CBSE Class 12 Physics Semiconductor Electronics Materials Devices and Simple Circuits Notes Set 03

1. Classification of Solids by Conductivity

On the basis of electrical resistivity ρ (or conductivity σ = 1/ρ), solids are divided into three groups:
(i) Metals: very low resistivity, ρ ≈ 10⁻² to 10⁻⁸ Ω m (σ ≈ 10² to 10⁸ S m⁻¹).
(ii) Semiconductors: resistivity in between, ρ ≈ 10⁻⁵ to 10⁶ Ω m.
(iii) Insulators: very high resistivity, ρ ≈ 10¹¹ to 10¹⁹ Ω m.

Semiconductors may be elemental (silicon Si, germanium Ge) or compound (inorganic such as GaAs, CdS, InP, and organic such as anthracene and conducting polymers). Most devices today are made from silicon. Unlike metals, the resistance of a semiconductor decreases as its temperature rises.

2. Energy Bands in Solids

In an isolated atom the electrons occupy sharp energy levels. In a crystal the atoms are so close that their outer orbits overlap and interact, so each level splits into a large number of very closely spaced levels. These form a nearly continuous energy band.
1. Valence band (VB): the band containing the valence electrons; at 0 K it is completely filled in semiconductors and insulators.
2. Conduction band (CB): the next higher band. Electrons in it are free to move and can conduct.
3. Energy band gap (E_g): the gap between the top of the valence band and the bottom of the conduction band. No electron can have an energy inside this forbidden gap.

(a) Metals: the conduction band is partly filled, or the conduction and valence bands overlap (E_g = 0). Plenty of electrons are available for conduction.
(b) Insulators: E_g > 3 eV (for diamond about 5.4 eV). Thermal energy cannot lift electrons across the gap, so the conduction band stays empty.
(c) Semiconductors: E_g < 3 eV (Si ≈ 1.1 eV, Ge ≈ 0.7 eV). At room temperature a few electrons gain enough thermal energy to cross the gap, giving a small conductivity. At 0 K a semiconductor behaves as an insulator.

3. Intrinsic Semiconductors

A pure semiconductor such as Si or Ge is called an intrinsic semiconductor. Each atom has four valence electrons and forms four covalent bonds with its neighbours.
1. When thermal energy breaks a covalent bond, an electron becomes free and leaves behind a vacancy with an effective charge +e. This vacancy is called a hole.
2. In an intrinsic semiconductor electrons and holes are always created in pairs, so
n_e = n_h = n_i, where n_i is the intrinsic carrier concentration.
3. A hole moves when an electron from a neighbouring bond jumps into it, leaving a new hole behind. So holes behave like positive charge carriers.
4. Under an applied field the total current is I = I_e + I_h (free electron current plus hole current).
5. Electron–hole pairs are also lost by recombination. At equilibrium the rate of generation equals the rate of recombination. As temperature increases, n_i increases and the resistivity falls.

4. Extrinsic Semiconductors

The conductivity of an intrinsic semiconductor is too low for most uses. It is increased greatly by adding a very small amount of a suitable impurity (about 1 atom in 10⁶ to 10⁸). This is called doping, the impurity atoms are called dopants, and the result is an extrinsic (doped) semiconductor. The dopant atom should be of roughly the same size as the host atom, so that the crystal structure is not distorted.

(i) n-type semiconductor: Si or Ge doped with a pentavalent impurity (arsenic As, antimony Sb, phosphorus P). Four of its five electrons form bonds; the fifth is loosely held (ionisation energy about 0.01 eV for Ge and 0.05 eV for Si) and becomes free easily. The impurity donates an electron, so it is called a donor.
1. Electrons are the majority carriers; holes are the minority carriers: n_e ≫ n_h.
2. The donor energy level lies just below the bottom of the conduction band.

(ii) p-type semiconductor: Si or Ge doped with a trivalent impurity (boron B, aluminium Al, gallium Ga, indium In). The impurity has only three valence electrons, so one bond has a vacancy, i.e. a hole. It accepts an electron from a neighbouring bond, so it is called an acceptor.
1. Holes are the majority carriers; electrons are the minority carriers: n_h ≫ n_e.
2. The acceptor energy level lies just above the top of the valence band.

(iii) Mass action relation: At thermal equilibrium, in any (intrinsic or extrinsic) semiconductor,
n_e n_h = n_i²
For an n-type sample with donor concentration N_D ≫ n_i: n_e ≈ N_D and n_h = \( \frac{n_i^2}{N_D} \). For a p-type sample with acceptor concentration N_A: n_h ≈ N_A.
Increasing the majority carriers decreases the minority carriers, because more of them recombine.

(iv) An extrinsic semiconductor is electrically neutral as a whole. In n-type material the free electrons are balanced by the fixed positive donor ions; in p-type material the holes are balanced by the fixed negative acceptor ions.

5. Formation of a p-n Junction

A p-n junction is formed when a p-type region and an n-type region are created in a single crystal (for example, by diffusing a donor impurity into one part of a p-type wafer).
1. Diffusion: Holes are many on the p-side and electrons are many on the n-side. Because of this concentration difference, holes diffuse from p to n and electrons diffuse from n to p. This gives a diffusion current from p to n.
2. Electrons leaving the n-side expose fixed positive donor ions near the junction; holes leaving the p-side leave fixed negative acceptor ions.
3. Depletion region: the thin layer (about 10⁻⁶ m wide) on both sides of the junction that contains only immobile ions and no free charge carriers.
4. The ions set up an electric field directed from the n-side to the p-side. This field pushes minority carriers across the junction, giving a drift current opposite to the diffusion current.
5. At equilibrium, diffusion current = drift current, and there is no net current across the junction.
6. Barrier potential (V₀): the potential difference across the depletion region, with the n-side at the higher potential. It opposes further diffusion of majority carriers. Typical values are about 0.7 V for Si and 0.3 V for Ge.

6. Semiconductor Diode and Biasing

A semiconductor diode is a p-n junction with metal contacts at its two ends. Its symbol is an arrowhead with a bar; the arrow (p-side, anode) points in the direction of conventional current in forward bias, and the bar is the n-side (cathode). It is a two-terminal device.

(i) Forward bias: p-side connected to the positive terminal and n-side to the negative terminal of a battery.
1. The applied voltage V opposes the barrier, so the effective barrier height falls to (V₀ − V).
2. The depletion layer becomes narrower.
3. Majority carriers cross the junction easily (minority carrier injection), and a large diffusion current flows, of the order of mA.

(ii) Reverse bias: p-side connected to the negative terminal and n-side to the positive terminal.
1. The applied voltage adds to the barrier, so the effective barrier height becomes (V₀ + V).
2. The depletion layer becomes wider.
3. Diffusion is almost stopped. Only a very small drift current due to minority carriers flows, of the order of μA. It is nearly independent of the applied voltage and is called the reverse saturation current.
4. At a high reverse voltage called the breakdown voltage (V_br), the reverse current increases suddenly. An ordinary diode is damaged if it is operated beyond this point.

7. V–I Characteristics of a p-n Junction Diode

The characteristic curve is a graph of the current through the diode against the voltage across it, drawn for both forward and reverse bias.

1. Forward bias: the current is very small until the voltage reaches the threshold (cut-in or knee) voltage, about 0.7 V for Si and 0.2–0.3 V for Ge. Beyond it the current rises steeply (exponentially) for a small increase in voltage.
2. Reverse bias: the current is very small (μA) and almost constant until breakdown.
3. The graph is not a straight line, so a diode does not obey Ohm's law. It conducts well in one direction only, which makes it useful as a rectifier.
4. Dynamic resistance: the ratio of a small change in voltage to the resulting small change in current,
r_d = \( \frac{\Delta V}{\Delta I} \)
It is small in forward bias (a few ohms) and very large in reverse bias.
5. An ideal diode has zero resistance in forward bias (acts as a closed switch) and infinite resistance in reverse bias (acts as an open switch).

8. Diode as a Rectifier

Rectification is the conversion of alternating voltage into a unidirectional (pulsating dc) voltage. It is possible because a diode lets current flow only when it is forward biased. The ac voltage is applied through a transformer, whose secondary gives the required voltage.

(i) Half-wave rectifier: one diode is connected in series with the transformer secondary and the load resistance R_L.
1. During the half cycle in which the anode is positive, the diode is forward biased and current flows through R_L.
2. During the other half cycle the diode is reverse biased and no current flows.
3. The output appears only in alternate half cycles. Its frequency equals the input frequency (50 Hz for a 50 Hz supply). The reverse breakdown voltage of the diode must be greater than the peak ac voltage of the secondary.

(ii) Full-wave rectifier (centre-tap): two diodes D₁ and D₂ are connected to the two ends A and B of a centre-tapped secondary; their outputs are joined and the load R_L is connected between this common point and the centre tap C.
1. When A is positive with respect to C, D₁ is forward biased and conducts while D₂ is reverse biased.
2. In the next half cycle B is positive with respect to C, so D₂ conducts and D₁ does not.
3. In both half cycles the current flows through R_L in the same direction, so output is obtained in both halves. The output frequency is twice the input frequency (100 Hz for a 50 Hz supply).
4. The voltage across each half of the secondary is half the full secondary voltage, and each diode conducts for one half cycle.

(iii) Filter: the rectified output is pulsating and contains ac ripple. A capacitor filter connected across R_L charges to the peak voltage when the output rises and discharges slowly through R_L when it falls. If the time constant C R_L is large compared with the time between peaks, the output becomes nearly steady dc, close to the peak voltage. A full-wave output is easier to smooth than a half-wave output because its peaks come twice as often.

9. Quick Formula Recap