Problem 1
An $n$+ well is doped into an $n$ substrate to create an ohmic contact. The interface between $n$ and $n$+ is at $x=0$. No bias voltage is applied.

(a) Draw the band diagram (conduction band, valence band, Fermi energy) around $x=0$.

(b) Draw the electric field and the charge density around $x=0$. Which way is the electric field pointing? Explain why the field points in this direction.

Problem 2
Holes are injected into a homogenously doped $n$-type semiconductor ($N_d = 10^{16} \text{cm}^{-3}$) at 300 K. In a Haynes-Shockley experiment, the minority carriers are observed to move 0.01 m in 100 μs when an electric field of 10000 V/m is applied.

(a) What is the mobility of the holes?

(b) It is possible to determine the diffusion constant of the holes from the mobility using the Einstein relation. What is the Einstein relation? What is the diffusion constant of the holes?

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

(b) How should this MOSFET be biased so that it is in the saturation regime?

(c) How does the drain current depend on the gate voltage in the saturation regime?

(d) A voltage is applied between the source and the body. What happens if a positive voltage is applied? What happens if a negative voltage is applied?

Problem 4
(a) Describe how a $pnp$ bipolar transistor works and explain the base transport factor and the emitter efficiency.

(b) How would you calculate the electron an hole contributions to the emitter current $I_{En}$ and $I_{Ep}$?

(c) Why does a heterojunction bipolar transistor have more gain than a bipolar transistor?