Solutions lecture 7

src/7_tight_binding.md

@@ -195,22 +195,22 @@ 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 $m^*=\hbar^2/2ta^2$.
 
-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! Th *band structure* of the electrons now consists of two **separate** bands. 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 *band structure* of the electrons then consists of two **separate** bands. The complete dispersion relation is called the *band structure*.
 
 
 ### Group velocity, effective mass, density of states
 
 *(here we only discuss electrons; for phonons everything is the same except for replacing $E = \hbar \omega$)*
 
-Question: what happens if we apply an external electric field to the crystal:
-
-![](figures/electric_field.svg)
-
-If electrons form bands, then in each band
-
-$$ H = E(k) + \tilde{U}(r),$$
-
-where $\tilde{U}(r) = -e|\mathbf{E}|r$ only includes slow variations of the electrostatic potential (the rapidly changing atomic potential is responsible for $E(k)$):
+> Question: what happens if we apply an external electric field to the crystal:
+>
+> ![](figures/electric_field.svg)
+> 
+> If electrons form bands, then in each band
+> 
+> $$ H = E(k) + \tilde{U}(r),$$
+> 
+> where $\tilde{U}(r) = -e|\mathbf{E}|r$ only includes slow variations of the electrostatic potential (the rapidly changing atomic potential is responsible for $E(k)$):
 
 To derive expressions for the velocity and mass, we recall from Hamiltonian mechanics:
 
@@ -223,7 +223,7 @@ v \equiv \frac{dr}{dt} &= \frac{\partial H(p, r)}{\partial p}\\
 F \equiv \frac{dp}{dt} &= -\frac{\partial H(p, r)}{\partial r}.
 \end{aligned}$$
 
-The connection to quantum mechanics is made by $p = \hbar k$. Analogously, the group velocity for electrons in a band structure is given by $v \equiv \hbar^{-1}\partial E/\partial k$ is the **group velocity** (same as for phonons).
+With $H=E$. The connection to quantum mechanics is made by $p = \hbar k$. Analogously, the group velocity for electrons in a band structure is given by $v \equiv \hbar^{-1}\partial E/\partial k$ is the **group velocity** (same as for phonons).
 
 Similarly, the **effective mass** is defined by:
 
@@ -231,7 +231,7 @@ $$m_{eff} \equiv F\left(\frac{dv}{dt}\right)^{-1} $$
 
 Substituting, we get
 
-$$m_{eff} = \hbar^2\left(\frac{d^2 E(k)}{dk^2}\right)^{-1}$$
+$$m_{eff} = \frac{F}{\frac{dv}{dt}} = \hbar \frac{\frac{dp}{dt}}{\hbar^{-1}\frac{d^2E}{dtdk}} = \hbar \frac{dk}{} = \hbar^2\left(\frac{d^2 E(k)}{dk^2}\right)^{-1}$$
 
 ```python
 pyplot.figure(figsize=(8, 5))