Problem 1
(a) Describe some methods that can be used to dope silicon with acceptors.
(b) Silicon is doped with acceptors to a concentration of $10^{17}$ 1/cm³. Plot the hole concentration as a function of temperature indicating the intrinsic, extrinsic and freeze-out regimes. At what temperature does the transition from intrinsic to extrinsic take place?
(c) What is the minority carrier concentration at 300 K?
(d) In the depletion region of a silicon $pn$-junction, the $p$-type semiconductor has an acceptor concentration of $10^{17}$ 1/cm³. What is the maximum value of the derivative of the electric field $\frac{dE}{dx}$?
For silicon at 300 K: $E_g = 1.12$ 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, $\epsilon_r = 11.9$.
Problem 2
(a) Draw the band diagram (conduction band, valence band, Fermi energy) of a p-Schottky diode.
(b) Why is the Fermi energy pinned to the middle of the semiconductor bandgap in a Schottky diode?
(c) Indicate $V_{bi}$ in your drawing. How does $V_{bi}$ depend on the Schottky barrier $\varphi_b$ and the acceptor doping?
(d) Which way does current flow when light falls on this Schottky diode?
Problem 3
(a) Draw a cross section of an $n$-channel JFET showing the source, drain, and gate contacts.
(b) Draw the band diagram (conduction band, valence band, Fermi energy) from the gate to the channel. Assume that there is a negative voltage on the gate and a positive voltage on the drain.
(c) Consider a JFET in the linear regime (the drain voltage is not so high as to cause pinch-off). How could you calculate the depletion width on the source side and on the drain side of the channel?
Problem 4
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) If the transistor is unbiased, which depletion region is wider? Why?
(b) For an unbiased transistor, is the maximum electric field larger in the emitter-base junction or in the base-collector junction? Why?
(c) Plot the minority carrier concentration in forward active mode.
(d) How could you calculate the voltage at which this transistor will experience punch-through?
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 |