Electronic Properties of Cerium Oxides:Towards an Effective Valence Model Hamiltonian

The primary aim of this thesis is to develop a minimal model Hamiltonian to describe the electronic properties of ceria (CeO$_{2}$) and its reduced phases. In order to do this, several energy scales of the problem were explored to determine their relative significance to the problem. These included the crystal electric field ($\Delta_{CEF}$), the spin-orbit coupling ($\lambda_{so}$), electron hopping ($t$), on-site Coulomb repulsion for the Ce $4f$ states ($U$), the reorganization energy ($\lambda_{0}$), the direct exchange ($J$) and the energy gap between the oxygen valence band level and the cerium $4f$ states ($\Gamma\epsilon$). Once the relative magnitudes of the various energy scales were determined, it was then possible to define a minimal set of degrees of freedom required to obtain a meaningful description of the system in the minimal model. The first task was to obtain baseline data for both CeO$_{2}$ and Ce$_{2}$O$_{3}$ as well as the metallic phases of Ce. Density Functional Theory (DFT) calculations were performed on these materials to obtain band structures as well as structural properties. The DFT results indicated that both the LDA and GGA functionals perform poorly for $\gamma$-Ce and Ce$_{2}$O$_{3}$. In the case of Ce$_{2}$O$_{3}$, both LDA and GGA give a metallic ground state contrary to the insulator observed in experiment. The first set of energy scales that were investigated included the crystal electric field and the spin-orbit coupling. Since these calculations, except for the presence of a crystal field, involve an isolated Ce ion, the energy scales associated with spin and charge fluctuations at a Ce site are ignored. In order to perform these calculations, it was necessary to determine the $f^{n}$ configuration at each Ce site in the various phases of the oxides. This was achieved by a simple empirical model called the bond valence model. The bond valence model provides a method for calculating site valences in crystals from bond length data alone. Apart from getting information about the $f^{n}$ configurations at the various Ce sites, the results of the bond valence analysis revealed that there was significant mixed valence in the different phases of the cerium oxies. In addition, it was possible to characterize the charge distribution in the local environment of the oxygen vacancies from the bond valence results. This analysis led to the important result that the charge prefers Ce sites farthest away from the oxygen vacancy. This is contrary to the widely accepted view that the extra charge resulting from oxygen vacancy formation localizes on Ce sites closest to the O vacancy. The LDA calculations which support the widely accepted view do not properly treat electron correlations. Thus this is an important result which emphasizes the role of strong electron correlations in the processes of oxidation and reduction in these materials. The results of the crystal field calculations were not quantitative because we could not find data on crystal field parameters which is required to evaluate the matrix elements. However, it is expected that the splitting of the $f$ manifold by the crystal fields will be smaller than that of the spin-orbit coupling energy which is of the order of $0.07\unit{eV}$. Results for the Hubbard $U$ parameter in Ce oxides were obtained from the literature but these vary widely and fall in the range $1.0 - 10.5\unit{eV}$. The reorganization energy reported from spectroscopic results of the intervalence transition band for reduced ceria is $1.4\unit{eV}$. Tight-binding calculations were performed for CeO$_{2}$ to obtain the energy band structure for this material. It was then possible to extract from these results the matrix elements for electron hopping. The sizes of the matrix elements obtained suggested that the direct $f$-$f$ hopping between Ce sites ($t_{ff}$) is of order $0.02\unit{eV}$ which is negligible compared to the indirect hopping via an O atom (i.e., $t_{eff}$) which is of order $0.2\unit{eV}$. The hopping matrix element between a Ce site and a neighbouring O atom ($t_{fp}$) is even higher $\sim 0.5\unit{eV}$. The role of multiple orbitals on a Ce site was explored to determine the minimal set of orbitals to include in the effective model Hamiltonian. The results indicated that the essential physics could be described well by replacing the $4f$ manifold with two orbitals on a Ce site. The magnitudes of the energy scales of the problem decrease in the order $U > \lambda_{0} > t_{fp}>t_{eff} >\lambda_{so} \gg t_{ff} \approx \Delta_{CEF} \approx J$ with a high uncertainty on the last two relationships. The results of the above analysis led us to the conclusion that the only relevant energy scales for the effective model Hamiltonian are the On-site Coulomb repulsion for the Ce $4f$ states ($U$) and electron hopping between a Ce site and a neighbouring O site ($t_{fp}$). A minimal model Hamiltonian was then constructed to take into account the spin and charge fluctuations on a Ce site due to electron hopping between the Ce site and neighbouring O atoms. In this description, the strongly correlated ground states of cerium oxides is then described by a variational wavefunction whose main feature is that double occupancy of a Ce $f$ orbital is prohibited