C, Si and Ge have same lattice structure. Why is C insulator while Si and Ge intrinsic semiconductors?

Worked Examples · Example 14.1

Q. C, Si and Ge have same lattice structure. Why is C insulator while Si and Ge intrinsic semiconductors?

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Concept understanding — Band Gap Energy

What is Band Gap Energy? — A First Look

Imagine you have a single atom. Its electrons live in specific, fixed energy levels — like rungs on a ladder. You can't put an electron halfway between two rungs; it's either on one rung or another.

Now bring two atoms close together. Their electron rungs interact and split into two slightly different energies. Bring a billion atoms together — as in a solid crystal — and those original rungs spread into continuous bands of allowed energies, separated by gaps where no electron can exist.

That gap — the forbidden region between two bands — is the band gap.

The Intuition: A Wall Between Two Rooms

Think of the valence band as the ground floor of a building — electrons here are tightly bound to atoms, not free to move. The conduction band is the first floor above — electrons here can roam freely through the crystal, carrying current.

The band gap is the height of the ceiling between these two floors. An electron needs exactly that much energy to jump from the valence band to the conduction band. If you give it less energy, it stays stuck on the ground floor. If you give it exactly the gap energy or more, it can leap up and become a mobile charge carrier.

Note

In a metal, the valence and conduction bands overlap — there is no gap. That's why metals conduct electricity so easily: electrons already have free states available at no energy cost.

The Precise Statement

Band gap energy ( $E_g$ ) is the minimum energy required to excite an electron from the top of the valence band to the bottom of the conduction band in a solid.

E_g = E_{\text{conduction band minimum}} - E_{\text{valence band maximum}}

It is measured in electron volts (eV). One eV is the energy gained by an electron when accelerated through a potential difference of 1 volt — a tiny but convenient unit for atomic-scale energies.

Why Does It Matter?

The band gap determines almost everything about how a material behaves electrically and optically:

Material type	Typical $E_g$	Behaviour
Conductor (metal)	$E_g = 0$ (bands overlap)	Electrons flow freely at room temperature
Semiconductor	$0.1 \text{ eV} < E_g < 3 \text{ eV}$	Conducts only when given energy (heat, light)
Insulator	$E_g > 3 \text{ eV}$	Almost no conduction at normal conditions

Tip

A quick rule of thumb: if a material is transparent to visible light, its band gap is larger than about 3.1 eV (the energy of violet light). Diamond ( $E_g \approx 5.5$ eV) is transparent; silicon ( $E_g \approx 1.1$ eV) is opaque and shiny.

A Concrete Example: Silicon

Silicon has a band gap of 1.12 eV at room temperature. This means:

An electron in the valence band needs at least 1.12 eV to jump to the conduction band.
Visible light photons have energies between 1.8 eV (red) and 3.1 eV (violet). So silicon absorbs most visible light — that's why solar cells are dark. …

Why this formula?

Band Gap Energy: Why the Formula Holds

The band gap energy $E_g$ is the energy difference between the top of the valence band and the bottom of the conduction band in a solid. The key formula is:

E_g = E_c - E_v

where $E_c$ is the minimum energy of the conduction band and $E_v$ is the maximum energy of the valence band.

But why does this simple difference matter? The answer lies in how electrons behave in a crystal.

The Origin of Energy Bands

In an isolated atom, electrons occupy discrete energy levels. When atoms come together to form a solid, their atomic orbitals overlap. According to the Pauli exclusion principle, no two electrons can occupy the same quantum state. So the discrete levels split into a continuum of closely spaced levels — an energy band.

The valence band is formed from the outermost (valence) atomic orbitals. The conduction band is formed from the next higher set of orbitals (typically the empty orbitals above the valence orbitals). Between these bands lies the band gap — a region of forbidden energies where no electron states exist.

Why the Formula $E_g = E_c - E_v$ Is Not Trivial

You might think: "Of course the gap is the difference between the bottom of one band and the top of another." But the real insight is that $E_c$ and $E_v$ are not arbitrary points — they are the extrema of the band structure.

In a periodic crystal, the electron energy $E(\mathbf{k})$ depends on the wavevector $\mathbf{k}$ . The valence band has its maximum at some $\mathbf{k}$ -point (often at $\mathbf{k}=0$ for direct-gap semiconductors), and the conduction band has its minimum at some $\mathbf{k}$ -point. The band gap is:

E_g = \min_{\mathbf{k}_c} E_c(\mathbf{k}_c) - \max_{\mathbf{k}_v} E_v(\mathbf{k}_v)

This is not just a difference — it's a minimisation over all possible electron momenta.

Why This Difference Determines Conductivity

The band gap controls whether a material is an insulator, semiconductor, or conductor because of the Fermi-Dirac distribution:

f(E) = \frac{1}{1 + e^{(E - E_F)/k_B T}}

At absolute zero, all states below the Fermi level $E_F$ are filled, and all above are empty. For an intrinsic semiconductor, $E_F$ lies in the middle of the band gap. The probability that an electron is thermally excited from the valence band to the conduction band is proportional to $e^{-E_g/2k_B T}$ .

Important

The band gap energy $E_g$ appears in the exponent of the carrier concentration formula:

$n = p = N_c N_v e^{-E_g/2k_B T}$

This is why a small change in $E_g$ causes a huge change in conductivity — it's an exponential dependence.

The Physical Meaning of $E_g$

The band gap is not just a number — it's the minimum energy required to:

Break a covalent bond in the crystal (creating an electron-hole pair)
Promote an electron from a bonding state to an antibonding state
Create a mobile charge carrier

For example, in silicon ( $E_g = 1.12$ eV at 300 K), a photon with energy greater than 1.12 eV can be absorbed, exciting an electron from the valence band to the conduction band. This is why silicon is used in solar cells — the band gap matches the solar spectrum. …

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Q.C, Si and Ge have same lattice structure. Why is C insulator while Si and Ge intrinsic semiconductors?