Carrier Concentration

The number of electrons and holes in a semiconductor determines its conductivity, switching speed, and leakage current. Every BJT datasheet parameter traces back to carrier concentration.

Core Concept

Carrier concentration tells you how many free electrons and holes exist per cubic centimetre in a semiconductor. In intrinsic silicon at 300K, this is ni = 1.5 × 10^10 cm^-3. That sounds large, but silicon has about 5 × 10^22 atoms/cm^3, so fewer than one in a trillion atoms contributes a carrier at room temperature.

The Fermi-Dirac distribution f(E) gives the probability that an electron occupies an energy state E at temperature T. At the Fermi level EF, the probability is exactly 0.5. In N-type silicon, EF rises above the midgap. In P-type, EF falls below the midgap. This shift directly changes how many electrons sit in the conduction band and how many holes exist in the valence band.

In doped semiconductors, the majority carrier concentration is simply the doping level (ND or NA) as long as doping is much greater than ni. The minority carrier concentration is found from the mass action law: n * p = ni^2. This relationship holds at thermal equilibrium regardless of doping, and it is the foundation for analysing every diode and BJT made from silicon.

Key Equations

Fermi-Dirac distribution:

f(E) = 1 / (1 + exp((E - EF) / kT)) where kT = 26 meV at 300K

Intrinsic carrier concentration (silicon):

ni^2 = Nc * Nv * exp(-Eg / kT) where Nc = 2.8 × 10^19 cm^-3, Nv = 1.04 × 10^19 cm^-3, Eg = 1.12 eV for Si

Mass action law:

n * p = ni^2 always holds at thermal equilibrium

Intrinsic Fermi level (midgap approximation):

Ei ≈ (Ec + Ev) / 2 + (kT/2) * ln(Nv / Nc)

Given:
  N-type silicon doped with ND = 10^17 cm^-3
  ni = 1.5 × 10^10 cm^-3 at T = 300K
  kT = 0.026 eV

Why this formula:
  When ND >> ni, the electron concentration equals ND.
  Hole concentration follows from the mass action law.

Formula:
  n = ND
  p = ni^2 / n
  EF - Ei = kT * ln(n / ni)

Substitution:
  n = 10^17 cm^-3
  p = (1.5 × 10^10)^2 / 10^17
  EF - Ei = 0.026 * ln(10^17 / 1.5 × 10^10)

Calculation:
  p = 2.25 × 10^20 / 10^17 = 2.25 × 10^3 cm^-3
  n / ni = 10^17 / 1.5 × 10^10 = 6.67 × 10^6
  ln(6.67 × 10^6) = 15.71
  EF - Ei = 0.026 × 15.71 = 0.408 eV

Final Answer:
  Electron concentration n = 10^17 cm^-3
  Hole concentration p = 2250 cm^-3
  Fermi level is 0.408 eV above the intrinsic Fermi level

Exam Tip: GATE often gives a doping level and asks for the Fermi level position. Use EF - Ei = kT * ln(ND / ni) for N-type and Ei - EF = kT * ln(NA / ni) for P-type. At 300K, kT = 0.026 eV. A common trap is forgetting to use ni = 1.5 × 10^10 cm^-3 for silicon, not 10^10 or 10^12. Using the wrong ni shifts your answer by many millivolts and causes wrong option selection.

Key Properties

• Intrinsic silicon at 300K: ni = 1.5 × 10^10 cm^-3, with n = p = ni.
• The Fermi-Dirac probability f(EF) = 0.5 at any temperature. This is a definition, not an approximation.
• Mass action law n * p = ni^2 holds only at thermal equilibrium. It does not apply under forward bias or illumination.
• Effective density of states for silicon: Nc = 2.8 × 10^19 cm^-3 (conduction band) and Nv = 1.04 × 10^19 cm^-3 (valence band).
• Carrier concentration doubles roughly every 10°C rise in temperature for silicon, which is why reverse leakage current increases with temperature.
• For degenerate semiconductors (very heavy doping, ND > 10^18 cm^-3), the Fermi level enters the conduction band and the semiconductor behaves like a metal.

Quick Revision

• ni (Si, 300K) = 1.5 × 10^10 cm^-3.
• f(EF) = 0.5 always, at any temperature.
• n * p = ni^2 at thermal equilibrium only.
• N-type: n ≈ ND, p = ni^2 / ND. P-type: p ≈ NA, n = ni^2 / NA.
• kT = 26 meV at 300K. This equals the thermal voltage VT used in diode equations.
• EF moves up with N-type doping and down with P-type doping.
• Carrier concentration increases with temperature, increasing leakage and reducing device stability.
• Exam trap: Applying n * p = ni^2 under forward bias. The law of mass action holds only at thermal equilibrium. Under injection, minority carrier concentration can exceed ni^2 / (majority carrier concentration).