Update 13_semiconductors_solutions.md

src/13_semiconductors_solutions.md

@@ -32,6 +32,8 @@ force is perpendicular to $\nabla_\mathbf{k} E$.
 A hole near the bottom of the conduction band will have the same chirality as an electron.
 The chirality would be just the opposite if we now consider the valence band (for both electrons and holes).
 
+
+
 ## Exercise 2: holes in Drude and tight binding model
 
 ### Subquestion 1
@@ -46,7 +48,11 @@ When considering equal concentrations, $\rho_{xy}=0$.
 
 ### Subquestion 3
 
-The effective masses will be of opposite sign. The group velocities will be the same.
+$$m_e = -\frac{\hbar^2}{2ta^2cos(ka)}$$
+$$v_e = -\frac{2tasin(ka)}{\hbar}$$
+$$m_h = -m_e$$
+$$v_h = v_e$$
+The effective masses will be of opposite sign, while the group velocities will be the same!
 
 ### Subquestion 4
 
@@ -55,7 +61,10 @@ $$ n_h = \int_{\varepsilon-2t}^{\varepsilon+2t} (1-f(\varepsilon)) g_h(\varepsil
 ### Subquestion 5
 
 
-## Exercise 3: 
+
+
+
+## Exercise 3: a 1D semiconductor
 
 ```python
 def dispersion(EG, tcb, tvb, N=100, kmax=np.pi/2):