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Fermi level

The Fermi level, also known as the Fermi energy, is a key concept in solid-state physics that denotes the energy at which the probability of occupation by a fermionic particle, such as an electron, is exactly one-half according to the Fermi-Dirac distribution function.^[1] At absolute zero temperature, it represents the highest energy level occupied by electrons in a system, forming the surface of the so-called "Fermi sea" where all states below this energy are fully occupied and those above are empty, due to the Pauli exclusion principle.^[2] This level arises from the quantum statistical mechanics described by Fermi-Dirac statistics, which govern the behavior of indistinguishable fermions.^[1] In metals, the Fermi level typically lies within the conduction band, resulting in a high density of states at this energy and enabling significant electrical conductivity even at low temperatures, as electrons near the Fermi level can readily respond to electric fields.^[3] For instance, in copper, the Fermi energy is approximately 7 eV, corresponding to a Fermi temperature of about 82,000 K, far exceeding room temperature, which underscores the degenerate nature of the electron gas in metals.^[2] The Fermi level also influences thermal properties, such as the low specific heat of metals at low temperatures, since only electrons within ~kT of the Fermi level (where k is Boltzmann's constant and T is temperature) can participate in thermal excitations.^[1] In semiconductors, the position of the Fermi level relative to the band edges is crucial for determining carrier concentrations and device behavior. In intrinsic semiconductors, it resides near the midpoint of the band gap, where electron and hole concentrations are equal and thermally generated, as given by n_i = (N_c N_v)^{1/2} exp(-E_g / 2kT), with E_g being the band gap energy.^[4] Doping shifts this level: in n-type materials, donor impurities raise the Fermi level closer to the conduction band edge, increasing electron concentration via E_F ≈ E_i + kT ln(N_D / n_i), while in p-type materials, acceptors lower it toward the valence band, boosting hole concentration as E_F ≈ E_i - kT ln(N_A / n_i).^[4] These shifts enable the tailoring of electrical properties in semiconductor devices like transistors and diodes.^[5] The Fermi level's role extends to interfaces and non-equilibrium conditions, where quasi-Fermi levels describe separate electron and hole distributions under applied bias or illumination, essential for understanding photovoltaic and optoelectronic phenomena.^[5] Overall, the Fermi level provides a unifying framework for interpreting the electronic structure and transport properties across diverse materials, from metals to insulators.^[6]

Fundamentals

Definition and origin

The concept of the Fermi level emerged from the early development of quantum statistics for identical particles obeying the Pauli exclusion principle. In 1926, Enrico Fermi formulated the statistical mechanics of an ideal gas of such particles in his seminal paper "Zur Quantelung des idealen einatomigen Gases," published in Zeitschrift für Physik, where he derived the distribution now known as Fermi-Dirac statistics for systems like electrons. Independently, Paul A. M. Dirac arrived at an equivalent formulation in his work "On the Theory of Quantum Mechanics" that same year, applying it to the quantization of the ideal gas.^[7] The Fermi level, denoted E_F, represents the highest occupied energy state in a fermionic system at absolute zero temperature (T = [0](/page/0) K), where all states below E_F are fully occupied and those above are empty, in accordance with the Pauli exclusion principle. In thermodynamic terms, it corresponds to the electrochemical potential, or chemical potential \mu, which governs the average occupancy of quantum states in the grand canonical ensemble; specifically, \mu = E_F at T = [0](/page/0). At finite temperatures, the Fermi level serves as an approximation for \mu(T), the energy at which the probability of state occupation is 50% according to the Fermi-Dirac distribution.^[1]^[8] For a three-dimensional free electron gas model, the Fermi energy can be derived by filling the available states in momentum space up to the Fermi wavevector k_F = (3\pi^2 n)^{1/3}, where n is the electron density. This yields the expression

E_F = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3},

with m the electron mass and \hbar the reduced Planck's constant; this formula establishes the scale of electron energies in metals, typically several electron volts./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/09%3A_Condensed_Matter_Physics/9.05%3A_Free_Electron_Model_of_Metals) While the terms are sometimes used interchangeably, the Fermi energy E_F strictly denotes the fixed energy value at T = 0, whereas the Fermi level more generally refers to the temperature-dependent chemical potential \mu(T) in contexts like solid-state systems, where thermal excitations cause slight shifts but \mu remains near E_F for degenerate electron gases.

Fermi-Dirac statistics

Fermi-Dirac statistics provide the quantum mechanical framework for describing the thermal distribution of identical fermions, which are particles with half-integer spin that obey the Pauli exclusion principle, preventing more than one particle from occupying the same quantum state. This statistical approach was independently developed by Enrico Fermi and Paul Dirac in 1926, building on the principles of quantum mechanics and indistinguishability of particles. Unlike the classical Maxwell-Boltzmann statistics applicable to distinguishable particles at low densities, Fermi-Dirac statistics account for the antisymmetric wavefunction of fermions under particle exchange, leading to a fundamentally different occupation probability for energy states. The core of Fermi-Dirac statistics is the distribution function f(E), which gives the average occupation number of a quantum state with energy E:

f(E) = \frac{1}{1 + \exp\left(\frac{E - \mu}{k_B T}\right)},

where \mu is the chemical potential, k_B is the Boltzmann constant, and T is the absolute temperature. This formula emerges from the grand canonical ensemble in quantum statistical mechanics, where the partition function for fermions is constructed by summing over antisymmetric states, ensuring compliance with the Pauli principle. In contrast to the exponential decay of the Maxwell-Boltzmann distribution f(E) \approx \exp\left(-\frac{E - \mu}{k_B T}\right) for classical dilute gases, the Fermi-Dirac form saturates at 1 for E < \mu and approaches 0 for E \gg \mu, reflecting the exclusion effect even at high temperatures. At absolute zero temperature (T = 0), the distribution simplifies to a step function:

f(E) = \theta(\mu - E),

where \theta is the Heaviside step function, fully occupying all states below the chemical potential \mu (identified as the Fermi energy E_F at T = 0) and leaving higher states empty. For finite but low temperatures (k_B T \ll E_F), the sharp step smears over an energy width of order k_B T, and the Sommerfeld expansion approximates integrals involving f(E) by expanding around the T = 0 limit. This expansion, developed by Arnold Sommerfeld in 1928, yields corrections to the chemical potential as

\mu(T) \approx E_F \left[1 - \frac{\pi^2}{12} \left(\frac{k_B T}{E_F}\right)^2 \right],

enabling precise calculations of thermodynamic properties like energy and pressure for degenerate systems. A key application of Fermi-Dirac statistics is in identifying degenerate Fermi gases, where quantum effects dominate classical behavior when the thermal energy is much smaller than the Fermi energy, i.e., E_F \gg k_B T. This degeneracy condition arises naturally from the filling of states up to E_F, leading to phenomena such as the pressure of a zero-temperature electron gas, which persists even without interactions due to the Pauli principle. In equilibrium, the chemical potential \mu serves as the Fermi level at low temperatures, parameterizing the distribution.

Application in solids

Band structure context

In crystalline solids, the electronic wavefunctions of electrons are described by the Bloch theorem, which states that the solutions to the Schrödinger equation in a periodic potential take the form of plane waves modulated by a periodic function with the same periodicity as the lattice. This theorem, proposed by Felix Bloch, leads to the formation of energy bands where electron energies are quantized into allowed ranges separated by forbidden gaps.^[9] In insulators and semiconductors, the periodic potential results in a filled valence band below a bandgap E_g and an empty conduction band above it, with E_g determining the material's insulating or semiconducting nature.^[10] The position of the Fermi level E_F within this band structure dictates the occupation of electronic states according to the Pauli exclusion principle, which prohibits more than one fermion from occupying the same quantum state. In metals, E_F lies within the conduction band, allowing electrons near E_F to contribute to conduction. In intrinsic semiconductors, E_F is located approximately at the mid-gap position, ensuring equal numbers of electrons in the conduction band and holes in the valence band at thermal equilibrium. This filling up to E_F at absolute zero temperature minimizes the total energy of the system under the constraints of quantum statistics.^[2]^[11] The electron density n in the solid is obtained by integrating the density of states g(E), derived from the band structure, with the occupation probability given by the Fermi-Dirac distribution f(E):

n = \int_0^\infty g(E) f(E) \, dE,

where g(E) accounts for the number of available states per unit energy interval, varying with the band dispersion relation. The Fermi-Dirac occupation f(E) briefly referenced here provides the probabilistic filling of states. This integral encapsulates how the band structure influences the total electron population.^[12] The Fermi level plays a crucial role in electrical conductivity by determining the availability of states for thermal excitation. In metals, modeled as a free electron gas, the carrier concentration is approximately

n \approx \frac{1}{3\pi^2} \left( \frac{2m E_F}{\hbar^2} \right)^{3/2},

where m is the electron mass and \hbar is the reduced Planck's constant; this relation highlights how E_F sets the scale for the number of conduction electrons, enabling metallic transport. In semiconductors, the position of E_F relative to the band edges governs the excitation of carriers across E_g, influencing the material's response to external stimuli.^[13]

Doping and position in materials

In intrinsic semiconductors, the Fermi level E_F is positioned approximately at the midpoint of the band gap, between the valence band maximum E_v and the conduction band minimum E_c, reflecting equal concentrations of electrons and holes.^[11] Doping introduces impurities to create extrinsic semiconductors, shifting E_F and altering carrier concentrations to enable controlled electrical properties.^[11] In n-type semiconductors, donor impurities (e.g., phosphorus in silicon) add extra electrons, increasing the electron concentration n \approx N_d, where N_d is the donor concentration. This shifts E_F toward E_c, making electron excitation to the conduction band more probable. The position is approximated by

E_F \approx E_c - kT \ln \left( \frac{N_c}{N_d} \right),

where kT is the thermal energy (≈0.026 eV at 300 K), and N_c is the effective density of states in the conduction band (≈2.8 × 10¹⁹ cm⁻³ for silicon).^[11] For example, in silicon doped with phosphorus at N_d = 10^{17} cm⁻³, E_F lies ≈0.146 eV below E_c.^[11] At higher doping levels (e.g., N_d > 10^{18} cm⁻³), E_F can enter the conduction band, leading to degenerate semiconductors with metallic-like behavior.^[11] In p-type semiconductors, acceptor impurities (e.g., boron in silicon) create holes by accepting electrons from the valence band, increasing the hole concentration p \approx N_a, where N_a is the acceptor concentration. This shifts E_F toward E_v. The position is approximated by

E_F \approx E_v + kT \ln \left( \frac{N_v}{N_a} \right),

where N_v is the effective density of states in the valence band (≈1.04 × 10¹⁹ cm⁻³ for silicon).^[11] For p = 10^{14} cm⁻³ in silicon, E_F lies ≈0.31 eV above E_v.^[11] High acceptor doping can similarly cause degeneracy within the valence band.^[11] These shifts determine the semiconductor type (n or p) and are essential for fabricating devices like p-n junctions in diodes, where the differing E_F positions in adjacent regions create a built-in potential for rectification.^[11]

Equilibrium properties

Chemical potential relation

In thermal equilibrium, the Fermi level, often denoted as \zeta or E_F, is identical to the electrochemical potential \bar{\mu} for electrons in a system, which remains constant throughout. The chemical potential \mu is defined as the change in the Gibbs free energy of the system when an additional electron is added (or removed) while keeping the temperature T, volume V, and total number of particles N constant, i.e., \mu = \left( \frac{\partial G}{\partial N} \right)_{T,V}. This equivalence arises because, at absolute zero temperature, E_F marks the highest occupied energy state, and at finite temperatures, it determines the occupation probability via the Fermi-Dirac distribution, aligning precisely with the thermodynamic role of \bar{\mu} in controlling particle exchange.^[14]^[15] In spatially non-uniform systems, such as heterostructures or interfaces, the local chemical potential \mu(\mathbf{r}) varies with position due to differences in composition or density, while the electrostatic potential \phi(\mathbf{r}) causes band bending. The electrochemical potential \bar{\mu} = \mu(\mathbf{r}) - e \phi(\mathbf{r}) (for electrons with charge q = -e) remains constant across the system in equilibrium, where e is the elementary charge. This constancy of the electrochemical potential (the Fermi level E_F) ensures no net current flows, as electrons redistribute to align \bar{\mu} throughout. Thus, the local chemical potential is \mu(\mathbf{r}) = E_F + e \phi(\mathbf{r}), reflecting how band bending via \phi(\mathbf{r}) shifts the effective energy landscape relative to the constant Fermi level.^[16]^[17]^[18] In closed systems where different phases or materials are in contact, the electrochemical potential \bar{\mu} is conserved and equalizes across interfaces, driving charge transfer until equilibrium is reached. This process minimizes the free energy by aligning the Fermi levels relative to a common reference, such as the vacuum level, preventing further diffusion or drift currents. For instance, in a metal-semiconductor junction, initial differences in \bar{\mu} lead to depletion or accumulation regions until the potentials balance.^[19]^[20] The connection to the work function \Phi in metals further illustrates this thermodynamic link: \Phi = -\mu / e, where \mu is the chemical potential of electrons referenced to the vacuum level just outside the surface. This expression quantifies the minimum energy required to remove an electron from the Fermi level to infinity, directly tying \mu to surface properties and enabling predictions of contact potentials between materials.^[21]^[22]

Temperature dependence

In metals, the chemical potential μ(T), which corresponds to the Fermi level at finite temperature, shows only a weak dependence on temperature due to the high degeneracy of the electron gas. The Sommerfeld expansion yields the approximation μ(T) ≈ E_F [1 - (π²/12)(kT/E_F)²], where E_F is the zero-temperature Fermi energy, k is the Boltzmann constant, and T is the temperature. This formula indicates a slight decrease in μ with rising T, typically less than 1% at room temperature for metals with E_F in the range of 5–10 eV.^[23] In intrinsic semiconductors, the Fermi level μ_i(T) lies near the center of the band gap, with its position given by μ_i(T) = (E_c + E_v)/2 + (kT/2) ln(N_v / N_c), where E_c and E_v are the conduction and valence band edges, respectively, and N_v and N_c are the effective densities of states in the valence and conduction bands. The logarithmic term introduces a linear temperature shift, which depends on the ratio of density-of-states effective masses (m_{dh}^* / m_{de}^) since N_v ∝ (m_{dh}^)^{3/2} and N_c ∝ (m_{de}^)^{3/2}; for example, in silicon where m_{de}^ > m_{dh}^*, resulting in N_c > N_v and ln(N_v / N_c) < 0, this shift moves μ_i slightly toward the valence band. Furthermore, the band edges shift with temperature owing to lattice expansion and electron-phonon coupling, generally narrowing the band gap E_g = E_c - E_v by about 10^{-4} eV/K, which causes μ_i to move toward the conduction band as T increases to maintain charge neutrality.^[24]^[25] For doped semiconductors, the Fermi level's temperature dependence reflects distinct regimes influenced by doping. At low temperatures (below ~100 K), carrier freeze-out pins μ near the dopant levels—close to the donor energy E_d (∼0.05 eV below E_c) for n-type materials—since thermal energy kT is insufficient to ionize most impurities, resulting in low free carrier density. In the intermediate extrinsic regime (∼100–400 K), full ionization occurs, and μ stabilizes at a doping-dependent position, such as μ ≈ E_c + kT ln(N_c / N_D) for n-type with donor concentration N_D, lying several kT below E_c for moderate doping. At high temperatures (above ∼400 K), intrinsic excitation dominates, and μ approaches the intrinsic value μ_i(T) as the intrinsic carrier density n_i exceeds the dopant density.^[26] Doping strongly influences the Fermi level by shifting it from the intrinsic position toward the majority carrier band, with the extent depending on impurity type and concentration. The degeneracy parameter η = (E - μ)/kT characterizes these behaviors; in non-degenerate semiconductors, η ≫ 1 (e.g., μ ≪ E_c for electrons), enabling classical Maxwell-Boltzmann statistics, whereas η ≈ 0 signals degeneracy near the band edges. Typical plots of μ(T) illustrate near-constancy in metals, a slow drift toward mid-gap in intrinsic semiconductors, and stepwise transitions in doped cases from dopant-pinned to intrinsic positions.^[27]

Measurement techniques

Voltage-based methods

Voltage-based methods for determining the Fermi level rely on measuring equilibrium potentials or voltage responses that arise from the alignment of electrochemical potentials in materials, particularly in contact with metals, electrolytes, or under magnetic fields. These techniques exploit the fact that the Fermi level, as the chemical potential of electrons at absolute zero, equilibrates across interfaces, manifesting as measurable voltages without net current flow. In practice, such methods provide indirect access to the Fermi level position relative to reference scales like the vacuum level or standard hydrogen electrode. The Kelvin probe technique measures the contact potential difference (CPD) between a vibrating reference electrode and the sample surface, which directly relates to the difference in work functions and thus Fermi levels of the two materials. The CPD, denoted as \Delta \Phi, is given by \Delta \Phi = \frac{\mu_1 - \mu_2}{e}, where \mu_1 and \mu_2 are the chemical potentials (Fermi levels at low temperature) of the sample and probe, respectively, and e is the elementary charge. This voltage is induced by the capacitive coupling from the probe's vibration, nullifying the electric field between the surfaces. The method, originally developed by Lord Kelvin in 1898 and refined into scanning Kelvin probe force microscopy (KPFM) for nanoscale resolution, allows mapping of local Fermi level variations, such as those due to surface states or doping gradients, with sensitivities down to millielectronvolts.^[28] In semiconductors, Mott-Schottky analysis extracts the flat-band potential from capacitance-voltage measurements at a semiconductor-electrolyte interface, linking it to the bulk Fermi level. By plotting the square reciprocal of the space-charge capacitance $1/C^2 versus applied voltage V, the x-intercept yields the flat-band potential V_{fb}, which corresponds to the position of the bulk Fermi level relative to the reference electrode potential through alignment with the electrolyte's redox level. This linear relation assumes a depleted space-charge region and negligible surface states, with the slope providing the doping density N_D or N_A. The technique, rooted in the 1939 theory by N.F. Mott and further developed by Schottky, is widely applied to characterize band edges in photoelectrochemical materials like TiO_2, where V_{fb} shifts with pH due to Fermi level alignment with the redox couple.^[29] The Hall effect enables inference of the Fermi level in metals and degenerate semiconductors by measuring the transverse Hall voltage V_H under a magnetic field, which yields the carrier density n = \frac{IB}{V_H e t}, where I is current, B is magnetic field, t is thickness, and e is charge. For free-electron-like systems, the Fermi energy is then calculated as E_F = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3}, relating the equilibrium carrier density directly to the occupied states up to E_F. In non-degenerate semiconductors, temperature-dependent Hall data combined with mobility allow positioning E_F relative to band edges via Fermi-Dirac statistics. This equilibrium method, discovered by Edwin Hall in 1879, is essential for quantifying degeneracy and has been used to determine E_F \approx 7 eV in metals like copper from room-temperature measurements.^[30] Electrochemical cells determine the Fermi level through the open-circuit voltage V_{oc} between a working electrode (e.g., semiconductor or metal) and a reference redox couple, where V_{oc} = \frac{\mu_{redox} - \mu_{electrode}}{e}, equilibrating the Fermi level of the electrode with the electrochemical potential of the solution. In a setup like a semiconductor-electrolyte junction with a ferrocene/ferrocenium couple, the measured V_{oc} positions \mu_{electrode} on the absolute potential scale, calibrated against the standard hydrogen electrode (SHE), which corresponds to -4.44 eV versus vacuum. This approach, formalized in semiconductor electrochemistry, reveals Fermi level shifts with doping or illumination and is crucial for assessing stability in batteries and solar cells.^[31]

Spectroscopic approaches

Spectroscopic approaches provide direct probes of the Fermi level by measuring the energy distribution of electrons or photons associated with occupied and unoccupied electronic states. These techniques leverage the photoelectric effect or tunneling processes to map the density of states (DOS) relative to the Fermi energy E_F, offering insights into band structures without relying on electrical transport. Ultraviolet photoelectron spectroscopy (UPS), inverse photoemission spectroscopy (IPES or BIS), angle-resolved photoemission spectroscopy (ARPES), and scanning tunneling spectroscopy (STS) are among the primary methods, each targeting specific aspects of the electronic structure around E_F. Ultraviolet photoelectron spectroscopy (UPS) measures the kinetic energies of photoelectrons emitted from occupied electronic states up to the Fermi level when a sample is irradiated with ultraviolet photons, typically from a helium discharge lamp (h\nu \approx 21.2 eV for He I). The spectrum reveals the valence band structure, with the Fermi edge marking the highest occupied state at T = 0 K, and the binding energy referenced to E_F. The secondary electron cutoff, the low-energy edge of the spectrum, allows determination of the work function \Phi, defined as the energy difference between the vacuum level and E_F:

\Phi = h\nu - E_{\text{cutoff}}

where E_{\text{cutoff}} is the minimum kinetic energy of secondary electrons. This technique is surface-sensitive (top 1-2 nm) and widely used for metals and semiconductors to establish E_F positioning relative to surface potentials. Seminal developments in UPS, including its application to valence states, trace back to early implementations in the 1970s. Inverse photoemission spectroscopy (IPES, also known as bremsstrahlung isochromat spectroscopy or BIS) complements UPS by probing unoccupied electronic states above E_F. Low-energy electrons (typically 5-50 eV) are incident on the sample, and those that decay into empty states emit photons of fixed energy, detected to map the conduction band DOS starting from E_F. The energy of the emitted photon corresponds to the difference between the incident electron energy and the binding energy of the unoccupied state, providing electron affinity and band gap information. IPES is particularly valuable for semiconductors, where it reveals the conduction band minimum relative to E_F, with energy resolution around 0.3-1 eV. A comprehensive review of IPES principles and applications highlights its role in studying unoccupied states since the 1980s. Angle-resolved photoemission spectroscopy (ARPES) extends UPS by measuring both the energy and momentum of photoelectrons, enabling mapping of band dispersions E(k) across the Brillouin zone with E_F set as the zero-energy reference. The binding energy is given by E_b = h\nu - E_k, where E_k is the kinetic energy, and the in-plane momentum k_\parallel is derived from the emission angle \theta via k_\parallel = \sqrt{2m E_k}/\hbar \cdot \sin\theta. Sharp features at E = 0 (Fermi surface crossings) confirm the E_F position, often calibrated using a metallic reference sample. ARPES is essential for visualizing how bands cross E_F in metals or lie within the gap in insulators, with resolutions down to 1-10 meV in energy and 0.01 Å^{-1} in momentum. High-resolution ARPES has been pivotal in studies of topological materials and superconductors. Scanning tunneling spectroscopy (STS), performed using a scanning tunneling microscope (STM), probes the local density of states (LDOS) near E_F by measuring the differential tunneling conductance dI/dV as a function of bias voltage V. At zero bias (V = 0), the tip and sample Fermi levels align, and dI/dV \propto \rho(E_F), the LDOS at E_F; peaks or features in dI/dV(V) indicate energy positions relative to this reference. This technique offers atomic-scale spatial resolution (sub-nm) and energy resolution (~1-10 meV), ideal for heterogeneous materials like graphene or quantum dots where E_F varies locally. STS has revealed E_F shifts due to doping or strain in 2D materials.

Non-equilibrium dynamics

Transport and injection

In biased p-n junctions, carrier injection under forward bias leads to non-equilibrium conditions where minority carriers are injected across the junction, altering local carrier concentrations and thereby shifting the local chemical potential μ. Electrons injected into the p-region increase the local electron density n, raising μ relative to its equilibrium value, while holes injected into the n-region similarly elevate the local hole density p, lowering μ in that region to maintain charge neutrality away from the junction. This process is modeled within the drift-diffusion framework, where current densities for electrons and holes are given by \mathbf{J}_n = q \mu_n n \mathbf{E} + q D_n \nabla n and \mathbf{J}_p = q \mu_p p \mathbf{E} - q D_p \nabla p, with spatial variations in n and p driving the deviations in μ.^[32]^[33] The Boltzmann transport equation provides a more fundamental description of these non-equilibrium dynamics, governing the evolution of the distribution function f(\mathbf{r}, \mathbf{k}, t) for carriers in phase space:

\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f - \frac{q}{\hbar} \mathbf{E} \cdot \nabla_{\mathbf{k}} f = \left( \frac{\partial f}{\partial t} \right)_{\text{coll}},

where the collision term accounts for scattering, often approximated via the relaxation time τ. Under bias, f deviates from the equilibrium Fermi-Dirac distribution, resulting in spatially varying effective Fermi levels that reflect local imbalances in carrier populations and energies. In steady-state conditions, this spatial variation of the effective Fermi level is solved iteratively with Poisson's equation to capture drift, diffusion, and electrostatic effects.^[32] In light-emitting diodes (LEDs) and semiconductor lasers, forward bias injects electrons and holes into the active region, splitting the effective Fermi level by an amount approximately equal to qV, where V is the applied voltage and q is the elementary charge. This splitting, denoted as ΔE_F = E_{F,n} - E_{F,p}, enables population inversion when ΔE_F exceeds the bandgap energy E_g, as the occupation probability for states in the conduction band near the band edge surpasses that in the valence band, favoring stimulated emission over absorption. For instance, in direct-bandgap materials like GaAs, this condition is achieved at biases where qV > E_g, leading to net optical gain essential for lasing.^[34]^[35] Thermoelectric effects further illustrate the role of the Fermi level in transport, particularly through the Seebeck coefficient S, which relates temperature gradients to voltage via the Mott formula derived from the Boltzmann equation in the low-temperature limit:

S = -\frac{\pi^2 k_B^2 T}{3e} \left. \frac{d \ln \sigma(E)}{dE} \right|_{E=E_F},

where σ(E) is the energy-dependent conductivity, k_B is Boltzmann's constant, T is temperature, and e is the electron charge. This expression links the position of E_F relative to band features—such as the density of states or mobility edges—to the sign and magnitude of S, with p-type materials exhibiting positive S when E_F lies below the valence band maximum and n-type showing negative S above the conduction band minimum, driving voltage gradients in response to heat flow.^[36]^[37]

Quasi-Fermi levels

In non-equilibrium conditions characterized by high carrier injection and low recombination rates, the distribution of electrons and holes in semiconductors deviates from a single Fermi-Dirac function but can be approximated using separate quasi-Fermi levels for each carrier type. The electron occupation probability is then given by f(E) \approx f_{\text{FD}}(E - E_{Fn}), where f_{\text{FD}} is the standard Fermi-Dirac distribution and E_{Fn} is the electron quasi-Fermi level. Similarly, the hole occupation probability is f_h(E) \approx 1 - f_{\text{FD}}(E - E_{Fp}), with E_{Fp} as the hole quasi-Fermi level. This concept was introduced to describe carrier distributions in p-n junctions under bias, enabling the extension of equilibrium statistical mechanics to steady-state non-equilibrium scenarios.^[38] The separation between the quasi-Fermi levels, \Delta E_F = E_{Fn} - E_{Fp}, equals the applied voltage times the elementary charge in steady state, \Delta E_F = q V_{\text{applied}}, assuming negligible voltage drops across neutral regions. This relation arises from the constancy of the product of electron and hole quasi-Fermi potentials across the device in forward bias. The approximation holds under conditions of low temperature, where the Fermi-Dirac tails are sharp, and short carrier lifetimes, which maintain distinct populations; it breaks down in regimes of strong scattering that prevent local thermalization of carrier distributions within each band.^[38]^[39] Quasi-Fermi levels are central to analyzing device performance, particularly in solar cells, where the power conversion efficiency \eta scales with the normalized splitting (E_{Fn} - E_{Fp}) / E_g, with E_g the bandgap; maximum efficiency occurs when this splitting approaches E_g under detailed balance. In recombination processes, rates such as Shockley-Read-Hall are expressed using quasi-Fermi levels, yielding R = (np - n_i^2) / \tau, where n and p derive from E_{Fn} and E_{Fp}, n_i is the intrinsic carrier concentration, and \tau the effective lifetime, facilitating compact modeling of non-radiative losses.^[40]^[41]

Technical considerations

Nomenclature variations

The term "Fermi level" originated in the context of Enrico Fermi's 1926 work on the quantization of the ideal monatomic gas, where it described the highest occupied energy state for fermions at absolute zero temperature under Fermi-Dirac statistics.^[42] This concept was initially applied to free electron gases in atomic physics. Its adoption in solid-state physics occurred in the late 1920s and 1930s, particularly following Felix Bloch's 1928 analysis of electron waves in periodic crystal lattices, which introduced band structures and extended the Fermi-Dirac framework to electrons in solids, leading to the notion of a filled Fermi sea within energy bands. A persistent source of inconsistency in the literature arises from the interchangeable or context-dependent use of "Fermi energy" and "Fermi level." The Fermi energy, often denoted E_F, strictly refers to the energy of the highest occupied state at T = 0 K, representing a fixed value for a given system.^[2] In contrast, the Fermi level corresponds to the chemical potential \mu (sometimes denoted \zeta or E_F) at finite temperatures, where it shifts slightly due to thermal excitation while remaining approximately constant for typical solid-state systems.^[2] This distinction is not always maintained; many texts and papers use E_F for both, especially in approximations where \mu \approx E_F for metals at room temperature, leading to terminological overlap across theoretical and experimental works.^[43] In device physics, the electrochemical potential is frequently termed the "Fermi potential," denoted \psi_F or \phi_F, to emphasize its role in relating carrier concentrations to applied voltages in semiconductors. It represents the electrostatic potential shift of the Fermi level relative to the intrinsic level, which is zero in intrinsic semiconductors. For p-type materials, \phi_F \approx \frac{[kT](/page/KT)}{[q](/page/Q)} \ln\left(\frac{N_A}{n_i}\right); for n-type, \phi_F \approx -\frac{[kT](/page/KT)}{[q](/page/Q)} \ln\left(\frac{N_D}{n_i}\right), where N_A and N_D are acceptor and donor densities, and n_i is the intrinsic carrier concentration.^[44]^[45] This contrasts with thermodynamic contexts, where it is simply the chemical potential \mu, highlighting a disciplinary adaptation for engineering applications involving electrostatic potentials. In surface chemistry, "Fermi energy" is commonly used to denote the energy reference for electron transfer at interfaces, such as in adsorption processes, but care must be taken to distinguish it from the vacuum level to avoid conflation with work function measurements.^[46]

Energy referencing conventions

In energy band diagrams for solids, a standard convention is to set the Fermi level E_F to zero, which simplifies the visualization of the relative positions of the conduction band minimum E_c and valence band maximum E_v with respect to the occupied states. This relative referencing highlights the band gap and doping-induced shifts without needing an absolute energy scale, as commonly practiced in semiconductor device analysis.^[47] For absolute energy positioning, the vacuum level E_\mathrm{vac} serves as a reference, defined such that the work function \Phi = E_\mathrm{vac} - E_F, where E_\mathrm{vac} is typically set to 0 and E_F is negative relative to it—the minimum energy required to remove an electron from E_F to vacuum just outside the surface.^[48] This convention links the internal electronic structure to external potentials, enabling comparisons across materials via parameters like electron affinity \chi = E_\mathrm{vac} - E_c. Internal referencing relative to band edges, such as E_c - E_F, is also prevalent, as it directly relates to carrier concentrations; for instance, in n-type semiconductors, E_c - E_F quantifies the distance from the Fermi level to the conduction band, influencing electron density without invoking external levels.^[46] However, the vacuum level reference introduces challenges due to its sensitivity to surface conditions. Surface reconstruction, involving atomic rearrangements or dipole formation, can shift the local E_\mathrm{vac} relative to the bulk, altering the apparent work function by several electron volts and causing misalignment errors in band diagrams.^[49] In heterostructures, this non-invariance exacerbates issues, as interface dipoles and charge transfer disrupt simple vacuum-level alignment, leading to inaccuracies in predicting barrier heights or offsets between dissimilar materials like GaAs/AlGaAs.^[50] To address these limitations in multi-material systems, internal referencing schemes are recommended, such as aligning bands using the chemical potential \mu (equivalent to E_F at zero temperature) or average valence band maximum positions derived from core-level spectroscopy.^[50] Band-aligned scales, which match internal energy levels across interfaces without relying on surface-sensitive vacuum references, provide more reliable offsets for device modeling in complex heterojunctions.^[48]

Nanoscale and discrete effects

In nanoscale systems, the continuum approximation of the Fermi level breaks down due to quantum confinement, leading to discrete energy levels that dominate the electronic structure. In quantum dots, these discrete levels result in a stepwise variation of the Fermi level as electrons are added one by one, particularly in the Coulomb blockade regime where the charging energy U = e^2 / (2C), with C being the dot capacitance, greatly exceeds the thermal energy k_B T. This blockade suppresses electron tunneling until the applied bias or gate voltage overcomes U, manifesting as sharp conductance peaks corresponding to alignment of individual levels with the Fermi energy of the leads.

U = \frac{e^2}{2C} \gg k_B T

Such behavior is prominent in semiconductor quantum dots fabricated from materials like GaAs, where confinement in all three dimensions quantizes the spectrum, and the addition of each electron shifts the chemical potential (Fermi level) by approximately U, reflecting the discrete nature of charge quantization. In two-dimensional systems, such as graphene, the linear dispersion of Dirac fermions further modifies the Fermi level position, which depends on the carrier density n and can be continuously tuned via electrostatic gating. The relation is given by

E_F = \hbar v_F \sqrt{\pi n},

where v_F is the Fermi velocity, approximately $10^6 m/s. This density-dependent tuning allows the Fermi level to cross the Dirac point at zero density, transitioning from electron to hole doping without a bandgap, enabling unique transport properties in graphene nanoribbons or constrictions where edge effects introduce additional quantization. Finite-size effects introduce broadening to these discrete levels, characterized by \Gamma \approx \hbar / \tau, where \tau is the lifetime of the states due to coupling to leads or phonons, causing the effective Fermi level to smear over multiple discrete states when \Gamma exceeds the level spacing \delta. In small quantum dots, this broadening, often on the order of meV from tunneling rates, blurs sharp steps in the density of states, but the discrete regime persists when k_B T and \Gamma remain below \delta. For instance, in single-electron transistors based on quantum dots, the gate voltage V_g electrostatically tunes the position of the dot's discrete levels relative to the Fermi level of the source and drain, revealing Coulomb blockade oscillations; this quantized tuning is essential for devices below approximately 10 nm, where the continuous Fermi sea model fails and level spacing \delta \gtrsim 10 meV dominates over thermal smearing.^[51]^[52]

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