A metal and a band insulator are distinct states of matter.
A suitable order parameter is the Drude weight, defined as the integral over frequency of the frequency-dependent conductivity at zero temperature.
How might a single material undergo a transition between a band insulator and a metal?
The schematic below (from Kittel) illustrates three distinct possibilities for the band structure and band fillings.
(a) Band insulator. The bands do not overlap and the lower band is full. This occurs if there is an integer number of electrons per primitive cell in the crystal.
(b) Metal I. The bands overlap and the system is a metal regardless of the number density of electrons.
(c) Metal II. The bands do not overlap. There is a non-integer number of electrons per primitive cell.
Suppose the number of electrons is fixed, (i.e. there is no chemical doping).
How can a metal-insulator transition occur when some physical parameter (such as pressure) changes?
Scenario A.
There is no structural phase transition associated with the metal-insulator transition.
A transition between (a) and (b) can occur. This means that at the transition the volume of the Fermi surface will be zero.
This may be an example of a Lifshitz transition (where there is a change in the topology of the Fermi surface), based on this 1959 paper by I.M. Lifshitz.
[Aside: I find this nomenclature confusing because there is also a Lifshitz point, where there is a phase transition between commensurate ordering (such as Neel antiferromagnet) and incommensurate ordering (such as a spiral antiferromagnet).
This is E.M. Lifshitz, co-author with Landau of A Course in Theoretical Physics, and I.M.'s brother.]
Scenario B.
The transition is accompanied by a structural phase transition, such as the doubling of the size of the primitive cell in the crystal.
One example of the latter is the Peierls transition in a one-dimensional metal with one electron per lattice site. Dimerising the lattice produces an energy gap at the Fermi wavevector leading to an insulator.
Another example is the transition from graphite to diamond that occurs at about 15 kbar of a pressure. The crystal structure changes and consequently there is a transition from a semi-metal to an insulator.
I have a few questions for readers. The discussion above involves basic solid state physics but I have not seen it clearly set out before.
1. Do I have the physics correct?
2. Do you know somewhere this is discussed?
3. For scenario A, are the metallic and insulating states adiabatically connected?
4. Do you know of any specific materials where scenario A. actually occurs?
Regarding #3, did not Neville Mott also point out the possibility for an exciton instability in between A and B?
ReplyDeletePerhaps it is my stupidity, but I do not think there is any example of a material with a transition between A and B as a function of temperature/pressure/field that occurs without either a change in the lattice or a phase boundary. So I really wonder about how physical this adiabiticity is.
Thanks for the comment. I think you are asking profound not stupid questions.
DeleteThe following subtleties may be relevant.
1. Just because there is no known material with a transition between (a) and (b) with no associated symmetry breaking, does not mean that it is not possible.
2. A phase boundary is not a sufficient condition for absence of adiabatic connection. A liquid and gas can be adiabatically connected.
Good points that you made.
DeleteAny suggestions for experimentalists on where to look to find a system that might have this transition without symmetry breaking? The field gating of these twisted 2D systems looks appealing, though the 2D Metal insulator adiabaticity may be different from the 3D one.
Lifshitz transition is a topological transition. But metal to insulator is a trivial phase transition. Isn't that?
ReplyDeleteThanks for the comment. There is some subtlety here. My understanding is the following. A Lifshitz transition is a transition in the topology of the Fermi surface, not a transition between two states with different types of topological order (e.g. trivial and non-trivial).
DeleteTi2O3 is a narrow gap semiconductor at low temperature and its band structure shifts to a metal as one increases the temperature due to chaning bond distances.
ReplyDelete