08.03.2019

**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**

(a) What determines the emitter efficiency? What determines the base transport factor? How are they related to the current transfer ratio $\alpha =\frac{I_c}{I_e}$?

(b) What is the Early effect. How could you change a transistor to decrease it?

(c) Draw the minority carrier concentration in a pnp ttransistor in the forward active regime.

(d) Consider what would happen if the mobility of the electrons in an npn bipolar transister was doubled. Explain how this would change the transistor properties.

**Problem 3**

(a) Draw an n-channel JFET.

(b) Explain how a JFET works. Explain whether the gate voltage should be positive or negative to operate in the saturation regime.

(c) What is the dominant current mechanism for the source-drain current? (tunneling, drift, diffusion, thermionic emission).

(d) If a JFET is biased in saturation, where is the largest electric field? How would you calculate the value of the electric field at the point where the electric field is largest?

**Problem 4**

Make a drawing of a laser diode and describe how it works. What determines the frequency that is emitted? Why is there a threshold current? How is it different from an ordinary light emitting diode?

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 |