Phase Transitions in Conductivity
Phase Transitions in Conductivity
To achieve the super state, the following can be effective criteria
Larger brillouin zones could create less Umklapp scattering
If Umklapp scattering produces new k-vectors then there might not be any changes
There should be a coupling between the phononic and electronic k vectors ( this is like the pi-band conduction in graphene)
Umklapp and defect scattering should be reduced
On the one side Kittle talks about collision time due to scattering from lattice and imperfections and the shifting of the centre of the Fermi-sphere and on the other side it talks about Umklapp scattering. The important question to ask is, whether if the phononic and electronic lattices are the same would the umklapp scattering be reduced for electronic conductivity ? There is another possibility that the two lattices are already the same. We know that it is not the case, and is only true under certain conditions. Perhaps it can be calculated how would that be possible in the absence of defects. The best result would be to calculate it in the real space as real entities.
Umklapp scattering results due to change in the momentum not due to change in the k-space deltas. It can also be due to defects in crystalline materials that changes the delta functions in a certain manner.
It important to know whether without Umklapp scattering due to a reduced fermi sphere, due to defects will there be bands that are formed. Essentially are there bands at different levels even without Umklapp scattering because
phi(x) = e^(ik.r)u(r)
A sum over this phi(r) would denote a fourier transform over the delta functions of the lattice. It seems the availability of space in the conduction band is a pre-requisite for this transition. There seems to be some relation with Fermi-energy. The other major observation is that out of all the high temperature cuprate superconductors, there is always the involvement of filled shell ( p,d or f) metals and one unfilled (p or d) metal along with CuO layer. The CuO layer is closely associated with the Cooper-pair formation.
One more important consequence is that the Fermi-surface and the Cooper Pair interact in a complex way and lead to d-symmetries which are not seen in other materials. This needs to be explored further. After looking through theory on the creation of Cooper-pairs and their movement it seems strongly correlated systems exhibit behaviours that are conducive for such systems and by analyzing the correlation functions of electrons we can arrive at an understanding of how correlation affects the this transition.
There seems to be some related theories related to modelling this phase transition,
Mott insulators and Hubbard model for spin and e- correlations.
Eliashberg theory for e- correlations and phonon interaction
Quantum spin liquid theory and other theories that model such e- correlations
The soluton to an electron travelling as per the time dependent schrodinger equation is given by
phi(r,t) = u(r)e^i(\kx-\omega*t)
for all times rho = |phi(r,t)|^2 = |u(r)|^2 as |e^i\theta| = 1
For metals only the k-points matter that have an effective overlap. For semiconductors the concepts of direct and indirect band gap comes into picture. It can be thought of as an oscillating fluid. However, I feel this has something to do with the transition. It would be just for understanding that there is a requirement for developing a fluid model and then determining when the state for the same changes.
Pairing strength of coupled electrons (Cooper Pair) must be such that the structure that is formed is conducive for the transporting e- which is good enough to hold at that temperature. Here I can only quote the YBCO example as given on wikipedia which gives us only a pointer
Stability provided by the structure + Pairing Energy + Coupling strength of the crystal < Destabilizing energy of temperature + Exchange Energy + e- repulsion