Semiconductor Physics: Carriers, Drift and Diffusion

Every electronic device is built on how electrons and holes behave in a semiconductor. This lesson covers the carrier physics that GATE EC tests directly.

Energy bands

• Band gap E_g: about 1.12 eV for Si, 0.66 eV for Ge, 1.42 eV for GaAs (at 300 K).
• Intrinsic semiconductor: n = p = n_i (about 1.5 × 10¹⁰ cm⁻³ for Si at 300 K; 10¹⁰ is often used in problems). n_i rises steeply with temperature: n_i² ∝ T³ e^(−E_g/kT).
• Extrinsic: donors (P, As) give an n-type material; acceptors (B) give p-type.
• Direct band gap (GaAs) materials emit light efficiently (LEDs, lasers); indirect (Si) do not.

Carrier concentrations

• Mass-action law: n p = n_i² (in equilibrium).
• n-type with N_D ≫ n_i: n ≈ N_D, p = n_i²/N_D.
• Fermi level position: n = n_i e^((E_F − E_i)/kT), so E_F − E_i = kT ln(n/n_i). In n-type material E_F moves towards the conduction band.
• Thermal voltage V_T = kT/q ≈ 25.9 mV at 300 K.

Current flow

• Drift: carriers pushed by a field: J = q(nμ_n + pμ_p)E.
• Conductivity σ = q(nμ_n + pμ_p); resistivity ρ = 1/σ.
• Mobility μ falls with doping (impurity scattering) and with temperature (lattice scattering). For Si, μ_n ≈ 1350 and μ_p ≈ 480 cm²/V·s (lightly doped), so electrons move faster.
• Diffusion: carriers move from high to low concentration: J_n = qD_n dn/dx, J_p = −qD_p dp/dx.
• Einstein relation: D/μ = kT/q = V_T.

Generation, recombination and the Hall effect

• Excess carriers decay with lifetime τ; diffusion length L = √(Dτ).
• Continuity equation combines drift, diffusion, generation and recombination.
• Hall effect: a current across a magnetic field creates a transverse voltage V_H = BI/(qnt). Its sign shows the carrier type; it measures concentration and mobility.

Worked example

Si doped with N_D = 10¹⁶ cm⁻³, μ_n = 1350 cm²/V·s, n_i = 10¹⁰ cm⁻³.

• n = 10¹⁶, p = 10²⁰/10¹⁶ = 10⁴ cm⁻³.
• σ = 1.6 × 10⁻¹⁹ × 10¹⁶ × 1350 = 2.16 S/cm; ρ = 0.463 Ω·cm.

Practice