3.1 Fermi-Dirac distribution

Fermi-Dirac Distribution Fundamentals

The Fermi-Dirac distribution describes the probability that an electron occupies a given energy state when a system is in thermal equilibrium. It's the starting point for calculating carrier concentrations in semiconductors, which in turn governs how every semiconductor device behaves.

What makes this distribution different from classical statistics is that electrons are fermions. They obey the Pauli exclusion principle: no two electrons can occupy the same quantum state at the same time. The classical Maxwell-Boltzmann distribution ignores this constraint entirely, so it breaks down whenever quantum effects matter.

Derivation of Fermi-Dirac Statistics

The Fermi-Dirac distribution is derived from statistical mechanics using the grand canonical ensemble. The key physical inputs are:

1. Electrons are indistinguishable from one another.
2. Each quantum state can hold at most one electron (Pauli exclusion).
3. The system is in thermal equilibrium with a fixed temperature and a reservoir that sets the chemical potential.

Maximizing the number of microstates subject to these constraints yields the Fermi-Dirac distribution function:

$f(E) = \frac{1}{1 + e^{(E - E_F) / k_B T}}$

where $E$ is the energy of the state, $E_F$ is the Fermi energy (chemical potential at equilibrium), $k_B$ is Boltzmann's constant, and $T$ is the absolute temperature.

Notice what this function tells you: at $E = E_F$, the exponential term equals 1, so $f(E_F) = 0.5$. The probability of occupation is exactly 50% at the Fermi energy, regardless of temperature.

Fermi-Dirac vs. Maxwell-Boltzmann Statistics

The Maxwell-Boltzmann distribution treats particles as distinguishable and places no limit on how many can share a state. The Fermi-Dirac distribution caps occupancy at one electron per state.

At low temperatures or high carrier densities (degenerate conditions), the two distributions diverge sharply. The Fermi-Dirac function saturates at 1 for states well below $E_F$, while Maxwell-Boltzmann would predict occupancies greater than 1, which is physically impossible for fermions.

Assumptions of the Fermi-Dirac Distribution

• The system is in thermal equilibrium: no net flow of energy or particles.
• Particles are indistinguishable fermions obeying the Pauli exclusion principle.
• The total number of particles is conserved.
• Energy states are treated as non-interacting (single-particle picture). Electron-electron interactions are not explicitly included.

If any of these assumptions break down (for example, under strong illumination or applied bias), you can no longer use a single Fermi level. Instead, you'd need quasi-Fermi levels, which is a topic for non-equilibrium conditions.

Fermi Energy and Fermi Level

The Fermi level is the single most important parameter for describing the electronic state of a semiconductor at equilibrium. Its position relative to the band edges tells you the carrier type, carrier concentration, and how the material will behave when contacted with other materials.

Definition of Fermi Energy

At absolute zero ($T = 0$ K), the Fermi energy $E_F$ is the energy of the highest occupied electron state. Every state below $E_F$ is filled; every state above is empty.

At finite temperature, the boundary between filled and empty states blurs. The Fermi-Dirac function smoothly transitions from $f \approx 1$ (well below $E_F$) to $f \approx 0$ (well above $E_F$), with the transition region spanning roughly $\pm 3k_BT$ around $E_F$. At room temperature ($T \approx 300$ K), $k_BT \approx 0.026$ eV, so this transition window is only about 0.15 eV wide.

The Fermi energy is still defined as the energy where $f(E) = 0.5$ at any temperature.

Fermi Level in Semiconductors

In semiconductors, the Fermi level lies within the bandgap, between the conduction band edge $E_C$ and the valence band edge $E_V$. Its exact position depends on doping:

• Intrinsic (undoped): The Fermi level sits near the middle of the bandgap. It's not exactly centered unless the effective density of states in the conduction and valence bands are equal ($N_C = N_V$), which is rarely the case. The intrinsic Fermi level $E_i$ is offset slightly toward the band with the smaller effective density of states.
• n-type (donor-doped): The Fermi level shifts toward the conduction band. Higher donor concentration pushes it closer to $E_C$.
• p-type (acceptor-doped): The Fermi level shifts toward the valence band. Higher acceptor concentration pushes it closer to $E_V$.

Temperature Dependence of Fermi Level

The Fermi level position changes with temperature, and the behavior depends on whether the semiconductor is intrinsic or extrinsic.

For intrinsic semiconductors, as temperature rises, more electron-hole pairs are generated, but the Fermi level stays near mid-gap. It shifts slightly due to the temperature dependence of the effective densities of states.

For extrinsic semiconductors, the picture is more nuanced:

1. At low temperatures (freeze-out regime), not all dopants are ionized, and the Fermi level sits close to the dopant energy level.
2. At moderate temperatures (extrinsic regime), dopants are fully ionized, and the Fermi level is near the relevant band edge (close to $E_C$ for n-type, close to $E_V$ for p-type).
3. At high temperatures (intrinsic regime), thermally generated carriers overwhelm the dopant contribution, and the Fermi level migrates back toward mid-gap.

This progression matters because it determines how carrier concentrations and device characteristics change with operating temperature.

Electron and Hole Concentrations

Carrier concentrations are what connect the abstract Fermi-Dirac distribution to measurable electrical properties like conductivity and current. The calculation combines two ingredients: the density of available states and the probability that each state is occupied.

Calculation Using the Fermi-Dirac Distribution

The electron concentration in the conduction band is found by integrating the product of the density of states and the occupation probability over all conduction band energies:

$n = \int_{E_C}^{\infty} g_C(E) \, f(E) \, dE$

Here $g_C(E)$ is the conduction band density of states (proportional to $\sqrt{E - E_C}$ for a parabolic band) and $f(E)$ is the Fermi-Dirac function.

The hole concentration in the valence band uses the probability that a state is unoccupied:

$p = \int_{-\infty}^{E_V} g_V(E) \, [1 - f(E)] \, dE$

These integrals don't have simple closed-form solutions in general. Under the Boltzmann approximation ($(E_C - E_F) \gg k_BT$), they simplify to:

$n = N_C \, e^{-(E_C - E_F)/k_BT}$

$p = N_V \, e^{-(E_F - E_V)/k_BT}$

where $N_C$ and $N_V$ are the effective densities of states in the conduction and valence bands. For silicon at 300 K, $N_C \approx 2.8 \times 10^{19} \text{ cm}^{-3}$ and $N_V \approx 1.04 \times 10^{19} \text{ cm}^{-3}$.

Intrinsic Carrier Concentration

In an intrinsic (undoped) semiconductor, every electron promoted to the conduction band leaves behind a hole, so $n = p = n_i$. Multiplying the expressions for $n$ and $p$ gives the mass action law:

$n \cdot p = n_i^2 = N_C N_V \, e^{-E_g / k_B T}$

Taking the square root:

$n_i = \sqrt{N_C N_V} \, e^{-E_g / (2 k_B T)}$

The intrinsic carrier concentration depends exponentially on the ratio of bandgap energy to thermal energy. For silicon at 300 K ($E_g \approx 1.12$ eV), $n_i \approx 1.5 \times 10^{10} \text{ cm}^{-3}$. For GaAs ($E_g \approx 1.42$ eV), $n_i$ is orders of magnitude smaller. This is why wide-bandgap semiconductors have much lower intrinsic carrier concentrations.

Extrinsic Carrier Concentration

Doping introduces either donors (which supply electrons) or acceptors (which supply holes). At room temperature, assuming full ionization:

• n-type: $n \approx N_D$ (the donor concentration). The minority hole concentration is $p = n_i^2 / N_D$.
• p-type: $p \approx N_A$ (the acceptor concentration). The minority electron concentration is $n = n_i^2 / N_A$.

These relations come directly from the mass action law ($np = n_i^2$), which holds at equilibrium regardless of doping. For example, if you dope silicon with $N_D = 10^{16} \text{ cm}^{-3}$ donors, the hole concentration drops to roughly $p = (1.5 \times 10^{10})^2 / 10^{16} \approx 2.25 \times 10^{4} \text{ cm}^{-3}$. The majority carriers vastly outnumber the minority carriers.

Fermi-Dirac Distribution Applications

Equilibrium Carrier Concentrations

Everything covered in the previous section on carrier concentrations is a direct application of the Fermi-Dirac distribution. In practice, you determine the Fermi level position from the doping and temperature, then use the distribution (or its Boltzmann approximation) to compute $n$ and $p$. These equilibrium values set the baseline for analyzing any device.

Density of States and Occupation Probability

The density of states $g(E)$ tells you how many quantum states are available per unit energy per unit volume. By itself, it says nothing about how many electrons actually occupy those states. The Fermi-Dirac function $f(E)$ provides the missing piece: the occupation probability.

The product $g(E) \cdot f(E)$ gives the electron energy distribution, which peaks at an energy slightly above $E_C$. Plotting this product is a useful way to visualize where most conduction electrons sit in energy. Similarly, $g(E) \cdot [1 - f(E)]$ gives the hole energy distribution near $E_V$.

Fermi-Dirac Distribution in Degenerate Semiconductors

A semiconductor becomes degenerate when it's doped so heavily that the Fermi level enters the conduction band (n-type) or valence band (p-type). At this point, the Boltzmann approximation fails because $(E_C - E_F)$ is no longer much greater than $k_BT$; in fact, it can be negative.

In degenerate semiconductors, you must use the full Fermi-Dirac integral (often written using the Fermi-Dirac integral of order 1/2, $\mathcal{F}_{1/2}$) rather than the simplified exponential expressions. Degenerate doping is common in modern devices: the source/drain regions of MOSFETs and the emitter of bipolar transistors are often doped above $10^{19} \text{ cm}^{-3}$, well into the degenerate regime.

Fermi-Dirac Distribution in Semiconductor Devices

Effect on p-n Junctions

In a p-n junction at equilibrium, the Fermi level must be constant throughout the device. Since the p-side Fermi level is near $E_V$ and the n-side Fermi level is near $E_C$, aligning them creates a built-in potential $V_{bi}$. This built-in potential is directly related to the doping levels through:

$V_{bi} = \frac{k_BT}{q} \ln\left(\frac{N_A N_D}{n_i^2}\right)$

The Fermi-Dirac distribution determines the carrier profiles on both sides of the junction, which in turn set the depletion width, junction capacitance, and I-V characteristics.

Role in Metal-Semiconductor Contacts

When a metal contacts a semiconductor, the Fermi levels of the two materials align at equilibrium. The difference between the metal's work function and the semiconductor's electron affinity determines whether you get a Schottky barrier (rectifying contact) or an ohmic contact (low-resistance contact).

The Fermi-Dirac distribution governs the electron energy distribution on both sides of the interface. This distribution determines how many carriers have enough energy to cross the barrier, which directly affects the contact resistance and current-voltage behavior.

Impact on Semiconductor Device Performance

Across all semiconductor devices, the Fermi-Dirac distribution influences:

• Carrier transport: The distribution of carriers in energy determines drift and diffusion currents.
• Recombination and generation: The occupation probabilities of trap states (described by $f(E)$) enter directly into Shockley-Read-Hall recombination rates.
• Threshold voltage and subthreshold behavior: In MOSFETs, the Fermi level position relative to the bands at the semiconductor surface determines inversion and accumulation conditions.

Getting the carrier statistics right is the foundation for any quantitative device analysis. The Fermi-Dirac distribution provides that foundation.