Classification of Metals, Conductors and Semiconductors

Where the chapter starts: why solid-state devices replaced vacuum tubes

• The building block of any electronic circuit is a device in which the flow of electrons can be controlled. Before the transistor (1948) these were mostly vacuum tubes (valves): the vacuum diode with two electrodes (anode or plate, and cathode), the triode with three (cathode, plate, grid), and the tetrode and pentode with four and five.
• In a vacuum tube the electrons are supplied by a heated cathode and are controlled by varying the voltage between the electrodes. The inter-electrode space must be evacuated, or the moving electrons would lose energy in collisions with air molecules. Electrons can flow only from cathode to anode, that is, in one direction — hence the name valve.
• Vacuum tubes are bulky, consume high power, work at high voltages (of the order of $100$ V), and have a limited life and low reliability.
• In a semiconductor device, by contrast, the supply and the flow of charge carriers are both inside the solid itself. No external heating and no evacuated space are needed. Such devices are small, consume low power, operate at low voltages, and have a long life and high reliability.
• Simple excitations — light, heat, or a small applied voltage — can change the number of mobile charges in a semiconductor, which is exactly the control the earlier valves obtained only with a heated cathode and a vacuum.
• This is why the chapter goes on to study junction diodes (two-electrode devices) and the circuits built from them, rather than valves.

Two ways of classifying solids

• On the basis of conductivity. Solids are grouped by the relative values of their conductivity $\sigma$ and resistivity $\rho = 1/\sigma$.

• These figures are indicative magnitudes and can fall a little outside the ranges. Resistivity is not the only criterion — the band picture and the temperature behaviour separate the three classes far more sharply.
• Semiconductors of interest: elemental (Si, Ge), compound inorganic (CdS, GaAs, CdSe, InP), organic (anthracene, doped phthalocyanines) and organic polymers (polypyrrole, polyaniline, polythiophene). The syllabus works only with the elemental ones.

On the basis of energy bands

• In an isolated atom, an electron's energy is fixed by the orbit it occupies — a set of discrete levels. When atoms come together to form a solid, the outer orbits of neighbouring atoms come very close and may overlap, so the motion of an electron in a solid is quite unlike that in a free atom.
• Inside the crystal no two electrons see exactly the same pattern of surrounding charges, so each has a slightly different energy. These closely spaced levels, varying almost continuously, form energy bands.
• The band that contains the energy levels of the valence electrons is the valence band; the band immediately above it is the conduction band. With no external energy supplied, all valence electrons sit in the valence band.
• If the lowest level of the conduction band lies below the highest level of the valence band, the two bands overlap and electrons pass freely into the conduction band — this is a metallic conductor. If there is a gap, the valence electrons stay bound and the conduction band is empty.

Counting the states in Si and Ge

• For Si the outermost orbit is $n = 3$; for Ge it is $n = 4$. Either way the outermost orbit holds $4$ electrons ($2s$ and $2p$), and its maximum capacity is $8$ ($2s + 6p$).
• A crystal of $N$ atoms therefore has $4N$ valence electrons and $8N$ available outer-orbit states. At the actual lattice spacing of Si and Ge, these $8N$ states split into two bands separated by the gap $E_g$: a lower band of $4N$ states, completely filled by the $4N$ valence electrons at absolute zero (the valence band), and an upper band of $4N$ states, completely empty at absolute zero (the conduction band).
• $E_C$ is the lowest energy level of the conduction band and $E_V$ the highest of the valence band. Above $E_C$ and below $E_V$ lie a very large number of closely spaced levels.

Energy band gap — the definition to quote

• The energy gap between the top of the valence band and the bottom of the conduction band, in which no allowed electron energy state exists, is called the energy band gap (or forbidden energy gap) of the solid:

$$E_g = E_C - E_V$$

• It may be large, small or zero depending on the material, and that single number decides whether the solid is an insulator, a semiconductor or a metal.

The three cases, as CBSE wants them drawn and described

Case I — metal (conductor)

• Either the conduction band is partially filled while the valence band is partially empty, or the conduction and valence bands overlap.
• Where they overlap, electrons move from the valence band into the conduction band with no energy cost at all; where the conduction band is partly filled, electrons in its lower levels move to higher levels within the same band.
• A very large number of electrons is therefore available for conduction. Resistance is low, conductivity high, and $E_g \approx 0$.
• Diagram in words: two shaded blocks drawn touching, or overlapping, with no blank strip between them.

Case II — insulator

• A large gap exists, $E_g > 3$ eV. The valence band is completely filled and the conduction band is completely empty.
• The gap is so wide that thermal excitation cannot lift electrons across it, so the conduction band stays empty even when a field is applied and no current flows.
• Diagram in words: two shaded blocks far apart, with a wide blank strip between them labelled $E_g > 3$ eV; the lower block full, the upper one empty.

Case III — semiconductor

• A finite but small gap exists, $E_g < 3$ eV. At $0$ K the arrangement is exactly that of an insulator: full valence band, empty conduction band.
• At room temperature, some valence electrons acquire enough thermal energy to cross this narrow gap into the conduction band, where they can move. Their number is small, so the resistance of a semiconductor is much higher than that of a metal, but nothing like as high as that of an insulator.
• Diagram in words: two shaded blocks with only a narrow blank strip between them, labelled $E_g < 3$ eV; at $T > 0$ K draw a few filled dots at the bottom of the conduction band and the same number of empty circles at the top of the valence band.

Two distinguishing features (the exact 3-mark answer)

• On the bands themselves: in a conductor the valence and conduction bands overlap (or nearly do), so there is effectively no forbidden gap; in an insulator the two bands are separated by a large gap; in a semiconductor they are separated by only a small gap.
• On the occupancy of the conduction band: a conductor's conduction band already holds a large number of electrons available for conduction; an insulator's conduction band is almost completely empty; a semiconductor's conduction band holds a small but non-zero number of thermally generated electrons.

Metal versus insulator (the 2-mark version)

• Metal: valence band completely filled, conduction band either partially filled with a practically zero gap or overlapping the valence band. Electrons need only a negligible amount of energy to reach the conduction band, so even a small applied field produces a current easily.
• Insulator: valence band completely filled, conduction band completely empty, separated by a very large forbidden gap ($E_g > 3$ eV). Applying a field cannot give electrons enough energy to jump this gap, so the conduction band stays empty and no current flows.

Why the band gap decides everything — C, Si and Ge

• Carbon (diamond), silicon and germanium all have four valence electrons and the same diamond-like lattice, yet carbon is an insulator while Si and Ge are semiconductors. The reason is the band gap:

• The four bonding electrons of C, Si and Ge lie respectively in the second, third and fourth orbit. The further out the orbit, the more weakly the electron is held, so the energy needed to free an electron is highest for C, less for Si, and least for Ge. Hence carbon has essentially no free electrons at room temperature while Si and Ge have a significant, if small, number.
• So: $(E_g)_C > (E_g)_{Si} > (E_g)_{Ge}$. This ordering is itself a standard MCQ. Sn belongs to the same group but is a metal, because its energy gap is zero.
• The property of C, Si and Ge that depends on this gap is their conductivity (equivalently, their resistivity). CBSE has set this as a one-word fill-in-the-blank.

Variation of resistivity with temperature

• Conductor: $\rho$ increases with temperature. The curve rises very steeply, almost vertically, as $T$ goes from near $0$ K to about $10$ to $20$ K, and then increases roughly linearly with $T$ up to a few hundred kelvin. Physically, the hotter lattice vibrates more and scatters the free electrons more, so conductivity falls.
• Semiconductor: $\rho$ decreases with temperature. The curve starts very high at low $T$ and falls off steeply and continuously, like a decaying exponential, as $T$ rises towards room temperature (about $300$ K). Physically, more electron-hole pairs are thermally generated, so more carriers are available and conductivity rises.
• Draw the two curves on separate axes; do not superpose them, and label which is which.

Standard results

• $\sigma = 1/\rho$; ranges as tabulated above.
• $E_g = E_C - E_V$; $E_g \approx 0$ for metals, $E_g < 3$ eV (about $0.2$ to $3$ eV) for semiconductors, $E_g > 3$ eV for insulators.
• $E_g$: C $5.4$ eV, Si $1.1$ eV, Ge $0.72$ eV, Sn $0$ eV. The energy an electron needs to jump the forbidden band in Si at room temperature is about $1.1$ eV.
• $N$ atoms give $4N$ valence electrons in $8N$ states, splitting into two bands of $4N$ states each.
• $\rho$ rises with $T$ for a conductor and falls with $T$ for a semiconductor.