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 $N_A= 10^{17}\exp\left(-\frac{x^2}{4\times 10^{-12}}\right)$ 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) Draw an $p$-channel MOSFET showing the source, drain, gate, and body contacts.(b) Draw the band diagram in accumulation for this MOSFET (conduction band, valence band, Fermi energy), and the electric field as a function of position along a line from the gate through the oxide into the body.
(c) How are the source-body and drain-body pn-junctions biased (forward or reverse) when the transitor is in saturation?
(d) How could the subthreshold current be calculated?
Problem 3
(a) Draw an n-channel MESFET.
(b) Draw the band diagram (conduction band, valence band, Fermi energy) from the gate into the n-channel for no applied bias voltage.
(c) How do the interface states affect the band diagram?
(d) Describe the current mechanism when the gate diode is forward biased.
(e) What voltages would you apply to the gate and drain to bias an n-channel MESFET in saturation?
Problem 4
(a) What would have a larger reverse saturation current, a light emitting diode or a solar cell? Why?
(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 happens when a solar cell heats up? How do these changes affect the functioning of a solar cell?
Solution
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 |