07.03.2018

**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 contration 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 $p$-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) What is the subthreshold current?

**Problem 3**

Explain how SRAM, DRAM and flash memories work.

**Problem 4**

In a bipolar transistor,

(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 transistor in the forward active regime.

(d) How can you calculate the collector current?

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 |