Problem 1
An n-type silicon wafer is uniformly doped with phosphorus at a concentration of $10^{15}$ cm^-3. Boron acceptors are then diffused into the wafer to form a concentration profile $N_A= 10^{17}\exp\left(-\frac{2x}{10^{-6}}\right)$ 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 $\left( N_D(x),\, N_A(x),\, n(x),\, p(x)\right)$ as a function of $x$.

(b) What is the concentration of holes at $x=1$ μm?

(c) What is the concentration of holes at $x=5$ μm?

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

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

For silicon: $E_g = 1.12$ eV, $N_c = 2.78 \times 10^{19}$ 1/cm³, $N_v = 9.84 \times 10^{18}$ 1/cm³, and $n_i= 1.5\times 10^{10}$ cm^-3.

Problem 2
A photodiode consists of semiconducting layers $n$+ / $n$ / $p$ / $p$+. No bias voltage is applied.

(a) Draw the band diagram (conduction band, valence band, Fermi energy).

(b) Draw the electric field and the charge density.

(c) Which way is does current flow when light falls on the photodiode?

Problem 3
(a) Draw an $n$-channel MOSFET showing the source, drain, gate, and body contacts.

(b) Draw the band diagram (conduction band, valence band, Fermi energy), and the electric field as a function of position in a MOS capacitor with a p-type substrate in accumulation.

(c) Consider a MOSFET in the linear regime (the gate is above the threshold to induce an inversion layer in the channel but the drain voltage is not so high as to cause pinch-off). How does the depletion width vary along the channel from source to drain?

(d) Explain what latch-up is in CMOS circuits.

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?