Electron mobility
Electron mobility, denoted as \mu_n, is a fundamental parameter in solid-state physics that quantifies the ease with which electrons drift through a conductor or semiconductor under the influence of an applied electric field, defined as the ratio of the electron drift velocity v_d to the electric field strength E, given by \mu_n = v_d / E.[1] The typical units for electron mobility are cm²/V·s, reflecting its measurement of velocity per unit field.[1] In semiconductors, electron mobility plays a critical role in determining electrical conductivity and device performance, as it directly influences the current density J_n = n q \mu_n E, where n is the electron concentration and q is the elementary charge.[1] For instance, in silicon at room temperature, electron mobility is approximately 1350 cm²/V·s, significantly higher than hole mobility at 480 cm²/V·s, enabling faster electron transport in n-type materials.[2] High electron mobility is essential for applications in high-speed electronics, such as field-effect transistors and solar cells, where materials like gallium arsenide exhibit values exceeding 8000 cm²/V·s due to reduced scattering.[3] Electron mobility is primarily limited by scattering mechanisms, including interactions with lattice phonons, ionized impurities, and defects, which reduce the mean free time between collisions and thus lower \mu_n.[1] Temperature dependence is notable, with mobility decreasing at higher temperatures due to increased phonon scattering, though this effect is less pronounced in heavily doped semiconductors.[1] In two-dimensional materials like graphene, mobilities can reach over 200,000 cm²/V·s at low temperatures, highlighting opportunities for strain engineering and doping to enhance performance in next-generation devices.[4]Fundamentals
Definition and Units
Electron mobility, denoted as \mu_e or simply \mu, quantifies the ease with which electrons respond to an applied electric field in a solid material, such as a metal or semiconductor. It serves as the proportionality constant between the average drift velocity v_d of electrons and the electric field strength E, formally defined by the relation v_d = \mu_e E. This definition captures the directed motion of charge carriers superimposed on their random thermal motion, providing a key parameter for understanding electrical transport properties.[5] The concept originated in early 20th-century solid-state physics as part of the Drude model, a classical theory of electron conduction in metals proposed by Paul Drude in 1900. In this model, electrons are treated as a free gas interacting with the ionic lattice, where mobility arises from the balance between accelerating electric forces and collisional scattering. The term "mobility" derives from its representation of the electrons' freedom of movement, distinguishing it from fixed lattice ions. A parallel quantity, hole mobility \mu_h, applies to positive charge carriers (holes) in semiconductors, which behave as if moving oppositely to electrons; in materials like silicon, \mu_h is typically about one-third of \mu_e due to differences in effective mass and scattering.[6][7] In the International System of Units (SI), electron mobility has dimensions of area per volt-second, yielding m²/(V·s), which emerges dimensionally from drift velocity (m/s) divided by electric field (V/m). For practical applications in semiconductor device physics, the cgs-derived unit cm²/(V·s) is prevalent, where values for electrons in silicon range up to approximately 1400 cm²/(V·s) at room temperature, and 1 m²/(V·s) equals 10⁴ cm²/(V·s). This unit choice facilitates comparison with experimental data on carrier transport.[8][2]Drift Velocity and Derivation
In the presence of an applied electric field \mathbf{E}, charge carriers such as electrons in a semiconductor or metal experience a net directed motion superimposed on their random thermal velocities. The thermal velocity arises from the random kinetic energy due to temperature, typically on the order of $10^7 cm/s for electrons at room temperature, resulting in no net current without a field. In contrast, the drift velocity \mathbf{v}_d is the average velocity acquired by the ensemble of electrons in response to the field, directed opposite to \mathbf{E} for electrons due to their negative charge, and is much smaller in magnitude under typical conditions.[7] This steady-state drift arises from a balance between the accelerating electric force and the decelerating effect of scattering events with lattice vibrations, impurities, or other obstacles. The electric force on an electron is -[e](/page/E!) \mathbf{E} (where [e](/page/E!) > 0 is the elementary charge magnitude), imparting an acceleration \mathbf{a} = - ([e](/page/E!) / m^*) \mathbf{E}, with m^* as the effective mass accounting for band structure effects in the solid. Scattering introduces a frictional drag, approximated in the relaxation time model as a momentum loss term proportional to the velocity, with relaxation time \tau representing the average time between collisions. In equilibrium, this balance yields the drift velocity \mathbf{v}_d = - ([e](/page/E!) \tau / m^*) \mathbf{E}.[1] To derive this formally, start with the semiclassical equation of motion for the average electron velocity \mathbf{v}: m^* \frac{d\mathbf{v}}{dt} = -e \mathbf{E} - \frac{m^* \mathbf{v}}{\tau}, where the second term on the right models the exponential decay of momentum toward zero after each scattering event, assuming carriers lose their drift momentum upon collision. For steady-state conditions, d\mathbf{v}/dt = 0, so -e \mathbf{E} = \frac{m^* \mathbf{v}_d}{\tau} \implies \mathbf{v}_d = -\frac{e \tau}{m^*} \mathbf{E}. The electron mobility \mu_n is defined as the proportionality constant between |\mathbf{v}_d| and E, giving \mu_n = e \tau / m^*, such that \mathbf{v}_d = -\mu_n \mathbf{E}. This derivation relies on the relaxation time approximation from the Boltzmann transport equation, where \tau is taken as constant.[9][7] The model assumes low electric fields where the linear response v_d \propto E holds, an isotropic material with uniform scattering, and neglect of magnetic fields or other external influences that could alter carrier trajectories.[1]Transport Relations
Relation to Current Density
The drift current density arises from the collective motion of electrons under an applied electric field, directly linking microscopic electron mobility to macroscopic charge transport in materials such as semiconductors and metals. When an electric field \mathbf{E} is applied, electrons experience a drift velocity \mathbf{v}_d = -\mu \mathbf{E}, where \mu is the electron mobility (positive by convention), leading to a net current due to the imbalance in electron flow. The resulting drift current density for electrons is expressed as \mathbf{J}_d = n e \mu \mathbf{E}, where n is the electron number density and e is the elementary charge ($1.602 \times 10^{-19} C). This formula quantifies how efficiently electrons contribute to current flow, with higher mobility yielding greater current for a given field and carrier density.[10][7] From experimental measurements of current density under controlled conditions, electron mobility can be determined by rearranging the relation: \mu = \frac{J_d}{n e E}, assuming isotropic conditions and known n and E. This inversion is particularly useful in device characterization, where applied fields generate measurable currents to infer material properties. In the context of Ohm's law, \mathbf{J} = \sigma \mathbf{E} (with \sigma as electrical conductivity), the drift current forms the primary contribution in low-field regimes for extrinsic semiconductors, bridging microscopic carrier dynamics to the material's overall resistive behavior.[10] For anisotropic materials, such as certain crystals or layered semiconductors where electron transport varies by direction due to band structure asymmetry, the scalar mobility is replaced by a second-rank mobility tensor \mu_{ij}. The drift current density then takes the vector form J_{d,i} = n e \sum_j \mu_{ij} E_j, capturing direction-dependent responses; for example, in hexagonal crystals like graphene derivatives, \mu_{xx} may differ significantly from \mu_{zz}. This tensorial description is essential for modeling transport in non-cubic lattices, ensuring accurate predictions of current flow in devices exploiting anisotropy.[11]Relation to Electrical Conductivity
In the classical Drude model of electrical conduction in metals, the electrical conductivity \sigma is directly related to electron mobility \mu through the formula \sigma = n e \mu, where n is the electron density and e is the elementary charge.[12] This expression arises from the drift velocity of electrons under an applied electric field, where the average drift speed v_d = \mu E leads to a current density J = n e v_d = n e \mu E, and thus \sigma = J / E = n e \mu. The model, proposed by Paul Drude in 1900, treats conduction electrons as a classical gas subject to random scattering, with mobility \mu = e \tau / m incorporating the relaxation time \tau between collisions and the electron mass m.[12] For semiconductors, the relation extends to account for both electrons and holes: the total conductivity is \sigma = e (n \mu_e + p \mu_h), where n and p are the electron and hole densities, respectively, and \mu_e and \mu_h are their respective mobilities.[13] In intrinsic semiconductors, n = p = n_i, simplifying to \sigma = e n_i (\mu_e + \mu_h), while in extrinsic cases, one carrier type dominates, such as \sigma \approx e n \mu_e for n-type materials. This formulation highlights how mobility influences charge transport efficiency, with higher \mu values yielding greater \sigma and correspondingly lower resistivity \rho = 1/\sigma, enabling better electrical performance in devices.[14] The dependence on effective mass m^* further explains variations across materials: in the quantum mechanical refinement, \mu \propto 1/m^*, as the effective mass accounts for lattice interactions in band structures, replacing the free electron mass in Drude's original expression./Solar_Basics/C._Semiconductors_and_Solar_Interactions/II._Conduction_in_Semiconductors/4%3A_Carrier_Drift_and_Mobility) Drude's classical approach was refined in 1928 by Arnold Sommerfeld using Fermi-Dirac statistics, which better describes the Pauli exclusion principle for degenerate electron gases in metals, while Felix Bloch's band theory incorporated periodic lattice potentials to validate the conductivity-mobility link quantum mechanically.[15] These advancements preserved the core relation \sigma = n e \mu but provided a more accurate foundation for understanding material-specific behaviors.Relation to Electron Diffusion
In semiconductors, the diffusion current arises from the random thermal motion of charge carriers, leading to a net flow from regions of higher concentration to lower concentration. For electrons, this is expressed as J_{\text{diff}} = -e D \nabla n, where e is the elementary charge, D is the diffusion coefficient, and n is the electron concentration.[7][16] This component of current is distinct from drift current but contributes to the total transport under non-uniform conditions. The Einstein relation connects the diffusion coefficient D to the electron mobility \mu through D = \frac{kT}{e} \mu, where k is Boltzmann's constant and T is the absolute temperature. This relation is derived by considering thermal equilibrium, where the net current is zero, balancing the drift current due to a built-in electric field and the diffusion current due to a concentration gradient. In equilibrium, the carrier concentration follows n = n_0 \exp\left(-\frac{eU}{kT}\right), leading to \nabla n = -\frac{e}{kT} n \nabla U, and setting the total flux j = e \mu n (-\nabla U) - e D \nabla n = 0 yields the Einstein relation.[17][7] Thermoelectric effects, such as the Seebeck effect, rely on this relation, as the Seebeck coefficient S, which measures the voltage generated by a temperature gradient, incorporates the ratio D / \mu = kT / e to describe the diffusion-driven separation of carriers. In non-degenerate semiconductors, S for electrons is approximately S = -\frac{k}{e} \left( \frac{E_c - E_F}{kT} + A \right), where A accounts for scattering, and the Einstein relation ensures consistency between thermal diffusion and mobility in the underlying transport equations.[16][7] The standard Einstein relation applies to non-degenerate semiconductors, where carrier statistics follow Maxwell-Boltzmann distributions. For degenerate cases, such as heavily doped materials where the Fermi level enters the conduction band, generalizations incorporate Fermi-Dirac integrals, modifying the relation to D = \frac{kT}{e} \mu \frac{\mathcal{F}_{1/2}(\eta)}{\mathcal{F}_{-1/2}(\eta)}, where \eta is the reduced Fermi energy and \mathcal{F}_j are Fermi-Dirac integrals.[18][7]Examples
In Traditional Semiconductors and Metals
In traditional semiconductors, electron mobility values provide a benchmark for carrier transport in established materials used in electronics. For intrinsic silicon at 300 K, the electron mobility is approximately 1400 cm²/(V·s), dominated by phonon scattering in lightly doped conditions.[7] Germanium exhibits higher mobility, with μ_e ≈ 3900 cm²/(V·s) in its intrinsic form at room temperature.[19] Gallium arsenide, a III-V compound semiconductor, shows even greater values, around 8500–8800 cm²/(V·s) for intrinsic material, enabling faster device speeds compared to silicon.[20] In metals, electron mobility is generally much lower due to the high density of conduction electrons (n ≈ 10^{22}–10^{23} cm^{-3}), despite high electrical conductivity. For copper, a prototypical conductor, μ_e ≈ 43 cm²/(V·s) at room temperature, limited primarily by electron-electron and electron-phonon interactions.[21] These mobility values are sensitive to material purity, where impurities introduce scattering that reduces μ_e, and to temperature, with phonon scattering causing a decrease as temperature rises.[7]| Material | Electron Mobility (cm²/(V·s)) | Conditions |
|---|---|---|
| Silicon (Si) | 1400 | Intrinsic, 300 K |
| Germanium (Ge) | 3900 | Intrinsic, room T |
| GaAs | 8500–8800 | Intrinsic |
| Copper (Cu) | 43 | Bulk, room T |