Problem 1
An intrinsic silicon sample is doped with donors from one side such that $N_D = N_0\exp (-x/a)$.

(a) Draw the band diagram (conduction band, valence band, Fermi energy) from $x=0$ to $x=3a$. There is no current flowing so the Fermi energy should be constant.

(b) Draw the electron and hole concentrations ($n$ and $p$) and the electric field. Explain why these quantities have the form you have drawn.

Problem 2
There are four mechanisms that typically cause currents to flow: thermionic emission, diffusion, drift, and tunneling.

Explain briefly which kind of current mechanisms are important in the following devices. For instance, in a MOSFET, there is tunneling at source, drain, and body tunnel contacts; drift is the dominant current mechanism for the drain current when the MOSFET is turned on while diffusion is the dominant mechanism for the drain current (sub-threshold current) when the transistor is turned off .

(a) pn diode
(b) Schottky diode
(c) JFET
(d) bipolar transistor
(e) thyristor

Problem 3
(a) Draw an $n$-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) How do you calculate the maximum frequency at which a MOSFET operates? About what frequency is this?

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) What limits the efficiency of a solar cell?