Low energy effective Hamiltonians for strongly-correlated organic metals

Strongly-correlated electronic materials can exhibit richly featured phase diagrams, including a Mott metal-insulator transition. In two particular classes of material, $\kappa-$ET$_2$X and $\beta'-$X[Pd(dmit)$_2$]$_2$, the conducting side of this phase transition may contain superconducting and non-Fermi liquid metallic states, as well as the more usual Fermi liquid metallic state. The insulating side of the transition is often antiferromagnetic. However, in some of these materials more exotic states emerge such as spin liquids and valence bond crystals. These unusual insulating states are related, among other considerations, to the subtle changes in geometry among the crystal lattices of these materials, and the implications for the electronic structure. These two materials both have a quasi-2D layered salt crystal structure whose layers can be described as an anisotropic triangular lattice. Each lattice site contains two organic units (ET or dmit) stacked into dimers stabilised by $\pi-\pi$ intermolecular bonding. The dimers donate (ET) or accept (dmit) one electron from their counterions in the salt crystal, resulting in half-filled electronic systems. One of the implications of strong electron correlation is that typical first-principles techniques in quantum chemistry, such as H\"uckel methods or density functional theory (DFT) using approximate exchange-correlation functionals, do not capture the qualitative physics of strongly-correlated systems. While DFT does not directly capture strongly-correlated phenomena, it is still effective at finding the parameters of low energy effective Hamiltonians describing these materials. Therefore, I studied the tight-binding and Hubbard models to explore the correlation between the model parameters and the electronic nature of each material. This work contributes to a greater goal of predicting from first principles calculations the low temperature states of these materials from the ground states of the relevant models. I calculated the effective Coulomb repulsion between electrons/holes, \umvn, and site energy for an isolated ET molecule {\it in vacuo}. $U_m^{(v)}=4.2 \pm0.1$ eV for 44 experimental geometries taken from a broad range of conformations, polymorphs, anions, temperatures, and pressures (the quoted `error' is one standard deviation). Hence \umv is essentially the same for all of the compounds studied. This shows that the strong (hydrostatic and chemical) pressure dependence observed in the phase diagrams of the ET salts is not due solely to \umvn. Therefore, if the Hubbard model is sufficient to describe the phase diagram of the ET salts there must be significant pressure dependence on the intramolecular terms in the Hamiltonian and/or the reduction of the Hubbard $U$ due to the interaction of the molecule with the polarisable crystal environment. The renormalised value of \umv is significantly smaller than the bare value of the Coulomb integral: $F_0=5.2\pm0.1$ eV across the same set of geometries. This emphasises (i) the importance of using the renormalised value of \umvn and (ii) that a site in the Hubbard model does not correspond to any real orbital in the ET molecule as the orbitals change significantly, even in the isolated molecule, as the charge fluctuates. The site energy (for holes), $\xi_m=5.0\pm0.2$ eV, varies only a little more than \umvn across the same set of geometries. However, the site energy plays a key role in understanding the role of disorder in ET salts in general and in particular the difference between the $\beta_L$ and $\beta_H$ phases of \bIn. The next parameterisation step I performed was an extension of the model to two sites, to calculate the interactions between holes in ET dimers for 23 experimental geometries taken from a range of materials in both the $\beta$ and $\kappa$ polymorphs. I find that the effective Coulomb interactions are essentially the same for all of the compounds studied. My parameterisation disagrees with similar reported parameterisations from the literature, which used both DFT and H\"{u}ckel methods. This is caused by the failure of an assumption made in previous calculations (which estimate the effective Coulomb interaction from the intra-dimer hopping integral). I subsequently use my parameterisation results to explain a number of phenomena caused by conformational disorder in these materials. Finally, I parameterised the tight-binding model for \kCN and eleven $\beta'$ dmit materials. The ratio of the tight-binding parameters, $t'/t$, provides a measure of the geometric frustration present in the lattice. The materials studied range in $t'/t$ from 0.7--1.5, with one outlying crystal, $\beta'$-Et$_3$Me$_1$N[Pd(dmit)$_2$]$_2$, which has $t'/t \sim 0.33 $ and has the typical antiferromagnetic insulating state. Of the materials with an unusual insulating state, the spin liquid $\beta'$-Et$_1$Me$_3$Sb[Pd(dmit)$_2$]$_2$ has $t'/t = 0.79$ while the valence bond crystal $\beta'$-Et$_1$Me$_3$P[Pd(dmit)$_2$]$_2$ has $t'/t = 0.87$. These parameter values are in the expected range for frustrated spins on an anisotropic triangular lattice in the model Hamiltonians that represent these materials. The presence of a glide plane symmetry in the layering direction of many of the dmit materials produces a band structure in which two bands cross the Fermi energy, one for each distinct layer. In some materials, the bands may have a small energy separation, implying the half-filled electronic system is actually half-filled on average, with each band having an effective doping away from half-filling. This effect is on the order of $1\%$, but can have implications for the ground state of the relevant model Hamiltonians