Band theory of solids

Model for Polyacetylene

1D Tight Binding Chain

• Initially let us make the assumption that all the Carbon are evenly spaced (not necessarily true)
• Equation for Atom $n$:

$\sum _{m}(\epsilon _{0}\delta _{mn}-(1-\delta _{mn})t)b_{n}=Eb_{i}\!$

• Here we have N Equations, where E is the eigenvalue we are trying to solve for.

• We can guess a solution in the form:

$b_{n}=ce^{ikan}\!$ , where $c$ is an arbitrary constant

• Now plugging this solution into the initial equation gives:

$\sum _{m}(\epsilon _{0}\delta _{mn}-(1-\delta _{mn})t)ce^{ikan}=Eb_{n}\!$

$\sum _{m}(ce^{ikan}\epsilon _{0}\delta _{mn}-(1-\delta _{mn})tce^{ikan})=Eb_{n}\!$

• There are only three significant case we are concerned about. This is due to that fact the hooping probability dies down exponentially the farther you move from the chosen starting atom. Therefore, we are only concerned about when we are looking at the same atom, $m=n$ or when we are looking at the two nearest atoms $m=n+1$ and $m=n-1$.

$\epsilon _{0}ce^{ikan}-tce^{ika(n+1)}-tce^{ika(n-1)}-Ece^{ikan}=0\!$

• Canceling $ce^{ikan}$,

$\epsilon _{0}-te^{ika}-te^{-ika}-E=0\!$

$E=\epsilon _{0}-t(e^{ika}-e^{-ika})\!$

• We can simplify using a trig identity:

$\cos(x)={\frac {e^{ix}+e^{-ix}}{2}}$,

to get:

$E=\epsilon _{0}-2t\cos {(ka)}\!$

• However we still need boundary conditions

...

$k_{m}={\frac {2\pi }{L}}m$

• So is this a metal or an insulator?
  • It has N k-states and N electrons, so only ${\frac {N}{2}}$ energy levels are full since you can put two electrons in each energy level (one spin up, one spin down).
  • Recall, that our assumption was that the Carbon was equally spaced. In that case then Carbon is metal. However, this assumption is not true.
  • The Carbon atoms undergo dimerization, which causes the distance between the Carbon atoms to vary periodically. Although the actual difference between the short and long bond is very small, recall that hopping probability is proportional to $e^{\frac {-a}{a_{0}}}$, so even a small difference in atomic spacing will have a large impact.

• Now we can repeat the same process as above using $t_{1}$ and $t_{2}$, where $t_{1}>t_{2}$. Note that we have doubled our atomic spacing, since now it contains two Carbon atoms, such that $a'=2a$.

RESULT

$E_{k}=\pm ((t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos {(ka')})^{\frac {1}{2}}\!$

• Now looking at when $k={\frac {\pi }{a'}}$, we get

$E_{k}=\pm (t_{1}^{2}+t_{2}^{2}-2t_{1}t_{2})^{\frac {1}{2}}=(t_{1}-t_{2})\!$

• Note that the case where there dimerization is the more general of the two cases, since we can recover our initial assumption (equal spacing).

• When a gap is opened (dimerization occurs), you push down the occupied energy levels, creating an insulator.

• Peierls Instability -- Molecules like to dimerize to lower energies to some equilibrium (similar to Lennard-Jones Potential)