Conductivity
Conductivity is a fundamental property of materials that measures the ease with which electric charge or heat can pass through them without the bulk motion of the material itself.[1] In physics and engineering, it primarily encompasses electrical conductivity, which governs the flow of electric current, and thermal conductivity, which determines the transfer of heat energy.[1] These properties are essential for classifying materials as conductors, semiconductors, or insulators and play critical roles in applications ranging from electrical wiring to heat exchangers.[1] Electrical conductivity (denoted as σ) quantifies a material's ability to conduct electric current and is defined by the relation between conduction current density \mathbf{J} and the applied electric field intensity \mathbf{E}, given by Ohm's law in the form \mathbf{J} = \sigma \mathbf{E}.[2] The SI unit of electrical conductivity is the siemens per meter (S/m), where 1 S/m = 1 A/(V·m).[2] Metals exhibit the highest values, often exceeding $10^7 S/m, due to the abundance of free electrons that serve as charge carriers; for instance, silver has the highest electrical conductivity among elements (rated at 105% on the International Annealed Copper Standard (IACS), where annealed copper is 100%), followed by copper (100%) and gold (70%).[3] In contrast, insulators like ceramics or plastics have very low conductivity, typically $10^{-12} to $10^{-15} S/m or less, while semiconductors fall in between ($10^{-6} to $10^4 S/m) and their conductivity can be tuned by doping or temperature.[2] Thermal conductivity (often denoted as κ or λ) measures a material's capacity to conduct heat through atomic or molecular interactions, such as lattice vibrations (phonons) in non-metals or free electron movement in metals, without any net displacement of the material.[4] It is defined by Fourier's law of heat conduction, \mathbf{q} = -\kappa \nabla T, where \mathbf{q} is the heat flux vector and \nabla T is the temperature gradient; the SI unit is watts per meter-kelvin (W/(m·K)).[4] Metals generally have high thermal conductivity—for example, copper at approximately 400 W/(m·K) at room temperature—owing to electron contributions, whereas non-metallic solids like diamond (1800–2200 W/(m·K) at room temperature) excel due to phonon transport, and insulators like wood or air have low values (less than 0.1 W/(m·K)).[5] The Wiedemann-Franz law relates electrical and thermal conductivities in metals, stating that their ratio is proportional to temperature, highlighting the shared role of electrons in both processes.[4]Overview
Definition and Scope
Conductivity is a fundamental material property that quantifies the ability of a substance to transmit energy or matter, such as electric current, heat, or fluids, through it. In physics, it is mathematically defined as the inverse of resistivity (ρ), expressed as σ = 1/ρ, where σ represents conductivity; this relationship highlights how conductivity measures the ease of flow in response to an applied driving force, such as an electric field or temperature gradient.[6] For electrical conductivity, this concept ties directly to Ohm's law, which states that the current density (J) is proportional to the electric field (E), with conductivity serving as the proportionality constant (J = σ E), assuming basic familiarity with the law's implication that higher conductivity implies lower opposition to current flow.[7] The scope of conductivity in physics primarily encompasses electrical and thermal variants, which describe the transmission of electric charge and heat, respectively, within materials. Electrical conductivity arises from the motion of charge carriers like electrons or ions under an electric field, while thermal conductivity involves the transfer of thermal energy through atomic vibrations (phonons) or free electrons without bulk material movement.[8][4] Extensions of the concept appear in other domains, such as ionic conductivity in electrolytes (governed by ion mobility), hydraulic conductivity for fluid permeation through porous media, and thermal diffusivity (related but distinct from thermal conductivity by incorporating material density and specific heat).[9][10] These broader applications share the core idea of transport efficiency but are not detailed here, as the focus remains on electrical and thermal conductivity in solid-state physics and materials science. In the International System of Units (SI), electrical conductivity is measured in siemens per meter (S/m), reflecting its role in current conduction per unit length and cross-section.[11] Thermal conductivity, conversely, uses watts per meter-kelvin (W/(m·K)), indicating the heat flux per unit area driven by a unit temperature gradient.[12] These units underscore conductivity's dimensional dependence on geometry and driving force, enabling standardized comparisons across materials. Temperature influences both types of conductivity, with effects varying by material class, though specifics are addressed elsewhere.[13]Historical Development
The concept of conductivity emerged from early experiments on electrical and thermal phenomena in the 18th and early 19th centuries. In the late 1700s, Charles-Augustin de Coulomb conducted pivotal experiments using a torsion balance to quantify the electrostatic forces between charged bodies, laying foundational observations for electrical conduction in metals and insulators.[14] Concurrently, Alessandro Volta's invention of the voltaic pile in 1800 provided a steady source of electric current, enabling systematic studies of conduction through various materials and demonstrating that electricity could flow continuously in metallic circuits.[15] For thermal conduction, Joseph Fourier's 1822 treatise Théorie Analytique de la Chaleur formalized the diffusion of heat through solids, introducing the heat equation and establishing conductivity as a material property proportional to heat flux and temperature gradient.[16] A major milestone came in 1827 when Georg Simon Ohm published Die galvanische Kette, mathematisch bearbeitet, introducing Ohm's law, which states that the voltage drop across a conductor is proportional to the current through it and its resistance (V = IR), thereby defining electrical conductivity σ as the reciprocal of resistivity.[17] This macroscopic relation later inspired microscopic interpretations, such as the Drude expression for electron conductivity σ = n e² τ / m, where n is electron density, e is charge, τ is relaxation time, and m is mass.[18] In 1853, Gustav Wiedemann and Rudolf Franz empirically discovered the Wiedemann-Franz law, revealing that the ratio of thermal conductivity κ to electrical conductivity σ is proportional to temperature (κ / σ ∝ T) for metals, suggesting a common carrier for both heat and electricity.[19] Key contributors refined these ideas in the mid-19th century. Gustav Kirchhoff's 1845 circuit laws generalized Ohm's work by applying conservation of charge and energy to networks of conductors, enabling analysis of conductivity in complex systems.[20] James Clerk Maxwell incorporated conductivity into his 1865 electromagnetic equations, modifying Ampère's law to include the conduction current density J = σ E, unifying electrical conduction with broader electromagnetic phenomena. The 20th century brought quantum mechanical advancements. In 1900, Paul Drude proposed a classical model treating conduction electrons as a gas scattering off ions, deriving expressions for both electrical and thermal conductivities that qualitatively matched experiments.[21] Arnold Sommerfeld refined this in 1927 by applying Fermi-Dirac statistics to the free electron gas, improving quantitative predictions for conductivity at low temperatures while retaining Drude's core framework.[22] Felix Bloch's 1928 theorem introduced wave-like electron states in periodic lattices, providing a quantum basis for conductivity in crystalline solids and explaining band structures that distinguish metals from insulators.[23]Electrical Conductivity
Basic Principles
Electrical conductivity arises from the motion of charge carriers, primarily electrons in solids, under an applied electric field. The foundational microscopic description is provided by the free electron model, proposed by Paul Drude in 1900, which treats conduction electrons as a classical gas of non-interacting particles that move freely except for occasional collisions with ions or impurities. In this model, electrons acquire a drift velocity in response to the field, leading to a net current. The average time between collisions, known as the relaxation time \tau, plays a central role in determining how effectively electrons respond to the field. The electrical conductivity \sigma is derived from the Drude model as \sigma = \frac{n e^2 \tau}{m}, where n is the density of free electrons, e is the electron charge, and m is the electron mass. This expression quantifies the material's ability to conduct current, with higher electron density or longer relaxation times yielding greater conductivity. From this, Ohm's law in microscopic form follows: the current density \mathbf{J} relates to the electric field \mathbf{E} by \mathbf{J} = \sigma \mathbf{E}. The derivation assumes that the drift velocity v_d = -\frac{e \tau}{m} E balances the acceleration from the field against collisional damping, resulting in a steady-state current \mathbf{J} = -n e \mathbf{v}_d. Several factors influence conductivity through their effect on the relaxation time \tau. Temperature typically decreases \sigma in metals because higher thermal energy increases phonon scattering, shortening \tau. Similarly, impurities and lattice defects introduce additional scattering centers, reducing \tau and thus lowering conductivity compared to pure, crystalline materials.[24] While the Drude model successfully explains Ohm's law and basic temperature dependence, it has notable limitations as a classical theory. For instance, it overestimates the electronic contribution to specific heat and fails to accurately predict the Hall coefficient's magnitude.[25] These shortcomings motivated quantum mechanical refinements, such as the Sommerfeld free electron gas model, which incorporates Fermi-Dirac statistics to better describe electron behavior near the Fermi level without altering the core conductivity formula significantly. The Wiedemann-Franz law connects electrical conductivity to thermal conductivity, stating that their ratio is proportional to temperature in metals, reflecting shared electron transport mechanisms.[4]Conductivity in Different Materials
Electrical conductivity varies significantly across different classes of materials, primarily due to differences in their electronic structures and the availability of charge carriers. Metals demonstrate exceptionally high electrical conductivity, attributed to the abundance of free electrons in their conduction bands that can move readily under an applied electric field.[9] For instance, copper exhibits a conductivity of approximately 5.96 × 10^7 S/m at 20°C.[26] Metals generally possess a positive temperature coefficient of resistivity, meaning their conductivity decreases as temperature rises because increased thermal vibrations scatter the free electrons more effectively.[27] Semiconductors display moderate conductivity that lies between that of metals and insulators, arising from a small band gap that allows some electrons to be thermally excited from the valence band to the conduction band./07:_The_Crystalline_Solid_State/7.01:_Molecular_Orbitals_and_Band_Structure/7.1.04:_Semiconductors-_Band_Gaps_Colors_Conductivity_and_Doping) In intrinsic semiconductors like pure silicon, the conductivity is low, around 4.35 × 10^{-4} S/m at room temperature, as it depends solely on the limited number of thermally generated electron-hole pairs.[28] Extrinsic semiconductors achieve higher conductivity through doping, where intentional addition of impurities introduces excess electrons (n-type) or holes (p-type), dramatically increasing the charge carrier concentration even at low dopant levels on the order of parts per million./07:_The_Crystalline_Solid_State/7.01:_Molecular_Orbitals_and_Band_Structure/7.1.04:_Semiconductors-_Band_Gaps_Colors_Conductivity_and_Doping) Unlike metals, semiconductors have a negative temperature coefficient of resistivity, with conductivity rising as temperature increases due to enhanced thermal excitation of carriers across the band gap.[29] Insulators are characterized by extremely low electrical conductivity, typically resulting from a large band gap that hinders electron promotion to the conduction band under normal conditions./Semiconductors/Band_Theory_of_Semiconductors) For example, glass shows conductivity below 10^{-12} S/m at room temperature, as the wide energy separation—often exceeding 4 eV—prevents significant charge carrier generation and flow.[28] Superconductors represent a unique class where electrical resistance drops to zero below a critical temperature, enabling dissipationless current flow.[30] This phenomenon is accompanied by the Meissner effect, in which the material completely expels magnetic fields from its interior, distinguishing type-I superconductors.[30] The Bardeen-Cooper-Schrieffer (BCS) theory explains this behavior microscopically: at low temperatures, electrons pair into Cooper pairs via lattice vibrations (phonons), allowing coherent motion without scattering and thus achieving perfect conductivity.[31] The following table provides representative electrical conductivity values (σ in S/m at approximately 20°C, unless noted) for common materials across these categories, illustrating the wide range of behaviors.[28]| Material | Conductivity (S/m) | Category |
|---|---|---|
| Silver | 6.30 × 10^7 | Metal |
| Copper | 5.96 × 10^7 | Metal |
| Aluminum | 3.77 × 10^7 | Metal |
| Silicon (intrinsic) | 4.35 × 10^{-4} | Semiconductor |
| Germanium (intrinsic) | 2.17 | Semiconductor |
| Glass | < 10^{-12} | Insulator |
| Fused quartz | ~10^{-16} | Insulator |