Phonons and the Isotopically Induced Mott transition - Physics

8 Ano**the**r possibility - **the** geometrical

isotope effect

From **the**se 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 **the**re is some evidence to suggest o**the**rwise. 41

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

hydrostatic pressure **and** **the**se 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** **the**refore 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 **the**re 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 lig**and**s. These results certainly warrant fur**the**r 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 underst**and** **the** role of phonons in **the** MIT. We **the**n introduced b**and**

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

in terms e-e interactions opening up gaps in o**the**rwise continuous b**and**s.

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 whe**the**r phonons

participate in **the** MIT.

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

**the**se 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 **the**n

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 o**the**r 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 **the**n 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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