05.10.2018

**Problem 1**

A p-type silicon wafer is uniformly doped with boron at a concentration of $10^{15}$ cm^{-3}. N-type doping is then introduced by diffusion. The doping concentration is $N_D= 10^{17}\exp\left(\frac{-x^2}{10^{-12}}\right)$ 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 $\left( N_D(x),\, N_A(x),\, n(x),\, p(x)\right)$.

(b) What is the concentration of electrons at $x=20$ μm?

(c) Draw the band diagram (valence band, conduction band, Fermi energy) assuming no voltage bias is applied.

(d) Draw the electric field as a function of $x$.

For silicon: $E_g = 1.12$ eV, $N_c = 2.78 \times 10^{25}$ 1/m³, and $N_v = 9.84 \times 10^{24}$ 1/m³.

**Problem 2**

(a) Draw an n-channel JFET.

(b) Explain how a JFET works.

(c) Where are there tunnel contacts in this device? What is the purpose of tunnel contacts? Draw the band diagram (valence band, conduction band, Fermi energy) of a tunnel contact.

(d) Why is a JFET slower than a MESFET?

**Problem 3**

In a silicon $pnp$ bipolar transistor, the emitter is doped to 10^{19} cm^{-3}, the base is doped to 10^{14} cm^{-3}, and the collector is doped to 10^{13} 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. (*n _{i}* = 1.5 × 10

(d) How can you calculate the collector current?

**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) If a semiconductor has an indirect bandgap, what consequence does this have for a solar cell?

Quantity | Symbol | Value | Units | |

electron charge |
| 1.60217733 × 10^{-19} | C | |

speed of light |
| 2.99792458 × 10^{8} | m/s | |

Planck's constant |
| 6.6260755 × 10^{-34} | J s | |

reduced Planck's constant | $\hbar$ | 1.05457266 × 10^{-34} | J s | |

Boltzmann's constant |
| 1.380658 × 10^{-23} | J/K | |

electron mass |
| 9.1093897 × 10^{-31} | kg | |

Stefan-Boltzmann constant | σ | 5.67051 × 10^{-8} | W m | |

Bohr radius |
| 0.529177249 × 10^{-10} | m | |

atomic mass constant |
| 1.6605402 × 10^{-27} | kg | |

permeability of vacuum | μ | 4π × 10^{-7} | N A | |

permittivity of vacuum | ε | 8.854187817 × 10^{-12} | F m | |

Avogado's constant |
| 6.0221367 × 10^{23} | mol |