 1928; Bloch, 1929). By the early 1930s, it was recognized that insulators with a small energy gap between the highest filled band and lowest empty band would be semiconductors due to thermal excitation of the electrons (Wilson, 1931a, 1931b; Fowler, 1933a, 1933b). More than fifteen years later the transistor was invented by Shockley, Brattain, and Bardeen. 

Although this band picture was successful in many respects, de Boer and Verwey (1937) reported that many transition-metal oxides with a partially filled d-electron band were nonetheless poor conductors and indeed often insulators. A typical example in their report was NiO. Concerning their report, Peierls (1937) pointed out the importance of the electron-electron correlation: Strong Coulomb repulsion between electrons could be the origin of the insulating behavior. According to Mott (1937), Peierls noted 

‘‘it is quite possible that the electrostatic interaction between the electrons prevents them from moving at all. At low temperatures the majority of the electrons are in their proper places in the ions. The minority which have happened to cross the potential barrier find therefore all the other atoms occupied, and in order to get through the lattice have to spend a long time in ions already occupied by other electrons. This needs a considerable addition of energy and so is extremely improbable at low temperatures.’’

These observations launched the long and continuing history of the field of strongly correlated electrons, particularly the effort to understand how partially filled bands could be insulators and, as the history developed, how an insulator could become a metal as controllable parameters were varied. This transition illustrated in Fig. 1 is called the metal-insulator transition (MIT). The insulating phase and its fluctuations in metals are indeed the most outstanding and prominent features of strongly correlated electrons and have long been central to research in this field." [For full text refer Masatoshi Imada et al, Reviews of Modern Physics, Vol. 70, No. 4, October 1998]

A lot of studies show that material exhibiting metal-to-insulator transition (i.e. Mott insulator) act as the parent materials for high temperature superconductors. When these Mott insulators or materials with metal-to-insulator transition is doped with oxygen (which is known for absorbing electrons), oxygen get some of the electrons from the materials. This somehow lead to high temperature super conductivity. Why? Nobody knows so far. (Is there an exception for this?). This need explanation.

What happens in this process also happens in the case of bilayer graphene that is twisted at a magic angle of 1.1 degrees. When rotated exactly at 1.1 degress, graphene becomes super conductor. What is the reason? No answer so far.