Classification of Solids: Conductivity & Energy Bands

From Vacuum Tubes to Solid State

Every electronic circuit is built from devices that give a controlled flow of electrons. Before the transistor (1948), that meant vacuum tubes (valves): the vacuum diode (anode/plate + cathode), triode (+ grid), tetrode and pentode. A heated cathode supplied electrons; voltages between electrodes controlled their flow through vacuum (needed so electrons don't lose energy colliding with air molecules). Current flows only cathode → anode — one way, hence 'valves'.

Their problems: bulky, high power consumption, high voltages (~100 V), limited life, low reliability.

The 1930s realisation: some solid-state semiconductors and their junctions let you control the number and direction of charge carriers — with light, heat or small voltages changing the number of mobile charges. The advantages invert every vacuum-tube weakness:

Vacuum tubes	Semiconductor devices
electrons from heated cathode	charge carriers generated within the solid
evacuated inter-electrode space	no vacuum needed
bulky, ~100 V, power-hungry	small, low voltage, low power
short life, unreliable	long life, high reliability

(Historical gem: a natural galena (PbS) crystal with a metal point contact served as a radio-wave detector before semiconductor physics was even understood.)

Classification I: By Conductivity

On the basis of resistivity $\rho$ (or conductivity $\sigma = 1/\rho$):

Class	Resistivity $\rho$ (ohm m)	Conductivity $\sigma$ (S/m)
Metals	$10^{-2}$ - $10^{-8}$	$10^{2}$ - $10^{8}$
Semiconductors	$10^{-5}$ - $10^{6}$	$10^{5}$ - $10^{-6}$
Insulators	$10^{11}$ - $10^{19}$	$10^{-11}$ - $10^{-19}$

(The values are indicative; resistivity alone is not the full story — the band picture below completes it.)

The semiconductor family

• Elemental: Si and Ge — the workhorses.
• Compound inorganic: CdS, GaAs, CdSe, InP…
• Organic: anthracene, doped phthalocyanines; organic polymers: polypyrrole, polyaniline, polythiophene — the seeds of polymer/molecular electronics (post-1990).

This chapter concentrates on Si and Ge; the concepts transfer to compound semiconductors.

[NEET Important] The resistivity ranges are direct one-markers. Anchor: metals below $10^{-2}$, insulators above $10^{11}$, semiconductors in the vast middle.

Classification II: By Energy Bands

In an isolated atom, electron energies are sharp Bohr levels. In a solid, atoms sit so close that outer orbitals overlap; each of the ~$10^{23}$ electrons sees a slightly different charge environment and takes a slightly different energy. The levels smear into energy bands:

• Valence band (VB): the band of valence-electron energies — highest level $E_V$.
• Conduction band (CB): the band above — lowest level $E_C$.
• Energy band gap: $E_g = E_C - E_V$, which may be large, small or zero.

The Si/Ge counting argument

A crystal of N atoms of Si (outer orbit n = 3) or Ge (n = 4) has 4N valence electrons but 8N available outer states (2s + 6p). At crystal spacing, these 8N states split into two bands of 4N states each, separated by $E_g$: at absolute zero the lower (valence) band is completely full, the upper (conduction) band completely empty.

The three cases (NCERT Fig. 14.2)

• Metals: CB and VB overlap (or the conduction band is partially filled) — electrons move freely; huge conductivity.
• Insulators: large gap, $E_g$ > 3 eV — thermal excitation cannot lift electrons across; CB stays empty; no conduction.
• Semiconductors: small gap, $E_g$ < 3 eV — at room temperature some electrons cross into the CB (leaving useful vacancies behind); modest conductivity.

The group-IV ladder — memorise

Element	$E_g$	Verdict
C (diamond)	5.4 eV	insulator
Si	1.1 eV	semiconductor
Ge	0.7 eV	semiconductor
Sn	0 eV	metal

Same column of the periodic table, four different electrical fates — the band gap decides everything.

[JEE Tip] Band edges $E_C$, $E_V$ drawn as lines are space-averaged levels (Points to Ponder) — bottom-of-CB and top-of-VB, not locations in the crystal. And the ordering $(E_g)_C > (E_g)_{Si} > (E_g)_{Ge}$ is NCERT Exercise 14.3's answer — asked constantly.

Solved Examples

Example 1: Why is C insulating but Si and Ge semiconducting? (NCERT Example 14.1)

Carbon, silicon and germanium share the same lattice structure. Explain the different behaviour.

Solution:

1. The four bonding electrons sit in the 2nd orbit (C), 3rd (Si), 4th (Ge).
2. The farther the orbit, the smaller the energy needed to free an electron: ionisation is hardest for C, easier for Si, easiest for Ge — mirrored in $E_g$: 5.4, 1.1, 0.7 eV.
3. So free-electron numbers at room temperature are negligible in C but significant in Si and Ge.
4. Takeaway: identical structure, different orbit radius → different band gap → insulator vs semiconductor.

Example 2: Classify by resistivity [Board Rapid]

Materials A, B, C have resistivities $10^{-6}$, $10^{2}$ and $10^{14}$ ohm m. Classify them.

Solution:

1. A: $10^{-6}$ ohm m — within $10^{-2}$-$10^{-8}$: metal.
2. B: $10^{2}$ — within $10^{-5}$-$10^{6}$: semiconductor.
3. C: $10^{14}$ — within $10^{11}$-$10^{19}$: insulator.
4. Takeaway: slot the exponent into the three ranges; boundaries are approximate but exam values sit safely inside.

Example 3: The 4N/8N band split [Board Conceptual]

A silicon crystal has N atoms. Account for the filling of its two outer bands at absolute zero.

Solution:

1. Each Si atom offers 4 valence electrons → 4N electrons; the outer orbit could hold 8 per atom → 8N states.
2. At crystal spacing the 8N states split into two bands of 4N states each, gap $E_g$ between.
3. At 0 K the 4N electrons exactly fill the lower band: valence band full, conduction band empty.
4. Takeaway: a full band conducts nothing (no empty states to move into) — which is why pure Si at 0 K is an insulator.

Example 4: Metal without overlap [Conceptual]

NCERT says a metal arises with overlapping bands OR a partially filled conduction band. Why does partial filling alone guarantee conduction?

Solution:

1. Conduction requires electrons to gain energy from a field, i.e. to move into nearby empty states.
2. In a partially filled band, empty levels sit immediately above filled ones — electrons accelerate freely.
3. In a full band there is nowhere to go (Pauli); in an overlap, VB electrons spill into CB states — same effect.
4. Takeaway: 'empty states adjacent to filled states' is the true criterion for metallic conduction.

Example 5: Thermal test of an insulator [NEET Numerical]

Estimate why room-temperature thermal energy cannot make diamond conduct. (kT at 300 K ≈ 0.026 eV.)

Solution:

1. Diamond's gap: $E_g$ = 5.4 eV; thermal quantum: ~0.026 eV.
2. Ratio: $\dfrac{5.4}{0.026} \approx 208$ — excitation probability ~$e^{-E_g/kT}$ is vanishingly small.
3. Conclusion: effectively zero electrons reach the CB — insulator. For Si (1.1 eV ≈ 42 kT) the exponential is tiny but nonzero: a weak semiconductor.
4. Takeaway: conduction is an exponential game in $E_g/kT$ — small gap changes make astronomical carrier differences.

Example 6: Photon threshold of a band gap [JEE Numerical]

What maximum wavelength of light can excite an electron across silicon's 1.1 eV gap?

Solution:

1. Condition: photon energy ≥ $E_g$: $\dfrac{1240}{\lambda(\text{nm})} \geq 1.1$.
2. Solve: $\lambda \leq \dfrac{1240}{1.1} \approx 1127$ nm.
3. Answer: ~1100 nm (near infrared) — silicon absorbs visible light easily, which is why it makes solar cells.
4. Takeaway: $\lambda_{max} = 1240/E_g$ nm — the bridge between this chapter and photon physics; appears in photodiode/LED problems too.

Example 7: Sn, the group-IV metal [Conceptual]

Tin sits below germanium in group IV yet is a metal. Reconcile this with the band picture.

Solution:

1. Down the group the valence orbit moves outward: C(2) → Si(3) → Ge(4) → Sn(5).
2. The band gap shrinks correspondingly: 5.4 → 1.1 → 0.7 → 0 eV.
3. Zero gap means valence and conduction bands touch/overlap — Sn conducts like a metal.
4. Takeaway: one periodic-table column spans insulator → semiconductors → metal purely through $E_g$.

Example 8: Vacuum-tube vs semiconductor checklist [Board Rapid]

List four advantages of semiconductor devices over vacuum tubes.

Solution:

1. Small size and no evacuated bulb or heated cathode (carriers generated within the solid).
2. Low operating voltage and low power consumption (tubes needed ~100 V).
3. Long life and high reliability.
4. Takeaway: NCERT's contrast list verbatim — plus the flexibility that light/heat/small voltages can control carrier numbers, impossible in a vacuum tube.

Example 9: Ordering the gaps (NCERT Exercise 14.3)

C, Si and Ge each have four valence electrons, with gaps $(E_g)_C$, $(E_g)_{Si}$, $(E_g)_{Ge}$. Which ordering is true?

Solution:

1. Values: C 5.4 eV, Si 1.1 eV, Ge 0.7 eV.
2. Ordering: $(E_g)_C > (E_g)_{Si} > (E_g)_{Ge}$.
3. Physical reason: bonding electrons in the 2nd, 3rd, 4th orbits respectively — the more distant, the easier to liberate.
4. Takeaway: the exact NCERT exercise answer, recycled endlessly by Boards and NEET.

Example 10: Where do organic semiconductors fit? [Conceptual]

Name the categories of semiconductors with examples, and the technology signalled by polymer semiconductors.

Solution:

1. Elemental: Si, Ge.
2. Compound inorganic: CdS, GaAs, CdSe, InP.
3. Organic: anthracene, doped phthalocyanines; organic polymers: polypyrrole, polyaniline, polythiophene.
4. Post-1990 devices from these signal polymer electronics and molecular electronics — NCERT's phrase for the flexible-display future.
5. Takeaway: the classification list is a straight recall question; GaAs is the most-quoted compound example.