Problem 1
A p-type silicon wafer is uniformly doped with boron at a concentration of $10^{15}$ cm-3. Linear n-type doping is then introduced with a concentration profile $N_D= 10^{17}\left(1-\frac{x}{3\times 10^{-6}}\right)$ 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. The donor doping goes to zero at a depth of 3 microns and remains zero for $x > 3\,\mu\text{m}$.
(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 holes at $x=1$ μm?
(c) What is the concentration of electrons 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
Consider an n-channel MESFET.
(a) Draw the band diagram from metal to the semiconductor at the gate and at the drain at zero bias. Indicate Schottky barrier in both drawings.
(b) Draw the band diagram from metal to the semiconductor at the gate when the transistor is biased in saturation. Indicate Schottky barrier in both drawings.
(c) What determines the Schottky barrier height?
(d) Describe the current mechanisms from source to drain, from gate to channel, and from the channel to the metal contact at the drain.
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. (ni = 1.5 × 1010 cm-3)
(d) How can you calculate the collector current?
Problem 4
(a) Why is there an optimal bandgap for a solar cell that is powered by sunlight?
(b) Can you use an indirect band gap semiconductor to make a solar cell? Explain why or why not.
(c) 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?
(d) What factors limit the efficiency of a solar cell?
Quantity | Symbol | Value | Units | |
| electron charge | e | 1.60217733 × 10-19 | C | |
| speed of light | c | 2.99792458 × 108 | m/s | |
| Planck's constant | h | 6.6260755 × 10-34 | J s | |
| reduced Planck's constant | $\hbar$ | 1.05457266 × 10-34 | J s | |
| Boltzmann's constant | kB | 1.380658 × 10-23 | J/K | |
| electron mass | me | 9.1093897 × 10-31 | kg | |
| Stefan-Boltzmann constant | σ | 5.67051 × 10-8 | W m-2 K-4 | |
| Bohr radius | a0 | 0.529177249 × 10-10 | m | |
| atomic mass constant | mu | 1.6605402 × 10-27 | kg | |
| permeability of vacuum | μ0 | 4π × 10-7 | N A-2 | |
| permittivity of vacuum | ε0 | 8.854187817 × 10-12 | F m-1 | |
| Avogado's constant | NA | 6.0221367 × 1023 | mol-1 |