Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are se

NCERT Exemplar · Q13

Q. Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are semiconductors. Why?

Andhra Pradesh BieapShort· 2mImportance★★★★★

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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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The electrical behaviour of group XIV elements (C, Si, Ge, Sn) is determined by the size of their band gap — the energy difference between the valence and conduction bands. Diamond (C) has a very large band gap ( $\approx 5.5\ \text{eV}$ ), making it an insulator. Si ( $\approx 1.1\ \text{eV}$ ) and Ge ( $\approx 0.7\ \text{eV}$ ) have moderate band gaps, making them semiconductors. Sn (grey tin) has a nearly zero or overlapping band gap, behaving as a conductor.

The key lies in how the band gap energy changes as you go down group XIV. Let’s understand why.

The Concept: Band Gap and Bonding

In a solid, atomic orbitals overlap to form energy bands. The valence band is filled with electrons; the conduction band is empty. The band gap ( $E_g$ ) is the energy an electron needs to jump from the valence band to the conduction band — to become free to conduct electricity.

For group XIV elements, each atom has four valence electrons. In the solid, these form covalent bonds. The strength of these bonds determines how wide the band gap is.

Bond strength and atomic size: As you go down the group (C → Si → Ge → Sn), atomic size increases. Larger atoms have valence electrons that are farther from the nucleus and more loosely held. This means the covalent bonds between atoms become weaker.
Weaker bonds → smaller band gap: A weaker bond means the energy required to break an electron free (to promote it to the conduction band) is smaller. So, the band gap decreases down the group.
The critical threshold: For electrical conduction, thermal energy at room temperature ( $k_B T \approx 0.025\ \text{eV}$ ) can promote a small number of electrons if the band gap is small enough (a few eV). If the band gap is huge, almost no electrons can jump — the material is an insulator. If the band gap is zero or negative (bands overlap), electrons flow freely — the material is a conductor.

E_g(\text{C}) \approx 5.5\ \text{eV},\quad E_g(\text{Si}) \approx 1.1\ \text{eV},\quad E_g(\text{Ge}) \approx 0.7\ \text{eV},\quad E_g(\text{Sn}) \approx 0\ \text{eV}

Now, let’s apply this to each element.

Step-by-Step Reasoning

Carbon (Diamond): Carbon is the smallest atom in the group. Its four valence electrons form extremely strong, directional covalent bonds (the diamond structure). This gives a very large band gap of about $5.5\ \text{eV}$ . At room temperature, thermal energy is far too small to excite electrons across this gap. Hence, diamond has almost no free electrons and is an insulator. …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

AP EAPCET 2026Set eng-2026-05-13-FN1 markMCQ
$Q. The minimum energy gap of a semi-conductor used in manufacturing LED is (A) 3 eV (B) 3.6 eV (C) 2.8 eV (D) 1.8 eV$
›Reveal solutionSolution
This tests the standard semiconductor-electronics fact (NCERT) that LED materials need $E_g \geq 1.8$ eV to emit visible light, since $E_g = hc/\lambda$ and the visible spectrum's red edge ( $\sim700$ nm) sets this lower bound.

Concept and Intuition

In an LED, forward biasing causes electron-hole recombination at the junction, and each recombination releases a photon whose energy roughly equals the semiconductor's band gap $E_g$ . For the emitted light to be visible (400–700 nm), the photon energy must be at least as large as the energy corresponding to the longest visible wavelength (red, $\sim700$ nm), since longer wavelength = lower photon energy. This is why ordinary Si or Ge junctions (small gaps) emit only infrared, while LED materials (GaAs, GaP, GaAsP etc.) are chosen with larger gaps.

Step-by-Step Solution

Photon energy and wavelength are related by $E = \dfrac{hc}{\lambda}$ .

The longest (least energetic) visible wavelength is about $\lambda \approx 700$ nm $=700\times10^{-9}$ m.

Compute the corresponding energy: $E = \dfrac{(6.63\times10^{-34})(3\times10^8)}{700\times10^{-9}} \approx 2.84\times10^{-19}$ J $\approx 1.77$ eV. …
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AP EAPCET 2025Set eng-2025-05-22-FN1 markMCQ
$Q. A cc camera is fabricated using a semiconducting material having a band gap of 3 eV. The wavelength of light it can detect is nearly (A) 210 nm (B) 546 nm (C) 413 nm (D) 345 nm$
›Reveal solutionSolution
Tests the photon-energy/band-gap relation for a photodetector; the threshold wavelength is $\lambda = hc/E_g$ .

Concept and Intuition

A semiconductor detector (like a CCD) can only detect photons whose energy is at least equal to the band gap $E_g$ , since an electron must be excited from the valence band to the conduction band. The longest wavelength (lowest energy) it can detect is the threshold wavelength corresponding exactly to $E_g$ .

Step-by-Step Solution

Use the convenient relation $E(\text{eV}) = \dfrac{1240 \text{ eV·nm}}{\lambda(\text{nm})}$ , derived from $E = hc/\lambda$ with $hc \approx 1240$ eV·nm.

Given $E_g = 3$ eV, solve for $\lambda$ : $\lambda = \dfrac{1240}{3} = 413.3$ nm.

This matches option (C), 413 nm.

Common Mistakes …
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AP EAPCET 2024Set ap-2024-05-16-FN1 markMCQ
$Q. Semiconductors suitable for solar cell fabrication have a band gap nearby (A) 0.015 eV (B) 1.5 eV (C) 15 eV (D) 150 eV$
›Reveal solutionSolution
Solar cells are made from semiconductors whose band gap is close to 1.5 eV, matched to the peak of the solar spectrum's usable photon energies.

Concept and Intuition

A semiconductor absorbs a photon and creates an electron-hole pair only if the photon's energy exceeds the band gap $E_g$ . Too small a gap wastes photon energy as heat; too large a gap means many solar photons don't have enough energy to be absorbed at all. The band gaps of practical solar-cell materials (silicon ~1.1 eV, GaAs ~1.4 eV) cluster near the value that gives the best trade-off, commonly quoted in textbooks as approximately 1.5 eV.

Step-by-Step Solution

Recall that a good photovoltaic material needs $E_g$ neither too small (excess thermalisation losses) nor too large (many photons below threshold, unabsorbed).

Real solar-cell semiconductors (Si, GaAs) have band gaps in the 1–1.5 eV range. …
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AP EAPCET 2024Set eng-2024-05-20-AN1 markMCQ
$Q. If the energy gap of a substance is 5.4 eV, then the substance is (A) Insulator (B) Conductor (C) p-type semiconductor (D) n-type semiconductor$
›Reveal solutionSolution
A band gap as large as 5.4 eV is far too wide for thermal excitation across it at room temperature — this is the signature of an insulator, not a semiconductor or conductor.

Concept and Intuition

In band theory, electrical conduction depends on how easily electrons can be promoted from the filled valence band to the empty conduction band. In conductors, the bands overlap (or the conduction band is partially filled), so no gap needs to be crossed. In semiconductors, a small gap (roughly under 3 eV) allows some thermal or doping-assisted excitation, giving moderate conductivity. In insulators, the gap is large (several eV), so essentially no electrons can be thermally promoted across it at ordinary temperatures, and the material does not conduct.

Step-by-Step Solution

Note the given band gap: $E_g = 5.4$ eV.

Compare against typical benchmarks: semiconductors like Si (1.1 eV) and Ge (0.7 eV) have gaps well under 3 eV; classic insulators like diamond have gaps around 5.4–5.5 eV. …
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AP EAPCET 2024Set eng-2024-05-21-FN1 markMCQ
$Q. The semiconductor used for fabrication of visible LEDs must at least have a band gap of (A) 0.6 eV (B) 1.2 eV (C) 1.8 eV (D) 0.9 eV$
›Reveal solutionSolution
The band gap sets the photon energy (and hence colour) an LED emits; to reach even the lowest-energy visible colour (red, ~700 nm), the gap must be at least about 1.8 eV.

Concept and Intuition

An LED emits photons with energy approximately equal to its semiconductor's band gap ( $E_g\approx hc/\lambda$ ). Visible light ranges from about 700 nm (red, lowest photon energy) to 400 nm (violet, highest photon energy). A material with too small a band gap would only emit infrared, not visible light — so for any visible-light LED to be possible at all, the minimum acceptable band gap corresponds to the least energetic visible colour, red.

Step-by-Step Solution

Longest visible wavelength (least energetic, sets the floor): $\lambda\approx700$ nm.

Photon energy: $E=\dfrac{hc}{\lambda}=\dfrac{1240\ \text{eV·nm}}{700\ \text{nm}}\approx1.77$ eV $\approx1.8$ eV. …
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AP EAPCET 2023Set eng-2023-05-15-AN1 markMCQ
$Q. The material used in the fabrication of infrared LED's is (A) silicon (B) germanium (C) gallium arsenide phospide (D) carbon dioxide$
›Reveal solutionSolution
This tests which semiconductor material family is used to fabricate LEDs; compound semiconductors like gallium arsenide phosphide are used, not elemental Si/Ge.

Concept and Intuition

An LED needs a direct band-gap semiconductor so that when an electron recombines with a hole across the junction, the energy is released efficiently as a photon rather than being lost as heat (phonons). Silicon and germanium are elemental semiconductors with an indirect band gap, so they recombine radiation very inefficiently and are not used for light-emission (they are used for diodes/transistors, not LEDs). Compound semiconductors formed from group III and group V elements — such as gallium arsenide (GaAs) and its phosphide-doped variants (GaAsP) — have direct band gaps and can be tuned by varying composition to emit anywhere from infrared through visible colours.

Step-by-Step Solution

Rule out silicon and germanium: both are elemental, indirect-band-gap semiconductors, used for diodes/transistors but not efficient light emitters.

Rule out carbon dioxide: this is a gas, not a semiconductor, and plays no role in LED fabrication. …
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AP EAPCET 2022Set eng-2022-07-06-FN1 markMCQ
$Q. The energy of a photon in a monochromatic light of wavelength 621 nm matches with the band gap of a semiconducting material. Then the minimum energy required to create an electron-hole pair from the semiconductor is [Take hc = 1242 eV - nm, where h is Planck's constant and c is speed of light in vacuum] (A) 3.4 eV (B) 1.7 eV (C) 2 eV (D) 2.2 eV$
›Reveal solutionSolution
A straightforward $E=hc/\lambda$ calculation, using the given $hc=1242$ eV·nm shortcut, gives exactly 2 eV — and since the problem tells us this photon energy matches the band gap, that band-gap energy is the minimum energy needed to create an electron-hole pair.

Concept and Intuition

In a semiconductor, an electron can be excited from the valence band to the conduction band (creating an electron-hole pair) only if it absorbs at least the band-gap energy $E_g$ . A photon with exactly this energy is the threshold case — the minimum energy photon capable of creating a pair. Since the problem states the photon's energy equals the band gap, computing the photon energy directly gives us $E_g$ , the minimum pair-creation energy.

Step-by-Step Solution

Photon energy: $E = \dfrac{hc}{\lambda}$ , using the convenient constant $hc = 1242$ eV·nm.

$E = \dfrac{1242}{621} = 2$ eV. …
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AP EAPCET 2022Set eng-2022-07-08-AN1 markMCQ
$Q. The class of materials having the largest band gap in the following is (A) Metals (B) Semi-metals (C) Semi-Conductors (D) Insulators$
›Reveal solutionSolution
Band gap size increases from metals (essentially zero) through semiconductors to insulators, which have the largest band gap — that is exactly what makes them insulators.

Concept and Intuition

In band theory of solids, electrical conductivity is governed by the energy gap between the filled valence band and the empty conduction band. Metals have overlapping or touching bands (zero or negligible gap), so electrons move freely — high conductivity. Semiconductors have a small but nonzero gap (~1 eV for Si/Ge), so only a few electrons are thermally promoted across it at room temperature. Insulators have a very large band gap (several eV, e.g., diamond ~5.5 eV), so essentially no electrons can be thermally excited into the conduction band — this large gap is the defining reason they don't conduct.

Step-by-Step Solution

Compare typical band gaps: metals ≈ 0 eV, semiconductors ≈ 0.7–1.5 eV, insulators ≈ several eV (often > 3 eV). …
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Q.Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are semiconductors. Why?

Concept understanding — Band Gap Energy

What is Band Gap Energy? — A First Look

The Intuition: A Wall Between Two Rooms

The Precise Statement

Why Does It Matter?

A Concrete Example: Silicon

Why this formula?

Band Gap Energy: Why the Formula Holds

The Origin of Energy Bands

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

Why This Difference Determines Conductivity

The Physical Meaning of $E_g$

The Concept: Band Gap and Bonding

Step-by-Step Reasoning

Method: Tracking How Bond Strength (and Hence Band Gap) Changes Down a Group

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Common Mistakes …

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Solution

Q.Sn, C, and Si, Ge are all group XIV elements. Yet, Sn is a conductor, C is an insulator while Si and Ge are semiconductors. Why?

The Concept: Band Gap and Bonding

Step-by-Step Reasoning

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