Solutions lecture 7

src/7_tight_binding.md

@@ -58,7 +58,7 @@ For a system of size $L = Na$ with periodic boundary conditions, we also have $u
 
 ### Electrons
 
-The following figure shows the inter-atomic potential with atoms placed at $ka$ with $k \in \z$:
+The following figure shows the inter-atomic potential with atoms placed at $ka$ with $k \in \Z$:
 
 ![](figures/lattice_potential.svg)
 
@@ -164,12 +164,18 @@ Now $\sum_p → \frac{L}{2\pi}\int_{-\pi/a}^{\pi/a}dk$, the integral is over a f
 Substitute the Ansatz into the equations of motion:
 
 $$
-E e^{-ikx_n} = E_0 e^{-ikx_n} - t e^{-ikx_n-ika} - te^{-ikx_n+ika},
+E Ae^{iEt/\hbar-ikna} = E_0 Ae^{iEt/\hbar-ikna} - t Ae^{iEt/\hbar-ik(n+1)a} - t Ae^{iEt/\hbar-ik(n-1)a},
 $$
-and after canceling the exponents we immediately get
+Since we're not interested in $Ae^{iEt/\hbar-ikna}=0$ we can omit them and we get:
+
+$$
+E = E_0 -te^{-ika} -te^{ika}
+$$
+
 $$
 E = E_0 - 2t\cos(ka),
 $$
+
 so we arrive at the dispersion relation:
 
 ```python
@@ -189,7 +195,7 @@ E = E_0 - 2t\cos{ka} \approx E_0 - 2t + t (ka)^2.
 $$
 Comparing this with $E=(\hbar k)^2/2m$, we see that the dispersion is similar to that of free electrons but with an effective mass given by $m^*=\hbar^2/2ta^2$.
 
-In the following lectures we will see that an electron dispersion usually has multiple options for $E(k)$, each called an energy band. The complete dispersion relation is also called a *band structure*.
+Notice that in this particular case we can occupy a continuous band from $E_0+2t$ to $E_0-2t$, in the next lecture we shall see that small deviations in the hopping between two adjacent atoms can change the continuity of the band! The \textit{band structure} of the electrons now consists of two **separate** bands. The complete dispersion relation is also called a *band structure*.
 
 
 ### Group velocity, effective mass, density of states
@@ -239,6 +245,8 @@ pyplot.xlabel('$ka$'); pyplot.ylabel('$m_{eff}$')
 pyplot.xticks([-pi, 0, pi], [r'$-\pi$', 0, r'$\pi$']);
 ```
 
+Notice that we can have a negative effective mass, which implies the electrons move opposite to the direction of the force.
+
 ### Density of states
 
 The DOS is the number of states per unit energy. In 1D we have