\chapter{Transport Computations: Simple Metals}
\label{chap:simple-metal}
\marginlabel{simple-metal}

\section{Introduction}
We are now in the position to start testing and applying the tools that have
been developed in the previous chapters to real materials and to start out we
will consider the so called simple metals, the monovalent alkali and noble
metals.  Our main goal in this chapter is not so much to obtain any realistic
numbers for the transport parameters of the simple metal, we do not have any
realistic scattering data to do that, but rather to test the code on a real
Hamiltonian with some simple assumptions about the scattering.  This is
important from the standpoint of obtaining an understanding of the behavior
of the transport function as we modify the scattering.  

In spite of their apparent simplicity a good theoretical description of the
thermoelectric power in the simple metals is still lacking.  In this respect
the alkali metals are somewhat better understood, except for Lithium which
along with the noble metals exhibits a positive thermoelectric power contrary
to all expectations.

The behavior of the thermopower is further complicated by the presence of a
phonon contribution to the low temperature thermopower.  In general there are
two contributions to the so called absolute thermopower of a given material,
one coming from the diffusion of electrons, (this is the one that we compute)
and the other one coming from phonons that scatter with the moving electrons.
The latter one is referred to as the phonon-drag thermopower and is mainly
present at low temperatures, often below 150K or so.  So far we have
completely ignored this contribution in our discussion and we do not intend to
take this into account in our computations except to note that its presence
limits the applicability of our calculations at lower temperatures

The phonon drag is however not to blame for the positivity of the thermopower
in the noble metals and Lithium, at higher temperature the thermopower is
completely of a diffusive nature and still it is positive.  Various proposals
have been put forth in order to obtain the correct sign
\cite{Barnard:Thermoelectricity}, many focusing on the direct influence of
the distortion of the Fermi surface in these materials and others using the
distortion as a means of obtaining anharmonic relaxation lifetimes at the
Fermi-level.

\section{Bandstructure calculations}
In table \ref{table:monovalentmetals} we have gathered some data on the
monovalent metals that is used in the band structure calculation.
\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Material & Crystal Structure & 
Electron Structure & Lattice Constant [au]   \\ \hline\hline
Li & BCC & 1s$^2$2s$^2$ & 6.635  \\ \hline
Na & BCC & [Ne]3s$^1$   & 8.111  \\ \hline
K  & BCC & [Ar]4s$^1$   & 10.072 \\ \hline
Rb & BCC & [Kr]5s$^1$   & 10.557 \\ \hline
Cs & BCC & [Xe]6s$^1$   & 11.609 \\ \hline
Cu & FCC & [Ar]3d$^{10}$4s$^1$        & 6.833 \\ \hline
Ag & FCC & [Kr]4d$^{10}$5s$^1$        & 7.223 \\ \hline
Au & FCC & [Xe]4f$^{14}$3d$^{10}$4s$^1$ & 7.709 \\ \hline
\end{tabular}
\caption{The electronic and crystal structure of the monovalent metals. Here 
  au denotes atomic units, i.e. Bohr radius.}
\label{table:monovalentmetals}
\end{center}
\end{table}
The computation of the band structure is done using the {\em lmtart} code of
Sergei Savrasov in the atomic spheres approximation using screened
wavefunctions in the basis.  The program furnishes both the Hamiltonian and
the overlap matrix for the valence orbitals of these materials along with all
the necessary space group information and Brillouin zone data.  The
computations are all done using twenty points per axis in the Brillouin zone
so we will have enough points to get a good computation of the transport
function.  In figures \ref{fig:monovalent_baredos1} and
\ref{fig:monovalent_baredos2} we have displayed the bare density of states as
computed from the LMTO Hamiltonian with the zero temperature Fermi level being
at zero energy.  For all the materials except Lithium the Hamiltonian is a
nine by nine matrix corresponding to s, p and d orbitals.  In Lithium we only
have s and p orbitals so the Hamiltonian is only four by four.  In all cases
the density of states exhibits a typical free electron square root behavior
at the band edge but as we move further away more bands start showing up and
we get more structure in the DOS.  We note that the density of states of the
last two elements in the alkali metal column is quite different from the other
alkali metals and in fact it looks strangely like a tight binding simple cubic
DOS in the vicinity of the Fermi level.  For the noble metals we see a filled
d-band 2-3 eV below the Fermi level and we note that the width of the band
decreases as we move down the noble metal column.
\begin{figure}[t]
\centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/baredos_li.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/baredos_na.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/baredos_k.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/baredos_rb.ps,angle=-90,width=0.45\linewidth}} \\
\end{tabular}
\caption{The bare densities of states for the simple monovalent metals.}
\label{fig:monovalent_baredos1}
\end{figure}
\begin{figure}[t]
\centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/baredos_cs.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/baredos_cu.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/baredos_ag.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/baredos_au.ps,angle=-90,width=0.45\linewidth}}
\end{tabular}
\caption{The bare densities of states for the simple monovalent metals.}
\label{fig:monovalent_baredos2}
\end{figure}

\section{Transport function}
As mentioned before we do not have any realistic scattering data for this
computation, the only data we have is the bandstructure, i.e. the Hamiltonian.
We therefore propose at first to compute the transport properties by simply
taking the imaginary part of the self energy to be a small frequency
independent constant.  Since the simple metals are weakly correlated one might
be tempted to believe that such a simple form for the scattering would be
sufficient for the computation of the transport parameters.  We should note
however that this will not allow us to obtain any real numbers for the
conductivities but the thermopower should in the small scattering limit be
more or less independent of the value of the scattering.  In this limit the
transport function should also scale as the inverse of the scattering and when
normalized should tend towards the bare transport function considered in
\article\ref{sec:smallscattering}.  We will thus be able to see if this
scaling holds and compare our computed thermopower to the experimental data.
This test will also give us a clue as to whether our initial assumption of a
frequency independent scattering is reasonable.  There is also another issue
that confronts us here and that is that even though we use a simple constant
scattering we have in principle no reason to assign the same scattering to the
s, p and d orbitals and the result will of course depend quite sensitively on
how they are assigned.

In figures \ref{fig:monovalent_orbdos1} and \ref{fig:monovalent_orbdos2} we
have displayed the density of states computed with a frequency independent
scattering equal to $-0.1eV$ in all orbitals.  In addition to the total DOS we
have also displayed the DOS for the individual orbitals.  We note that for Rb
and Cs the spectral density of the p-orbitals becomes negative.  This can be
attributed to the non-orthogonality of the basis and the ambiguous definition
of the individual orbital spectral densities as discussed in
\article\ref{sec:no_mbt}.  Nevertheless the individual spectral densities can
be quite useful in practice and they can give us an indication of how the
transport function reacts to changes in the scattering assigned to individual
orbitals.  For all the materials we see that the spectral density of the s
orbital is decreasing around the Fermi surface and the p orbital is increasing
except in the case of Rb and Cs.  For the noble metals we see the tail of d
orbitals that lie below the Fermi surface.
\begin{figure}[t]
  \centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/orbdos_li.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/orbdos_na.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/orbdos_k.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/orbdos_rb.ps,angle=-90,width=0.45\linewidth}} \\
\end{tabular}
\caption{The orbital spectral densities computed with a constant scattering 
  of 0.1eV in each orbital.}
\label{fig:monovalent_orbdos1}
\end{figure}
\begin{figure}[t]
  \centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/orbdos_cs.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/orbdos_cu.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/orbdos_ag.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/orbdos_au.ps,angle=-90,width=0.45\linewidth}} 
\end{tabular}
\caption{The orbital spectral densities computed with a constant scattering 
  of 0.1eV in each orbital.}
\label{fig:monovalent_orbdos2}
\end{figure}
%---------------------------------------------------------------------------

In figures \ref{fig:monovalent_phisc1} and \ref{fig:monovalent_phisc2} we have
displayed the scaled transport function computed with the same choice of self
energy as used in figures \ref{fig:monovalent_orbdos1} and
\ref{fig:monovalent_orbdos2} along with the corresponding calculation of the
bare transport function from \article\ref{sec:smallscattering} (see equation
(\ref{eq:BareTransportFunction})).  We have here used three different choices
for the lifetime; the simplest one is taking the lifetime to be a constant
$1/2\pi$, but we can also compute it using equation
(\ref{eq:BareTransportLifetime}) and use the full momentum dependent version,
finally the third case uses the Brillouin zone average of the momentum
dependent lifetime.  We see that the full computation compares quite favorably
with bare computation using the momentum dependent lifetime and this gives us
further evidence for the accuracy of the full computation.  The average
lifetime computation is not too far off from the full result, which indicates
that our previous assumption of slow variation in the imaginary part of the
eigenvalues is quite good.
%---------------------------------------------------------------------------
\begin{figure}[htb]
\centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/phisc_li.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phisc_na.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/phisc_k.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phisc_rb.ps,angle=-90,width=0.45\linewidth}} \\
\end{tabular}
\caption{The scaled transport function computed with a scattering of 0.1 in
  each orbital.  Also shown is the bare transport function.  Bare 1 is
  computed with a momentum dependent lifetime, Bare 2 is computed with the
  average lifetime of each orbital and Bare 3 is computed with a constant
  lifetime of $1/2\pi$.}
\label{fig:monovalent_phisc1}
\end{figure}
\begin{figure}[htb]
\centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/phisc_cs.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phisc_cu.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/phisc_ag.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phisc_au.ps,angle=-90,width=0.45\linewidth}} 
\end{tabular}
\caption{The scaled transport function computed with a scattering of 0.1 in
  each orbital.  Also shown is the bare transport function.  Bare 1 is
  computed with a momentum dependent lifetime, Bare 2 is computed with the
  average lifetime of each orbital and Bare 3 is computed with a constant
  lifetime of $1/2\pi$.}
\label{fig:monovalent_phisc2}
\end{figure}
%---------------------------------------------------------------------------

It is worth noting that for the all the alkali metals except Rubidium the
spectral density and transport function have a positive slope at the
Fermi-level -- the positive slope of the transport function indicating that
the low-temperature thermopower is negative or electron-like.  For the noble
metals the slope of the density of states is negative, the Fermi-level being
in the upper tail of the d-bands, whereas the slope of the transport function
is positive.  Quite often the slope of the density of states is used as an
indicator of the sign of the thermopower, and in this case the prediction for
the noble metals would give the correct sign, i.e.~positive or hole-like.  The
transport function that we obtain here however indicates a negative
thermopower contradicting experiment.  The reason for this difference in the
slopes of the density of states and the transport function is due to the fact
that the transport function is essentially equal to the density of states
weighted by the squared velocity.  If we now look back at figure
\ref{fig:monovalent_orbdos2} we see that for all the noble metals the density
of states for the p-bands is increasing with frequency at the Fermi-level.
Thus even though the density of states for the d-bands is much greater than
the p-band density of states the p-band contributes much more to the transport
function.  This is due to the fact that the electrons in the p-band are
extended and have much greater velocity than the more localized electrons in
the d-band.  Thus it would seem that in order to effect a change in the sign
of the thermopower for the noble metals we would have to cut off the
contribution from the p-band and probably the s-band also by assigning a much
greater scattering to those bands.

\section{Thermopower}
In figures \ref{fig:monovalent_phi1} and \ref{fig:monovalent_phi2} we have
displayed the transport functions computed with three different scattering
scenarios using always frequency independent scattering but assigning
different scattering to different orbitals.  The values used for the
scattering rates are displayed in table \ref{table:scenarios}.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Scenario  & s-band scattering & p-band scattering & d-band scattering \\\hline
  1       &      -0.1  &     -0.1   &    -0.1    \\\hline
  2       &      -0.5  &     -0.1   &    -0.1    \\\hline
  3       &      -0.1  &     -0.5   &    -0.1    \\\hline
\end{tabular}
\caption{The three different scattering scenarios used in the computation of
  the transport parameters.}
\label{table:scenarios}
\end{table}

%---------------------------------------------------------------------------
\begin{figure}[htb]
\centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/phi_li.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phi_na.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/phi_k.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phi_rb.ps,angle=-90,width=0.45\linewidth}} \\
\end{tabular}
\caption{The transport functions computed with the three different scattering
  scenarios outlined in table \ref{table:scenarios}. } 
\label{fig:monovalent_phi1}
\end{figure}
\begin{figure}[htb]
\centering
\begin{tabular}{c@{\quad}c}
\mbox{\epsfig{file=Figures-7/phi_cs.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phi_cu.ps,angle=-90,width=0.45\linewidth}} \\
\mbox{\epsfig{file=Figures-7/phi_ag.ps,angle=-90,width=0.45\linewidth}} &
\mbox{\epsfig{file=Figures-7/phi_au.ps,angle=-90,width=0.45\linewidth}} 
\end{tabular}
\caption{The transport functions computed with the three different scattering
  scenarios outlined in table \ref{table:scenarios}. } 
\label{fig:monovalent_phi2}
\end{figure}
We note that the scattering is smallest in scenario number 1 and thus the
magnitude of the transport function is generally greatest there.  We see
however that at certain frequencies an increase in the scattering actually
increases the magnitude of the transport function.  This is due to the
broadening of the underlying band as the scattering is increased (see
\article\ref{subsec:analytic}).

In table \ref{table:thermopower_0C} we see a comparison of the experimental
and the computed thermopower at zero degrees Celsius using the three different
scattering scenarios mentioned above.  In all cases the thermopower is linear
in temperature at low temperatures as is to be expected when there is no
change in the scattering with temperature and the data is consistent with the
Fermi-liquid computation for the thermopower given in appendix
\ref{app:fermi-liq}.
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Material   &  S (exp)  & S (\#1) & S (\#2) & S (\#3) & 
$\gamma_{exp}$ & $\gamma_{calc}$ & $m^{*}/m_{0}$\\ \hline 
Lithium   &  10.6  & -1.67 & -4.81 & 0.20  & 4.2 & 2.70 & 1.56  \\\hline
Sodium    &  -5.8  & -3.03 & -4.31 & -1.56 & 3.5 & 2.69 & 1.30  \\\hline
Potassium &  -12.9 & -2.77 & -5.32 & -2.38 & 4.7 & 4.22 & 1.11  \\\hline
Rubidium  &  -9.5  & -1.94 & -3.14 & -1.45 & 5.8 & 5.36 & 1.08  \\\hline
Cesium    &  -0.9  & -2.37 & -5.35 & -2.97 & 7.7 & 6.51 & 1.18  \\\hline
Copper    &  1.70  & -2.73 & -2.02 & -0.96 & 1.6 & 1.62 & 0.99  \\\hline
Silver    &  1.38  & -1.02 & -1.07 & -0.15 & 1.6 & 1.24 & 1.29  \\\hline
Gold      &  1.79  & -1.60 & -0.80 & -0.71 & 1.6 & 1.53 & 1.05  \\\hline
\end{tabular}
\caption{Computed and experimental values for thermopower of the monovalent 
  metals measured in $\mu V/K$.  The three different computed values
  correspond to the scattering scenarios outlined in table 
  \ref{table:scenarios}.  The experimental values  are taken from 
  \cite[page 225]{Blatt:TEP}. 
  The values of the linear coefficient of specific heat, $\gamma$, are 
  measured in 10$^{-4}$cal/(mole K$^2$) and the experimental data is taken 
  from \cite[chapter 2]{Ashcroft}.  The effective mass is computed as 
  $\gamma_{exp}/\gamma_{calc}$.}
\label{table:thermopower_0C}
\end{table}
We see that the different scattering scenarios produce different values of the
thermopower and although we do obtain the correct sign for the simpler alkali
metals the actual values are quite off.  It is however clear that one might in
principle be able to tune the values of the scattering in each orbital to
reach the correct values.  For the other materials, except Lithium, we are not
able to reproduce the correct sign of the thermopower and it is likely that
one would need to introduce a more complex, frequency dependent, self energy
in order to do so.

In order to assess the quality of the underlying local-density-approximation
of the bandstructure we have also displayed in the table
\ref{table:thermopower_0C} the linear coefficient of specific heat,
$\gamma$, both experimental values and as computed from the data.  This
coefficient can easily be extracted from the original bandstructure using the
equation \cite{Ashcroft}
\begin{equation}
        \gamma = \frac{\pi^2}{3}\kb^2 \rho_{tot}(E_f),
\end{equation}
where $\rho_{tot}(E_f)$ is the total density of states at the Fermi-level,
including spin.  The ratio of the experimental and the computed $\gamma$'s
defines the effective mass of the charge carriers at the Fermi-level and as we
can see the effective mass is not very far from unity, indicating that the
bandstructure computation is quite accurate.  The small discrepancy can be
attributed to mass enhancement due to electron-phonon interactions
\cite{Savrasov:1996}.  It is interesting to note that some of the lowest
values of the mass enhancement are found for the noble metals where our
computation of the thermopower is the worst.

\section{Conclusion}
As we have seen from the computations outlined in this chapter the evaluation of
the thermopower of even the simplest metals requires a detailed knowledge of
the scattering mechanism at the Fermi-level.  The simplified calculations done
here, using essentially constant lifetimes, tend to draw out the role of the
bandstructure in the behavior of the thermopower as the lifetime dependency
cancels out.  It is seen that there is significant discrepancy between our
simple theory and experiment, especially for the noble metals and it is clear
that one must have a more realistic model for the self energies of these
metals.  On the other hand, as we see from the evaluation of the linear
coefficient of specific heat, the numerical value of the density of states is
fairly accurate and thus any changes in the scattering should not change
appreciably the density of states at the Fermi-level.



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