Intrinsic Semiconductors

An intrinsic semiconductor is a perfectly pure crystal of silicon or germanium with no added impurities. Every electron-hole pair in a 1N4007 diode near the depletion edge is generated by the same thermal excitation mechanism that governs intrinsic material.

Core Concept

In a pure silicon crystal at absolute zero, all four valence electrons of every atom are locked in covalent bonds and no current can flow. At room temperature (300K), thermal energy breaks some of these bonds. Each broken bond frees one electron to the conduction band and leaves behind a hole in the valence band. This process is called thermal generation.

The freed electron and hole can each contribute to current flow. Electrons drift toward the positive terminal of an applied field; holes drift toward the negative terminal. In intrinsic silicon, the electron concentration n equals the hole concentration p, and both equal the intrinsic carrier concentration ni. For silicon at 300K, ni = 1.5 × 10^10 per cm³.

Thermal generation is balanced by recombination, where a free electron falls back into a hole and releases energy (as heat in silicon, or as a photon in GaAs). At equilibrium, the generation rate equals the recombination rate, which fixes ni at a given temperature. Raising the temperature sharply increases ni because more bonds break. This is why reverse leakage current in a 1N4007 doubles for every 10°C rise.

Key Equations

Intrinsic carrier concentration: ni = sqrt(Nc * Nv) * exp(-Eg / (2*k*T))

where Nc and Nv are the effective density of states in the conduction and valence bands respectively, Eg is the bandgap energy, k is Boltzmann's constant, and T is the absolute temperature in Kelvin.

Mass action law: n * p = ni^2 (holds at thermal equilibrium, for both intrinsic and doped material).

For intrinsic material: n = p = ni

Standard values: ni(Si, 300K) = 1.5e10 /cm³ and ni(Ge, 300K) = 2.4e13 /cm³

Conductivity: σ = q * ni * (µn + µp) where µn = 1350 cm²/V·s and µp = 480 cm²/V·s for silicon at 300K.

Given:
  Intrinsic silicon at 300K
  ni   = 1.5e10 /cm³
  µn   = 1350 cm²/V·s
  µp   = 480  cm²/V·s
  q    = 1.6e-19 C
  Find: Intrinsic conductivity σ and resistivity ρ

Why this formula:
  Both electrons and holes carry current; σ = q*ni*(µn + µp).

Formula:
  σ = q * ni * (µn + µp)
  ρ = 1 / σ

Substitution:
  σ = 1.6e-19 * 1.5e10 * (1350 + 480)
    = 1.6e-19 * 1.5e10 * 1830

Calculation:
  Step 1: 1.6e-19 * 1.5e10 = 2.4e-9
  Step 2: 2.4e-9 * 1830 = 4.392e-6 S/cm
  σ = 4.39e-6 S/cm

  ρ = 1 / 4.39e-6
    = 2.28e5 Ω·cm
    = 228,000 Ω·cm

Final Answer:
  σ = 4.39 × 10^-6 S/cm
  ρ = 2.28 × 10^5 Ω·cm  (about 2280 Ω·m)
  This confirms silicon is a poor conductor at room temperature.

Exam Tip: The mass action law n*p = ni^2 is one of the most tested relations in semiconductor GATE problems. It applies to BOTH intrinsic AND doped semiconductors at thermal equilibrium. If a problem gives you the donor concentration Nd and asks for minority carrier concentration p, use p = ni^2 / Nd. Students who forget this law and assume p = ni in doped material get the answer wrong by many orders of magnitude.

Key Properties

• In intrinsic silicon at 300K, n = p = ni = 1.5 × 10^10 per cm³. For germanium, ni = 2.4 × 10^13 per cm³.
• The mass action law n*p = ni^2 holds at thermal equilibrium for any semiconductor, intrinsic or doped.
• Electron mobility in silicon (µn = 1350 cm²/V·s) is about 2.8 times higher than hole mobility (µp = 480 cm²/V·s), so electrons carry more current per unit field.
• Intrinsic carrier concentration ni increases exponentially with temperature, approximately doubling every 10°C rise in silicon near room temperature.
• The Fermi level in intrinsic silicon (Ei) lies very close to the middle of the bandgap, slightly below midgap because Nv > Nc in silicon.
• The resistivity of intrinsic silicon (about 2.3 × 10^5 Ω·cm) is millions of times higher than copper (1.7 × 10^-6 Ω·cm) but far lower than SiO2 (> 10^15 Ω·cm).
• Reverse leakage current in a pn junction diode is proportional to ni, so it approximately doubles for every 10°C increase in temperature.

Quick Revision

• Intrinsic semiconductor: no impurities, n = p = ni.
• ni(Si, 300K) = 1.5 × 10^10 /cm³; ni(Ge, 300K) = 2.4 × 10^13 /cm³.
• Mass action law: n * p = ni^2 at thermal equilibrium.
• Conductivity: σ = q * ni * (µn + µp).
• µn = 1350 cm²/V·s, µp = 480 cm²/V·s for silicon at 300K.
• ni increases exponentially with T; reversal leakage doubles every ~10°C.
• Fermi level in intrinsic Si (Ei) sits near midgap.
• Exam trap: Students apply n = p = ni to a doped semiconductor. The mass action law gives p = ni^2/Nd for n-type doped material, which is far smaller than ni, not equal to it.