Energy Bands Classification

14.1 Introduction — From Vacuum Tubes to Solid-State Electronics

Electronic devices are simply ways to control the flow of electrons. Before 1948 this control was achieved with bulky vacuum tubes (valves) — diodes, triodes, tetrodes, pentodes — in which a heated cathode boiled off electrons that travelled through an evacuated glass envelope toward an anode. Vacuum tubes worked, but they were large, fragile, hot, slow and power-hungry.

The 1948 invention of the transistor at Bell Labs (Bardeen, Brattain & Shockley — Nobel 1956) launched the solid-state revolution. In a semiconductor, mobile electrons live inside the crystal itself; they are summoned by simple stimuli — light, heat, a small voltage — without any vacuum or hot filament. The result: tiny, robust, fast, low-power devices that today form the building blocks of every smartphone, computer, sensor and solar panel.

14.2 Classification — Conductors, Semiconductors, Insulators

By electrical resistivity (ρ) or conductivity (σ = 1/ρ)

Semiconductors of interest in this chapter:

• Elemental: Silicon (Si) and Germanium (Ge) — group IV elements with 4 valence electrons.
• Compound (inorganic): GaAs, CdS, InP, CdSe.
• Organic: anthracene, doped phthalocyanines, conducting polymers like polyaniline.

By energy bands

In an isolated atom, electrons occupy discrete energy levels. When N atoms come together to form a crystal, the outer-shell levels overlap with neighbours and split into N closely-spaced levels. With N ~ 10²³, this dense set of levels behaves like a continuous energy band.

The two bands of greatest importance:

• Valence band (VB): contains the valence electrons. Filled (or nearly filled) at T = 0 K.
• Conduction band (CB): the next-higher band. Empty (or nearly empty) at T = 0 K.

Between the top of the VB (energy E_V) and the bottom of the CB (energy E_C) lies a forbidden region called the energy band gap E_g = E_C − E_V. The size of this gap, plus the band's filling, determines whether the material conducts.

14.3 Intrinsic Semiconductors

An intrinsic semiconductor is pure Si or Ge with no impurities. Both have 4 valence electrons; in the crystal each atom forms covalent bonds with four neighbours, sharing electrons in pairs.

Electrons and holes

At T = 0 K, all bonds are intact and the crystal is an insulator. Above 0 K, thermal vibrations occasionally break a bond, releasing a free electron into the conduction band and leaving an electron vacancy behind. This vacancy is called a hole and behaves like a positive charge carrier with an effective charge +e.

Hole motion is just rearrangement of bound electrons: an electron from a neighbouring bond fills the hole, creating a new hole at its old location. The net effect is as if a positive particle moved.

For an intrinsic semiconductor:

At room temperature, n_i is small (~10¹⁶ /m³ for Si), so intrinsic Si conducts only weakly. To make useful devices we need many more carriers — that's what doping does.

14.4 Extrinsic Semiconductors — Doping

The number of charge carriers in pure Si or Ge can be enormously increased by adding tiny, controlled amounts (parts-per-million) of selected impurities. This process is called doping; the impurities are dopants; the result is an extrinsic (or impurity) semiconductor. To preserve the lattice structure, the dopant atoms must be roughly the size of Si/Ge.

Two categories matter for tetravalent Si/Ge:

• Pentavalent (group V): As, Sb, P — they have 5 valence electrons → donor.
• Trivalent (group III): B, Al, In, Ga — they have 3 valence electrons → acceptor.

n-type (donor doping)

When a pentavalent atom replaces a Si atom, four of its electrons form covalent bonds with neighbours; the 5th is loosely bound and easily liberated (ionisation energy only ~0.05 eV for P in Si — far less than E_g ≈ 1.1 eV). Each donor contributes one electron to the conduction band:

p-type (acceptor doping)

A trivalent dopant has only 3 valence electrons. Its 4th bond with a neighbouring Si atom has a missing electron — i.e., a hole. A small thermal kick lets a nearby valence electron jump in, leaving a hole somewhere else. Each acceptor contributes one hole to the valence band:

Worked Example 14.2 (NCERT)

Interactive — Band-Gap Explorer