@@ -229,3 +229,26 @@ $n_i$ is the **intrinsic carrier concentration**, and for a pristine semiconduct
2. Answer the same last 3 questions for a hole in the valence band.
3. We consider an electron in a 2D semiconductor near the bottom of the conduction band, which is described by an energy dispersion $E=E_{G}+\frac{\hbar^2}{2m_{eff}}(k_x^2+k_y^2)$. Suppose we turn on a magnetic field $B$ in the $z$-direction. Write down the equation of motion for this electron (you may neglect 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 $\nabla E$?
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
1. Recall from the [Drude model](/3_drude_model/) that electrons give rise to a negative Hall coefficient. Explain why the Hall coefficient is positive if holes are the charge carriers in a material.
2. What would be the Hall coefficient if both carriers with equal concentration are present? Assume that both electrons and holes can move freely and have the same scattering time.
Recall that the dispersion relation of a [1D single orbital tight binding chain](/7_tight_binding/) is given by $$E(k)=\varepsilon + 2t \cos(ka),$$ where $a$ is the lattice constant and $\varepsilon$ and $t$ are tight binding parameters.
3. What is the group velocity and effective mass of this band for holes compared to that of electrons?
4. Give an integral expression of the hole concentration in this band given the chemical potential $\mu$ and temperature $T$.
5. Show that the sum of the electron and hole concentration in this band is constant as a function of the temperature.
#### Exercise 3: a 1D semiconductor
Suppose we have a 1D semiconductor with a conduction band described by $$E_{cb} = E_G - 2 t_{cb} [\cos(ka)-1],$$ and a valence band described by $$E_{vb} = 2 t_{vb} [\cos(ka)-1].$$ Furthermore, the chemical potential is set at $0 < \mu < E_G$.
1. Derive an expression for the group velocity and effective mass for electrons in the conduction bands and holes in the valence band.
Assume that the Fermi level is far away from both bands. That is, $|E - \mu| \gg k_B T$. In that case, it is acceptable to approximate the bands for low $k$.
2. Why is it acceptable? Write down an approximate expression of these bands.
3. Write down an expression for the density of states _per unit length_ for both bands using the approximated expressions. Compare with the actual density of states per unit length.
4. Calculate the electron density in the conduction band and the hole density in the valence band.
5. What would the chemical potential $\mu$ be in case of an intrinsic semiconductor?
