Problem 1
In the Boltzmann approximation, the Fermi function is approximated by an exponential function.

(a) What condition must be met so that the Boltzmann approximation is valid?

(b) A semiconductor has a band gap of 2.3 eV and the electrons have a much lighter effective mass than the holes. Draw the density of states for this semiconductor. Label the valence band edge $E_v$ and the conduction band edge $E_c$. Draw the Fermi function into this diagram.

(c) Consider the semiconductor from part (b). Will the Fermi energy increase or decrease as the temperature increases? Why?

Problem 2
(a) A Schottky diode is formed by pressing a metal tip against a silicon wafer. When a light falls on the diode, it works like a solar cell and current flows between the metal tip and the wafer. Explain how you can tell if this is an n-doped or a p-doped wafer from this experiment.

(b) Explain what thermionic emission is.

(c) When is tunneling the dominant current mechanism in a Schottky diode?

(d) Why does the Schottky diode respond faster than the pn diode?

Problem 3

(a) Draw a cross section of an n-channel JFET biased in the saturation regime. Include the depletion region in the drawing.

(b) Put + and - signs in the drawing to indicate where the positive and negative charge is located.

(c) Where is the highest electric field in the JFET and which way is it pointing?

(d) If you shine light on the JFET, how will this affect the gate current and the drain current?

Problem 4
In a silicon $pnp$ bipolar transistor, the emitter is doped to 10¹⁹ cm^-3, the base is doped to 10¹⁴ cm^-3, and the collector is doped to 10¹³ cm^-3.

(a) Why is the transistor doped this way?

(b) What is the equilibrium concentration of minority carriers in the collector? How would you calculate the concentration of minority carriers in the collector at the edge of the depletion zone? For Si, $n_i = 2.66\times 10^{9}$ cm^-3.

(c) Explain what the Early effect is and how it modifies the current gain $\alpha = \frac{I_C}{I_E}$.