Semiconductor Crystals Questions and Answers

Answer: B

The Hall resistivity rises linearly with the magnetic field, the straight line through the origin in (B).

In steady state the Hall field balances the magnetic force on the carriers, giving \(E_{y} = v_{x}B = J_{x}B/(ne)\). So

\[ \rho_{xy} = \frac{E_{y}}{J_{x}} = \frac{B}{ne} \]

which is proportional to \(B\) and zero when there is no field. In a non-magnetic material there is no anomalous Hall contribution from a spontaneous magnetisation, so the line passes through the origin.

A constant (A) would mean a Hall signal at zero field; a falling curve (C) has the dependence upside down; and a curve that saturates into a plateau (D) resembles the quantum Hall or anomalous Hall behaviour, not the classical Hall effect asked about.

Answer: D

The bandgaps satisfy E_A > E_C > E_B.

For an intrinsic semiconductor the conductivity varies as

\[ \begin{aligned} \sigma &\propto e^{-E_{g}/2k_{B}T} \\ \ln\sigma &= \text{const} - \frac{E_{g}}{2k_{B}}\cdot\frac{1}{T} \end{aligned} \]

so on a plot of ln σ against 1/T the slope is -E_g/2k_B. The steeper the line, the larger the bandgap. Line A is the steepest, C next and B the least steep.

The other orders mis-rank the slopes: putting C first, B first, or B above C would all require B or C to fall faster than the drawing shows.

Answer: A

The equilibrium hole concentration is \(2.25 \times 10^{3}\ cm^{-3}\).

With all donors ionised, the electron concentration in the n-type silicon is the donor concentration, \(n = 10^{17}\ cm^{-3}\). At equilibrium the mass-action law fixes the product of electron and hole concentrations:

\[ \begin{aligned} np &= n_i^{2} \\ p &= \frac{(1.5 \times 10^{10})^{2}}{10^{17}} \\ &= \frac{2.25 \times 10^{20}}{10^{17}} = 2.25 \times 10^{3} \end{aligned} \]

Holes are the minority carriers, so their number is tiny compared with the \(10^{10}\) of intrinsic silicon: doping raises one carrier type and suppresses the other. The remaining options do not follow from \(n_i^2/N_D\) with these numbers.