Occupation Probability (f(E)) is the statistical function giving the probability that a quantum energy state at energy E is occupied by an electron — described by the Fermi-Dirac distribution for fermions, it governs how many of the available quantum states in a semiconductor are actually filled with electrons and thus how many carriers participate in conduction.

What Is Occupation Probability?

• Definition: f(E) = 1 / (1 + exp((E - E_F)/kT)), where E_F is the Fermi energy, k is Boltzmann's constant, and T is absolute temperature. The function gives a value between 0 (empty) and 1 (filled) for each energy state.
• Fermi Energy Significance: f(E_F) = 0.5 exactly — the Fermi energy is defined as the energy at which the probability of occupation is exactly 50%. States well below E_F have f ≈ 1 (almost certainly filled); states well above E_F have f ≈ 0 (almost certainly empty).
• Temperature Effect: At T = 0K, f(E) is a perfect step function — all states below E_F are filled, all above are empty. At finite temperature, the step is smeared over an energy range of approximately 4kT (about 100meV at room temperature), allowing some electrons to thermally excite above E_F.
• Pauli Exclusion Origin: The hard limit f(E) ≤ 1 arises from the Pauli exclusion principle — each quantum state can hold at most two electrons (spin up and spin down), preventing the classical pile-up of arbitrarily many particles in a single low-energy state.

Why Occupation Probability Matters

• Carrier Concentration Calculation: Electron density n = integral of g(E)f(E)dE from E_C to infinity, where g(E) is the density of states. The product of available states and their occupation probability gives the actual carrier density — the fundamental calculation underlying all semiconductor device analysis.
• MOSFET Switching: A MOSFET switches by moving energy bands up or down relative to E_F through gate voltage, changing the occupation probability of conduction band states from approximately zero (OFF state, bands above E_F) to approximately one (ON state, bands aligned with E_F). The switching sharpness is limited by how sharply f(E) transitions — ultimately setting the kT/q = 60mV/decade subthreshold swing limit.
• Degenerate Doping Effects: At source/drain doping concentrations above approximately 3x10^18 cm-3 in silicon, the Fermi level enters the conduction band and occupation probabilities near E_C can no longer be approximated as small — the full Fermi-Dirac integral must be used, and classical Maxwell-Boltzmann carrier statistics underestimates actual carrier density.
• Contact Resistance Modeling: The occupation probability function at the interface between a metal and a heavily doped semiconductor determines the carrier injection and extraction rates, governing ohmic contact behavior and the minimum achievable contact resistance.
• Quantum Dot and Single-Electron Devices: In quantum dots with discrete energy levels, the occupation probability of individual levels determines charging state — the basis of single-electron transistors and charge-based quantum computing qubits.

How Occupation Probability Is Applied in Practice

• Fermi-Dirac Integrals: Carrier density integrals involving f(E) over the parabolic density of states give the Fermi-Dirac integrals F_j(eta) — tabulated functions used in TCAD and compact models when degenerate conditions are encountered.
• Quasi-Fermi Level Generalization: Under non-equilibrium conditions, f(E) is replaced separately for electrons and holes by their respective quasi-Fermi levels E_Fn and E_Fp — each carrier species has its own occupation probability function that drives carrier density and current independently.
• Thermal Noise Analysis: Thermal fluctuations in occupation probabilities of conduction states produce Johnson-Nyquist noise — the mean square noise current in a resistor is directly related to the variance in occupation probability of electronic states at E_F.

Occupation Probability is the statistical foundation that connects quantum mechanical energy states to measurable electrical carrier concentrations — the Fermi-Dirac function is the universal lens through which band structure, doping, temperature, and applied voltage all combine to determine how many electrons are available for conduction, making it an indispensable building block for every quantitative semiconductor device model from basic diode equations to full quantum transport simulation.