Problem 1
A p-type silicon wafer is uniformly doped with boron at a concentration of $10^{15}$ cm^-3. N-type doping is then introduced by diffusion. The doping concentration is $N_D= 10^{17}\exp\left(\frac{-x^2}{10^{-12}}\right)$ cm^-3. Here $x$ is the distance from the surface of the wafer measured in meters where $x=0$ is the surface of the wafer.

(a) Sketch the concentration of donors, acceptors, electrons, and holes $\left( N_D(x),\, N_A(x),\, n(x),\, p(x)\right)$.

(b) What is the concentration of electrons at $x=20$ μm?

(c) Draw the band diagram (valence band, conduction band, Fermi energy) assuming no voltage bias is applied.

(d) Draw the electric field as a function of $x$.

For silicon: $E_g = 1.12$ eV, $N_c = 2.78 \times 10^{25}$ 1/m³, and $N_v = 9.84 \times 10^{24}$ 1/m³.

Problem 2
(a) Draw an n-channel JFET.

(b) Explain how a JFET works.

(c) Where are there tunnel contacts in this device? What is the purpose of tunnel contacts? Draw the band diagram (valence band, conduction band, Fermi energy) of a tunnel contact.

(d) Why is a JFET slower than a MESFET?

Problem 3
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) Plot the minority carrier concentration in forward active mode.

(c) Calculate the equilibrium electron concentration in the collector. (n_i = 1.5 × 10¹⁰ cm^-3)

(d) How can you calculate the collector current?

Problem 4
(a) Describe how a solar cell works.

(b) The depletion region of a solar cell has a certain thickness in the dark. What determines this thickness? What happens to the depletion width when light falls on the solar cell?

(c) If a semiconductor has an indirect bandgap, what consequence does this have for a solar cell?