@@ -230,7 +230,7 @@ $n_i$ is the **intrinsic carrier concentration**, and for a pristine semiconduct
1. Consider the top of the valence band of a semiconductor (see [above](#semiconductors-materials-with-two-bands)). Does an electron near the top of the valence band have a positive or a negative effective mass? Does the electron's energy increase or decrease as $k$ increases from 0? Does the electron have a positive or negative group velocity for $k>0$?
2. Answer the same last 3 questions for a hole in the valence band.
3. We now consider an electron in a 2D semiconductor near the bottom of the conduction band described by an energy dispersion $E=E_{G}+\frac{\hbar^2}{2m_{eff}}(k_x^2+k_y^2)$. The electron's velocity is given by $\mathbf{v}=\nabla E/\hbar$. Suppose we turn on a magnetic field $B$ in the $z$-direction. Write down the equation of motion for this electron (neglecting collisions). What is the shape of the motion of the electron? What is the characteristic 'cyclotron' frequency of this motion? What is the direction of the Lorentz force with respect to $\nabla E$?
3. We now consider an electron in a 2D semiconductor near the bottom of the conduction band described by an energy dispersion $E=E_{G}+\frac{\hbar^2}{2m_{eff}}(k_x^2+k_y^2)$. The electron's velocity is given by $\mathbf{v}=\nabla_\mathbf{k} E/\hbar = \frac{1}{\hbar}(\frac{\partial E}{\partial k_x}\mathbf{\hat{x}} + \frac{\partial E}{\partial k_y}\mathbf{\hat{y}})$. Suppose we turn on a magnetic field $B$ in the $z$-direction. Write down the equation of motion for this electron (neglecting collisions). What is the shape of the motion of the electron? What is the characteristic 'cyclotron' frequency of this motion? What is the direction of the Lorentz force with respect to $\nabla_\mathbf{k} E$?
4. Suppose we now consider a hole near the bottom of the conduction band and turn on a magnetic field $B$ in the $z$-direction. Is the direction of the circular motion (i.e., the chirality) of the hole the same as that of the electron? Would the chirality change if we instead consider a hole (or electron) near the top of the valence band?
#### Exercise 2: holes in Drude and tight binding model
