Phonons and the Isotopically Induced Mott transition - Physics

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Phonons and the Isotopically Induced Mott transition - Physics

8 Another possibility - the geometrical

isotope effect

From these results we can conclude that phonons offer a plausible means to account

for the observed MIT upon deuteration in the CT’s in the form of frustration. However,

throughout this analysis we have assumed that the lattice parameters remain

unaffected by isotopic variation but there is some evidence to suggest otherwise. 41

Although we have not dealt with it specifically, this transition can also be driven by

hydrostatic pressure and these materials are highly sensitive to pressure variations

with insulating, superconducting, and metallic phases being observed over a rang of

a few hundred bars. 17 In terms of the Hubbard framework increasing pressure

increases the HOMO overlap and therefore W.

Due to this pressure sensitivity, changing the lattice parameters by even a small

amount may be enough to drive this transition. Watanabe et. al. 41 used X-ray

diffraction to examine how the lattice parameters in the crystal change with

deuteration. They find that there is up to a 0.11% change that may be associated

with changes in the bond lengths on the ethylene groups and the hydrogen bonding

to the anionic ligands. These results certainly warrant further investigation as a

possible source of the isotopically induced MIT.

9 Summary and conclusion

In Chapter 2 we introduce the concept of strongly correlated systems and why we

would like to understand the role of phonons in the MIT. We then introduced band

theory and the concept of the onsite Coulomb repulsion to account for this transition

in terms e-e interactions opening up gaps in otherwise continuous bands.

Chapter 3 introduces a family of quasi-2D organic metals - the charge transfer salts –

in which this transition can be driven by both isotopic and chemical substitution as

well as by hydrostatic pressure. We explain how the observed vibrational spectra

change upon isotopic substitution and ask the question of whether phonons

participate in the MIT.

We then outline the dimer Hubbard model as a minimal model that is often applied to

these quasi-2D strongly correlated systems and explain how it can be modified to

account for intramolecular phonons through reduction to a two-site model where

each site now represents a monomer. The extended Hubbard Hamiltonian is then

applied to this system. The onsite Coulomb repulsion and interdimer hopping

integrals are expressed as functions of the intermolecular (intradimer and molecular

interdimer) parameters and it is noted that the offsite Coulomb repulsion changes the

value of U d significantly from ~2t m to ~10t m using values for the monomer parameters

calculated by other authors.

In Chapter 5 we introduce the Holstein model as a minimal model to account for e-ph

coupling and apply a modified Lang Firsov transformation with squeezing to obtain

an effective variational Hamiltonian in which the phononic degrees of freedom have

been averaged out by applying a variational squeezed state wavefunction. This

Hamiltonian is then compared to an exact wavefunction to test the parameter range

over which this trial wavefunction is a suitable approximation.

Electron-electron interactions are introduced into this effective Hamiltonian in

Chapter 6 with the Hubbard-Holstein Hamiltonian. Averaging out the phononic

degrees of freedom we obtain what is effectively a polaronic Hubbard model with a

constant phonon energy offset. The parameter range over which this model is tested

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