Energy bands in solids

Take a single silicon atom and its electrons sit at sharp, well-separated energy levels, like rungs on a ladder. Pack $10^{23}$ such atoms into a centimetre cube of solid and something strange happens: the Pauli exclusion principle will not let any two electrons share an identical state, so each once-sharp rung smears into a dense band of closely spaced levels. This band picture — a valence band, a conduction band, and the forbidden gap $E_g$ between them — is the single idea that separates a copper wire that conducts effortlessly, a diamond that never conducts, and a silicon chip that can be switched between the two. Every diode, transistor, and solar cell in modern electronics is built on getting this picture right.

The core insight

Atomic energy levels widen into continuous bands when atoms bond into a solid, and the width of the forbidden gap $E_g$ between the filled valence band and the empty conduction band decides whether the solid is a conductor, an insulator, or a semiconductor.

The core idea

From levels to bands. In an isolated atom, electrons sit at discrete, well-defined energies. Bring $\sim10^{23}$ atoms per cm $^3$ together and their electron clouds interact; the exclusion principle forbids identical states, so each atomic level splits into as many closely spaced sub-levels as there are interacting atoms. These sub-levels merge into a quasi-continuous energy band. The highest band that is normally occupied at absolute zero is the valence band; the next band up, where electrons can move freely and carry current, is the conduction band. The energy range between them that no electron in the pure solid can occupy is the forbidden energy gap, $E_g$ .

Three classes, one picture. The three material types differ only in how these bands sit relative to each other:

Conductors: the valence and conduction bands overlap, or the conduction band is already partly filled at 0 K, so free carriers exist with no energy investment — effectively $E_g \approx 0$ .
Insulators: a full valence band, an empty conduction band, and a wide gap, typically $E_g > 3\ \text{eV}$ (diamond, $E_g \approx 5.4\ \text{eV}$ ).
Semiconductors: the same full-valence, empty-conduction structure as an insulator at 0 K, but with a narrow gap (silicon $E_g \approx 1.1\ \text{eV}$ , germanium $E_g \approx 0.7\ \text{eV}$ ), small enough that room-temperature thermal energy lifts a workable number of electrons across.

Why temperature matters. Room-temperature thermal energy is of order

k_BT \approx 0.026\ \text{eV},

where: $k_B = 8.62\times10^{-5}\ \text{eV/K}$ is the Boltzmann constant and $T \approx 300\ \text{K}$ . Comparing $E_g$ to this single number tells you, at a glance, which class a solid falls into: $E_g \approx 0$ gives a conductor, $E_g$ a few times $k_BT$ (silicon, germanium) gives a semiconductor, and $E_g \gg k_BT$ by two orders of magnitude (diamond) gives an insulator. Each electron that does jump the gap leaves behind a hole — a missing bond that behaves like a mobile positive carrier — so a semiconductor conducts through electron-hole pairs, and more heat means more pairs, which is why a semiconductor's resistance falls as it warms, the opposite of a metal.

Figure

Three energy band diagrams side by side for a conductor an insulator and a semiconductor showing the valence band conduction band and the forbidden gap between them

Derivation AWhy a small change in Eg swings a solid from semiconductor to insulator

The goal of this derivation is to show, with nothing beyond the ratio $E_g/k_BT$ already used in the syllabus and one extra idea from thermal physics — that the chance of an electron picking up enough random thermal energy to jump a gap $E_g$ falls off exponentially as $\exp(-E_g/k_BT)$ — exactly why silicon ( $E_g = 1.1\ \text{eV}$ ) conducts weakly at room temperature while diamond ( $E_g = 5.4\ \text{eV}$ ) conducts essentially not at all, even though the two gaps differ by only a factor of five.

Step 1. State the physical fact being used. Thermal energy is not delivered to every electron equally; it arrives in random jostles from lattice vibrations, and the fraction of electrons with an energy at least $E_g$ above the valence band top follows the Boltzmann factor $\exp(-E_g/k_BT)$ — this is the same exponential suppression that governs any thermally activated process, so no new physics beyond "hotter means more high-energy electrons, but rare ones are exponentially rarer" is required.

Step 2. Write the number of thermally excited carriers as proportional to this factor: $n \propto \exp(-E_g/k_BT)$ . A useful trick is to compare two materials by taking the ratio of their carrier counts, which removes the (unknown) constant of proportionality: $n_1/n_2 = \exp[-(E_{g1}-E_{g2})/k_BT]$ .

Step 3. Plug in silicon and diamond at room temperature, $k_BT \approx 0.026\ \text{eV}$ . The exponents themselves are $E_g/k_BT = 1.1/0.026 \approx 42$ for silicon and $5.4/0.026 \approx 208$ for diamond — so $n_{\text{Si}}/n_{\text{diamond}} = \exp(208-42) = \exp(166)$ , a number with roughly 72 zeros. Even though $E_g$ only changed by a factor of five, the carrier population changes by a factor with 72 digits.

Step 4. Interpret. This is why a "small" difference in gap width is decisive rather than incidental: the exponential means silicon's population of conduction electrons, though still tiny compared to a metal's, is unimaginably larger than diamond's, which is for all practical purposes zero. The qualitative rule "small gap gives semiconductor, large gap gives insulator" is not a rough guideline — it follows directly from how steeply an exponential falls.

Worked example

Question: Germanium has $E_g \approx 0.7\ \text{eV}$ and silicon has $E_g \approx 1.1\ \text{eV}$ . Using the Boltzmann-factor reasoning above, estimate roughly how many times more thermally excited carriers germanium has than silicon at room temperature ( $k_BT \approx 0.026\ \text{eV}$ ), and say what this predicts about their relative conductivities.

Step 1. Set up the ratio from Step 2 of the derivation: $n_{\text{Ge}}/n_{\text{Si}} = \exp[-(E_{g,\text{Ge}} - E_{g,\text{Si}})/k_BT]$ .

Step 2. Compute the exponent: $E_{g,\text{Ge}} - E_{g,\text{Si}} = 0.7 - 1.1 = -0.4\ \text{eV}$ , so $-(E_{g,\text{Ge}}-E_{g,\text{Si}})/k_BT = 0.4/0.026 \approx 15.4$ .

Step 3. Evaluate: $n_{\text{Ge}}/n_{\text{Si}} = \exp(15.4) \approx 5\times10^{6}$ .

Step 4. Interpret the answer. Germanium's smaller gap, only $0.4\ \text{eV}$ narrower than silicon's, is predicted to give it roughly five million times more thermally excited carriers at room temperature. This matches the real-world behaviour that intrinsic germanium is a noticeably better room-temperature conductor than intrinsic silicon, which is exactly why early transistors used germanium before silicon's other advantages (stability, abundance, oxide quality) made it the industry standard.

Common traps

Thinking an insulator has "no conduction band" — it has one, it is simply empty and separated by a wide gap; the band exists, it is just inaccessible at ordinary temperatures.
Confusing "small gap" with "no gap" — semiconductors always have a genuine forbidden gap; they differ from insulators in degree, not in kind.
Assuming a semiconductor behaves like a metal as temperature rises — a semiconductor's resistance falls with heat (more electron-hole pairs), while a metal's rises (more lattice scattering).
Treating the exponential $\exp(-E_g/k_BT)$ derivation above as required syllabus content — the ISC syllabus itself asks only for the qualitative picture and the ratio $E_g/k_BT$ ; the exponential form is background intuition, not an examinable formula.

Ask the tutor: Want to see how doping adds its own energy levels inside this gap to turn a semiconductor into a working p-n junction — ask the AI tutor to walk through it next.

Why does the exclusion principle force levels to split into bands at all?

Because no two electrons in the same system can occupy an identical quantum state. When $10^{23}$ atoms bond together, each atom's originally identical level would, if left alone, want to place many electrons in exactly the same state — the exclusion principle forbids this, so the levels shift apart very slightly from one another, and with so many atoms interacting, those tiny shifts pack together into a near-continuous band rather than staying as separate lines.

Does a bigger energy gap always mean a worse conductor?

Yes, among materials with the same basic full-valence-band structure, a larger $E_g$ means exponentially fewer thermally excited carriers, so weaker intrinsic conduction — this is exactly the exponential relationship shown in the derivation above. The one case this does not apply to is a true conductor, where the bands overlap and there is no gap to speak of, so the comparison of gap sizes is meaningless there.

If silicon has a gap, why is doped silicon used in nearly every real electronic device instead of relying on this thermal excitation?

Because thermal excitation alone gives silicon only a very small number of carriers, as the worked example shows for germanium versus silicon — usable, but weak and strongly temperature-dependent. Doping deliberately adds impurity atoms that donate extra free electrons or extra holes, giving a much larger and more controllable carrier population than waiting on random thermal jumps across $E_g$ . This band picture is the foundation doping builds on, not the final answer for device-grade conductivity.

Energy bands in solids

The core idea

Worked example

Common traps

Before this

Where this leads