Energy Bands and Carrier Transport

Energy bands and carrier transport explain how semiconductors conduct — the physics beneath every diode, BJT and MOSFET. GATE EC tests drift/diffusion, mobility and conductivity (1–2 marks) and it underlies the whole Electronic Devices section (~9 marks).

Key results

• Bandgap E_g: Si ≈ 1.12 eV, Ge ≈ 0.66 eV, GaAs ≈ 1.42 eV (direct). Conductor: overlapping bands; insulator: large gap; semiconductor: moderate gap.
• Drift current: J_drift = q(nμ_n + pμ_p)E; conductivity σ = q(nμ_n + pμ_p).
• Diffusion current: J_diff = qD_n(dn/dx) − qD_p(dp/dx) (electrons diffuse down their gradient).
• Einstein relation: D/μ = kT/q = V_T ≈ 0.026 V at 300 K.
• Mobility falls with temperature (lattice scattering) and heavy doping (impurity scattering).

Quantity	Expression
Conductivity	σ = q(nμ_n + pμ_p)
Drift current density	J = σE
Einstein relation	D_n/μ_n = D_p/μ_p = kT/q
Thermal voltage V_T	kT/q ≈ 26 mV at 300 K
Si bandgap	≈ 1.12 eV

Exam Tricks & Tips

• 🎯 Einstein relation D/μ = kT/q = V_T ≈ 0.026 V at 300 K links diffusion and mobility — memorise 26 mV.
• 🎯 Drift ∝ E-field; diffusion ∝ concentration gradient. Both electron and hole currents add (same direction for conventional current).
• 🎯 Electron mobility > hole mobility in Si (~1350 vs ~480 cm²/V·s), so n-channel devices are faster.
• 🎯 GaAs is a DIRECT bandgap (good for LEDs/lasers); Si and Ge are INDIRECT (poor light emitters).
• 🎯 Conductivity rises with doping (more carriers) even though mobility drops slightly.
• ❌ Common mistake: forgetting that both carriers contribute to σ = q(nμ_n + pμ_p) — using only majority carriers.

Expected exam pattern

1–2 marks: compute conductivity/resistivity, diffusion coefficient via Einstein relation, or identify direct/indirect bandgap and its consequence.

Quick recap

σ = q(nμ_n+pμ_p); J_drift=σE, J_diff∝gradient. Einstein: D/μ=kT/q=V_T≈26 mV. Si E_g≈1.12 eV; GaAs direct, Si/Ge indirect. Electron mobility > hole mobility.

Cover the answer, recall, then check. 12 cards on energy bands and carrier transport for GATE EC.

Q1. State the conductivity of a semiconductor.
A1. σ = q(n·μ_n + p·μ_p), summing electron and hole contributions.

Q2. State the Einstein relation.
A2. D/μ = kT/q = V_T ≈ 0.026 V (26 mV) at 300 K.

Q3. What drives drift current versus diffusion current?
A3. Drift is driven by the electric field (J = σE); diffusion is driven by the carrier concentration gradient.

Q4. Give the bandgap of Si, Ge, and GaAs.
A4. Si ≈ 1.12 eV, Ge ≈ 0.66 eV, GaAs ≈ 1.42 eV.

Q5. Which of Si, Ge, GaAs has a direct bandgap and why does it matter?
A5. GaAs is direct → efficient light emission (LEDs/lasers); Si and Ge are indirect and are poor emitters.

Q6. How does mobility change with temperature (near room temp)?
A6. It decreases as temperature rises (increased lattice/phonon scattering).

Q7. How does mobility change with doping?
A7. It decreases at high doping due to ionized-impurity scattering.

Q8. Compare electron and hole mobility in silicon.
A8. Electron mobility (1350 cm²/V·s) > hole mobility (480 cm²/V·s), so n-channel devices are faster.

Q9. What is the thermal voltage V_T at 300 K?
A9. kT/q ≈ 25.9 mV ≈ 0.026 V.

Q10. Write the drift current density.
A10. J_drift = q(n·μ_n + p·μ_p)·E = σE.

Q11. Distinguish conductor, insulator, semiconductor by bands.
A11. Conductor: overlapping/partially filled band; insulator: large bandgap; semiconductor: moderate bandgap (~1 eV).

Q12. In which direction do electrons diffuse?
A12. From high to low concentration (down the gradient); the resulting conventional current is opposite to electron flow.

Everything a semiconductor does — conduct, switch, amplify — traces back to how electrons occupy energy bands and how they move when nudged by a field or a concentration gradient. This is the physical bedrock of every device in the syllabus. GATE mines it for carrier-transport calculations (drift, diffusion, mobility) and conceptual band-gap questions.

Core concept: Electrons in a solid occupy allowed energy bands separated by a forbidden gap; conduction needs carriers in the conduction band, and current flows by drift (field-driven) and diffusion (gradient-driven).

The theory

Beginner — the band picture

Isolated-atom levels broaden into bands in a crystal. The valence band (filled) and conduction band (empty at 0 K) are separated by the band gap Eg:

• Insulator: large Eg (>5 eV).
• Semiconductor: moderate Eg (Si ≈ 1.12 eV, Ge ≈ 0.66 eV, GaAs ≈ 1.42 eV at 300 K).
• Conductor: overlapping bands.
  Thermal energy promotes electrons across Eg, leaving holes in the valence band; both carry current.

Intermediate — drift and diffusion

Two transport mechanisms:

• Drift: an electric field E exerts force, giving drift velocity vd = μE (μ = mobility). Drift current density Jdrift = q(n·μn + p·μp)·E, so conductivity σ = q(n·μn + p·μp) and resistivity ρ = 1/σ.
• Diffusion: carriers flow down a concentration gradient. Jdiff = qDn·(dn/dx) − qDp·(dp/dx) (signs from carrier charge). Diffusion coefficient D links to mobility by the Einstein relation D/μ = kT/q = VT ≈ 25.85 mV at 300 K.

Advanced — mobility, velocity saturation, recombination

Mobility falls with temperature (lattice scattering, μ ∝ T^(−3/2)) and with doping (impurity scattering). At high fields drift velocity saturates (~10⁷ cm/s in Si) — Ohm's law breaks down. Excess carriers recombine with a characteristic lifetime τ, and diffusion length L = √(Dτ) sets how far minority carriers travel before recombining — a number that governs diode and BJT behaviour. The continuity equation combines drift, diffusion, generation, and recombination into one master transport equation.

Worked example

An n-type Si sample has n = 10¹⁶ cm⁻³, μn = 1350 cm²/V·s. Find its resistivity (ignore holes).
σ = q·n·μn = (1.6×10⁻¹⁹)(10¹⁶)(1350) = 1.6×1350×10⁻³ = 2.16 (Ω·cm)⁻¹.
ρ = 1/σ = 0.463 Ω·cm.
Diffusion coefficient by Einstein: Dn = μn·VT = 1350×0.02585 ≈ 34.9 cm²/s.

GATE relevance

1–2 mark NAT/MCQ: conductivity/resistivity from doping and mobility, Einstein relation (D from μ), diffusion length, band-gap dependence of intrinsic behaviour, and drift-vs-diffusion current. Foundational for PN junction and transistor problems.

Exam tricks and shortcuts

• VT = kT/q ≈ 26 mV at room temperature — the single most-used constant; Einstein D = μ·VT.
• σ = q(nμn + pμp); in strongly doped material one carrier dominates.
• Diffusion length L = √(Dτ); mobility drops as temperature or doping rises.
• Mnemonic: "Drift follows the Field, Diffusion follows the Gradient."

Key formulas

• Drift current density: J_drift = q(n μ_n + p μ_p)E; conductivity σ = q(nμ_n + pμ_p).
• Diffusion current: J_n = qD_n(dn/dx); J_p = −qD_p(dp/dx).
• Einstein relation: D/μ = kT/q = V_T ≈ 26 mV at 300 K.
• Resistivity: ρ = 1/σ.
• Mobility μ decreases with temperature and doping (scattering).

Current flows by drift (field-driven) and diffusion (gradient-driven); the Einstein relation links the two coefficients.

Worked Example

Problem: A silicon sample is uniformly doped n-type with donor concentration N_d = 10¹⁶ cm⁻³. The electron mobility is μₙ = 1350 cm²/(V·s). Assuming complete ionization, find the conductivity and the resistivity of the sample. (q = 1.6 × 10⁻¹⁹ C.)

Solution:
With complete ionization and an n-type sample far above intrinsic level, the free electron concentration equals the donor concentration:
n ≈ N_d = 10¹⁶ cm⁻³.
Hole conduction is negligible here (n ≫ p), so drift is dominated by electrons.

Conductivity (electron drift):
σ = q·n·μₙ
σ = (1.6 × 10⁻¹⁹ C)(10¹⁶ cm⁻³)(1350 cm²/V·s)
σ = 1.6 × 10⁻¹⁹ × 1.35 × 10¹⁹
σ = 1.6 × 1.35 = 2.16 (Ω·cm)⁻¹.

Resistivity is the reciprocal:
ρ = 1/σ = 1/2.16 ≈ 0.463 Ω·cm.

Answer: σ ≈ 2.16 (Ω·cm)⁻¹ and ρ ≈ 0.46 Ω·cm.