\chapter{Introduction}
\label{chap:intro}

In the last decade or so our understanding and ability to tackle problems of
strongly correlated electron systems has dramatically increased.  This is in
no small part due to the development of dynamical mean field theory
\cite{Georges:1996} (DMFT) which allows us to map models of strongly
correlated lattice electrons onto an Anderson impurity model subject to a self
consistency condition much like we solve the Ising model within mean field
theory by mapping it onto a single spin interacting with the self consistent
field of its neighbors.  Since its introduction the DMFT has been
successfully applied to describe many aspects of the physics of three
dimensional strongly correlated materials such as the Mott metal insulator
transition in transition metal oxides.

The work in this thesis is a direct application of the DMFT to the problem of
computing and understanding transport properties in strongly correlated
electron systems.  The experimental motivation for the work is in large part
provided by the work of Y. Tokura's group which has demonstrated that within
the class of ternary perovskites one can control various parameters such as
the orbital degeneracy, bandwidth and carrier concentration to provide an
experimental realization of the density driven Mott transition.

With this goal in mind we will in the first part of the thesis look at the
grandfather of all strongly correlated models, the Hubbard model, in the limit
of infinite interactions.  This allows us to introduce the tools of the DMFT
and the mapping to the impurity model.  We then employ one of the various
methods for solving for the dynamical properties of the impurity model, the
so called non-crossing approximation (NCA) and outline how this method is
implemented numerically.  We then proceed to compute the transport properties
of this model with special attention to the thermoelectric properties.  This
brings us into the realm of thermoelectric devices and their performance and
we introduce here the thermoelectric coefficient of performance.

The thermoelectric properties of materials are interesting to consider for a
variety of reasons.  One is the practical application of thermoelectric
phenomena for heating or cooling purposes -- already there are small
commercial heater/coolers available but we have yet to see such appliances in
the kitchen, for that the efficiency of the thermoelectric devices still has
to improve to catch up with the compressor based alternatives. Another use is
in power generation and one could imagine that such devices could make use of
the waste heat generated by fossil fuel engines.  One of the main applications
though, would be in space travel and satellites where such generators are used
with nuclear heat sources to provide electricity.  For this reason much of the
attention that thermoelectricity receives is focused on ways to improve the so
called thermoelectric performance.

From a theoretical standpoint the thermoelectricity is also quite interesting
since it provides insight into the nature of the carriers in the material --
we will see that there is an intimate connection between the entropy of
carriers and their thermoelectric power and thus the thermopower can tell us
whether the carriers in the system are coherent, i.e. in a delocalized state
with small entropy, or localized and incoherent.  Compared to the electrical
resistivity the thermoelectric power is also a much more sensitive probe of
the topology of the Fermi-surface and the scattering experienced by carriers
around the Fermi-surface.

As a practical example of some of these issues we will consider the transport
properties of La$_{1-x}$Sr$_{x}$TiO$_3$ close to the Mott transition, i.e. at
small $x$.  Here we will also employ another method for solution of the DMFT
equations, namely the so called iterated perturbation theory (IPT).  This
method complements the NCA quite well since the NCA is known to work well at
high temperatures and not so well in the low temperature regime whereas the
opposite holds for the IPT.

The second half of the thesis is somewhat motivated by the success of
the numerical computations in the first half and anticipates the
further application of the DMFT for real materials.  Here the main
goal is to develop a computational tool that allows for the
computation of transport properties in real strongly correlated
materials.  The DMFT starts out with the non-interacting or bare
Hamiltonian of the system in question, then an interaction term is
introduced, such as the on-site density-density interaction which is
well known from the Hubbard model, and from there the DMFT furnishes a
self energy for the system.  Thus the ingredients that we have for our
computation of the transport properties are the bare Hamiltonian which
we obtain from first-principles band structure calculations and the
self energy which is furnished by the DMFT.  The bulk of the second
half is devoted to the details of this computation and it's numerical
implementation.  

Towards the end we will look at two simple applications of this
procedure.  The first example we will consider are the simple metals,
i.e.~alkali and noble metals with some simple assumptions about the
scattering around the Fermi-level.  In the second example we revisit
the La$_{1-x}$Sr$_{x}$TiO$_3$ system with a more realistic model than
before and consider the behavior of the system as a function of doping
and interaction strength.

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