Electrical Properties and Band Theory

• Explain band formation from atomic orbitals using MO/band theory
• Classify conductors, semiconductors, and insulators by band gap
• Describe intrinsic semiconductors and how temperature affects conductivity
• Explain n-type and p-type doping with correct group 14/15/13 logic
• Connect semiconductor properties to LED, solar cell, and rectifier applications
• Briefly explain diamagnetic, paramagnetic, and ferromagnetic behaviour

Level 1 — Foundations

Band Theory Origin
N atoms combine → N molecular orbitals → these are so closely spaced they form a continuous "band."

• Valence Band (VB): Highest filled energy band (contains bonding electrons)
• Conduction Band (CB): Next higher band (normally empty or partially filled)
• Band Gap (Eg): Energy gap between top of VB and bottom of CB

Three Classes of Solids

Intrinsic Semiconductor
Pure Si or Ge at room temperature: a few electrons have enough thermal energy to jump from VB to CB, leaving holes in VB. Both electrons (in CB) and holes (in VB) conduct. Number of electrons = number of holes (intrinsic carrier concentration).

Level 2 — JEE Depth

Why Metal Conductivity Decreases with Temperature
In metals, VB and CB overlap → plenty of free electrons at any T. As T rises, lattice vibrations increase → more electron-phonon scattering → resistivity increases → conductivity decreases.

Why Semiconductor Conductivity Increases with Temperature
In Si or Ge, more thermal energy → more electrons jump the band gap → more carriers → conductivity increases with T. This is opposite to metals — a classic JEE question.

Band Gap and Colour
Eg determines minimum photon energy absorbed:
E_photon = hf = hc/λ ≥ Eg
For Si (Eg = 1.1 eV = 1.76×10⁻¹⁹ J): λ_max = hc/Eg = 1130 nm (near-infrared → Si absorbs IR, useful for solar cells)
For diamond (Eg = 5.4 eV): λ_max = 230 nm (UV) → transparent to visible light → colourless

Extrinsic Semiconductors (Doping)

n-type (donor doping):

• Si (Group 14) doped with P, As, or Sb (Group 15)
• Dopant has 1 extra valence electron that cannot fit into Si's 4-bond structure
• Extra electron occupies a donor level just below CB → easily thermally excited to CB
• Majority carriers: electrons (negative)
• Minority carriers: holes

p-type (acceptor doping):

• Si doped with B, Al, or Ga (Group 13)
• Dopant has 1 fewer valence electron → creates an empty orbital = hole
• Hole level (acceptor level) just above VB → electrons from VB easily excited into it → holes in VB
• Majority carriers: holes (positive)
• Minority carriers: electrons

Devices (Brief)

p-n Junction Rectifier:
At junction, electrons from n-side recombine with holes from p-side → depletion region. Forward bias: reduces barrier, allows current. Reverse bias: widens barrier, blocks current. Converts AC to DC.

LED (Light Emitting Diode):
Forward-biased p-n junction: electrons from CB recombine with holes in VB, releasing energy as photon.
E_photon = Eg (for direct-gap semiconductors like GaAs, GaN).
Photon wavelength λ = hc/Eg determines LED colour:

• GaN (Eg ≈ 3.4 eV): blue/UV LEDs
• GaAs (Eg ≈ 1.4 eV): infrared LEDs
• GaP (Eg ≈ 2.3 eV): green LEDs

Solar Cell:
Photon absorbed → electron-hole pair generated → p-n junction built-in field separates them → external current flows.

Magnetic Properties (Brief)

JEE Traps

• Silicon's conductivity INCREASES with T (more carriers); copper's DECREASES (more scattering)
• n-type is doped with Group 15 (more electrons), p-type with Group 13 (fewer electrons = holes) — remember by number
• In n-type, electrons are majority carriers but the semiconductor itself is electrically neutral overall
• Ge has smaller band gap than Si → Ge has higher intrinsic conductivity at room temperature

Example 1: Why Silicon's Conductivity Increases with Temperature

Question: Explain using band theory why Si conductivity increases with T,
unlike copper where conductivity decreases with T.

Silicon (semiconductor, Eg = 1.1 eV):
  At 0 K: VB completely full, CB completely empty → no conduction
  At 300 K: thermal energy kT ≈ 0.026 eV
    A fraction of electrons have energy fluctuations >> kT (Maxwell-Boltzmann tail)
    Some electrons cross the 1.1 eV gap → free to conduct in CB
    Each electron leaves a hole in VB → hole also conducts
  At 400 K: more electrons thermally excited → more carriers → higher conductivity

  Equation: n_i ∝ e^(-Eg/2kT)
  As T increases → exponent less negative → n_i increases exponentially

Copper (conductor, overlapping bands):
  At 300 K: ~10²⁸ free electrons/m³ (always plenty; no gap to cross)
  As T increases: lattice atoms vibrate more vigorously
  → More electron-lattice collisions (scattering)
  → Mean free path of electrons decreases
  → Resistivity ρ ∝ T → conductivity σ = 1/ρ decreases

Conclusion: Semiconductors are carrier-limited at low T (conductivity increases with T);
metals are scattering-limited (conductivity decreases with T).

Example 2: n-type Doping of Germanium with Arsenic

Question: Ge doped with As — draw energy band and identify carrier type

Pure Ge: Eg = 0.7 eV, VB full, CB empty at 0 K

Doping with As (Group 15, 5 valence electrons):
  As replaces Ge in lattice, 4 electrons form covalent bonds with neighbours
  5th electron is NOT part of the bonding structure
  This extra electron is loosely held (ionisation energy only ~0.01 eV in Ge)

Energy band diagram:
  _______________  CB (conduction band, empty)
       ×              ← donor level (As⁺ + e⁻), just 0.01 eV below CB
  _______________  
  
  Eg = 0.7 eV     ← band gap
  
  _______________  VB (valence band, full)

At room temperature (kT = 0.026 eV >> 0.01 eV):
  ALL donor electrons easily thermally excited into CB
  → Each As atom contributes one free electron to CB

Result:
  Carrier type: n-type (majority carriers = electrons)
  Conductivity >> intrinsic Ge at same T
  Material is electrically neutral overall (As⁺ fixed in lattice)

Answer: n-type semiconductor with electrons as majority carriers,
holes as minority carriers. Band diagram shows donor level just below CB.

• Q1: What is the band gap of silicon, and what type of photon can it absorb?
• Q2: Germanium is doped with gallium. Is it n-type or p-type? What are the majority carriers?
• Q3: Why does a copper wire get more resistive when heated, but a silicon crystal becomes more conductive?
• Q4: In a p-n junction LED made from GaN (Eg = 3.4 eV), what colour light is emitted? (h = 6.63×10⁻³⁴ J·s, c = 3×10⁸ m/s)
• Stretch: Q5: Two semiconductor samples A (Eg = 0.3 eV) and B (Eg = 1.5 eV) are at 300 K. Which has more intrinsic carriers? Which is more useful as a solar cell? Explain why a large Eg is not ideal for solar cells even though it means a cleaner switch-off.

NCERT Chapter 1 link: Section 1.10 "Electrical Properties" and brief mention in 1.11 "Magnetic Properties." Band theory is introduced conceptually; doping types and device applications are covered. The conductor/semiconductor/insulator table is directly in NCERT.

Exam connections: JEE Mains: identify n-type/p-type from dopant, band gap comparison, temperature effect on conductivity. JEE Advanced: LED wavelength calculation from Eg, solar cell energy conversion efficiency logic, magnetic property classification of transition metal compounds using d-electron configuration.

Study strategy: The key conceptual pair: "metals — conductivity falls with T; semiconductors — conductivity rises with T." Know the Group 13/15 doping rule cold. Practise LED wavelength from band gap as a one-step E = hc/λ conversion.

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