Key Points

Solids are classified based on resistivity ($\rho$) or conductivity ($\sigma$). Metals have low resistivity ($\rho \sim 10^{-2} - 10^{-8} \Omega\text{m}$), insulators have high resistivity ($\rho \sim 10^{11} - 10^{19} \Omega\text{m}$), and semiconductors are intermediate ($\rho \sim 10^{-5} - 10^{6} \Omega\text{m}$).

In solids, atomic energy levels form bands. The valence band contains valence electrons, and the conduction band is for free electrons. The energy difference between the top of the valence band ($E_V$) and the bottom of the conduction band ($E_C$) is the energy gap ($E_g$).

The energy gap ($E_g$) determines electrical properties. For insulators, $E_g > 3 \text{ eV}$. For semiconductors, $E_g < 3 \text{ eV}$ (e.g., Si is $1.1 \text{ eV}$, Ge is $0.7 \text{ eV}$). For metals, the valence and conduction bands overlap, so $E_g \approx 0$.

A pure semiconductor is called an intrinsic semiconductor. In it, the number of free electrons ($n_e$) equals the number of holes ($n_h$), and this is known as the intrinsic carrier concentration ($n_i$). Thus, $n_e = n_h = n_i$.

The conductivity of a semiconductor is increased by adding impurities, a process called doping. The resulting material is an extrinsic semiconductor. The impurity atoms are called dopants.

Formed by doping a pure semiconductor (Si or Ge) with a pentavalent impurity (like As, P). Electrons become the majority carriers and holes are the minority carriers, so $n_e \gg n_h$. The impurity is called a donor.

Formed by doping a pure semiconductor (Si or Ge) with a trivalent impurity (like B, Al, In). Holes become the majority carriers and electrons are the minority carriers, so $n_h \gg n_e$. The impurity is called an acceptor.

In any semiconductor at thermal equilibrium, the product of the concentration of electrons and holes is constant and equals the square of the intrinsic carrier concentration. The formula is $n_e n_h = n_i^2$.

A p-n junction is formed where p-type and n-type semiconductors meet. Due to diffusion of charge carriers, a depletion region is formed at the junction, which is devoid of free charges and has an associated barrier potential.

In forward bias, the p-side is connected to the positive terminal and the n-side to the negative terminal of a battery. This reduces the barrier potential, narrows the depletion region, and allows a large current (in $\text{mA}$) to flow.

In reverse bias, the p-side is connected to the negative terminal and the n-side to the positive terminal. This increases the barrier potential, widens the depletion region, and allows only a very small leakage current (in $\mu\text{A}$) to flow.

The current-voltage graph shows that in forward bias, current increases exponentially after a threshold voltage (cut-in voltage, $\sim 0.7 \text{V}$ for Si). In reverse bias, a small, constant reverse saturation current flows until breakdown voltage is reached.

A rectifier is a device that converts alternating current (AC) into direct current (DC). A p-n junction diode acts as a rectifier because it allows current to flow in only one direction (when forward biased).

A half-wave rectifier uses a single diode to allow only one half of the AC cycle (positive or negative) to pass through. If the input AC frequency is $f$, the output frequency is also $f$.

A full-wave rectifier uses two or more diodes to convert both halves of the AC cycle into a pulsating DC output. If the input AC frequency is $f$, the output frequency is $2f$.

To get a steady DC output from the pulsating output of a rectifier, a filter circuit is used. A common filter is a capacitor connected in parallel with the load, which smooths out the voltage ripples.