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Copyright, 1995-2017, all rights reserved, Mark Jarrell (Dept.of
Physics and Atronomy, Louisiana State University, LA 70803). This
material may not be reproduced for profit, modified or published in any
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\title{Chapter 7: The Electronic Band Structure of Solids}
\author{Bloch \& \ Slater}
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\maketitle
\tableofcontents
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%\Large
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\textwidth,clip=true]{banddos.pdf}}
\caption[]{\em{The additional effects of the lattice potential
can have a profound effect on the electronic density of states (RIGHT)
compared to the free-electron result (LEFT).} }
\label{fig:banddos}
\end{figure}
In the last chapter, we ignored the lattice potential and considered
the effects of a small electronic potential $U$. In this chapter we will
set $U = 0$, and consider the effects of the ion potential $V(\r )$.
As shown in Fig.~\ref{fig:banddos}, additional effects of the lattice potential
can have a profound effect on the electronic density of states
compared to the free-electron result, and depending on the location of
the Fermi energy, the resulting system can be a metal, semimetal, an insulator,
or a metal with an enhanced electronic mass.
\section{ Symmetry of $\psi (\r )$ }
From the symmetry of the electronic potential $V(\r )$ one may infer some of
the properties of the electronic wave functions $\psi (r)$.
Due to the translational symmetry of the lattice $V(r)$ is periodic
\beq
V(\r ) = V(\r + \r _{n}), \qquad \r _n = n_{1}\a _{1} + n_{2}\a _{2} +
n_{3}\a _{3}
\eeq
and may then be expanded in a Fourier expansion
\beq
V(\r ) = \sum_{\G} V_{\G}e^{i\G \cdot \r}, \qquad \G = h\g _{1} +k\g
_{2} +l\g _{3} \,,
\eeq
which, since $\G \cdot \r _n = 2\pi m \quad (m \ \in \ \CZ )$ guarantees
$V(\r ) = V(\r + \r _{n})$. Given this, and letting $\psi (\r) = \sum _{\k}
C_{\k}e^{i\k \cdot \r}$ the Schroedinger equation becomes
\beq
H\psi (\r ) = \left[ -\frac{\hbar ^2}{2m}\nabla ^2 + V(\r ) \right]
\psi = E \psi
\eeq
\beq
\Rightarrow \sum _{\k} \frac{\hbar ^{2}k^2}{2m}C_{\k}e^{i\k \cdot \r} +
\sum _{\k ^{\prime }\G} C_{\k ^{\prime }}V_{\G}e^{i(\k ^{\prime
} + \G )\cdot \r} = E\sum _{\k} C_{\k}e^{i\k \cdot \r}, \qquad
\k ^{\prime } \rightarrow \k - \G
\eeq
\begin{wrapfigure}{l}{0.35\textwidth}
\includegraphics[width=0.34\textwidth,clip=true]{lattice.pdf}
\label{fig:lattice}
\protect\caption{\emph{The potential acts to couple each $C_{\k}$ with its
reciprocal space translations $C_{\k +\G}$ (i.e.\ $x\to x$, $\bullet\to\bullet$,
and $\bigcirc\to\bigcirc$) and the problem decouples into
$N$ independent problems for each $\k$ in the first BZ.}}
\end{wrapfigure}
or
\beq
\sum _{\k} e^{i\k \cdot \r}\left\{ \left( \frac{\hbar
^{2}k^2}{2m}-E\right) C_{\k} + \sum _{\G} V_{\G}C_{\k
-\G}\right\} = 0 \forall \r
\eeq
Since this is true for any $\r $, it must be that
\beq
\left( \frac{\hbar ^2 k^2}{2m}-E \right) C_{\k} +\sum _{\G} V_{\G}C_{\k
-\G} = 0, \qquad \forall \k
\eeq
Thus the potential acts to couple each $C_{\k}$ only with its reciprocal space
translations $C_{\k +\G}$ and the problem decouples in to $N$ independent
problems for each $\k $ in the first BZ. I.e., each of the $N$ problems has a
solution which is a sum over plane waves with wave vectors that differ only by
$\G $. Thus the eigenvalues may be indexed by $\k$.
\beq
E_{\k} = E(\k ), \qquad \mbox{I.e. $\k $ is still a good q.n.!}
\eeq
We may now sum over $\G$ to get $\psi_{\k}$ with the eigenvector sum restricted
to reciprocal lattice sites $\k , \k +\G ,\ldots $
\beq
\psi_{\k}(\r ) = \sum _{\G}C_{\k -\G}e^{i(\k -\G )\cdot \r} = \left(
\sum _{\G}C_{\k -\G}e^{-i\G \cdot \r}\right) e^{i\k \cdot \r}
\eeq
\beq
\psi_{\k}(\r ) = U_\k(\r )e^{i\k \cdot \r}, \qquad \mbox{where } U_\k(\r ) = U_\k(\r + \r _n)
\eeq
Note that if $V(\r ) = 0$, $U(\r ) = \frac{1}{\sqrt{V}}$. This result is
called {\it Bloch's
Theorem}; i.e., that $\psi $ may be resolved into a plane wave and a periodic
function. Its consequences as follows:
\begin{eqnarray}
\psi _{\k +\G}(\r ) & = & \sum _{\G'}C_{\k +\G -\G'}
e^{-i(\G'-\k-\G) \cdot \r} =
\left( \sum _{\G''}C_{\k -\G''}e^{-i\G''\cdot\r}\right)e^{i\k\cdot \r} \nonumber\\
& = & \psi_{\k }(\r ), \qquad \mbox{where } \G'' \equiv
\G ^{\prime} - \G
\end{eqnarray}
I.e., $\psi _{\k +\G }(\r ) = \psi _{\k }(\r)$ and as a result
\begin{eqnarray}
H \psi _{\k } = E(\k ) \psi _{\k } & \Rightarrow & H \psi_{\k +\G} =
E(\k +\G ) \psi_{\k +\G } \\
& = & H \psi _{\k } = E(\k +\G ) \psi_{\k +\G }
\end{eqnarray}
Thus $E(\k + \G) = E(\k): E(\k)$ is periodic then since both $\psi _{\k }(\r
)$ and $E(\k )$ are periodic in reciprocal space, one only needs knowledge of
them in the first BZ to know them everywhere.
\section{ The nearly free Electron Approximation. }
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{fig3.pdf}}
\caption[]{\em{For small $V_\G$, we may approximate the band structure
as composed of $N$ parabolic bands. Of course, it is sufficient to consider
this in the first Brillouin zone, where the parabola centered at finite
$\G$ cross at high energies. To understand the effects of the perturbation
$V_\G$ consider this special $\k $ at the edge
of the BZ. where the paraboli cross.}}
\label{fig:fig3}
\end{figure}
If the potential is weak, $V_{\G }\approx 0 \quad \forall \G $, then we may
solve the $V_{\G } = 0$ problem, subject to our constraints of periodicity,
and treat $V_{\G }$ as a perturbation.
When $V_{\G } = 0$, then
\beq
E(\k ) = \frac{\hbar ^{2}\k ^2}{2m} \qquad \mbox{free electron}
\eeq
However, we must also have that (if $V_{\G } \neq 0$)
\beq
E(\k ) = E(\k +\G ) \approx \frac{\hbar ^2}{2m}|\k +\G |^2
\eeq
I.e., the possible electron states are not restricted to a single parabola, but
can be found equally well on paraboli shifted by any $\G $ vector. In 1-d,
since $E(\k ) = E(\k +\G )$, it is sufficient to represent this in the first
zone only. For example in a 3-D cubic lattice the energy band structure along
$k_x (k_y = k_z = 0)$ is already rather complicated within the first zone.
(See Fig.\ref{fig:fig4}.)
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{fig4.pdf}}
\caption[]{\em{The situation becomes more complicated in three
dimensions since there are many more bands and so they can cross the first zone
at lower energies. For example in a 3-D cubic lattice the energy band
structure along $k_x (k_y = k_z = 0)$ is already rather complicated within the
first zone.}}
\label{fig:fig4}
\end{figure}
The effect of $V_G$ can now be discussed. Let's return to the 1-d problem and
consider the edges of the zone where the paraboli intersect. (See
Fig.~\ref{fig:fig3}.) An electron state with $\k = \frac{\pi }{a}$ will involve
{\it at least} the two $\G $ values $G = 0, \frac{2\pi }{a}$. Of course,
the exact solution must involve all $\G $ since
\beq
\left( \frac{\hbar ^{2}\k ^2}{2m} - E_{\k } \right) C_{\k } +
\sum _{\G }V_{\G }C_{\k -\G } = 0
\eeq
We can generally take $V_0=0$ since this just sets a zero for the potential.
Then, those $\G$ for which $E_{\k }= E_{\k -\G} \approx
\frac{\hbar ^{2}\k ^2}{2m}$ are going to give the largest contribution since
\begin{eqnarray}
C_{\k} &=& \sum_{\G}V_\G \frac{C_{\k -\G}}{\frac{\hbar^{2}\k ^2}{2m}-E_{\k-\G}} \\
C_{\k} &\sim & V_{\G _1} \frac{C_{\k -\G _1}}{\frac{\hbar ^{2}\k
^2}{2m} - E_{\k-\G_1}} \\
C_{\k -\G _1} &=&\sum _{\G} V_{\G}\frac{C_{\k -\G _1-\G }}{\frac{\hbar
^{2}\k ^2}{2m} - E_{\k-\G-\G_1}} \\
C_{\k -\G _1} &\sim & V_{-\G_1} \frac{C_{\k}}{\frac{\hbar ^{2}\k
^2}{2m} - E_{\k}}
\end{eqnarray}
Thus to a first approximation, we may neglect the other $C_{\k -\G}$,
and since $V_\G=V_{-\G}$ (so that $V(\r)$ is real)
$|C_{\k}|\approx |C_{\k -\G _1}| \gg $ other $G_{\k -\G }$
\beq
\psi _{\k}(\r ) = \sum _{\G}C_{\k -\G}e^{i(\k -\G )\cdot \r} \sim
\left\{ \begin{array}{ll}
(e^{iGx/2} + e^{-iGx/2} )\sim \cos \frac{\pi x}{\a} \\
(e^{iGx/2} - e^{-iGx/2} )\sim \sin \frac{\pi x}{\a}
\end{array} \right.
\eeq
The corresponding electron densities are sketched in Fig.~\ref{fig:densities}.
Clearly $\rho _+ $ has higher density near the ionic cores, and will be more
tightly bound, thus $ E_+ < E_-$. Thus a gap opens in $E_k$ near
$ k = \frac G2 $.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{densities.pdf}}
\caption[]{\em{$ \rho _+ \sim \cos^2(\pi x/a)$ has higher density near the
ionic cores, and will be more tightly bound, thus $ E_+
< E_-$. Thus a gap opens in $E_k$ near $ k = \frac G2 $. }}
\label{fig:densities}
\end{figure}
\subsection{The Origin of Band Gaps}
Now let's reexamine this gap at $\k \ = \G_1/2$ in a quantitative manner.
Start with the eigen value equation shifted by $\G $.
\beq
C_{\k -\G }\left( E_{\k} - \frac{\hbar ^2}{2m}|\k -\G |^2\right) \ = \sum _{\G
^{\prime }} V_{\G ^{\prime }}C_{\k -\G -\G ^{\prime }} = \sum
_{\G ^{\prime }} V_{\G ^{\prime}-\G }C_{\k -\G ^{\prime }}
\eeq
\beq
C_{\k -\G } = \frac{\sum _{\G ^{\prime }} V_{\G ^{\prime }-\G }C_{\k
-\G ^{\prime }}}{\left( E_\k - \frac{\hbar ^2}{2m}|\k -\G
|^2\right) }
\eeq
To a first approximation $(V_{\G} \simeq 0)$ let's set $E = \frac{\hbar
^{2}\k ^2}{2m}$ (a free-electron energy) and ignore all but the largest
$C_{\k -\G}$; i.e., those for which the denominator vanishes.
\beq
\k ^2 = |\k -\G |^2\,,
\eeq
or in 1-d
\beq
\k ^2 = (\k -\frac{2\pi}{\a})^2 \qquad or \;\; \k \ = -\frac{\pi}{\a}
\eeq
This is just the Laue condition, which was shown to be equivalent to the
Bragg condition. I.e., the strongest perturbation to the free-electron picture
occurs for states with energies at the edge of the first B.Z.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\textwidth,clip=true]{fig8.pdf}}
\caption[]{\em{We can satisfy the condition $E_{\k} \simeq \E_{\k -\G}$
only for $\k$ on the edge of the B.Z.. Here the lattice potential
strongly perturbs the electronic states (i.e.\ more than one $C_{\k-\G}$
is finite).}}
\end{figure}
Thus the equation above also tells us that $C_{\k}$ and $\ C_{\k -\G_1}$ are
the most important coefficients (if this electronic state was
unperturbed, only $C_{\k}$ would be important).
Thus approximately for $V_{\G} \sim \ 0, V_0 \equiv 0$ and for
$\k$ near the zone boundary
\beq
\G = 0 \qquad C_{\k} \left\{ E - \frac{\hbar ^2k^2}{2m}\right\}
= V_{\G_1}C_{\k -\G_1}
\eeq
\beq
\G = \G_1 \qquad C_{\k -\G_1} \left\{ E - \frac{\hbar ^2|\k
-\G_1|^2}{2m}\right\} = V_{-\G_1}C_{\k},
\eeq
Again, ignore all other $C_{\G}$. This is a secular equation which has a
nontrivial solution iff
\beq
\left| \begin{array}{cc}
\left( \frac{\hbar ^2\k ^2}{2m} - E\right) & V_{\G_1} \\
V_{-\G_1} & \left( \frac{\hbar ^2|\k -\G_1 |^2}{2m} - E\right)
\end{array}\right| \ = 0
\eeq
or
\beq
\left| \begin{array}{cc}
E_{\k}^0 - E & V_{\G_1} \\
V_{-\G_1} & E_{\k -\G_1}^0 - E
\end{array}\right| \ = 0
\eeq
\begin{displaymath}
(V_{-\G} = V_{\G}^*, \quad \mbox{so that} V(\r ) \in \Re )
\end{displaymath}
\beq
(E_{\k}^0 - E)(E_{\k -\G_1}^0 - E) - |V_{\G_1}|^2 = 0
\eeq
\beq
E_{\k}^0E_{\k -\G_1}^0 - E\lep E_{\k}^0 + E_{\k -\G_1}^0\rip + E^2 - |V_{\G_1}|^2 = 0
\eeq
\beq
E^{\pm } = \frac 12 \left( E_{\k -\G_1}^0 + E_{\k }^0\right) \ \pm \
\left\{ \frac 14 \left( E_{\k -G}^0 - E_{\k }^0\right) ^2 +
|V_{\G_1}|^2 \right\} ^{\frac 12}
\eeq
At the zone boundary, where $E_{\k -\G_1}^0 = E_{\k}^0$, the gap is
\beq
\Delta E = E_+ - E_- = 2|V_{\G_1}|
\eeq
And the band structure looks something like Fig.~7.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{band.pdf}}
\caption[]{}
\end{figure}
Within this approximation, the gap, or forbidden regions in which there are no
electronic states arise when the Bragg condition (${\bf k}_f - {\bf k}_0 = \G$)
is satisfied.
\beq
|-\k | \approx \ |\k + \G |
\eeq
The interpretation is clear: the high degree of back scattering for these
$\k $-values destroys the electronic states.
Thus, by treating the lattice potential as a perturbation to the free
electron problem, we see that gaps arise due to enhanced electron-lattice
back scattering for $\k$ near the zone edge.
However, in chapter one, we considered band structure qualitatively and
determined that gaps could arise from perturbing about the atomic limit.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{bandgaps.pdf}}
\caption[]{Band gaps in the electronic DOS naturally emerge when
perturbing around the atomic limit. As we bring more atoms together (left)
or bring the atoms in the lattice closer together (right), bands form
from mixing of the orbital states. If the band broadening is small
enough, gaps remain between the bands.}
\end{figure}
This in fact, is another natural way of constructing a band structure theory.
It is called the tight-binding approximation.
\section{ Tight Binding Approximation}
\begin{figure}[htb]
%\centerline{\psfig{figure=valence.eps,height=2.7in}}
\centerline{\includegraphics[width=0.6\textwidth,clip=true]{valence.eps}}
\caption[]{{\em In the tight-binding approximation, we generally ignore
the core electron dynamics and consider only the ionic core
potential. For now let's assume that there is only one valence
orbital $\phi _i$ on each atom.}}
\end{figure}
In the tight-binding approximation, we generally ignore
the core electron dynamics and treat consider only the ionic core
potential. For now let's assume that there is only one valence
orbital $\phi _i$ on each atom. We will also assume that the atomic problem
is solved, and perturb around this solution. The atomic problem has
valence eigenstates $\phi _i$, and eigen energies $E_i$. The unperturbed
Schroedinger equation for the nth atom is
\beq
H_A(\r - \r _n) \cdot \phi _i(\r - \r _\nn) = E_i\phi _i(\r -\r _\nn)
\eeq
There is a weak perturbation $v(\r - \r _\nn)$ coming from the atomic
potentials of the other atoms $\r _\mdm \neq \r _\nn$
\beq
H = H_A + v = -\frac{\hbar ^2\nabla ^2}{2m} + V_A(\r -\r _\nn) + v(\r -\r
_\nn)
\eeq
\beq
v(\r -\r _\nn) = \sum _{\mdm \neq \nn }V_A(\r -\r _{\mdm})
\eeq
We now seek solutions of the Schroedinger equation indexed by $\k $ (Bloch's
theorem)
\beq
H\psi _{\k}(\r ) = E(\k )\Psi_{\k}(\r )
\eeq
\beq
\Rightarrow \quad \int \psi ^* \qquad \Rightarrow \quad E(\k ) =
\frac{\left< \psi _{\k}|H|\psi _{\k}\right> }{\left<
\psi _{\k}|\psi _{\k}\right> }
\eeq
where
\beqa
\left< \psi _{\k}|\psi _{\k} \right> &\equiv & \int d^3\r \psi
_{\k}^*(\r )\psi _{\k}(\r ) \nonumber \\
\left< \psi _{\k}|H|\psi _{\k}\right> & \equiv & \int d^3\r \psi
_{\k}^*(\r )H\psi _{\k}(\r )
\eeqa
Of course, this problem is almost hopelessly complicated. We cannot solve for
$\psi _{\k}$. Rather, we will solve for some $\phi _{\k} \simeq \ \psi _{\k}$
where the parameters of $\phi _{\k}$ are determined by minimizing
\beq
\frac{\left< \phi _{\k}|H|\phi _{\k}\right> }{\left< \phi _{\k}|\phi
_{\k}\right> } \geq \ E(\k ).
\eeq
This is called the Raleigh-Ritz variational principle.
Consistent with our original motivation, we will approximate $\psi _{\k}$ with
a sum over atomic states.
\beq
\psi_{\k} \simeq \ \phi _{\k} = \sum _\nn a_n\phi _i(\r -\r _\nn) = \sum
_\nn e^{i\k \cdot \r _\nn}\phi _i(\r -\r _\nn)
\eeq
\begin{displaymath}
\psi _{\k}(\r ) = \U_{\k}(\r )e^{i\k \cdot \r}, \qquad \psi _{\k}(\r )
= \psi _{\k +\G }(\r )
\end{displaymath}
Where $\phi _{\k}$ must be a Bloch state $\phi _{\k +\G } = \phi _{\k}$ which
dictates our choice $\a_n = e^{i\k \cdot \r _n}$. Thus at this level of
approximation we have {\it no free parameters} to vary to minimize
$\left< \phi_{\k}|H|\phi _{\k}\right> /\left< \phi _{\k}|\phi _{\k}\right>
\approx E(\k)$.
Using $\phi _{\k}$ as an approximate state the energy denominator $\left< \phi
_{\k}|\phi _{\k}\right> $, becomes
\beq
\left< \phi _{\k}|\phi _{\k}\right> \ = \sum _{\nn ,\mdm}e^{i\k \cdot
(\r _{\nn}-\r_{\mdm})}\int d^3\r \phi _i^*(\r -\r
_{\mdm})\phi _i(\r -\r _{\nn})
\eeq
Let's imagine that the valance orbital of interest, $\phi _i$, has an
very small overlap with adjacent atoms
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\textwidth,clip=true]{valence2.pdf}}
\caption[]{In the tight binding approximation, we assume that
the atomic orbitals of adjacent sites have a very small overlap
with each other.}
\end{figure}
so that
\beq
\left< \phi _{\k}|\phi _{\k}\right> \ \simeq \ \sum _{\nn}\int d^3\r
\phi_i^*(\r -\r _{\nn})\phi _i(\r -\r _{\nn}) = N
\eeq
The last identity follows since $\phi _i$ is normalized.
The energy for our approximate wave function is then
\beq
E(\k ) \approx \ \frac 1N\sum _{\nn ,\mdm}e^{i\k \cdot (\r _{\nn}-\r
_{\mdm})}\int d^3\r \phi _i^*(\r -\r _{\mdm})\left\{ E_i + v(\r
-\r _{\nn})\right\} \phi _i(\r -\r _{\nn}) \,.
\eeq
Again, in the first part (involving $E_i$), we may neglect orbital overlap.
For the second term, involving $v(\r -\r _{\nn})$, the overlap should be included,
but only to the nearest neighbors of each atom (why?). In the simplest case,
where the orbitals $\phi _i$, are s-orbitals, then we can use this symmetry to
reduce the complexity of the problem to just two more integrals since the
hybridization ($B_i$) will be the same in all directions.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\textwidth,clip=true]{figa.pdf}}
\caption[]{A simple cubic tight binding lattice composed of
s-orbitals, with overlap integral $B_i$.}
\end{figure}
\beq
A_i = -\int \phi _i^*(\r -\r_\nn)v(\r -\r _{\nn})\phi _i(\r -\r
_{\nn}) d^3\r \qquad \mbox{ren. } E_i
\eeq
\beq
B_i = -\int \phi_i^*(\r -\r_\mdm)v(\r -\r _{\nn})\phi_i(\r -\r
_{\nn})d^3\r
\eeq
$B_i$ describes the hybridization of adjacent orbitals.
\beq
A_i; B_i > 0, \quad \hbox{since } v(\r -\r _{\nn}) < 0
\eeq
Thus
\beq
E(\k ) \simeq E_i - A_i - B_i\sum _{\mdm}e^{i\k (\r _{\nn}- \r
_{\mdm})} \qquad \mbox{sum over } \mdm \mbox{ n.n. to } \nn
\eeq
Now, if we have a cubic lattice, then
\beq
(\r _{\nn}-\r _{\mdm}) = (\pm a,0,0) (0,\pm a, 0) (0,0,\pm a)
\eeq
so
\beq
E(\k ) = E_i - A_i - 2B_i\{ \cos k_xa + \cos k_ya + \cos k_za\}
\eeq
Thus a band centered about $E_i - A_i$ of width $12 B_i$ is formed. Near the
band center, for $\k $-vectors near the center of the zone we can expand the
cosines $\cos ka \simeq \ 1- \frac 12\left(ka\right) ^2 + \cdot \cdot \cdot $
and let $k^2 = k_x^2 + k_y^2 +k_z^2 $, so that
\beq
E(k) \simeq \ E_i - A_i + B_ia^2k^2 %((*****))
\eeq
The electrons near the zone center act as if they were free with a
renormalized mass.
\beq
\frac{\hbar ^2k^2}{2m^*} = B_ia^2k^2 , \qquad \mbox{i.e. } \frac 1{m^*}
\propto \mbox{curvature of band}
\eeq
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\textwidth,clip=true]{cube.pdf}}
\caption[]{{\em Electronic states for a cubic lattice near the
center of the B.Z. act like free electrons with a renormalized mass.
Hence, if the band is partially filled, the Fermi surface will be
spherical.}}
\end{figure}
For this reason, the hybridization term $B_i$ is often associated with kinetic
energy. This makes sense, from its origins of wave function overlap and thus
electronic transfer.
The width of the band, 12 $B_i$, will increase as the electronic overlap
increases and the interatomic orbitals (core orbitals or valance f and d
orbitals) will tend to form narrow bands with high effective masses
(small $B_i$).
%$\uparrow \downarrow $) until we run out of electrons.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\textwidth,clip=true]{figb.pdf}}
\caption[]{In the tight-binding approximation, bands form from
overlapping orbitals states (states of the atomic potential). The
bandwidth is proportional to the hybridization $B$ ($12B$ for a SC
lattice). More localized, compact, atomic states tend to form
narrower bands.}
\end{figure}
The bands are filled then by placing two electrons in each band state (
with spins up and down).
A metal then forms when the valence band is partially full. I.e., for Na with
a $1s^2 2s^2 2p^6 3s^1$ atomic configuration the 1s, 2s and 2p orbitals evolve
into (narrow) filled bands, but the $3s^1$ band will only be half full, and
thus it evolves into a metal. Mg $1s^2 2s^2 2p^6 3s^2$ also metal since the p
and s band overlaps the unfilled d-band.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\textwidth,clip=true]{split.pdf}}
\caption[]{C (diamond) with atomic configuration of $1s^22s^22p^2$. Its
valance s and p states form a strong $sp^3$ hybrid band which is
split into a bonding and anti-bonding band.}
\label{fig:split}
\end{figure}
There are exceptions to this rule. Consider C with atomic configuration
of $1s^22s^22p^2$. Its valance s and p states form a strong $sp^3$ hybrid
band which is further split into a bonding and anti-bonding band. (See
Fig.\ref{fig:split}). Here, the gap is not tied to the periodicity of the
lattice, and so an amorphous material of C may also display a gap.
The tight-binding picture can also explain the variety of features seen in the
DOS of real materials. For example, in Cu (Ar)3d$^{10}$4s the d-orbitals are
rather small whereas the valence s-orbitals have a large extent .
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{orbitals.pdf}}
\caption[]{Schematic DOS of Cu $3d^{10}4s^1$. The narrow d-band
feature is split due to crystal fields.}
\label{fig:orbitals}
\end{figure}
As a result the s-s hybridization $B_i^{ss}$: is strong and the $B_i^{dd}$ is
weak.
\beq
B_i^{dd} \ll \ B_i^{ss}
\eeq
In addition the s-d hybridization is inhibited by the opposing symmetry of the
s-d orbitals.
\beq
B_i^{sd}= \ \int \phi _i^s(\r -\r _1)v(\r -\r _2)\phi _i^d(\r -\r _2) d^3\r \ll B_i^{ss}
\eeq
where $\phi _i^s$ is essentially even and $\phi _i^d$ is essentially odd. So
$B_i^{sd} \ll \ B_i^{ss}$. Thus, to a first approximation the s-orbitals
will form a very wide band of mostly s-character and the d-orbitals will form a
very narrow band of mostly d-character. Since both the s and d bands are
valance, they will overlap leading to a DOS with both d and s features
superimposed.
\section{ Photo-Emission Spectroscopy }
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{XPS.pdf}}
\caption[]{{\em {\bf XPS Experiment:} By varying the voltage one may
select the kinetic energy of the electrons reaching the counting detector.}}
\end{figure}
The electronic density of electronic states (especially for occupied
states), and to a less extent band structure, are very important for
illuminating the interesting physics of materials. As we saw in Chap.~6,
an enhanced DOS at the Fermi surface indicates an enhanced electronic
mass, and if $D(E_F)=0$, we have an insulator (semiconductor). The
effective electronic mass also varies inversely with the curvature of the
bands. The density of states away from the Fermi surface can allow us
to predict the properties of the material upon doping, or it can yield
information about core-level states. Thus it is important to be able to
measure $D(E)$. This may be done by x-ray photoemission (XPS), UPS or
PS in general. The band dispersion $E(\k)$ may also be measured using
angle-resolved photoemission (ARPES) where angle between the incident
radiation and the detector is also measured.
The basic idea is that a photon (usually an x-ray) is used to knock an
electron out of the system (See figure \ref{fig:pes1}.)
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\textwidth,clip=true]{pes1.pdf}}
\caption[]{{\em Let the binding energy be defined so that $E_b > 0$,
$\phi =$ work function, then the detected electron intensity
$I(E_{kin}-\hbar\omega -\phi) \propto D(-E_b)f(-E_b)$}}
\label{fig:pes1}
\end{figure}
Of course, in order for an electron at an energy of $E_b$ below the
Fermi surface to escape the material, the incident photon must have
an energy which exceeds $E_b$ {\em{and}} the work function $\phi$ of the
material. If $\hbar\omega > \phi$, then the emitted electrons
will have a distribution of kinetic energies $E_{kin}$, extending from
zero to $\hbar\omega -\phi$. From Fermi's golden rule, we know that the
probability per unit time of an electron being ejected is proportional to
the density of occupied electronic states times the probability (Fermi
function) that the electronic state is occupied
\beqa
I(E_{kin})&=&\frac{1}{\tau(E_{kin})}\propto D(-E_b) f(-E_b)\nonumber \\
&\propto& D(E_{kin}+\phi-\hbar\omega) f(E_{kin}+\phi-\hbar\omega)
\eeqa
Thus if we measure the energy and number of ejected particles, then we
know $D(-E_b)$.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.8\textwidth,clip=true]{pes3.pdf}}
\caption[]{Left: Origin of the background in $I(E_{kin}$. Right:
Electrons excited deep within the bulk scatter so often that they
rarely escape. Thus, most of the signal $I$ originates at the surface, \
which must be clean and representative of the bulk.}
\label{fig:pes3}
\end{figure}
There are several problems with this procedure. First some of the photon
excited particles will scatter off phonons and electronic excitations within
the material. Since these processes can occur over a very wide range of
energies, they will produce a broad featureless background in $N(E_{\k})$.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\textwidth,clip=true]{PE.eps}}
\caption[]{\em{In Photoemission, we measure the rate of ejected
electrons as a function of their kinetic energy. The raw data contains
a background. Once this is subtracted off, the subtracted data
is proportional to the electronic density of states convolved with a
Fermi function $I(E_{kin})\propto D(E_{kin}+\phi-\hbar\omega) f(E_{kin}+\phi-\hbar\omega)$. }}
\label{fig:PE}
\end{figure}
Second, due to these secondary scattering processes, it is very unlikely
that an electron which is excited deep within the bulk, will ever escape
from the material. Thus, we only learn about $D(E)$ near the surface of
the material. Therefore it is important for this surface to be ``clean''
so that it is representative of the bulk. For this reason these experiments
are often carried out in ultra-high vacuum conditions.
We can also learn about the electronic states $D(E)$ above the Fermi surface,
$E > F_F$, using {\it Inverse Photoemmision}. Here, an electron beam is
focused on the surface and the outgoing flux of photons are measured.
Inverse photoemission has significantly less energy resolution (roughly 100 meV)
than PES, which with a laser light source, can be as good as several meV.
%\begin{figure}
%\centerline{\includegraphics[width=0.4\textwidth,clip=true]{photoem.pdf}}
%\caption[]{{\em {\bf BIS} $E_{kin} = \hbar \omega \ -E_b - \phi , \E_b
%= \hbar \omega \ - E_{kin} - \phi $}}
%\label{fig:photoem}
%\end{figure}
\section{ Anderson Localization }
\begin{wrapfigure}{l}{0.35\textwidth}
\includegraphics[width=0.34\textwidth,clip=true]{typesofdisorder.png}
\protect\caption{\emph{Examples disorder, including substitution, interstitial,
vacancies. In addition, not shown, there are external disorder potentials,
amorphous systems, etc.
}\label{fig:typesofdisorer}
}
\end{wrapfigure}
In this chapter, we have seen that electrons do not scatter off a perfect periodic
lattice, unless the Bragg condition is satisfied opening a band gap. If not,
according to Bloch, they form extended states composed of a plane wave multiplied
by a periodic function. So, the electronic wavefunction (and charge!) is spread over
the entire system with equal amplitude on each site! Such states are characteristic
of a metal and are called ``extended states''.
On the other hand, we know that the impact of disorder on such a system can
be significant. As illustrated in Fig.~\ref{fig:typesofdisorer} there are
many different types of disorder, including substitutional disorder, as in the replacement
of Si by B or P in Si semiconductors used in your laptop which is responsible for
roughly one to four trillion dollars of the US economy. Disorder is not a
nuisance, rather it is very often used to tune or control the properties of
materials.
In fact very strong disorder can even destroy the metal which contains it. This
means that Bloch's theorem, which was derived in the absence of disorder, breaks
down and the extended states that are spread over the entire system become exponentially
localize states centered at one position in the material. In the most extreme
limit, this is obviously true. Consider a single orbital that you pull down
in energy so that it falls below (or above) the continuum in the density of states.
Clearly, such a state cannot hybridize with other states since there are none
at the same energy. Thus, any electron on this orbital is localized, and the
electronic DOS at this energy will be a delta function.
\begin{wrapfigure}{r}{0.55\textwidth}
\includegraphics[width=0.54\textwidth,clip=true]{increasedisorder.pdf}
\protect\caption{\emph{A periodic potential (left) leads to extended
states; whereas, strong disorder will lead to exponentially
localized states (right).
}\label{fig:increasedisorder}
}
\end{wrapfigure}
Anderson has shown that other types of disorder can lead to the localization
of electronic states, as illustrated in Fig.~\ref{fig:increasedisorder}.
Mott argued that the extended states would be separated from the localized
states by a sharp mobility (localization) edge in energy. His argument
is that scattering from disorder is elastic, so that the incoming wave and
the scattered wave have the same energy. On the other hand, nearly all
scattering potentials will scatter electrons from one wavevector to all others, since the
scattering potentials are local or nearly so. If two states, corresponding to
the same energy and different wavenumbers exist, then the scattering potential
will cause them to mix, causing both to become extended.
\begin{wrapfigure}{l}{0.5\textwidth}
\includegraphics[width=0.49\textwidth,clip=true]{blocks.pdf}
\protect\caption{\emph{To understand localization, divide a system up
into blocks. The average spacing of the energy levels of a block is
$\Delta$ and the Fermi golden rule width of the levels is $\Gamma$.
If $\Gamma \gg \Delta$ then we have a metal and if $\Gamma \ll \Delta$,
an insulator).
}\label{fig:blocks}
}
\end{wrapfigure}
These ideas have existed in one form or another for over fifty years! The main
remaining challenge is to develop a complete theory of localization. This has
been hampered by the lack of a clearly identified order parameter, akin to $m$
in the theory of magnetism! Recently, there has been significant progress along
these ideas, with the local typical density of states identified as the order
parameter.
\begin{wrapfigure}{r}{0.45\textwidth}
\includegraphics[width=0.44\textwidth,clip=true]{ldos.pdf}
\protect\caption{\emph{The local DOS in a metal (left) is a continuum; whereas,
that in an insulator is composed of a series of delta functions (right)).
}\label{fig:ldos}
}
\end{wrapfigure}
To see this, imagine dividing the system up into blocks, as illustrated in
Fig.~\ref{fig:blocks}. Here the average level spacing of the states in a block is
$\Delta$ and their average Fermi golden rule width is $\Gamma$. If
$\Gamma \gg \Delta$ then we have a metal since the states at this energy have a
significant probability of escaping from this block, and the next one.
Alternatively if $\Gamma \ll \Delta$ the escape probability of the electrons
is low, so that an insulator forms. So what does this mean in terms of the
local electronic density of states that are measured (i.e., via STM) at one site in
the system? If I measure the DOS at any site in a metal, it must be a continuum,
at least at the energy at question, since all states at that energy must be
extended and therefore accessible (c.f.\ Fig.~\ref{fig:ldos}). On the other
hand, in a insulator, at any one site, the DOS would ``see'' only the states that
are accessible (within the localization length) and since the number of these
states is necessarily finite, the DOS at any one site would be composed of a set
of delta functions. Of course, if I go to another site, then the distribution of
the delta functions will be different, so if I average over all $10^{23}$ sites, then I
again recover a continuum. I.e., the arithmatically averaged DOS is a continuum
in both a metal and an insulator.
On the other hand, the typical value of the local DOS in a metal is very different
than the typical value in an insulator. Consider again the local DOS in the metal
and insulator illustrated in Fig.~\ref{fig:ldos}. In the metal, at any one energy
the DOS at each site is a continuum. It will change as one goes from site to
site, but the typical value, as the average value, will be finite. now
reconsider the local DOS in the insulator. It is composed of a finite number
of delta functions. For any energy in between the delta functions, the
local DOS is zero. Since the number of delta functions is again finite,
the typical value of the local DOS is zero. As a result, the order parameter
for the Anderson metal-insulator transition is the typical local DOS, which
is zero in the insulator and finite in the metal.
\begin{wrapfigure}{l}{0.50\textwidth}
\includegraphics[width=0.49\textwidth,clip=true]{lognormal.pdf}
\protect\caption{\emph{The distribution of the local density of states in
a single-band Anderson model with disorder strength $\gamma$. Near the
localization transition, $\gamma=16.5$ the distribution becomes log-normal
(see also the inset), while for values well below the transition, $\gamma=3$ is shown,
the distribution is normal.\cite{schubert}.
}\label{fig:lognormal}
}
\end{wrapfigure}
A stronger statement is also possible. Early on, Anderson realized that the
distribution of the density of states in a strongly disordered metal would be
strongly skewed towards smaller values. More recently, this distribution
has been demonstrated to be log normal. Perhaps the strongest demonstration
of this fact was by the Vollhardt group\cite{schubert}, who illustrated the the
DOS near the transition has a log-normal distribution (Fig.~\ref{fig:lognormal})
over 10 orders of magnitude! Furthermore, one may also show that the
typical value of a log-normal distribution is the geometric average (the
geometric average of A and B is $\sqrt{AB}$) which is particularly easy
to calculate and can serve as an order parameter.
\begin{thebibliography}{00}
%% \bibitem must have the following form:
%% \bibitem{key}...
%%
\bibitem{schubert} Gerald Schubert, Jens Schleede, Krzysztof Byczuk, Holger Fehske,
and Dieter Vollhardt Phys. Rev. B 81, 155106 – Published 8 April 2010.
\end{thebibliography}
\edo