2. Historical Perspective

  • 1926 quantum mechanics

  • 1928-1931 Bloch/Wilson/Peierls move on to solids and invent band theory. Convincing explanation for metallic, insulating, and semiconducting behaviour in solids.

  • 1937 Bristol conference on electrical conduction mechanisms - data presented for series of transition metal oxides (most famous example NiO) which seems to imply drastic breakdown in band theory. First identification of strongly correlated materials.

  • 1949-1954 Series of paper by Mott provides conceptual explanation.

  • 1960s Hubbard models. New types of strongly correlated materials identified.

  • 1970s Advent of proper computers + density functional theory. People try to do real band theory calculations. They don't work too well.

  • 1980s People discover high Tc superconductivity and suddenly transition metal oxides become fashionable again. People try to do band theory calculations. They don't work too well.

  • 1990s People try to do band theory calculations. Some of them work a little better. Why?





Let's look at some historical aspects, just so we get a feel for what the problem is. After Scrodinger, Heisenberg, Bohr, Born, Dirac and Jordan finish inventing quentum theory, people like Bloch, Wilson and Rudolf Peierls start applying it to solids, and they come up with band theory. They found that, because the phase of the wave function can vary from site to site without violating the boundary condition, bands of allowed energy states exist that may be spread over several eV. They introduced the wave vector k that labels the individual levels in a band and describes the frequency and direction of the phase modulation. And essentially, they showed that a material can be classified as a metal or an insulator according to whether the highest occupied electron level lies within a band or at the bottom of a band gap.

A problem arose in some conference in 1937. De Boer and Verwey presented some conductivity data for some insulating transition metal oxides, and said "Look, this proves that Bloch/Wilson band theory breaks down.." Why? Well let's take the paradigmatic one: nickel oxide (NiO). The valence band is made up of 3d states. What can split the 3d band? In those days the answer was the crystal field. NiO is cubic so the d manifold splits into an lower triplet and an upper doublet, which for group theoretical reasons are known as the t2g and eg states respectively. There are 8 d electrons so we fill up the bands 2,4,6,8 and the Fermi level is right in the middle of the eg states. So, this must be a metal, according to Bloch and Wilson. Rudolf Peierls comments that this must be due to 'strong correlations' between the electrons which causes them to become localized, hence completely violating the band theory picture which is taken to refer to completely delocalized electrons. Professor Nevill Mott, who was the chairman, goes away to think about it. Thirteen years later in 1949 he begins to publish a series of articles which, conceptually at least, solves the problem.

As you may know, he considered a crystal in which you can vary the lattice constant. Actually a one-dimensional lattice of hydrogen atoms but you can think of any metallic crystal you want. Itinerant electrons can move from site to site in a metal. Increase the lattice constant up to a couple of miles. We then just have an array of independent atoms sitting around in space. So somewhere along the way, there must be transition point between localized and itinerant behaviour. In the 1D hydrogen metal, it must be the case that short-ranged charge fluctuations occur, leading in particular to configurations with two electrons with opposite spins on the same site. These fluctuations cost some energy, and therefore short-ranged correlations tend to suppress the formation of delocalized states and this tends to make the electrons prefer to be localized on each site. Now, Mott showed that we can expect complete localization when some characteristic Coulomb energy (U) is of the order of the band width, defined through the overlap between orbitals. Now, if this happens, only activated conductivity is possible, which means that a gap must have arisen in the one-electron density of states near the Fermi level. Fine.

In the 1960s people wanted to study this and they developed various simple models which lead to localized and itinerant behaviour in the appropriate limits. For example, the Hubbard model, which Peter talked about. You have a second-quantized Hamiltonian with a parameter U, which approximates the strong on-site Coulomb interaction and a parameter t, which is a kinetic term describing the motion of electrons between neighbouring sites. And people have managed to extract lots of important information from this.

I should mention that one thing that wasn't realized in the 1940s was that many of these interesting materials like NiO were antiferromagnetic: they have local magnetic moments which order below a certain temperature. When this was realized, Slater came up with a different idea to Mott for explaining the insulating behaviour, as follows: when we form an antiferromagnet, we double the size of the unit cell, hence we halve the size of the first Brillouin zone, and gaps open up at the new zone boundaries. See any second year solid-state course. (This is in fact not true).

So when computers arrive, and Hohenberg, Kohn and Sham invent density functional theory, and people started trying to do proper calculations and they predict NiO and other antiferromagnetic insulators to be non-magnetic metals. Then they discover spin polarization and they predict them to be ferromagnetic metals. So band theory breaks down and leads to lots of confusion. Then in 1987 high Tc cuprates arrive, which in their undoped state turn out to be two-dimensional antiferromagnetic Mott insulators. People really want to understand these so they do band theory calculations and they find ferromagnetic metals. So there really does appear to be a big problem.