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A dielectric is a material in which an electric field produces electric polarization while transport of electric charge is sufficiently small for the intended use. An insulating solid in a capacitor is an example; real dielectrics can still have dielectric loss and nonzero electrical conductivity.

The important distinction is between rearranging bound electric charge and transporting mobile charge carriers through the specimen. Bound electric charges can move a little within a chemical atom or molecule, or an existing electric dipole can turn. Their averaged electric dipole moments produce electric polarization even if there is negligible steady electric current between the electrodes. A changing electric polarization does give a bound current; the absence of steady conduction does not mean that the material is electrically unresponsive.^[ref]

For uniform electric polarization $\mathbf{P}=P\hat{\mathbf{x}}$ in a rectangular slab, polarization density gives zero bound volume charge density in the interior and opposite bound surface charge densities $-P,+P$ on the left and right faces. The electric charge has been redistributed, not created. More generally, the divergence theorem makes the cancellation explicit:Here $\mathbf{n}$ is the outward normal vector on the material boundary. This accounts for both bound volume charge density and bound surface charge density; omitting either can falsely suggest that an initially neutral dielectric acquired net electric charge.

An electrical insulator between capacitor plates supports a useful combination: appreciable electric polarization, little leakage electric current, and tolerance of the intended electric field. This lets a small parallel-plate capacitor store separated electric charge and recoverable capacitor energy. The comparison in dielectric capacitor response explains why specifying fixed electric charge or fixed electric potential difference changes the answer. High relative permittivity alone does not establish good insulation or a large usable energy density; dielectric loss and dielectric breakdown matter as well.

An electrical insulator has sufficiently small transport of electric charge for its operating conditions. Insulation is a practical requirement involving electric potential difference, temperature, geometry and duration, rather than a universal claim of zero electrical conductivity. Electric polarization can be substantial in an electrical insulator; dielectric breakdown can end the insulating regime.

For an idealized uniform slab with small constant electrical conductivity $\sigma$, the Ohmic conductor model predicts leakage electrical resistance $R=d/(\sigma A)$ in parallel with capacitance $C=ϵA/d$. An isolated capacitor then hasThis is a derived model of slow discharge, not an assertion that every leakage mechanism obeys Ohm's law. Its time scale matches charge relaxation. The fixed-charge comparison in dielectric capacitor response needs an observation time short compared with $RC$, but long enough for electric polarization to settle. During a maintained voltage, leakage consumes power $V^2/R$ as Joule heating; delayed electric polarization can add dielectric loss under changing electric fields.^[ref]

Large relative permittivity alone does not rule out either mechanism.^[ref]

A change between a material behaving as an electrical conductor and as an electrical insulator. The distinction concerns the limiting electronic response as temperature tends to zero; finite thermally activated electrical conductivity alone does not establish a metallic state. A Peierls transition opens a band gap through a change in the crystal structure.

polysulfur has a chain of S–S sigma bonds with filled lone-pair orbitals. Its occupied states do not supply a partly filled delocalized band, so the chain is an electrical insulator.

In polythiazyl, an approximately planar S–N chain permits overlap of orbitals perpendicular to the backbone. In an elementary pi bond electron picture, nitrogen supplies one electron and sulfur two to this manifold: three pi electrons per SN unit, rather than a completely filled, isolated two-centre bonding pair. The extended states are therefore not all fully occupied. More realistic descriptions include the crystallographic repeat, interchain interactions and overlapping bands crossing the Fermi level; the odd electron count of a chosen chemical repeat is suggestive, not by itself a proof of metallicity.

The essential distinction is occupied, localized bonding/lone-pair states in sulfur chains versus available extended states at the Fermi level in crystalline sulfur nitride. The measured high conductivity along the polymer fibres and its metallic temperature dependence provide the experimental basis, alongside the structural evidence for the chain. Conductivity depends on order and defects as well as chemical composition.^[ref]

At a valley $K$, $\gamma (K)=1+2\cos (2\pi /3)=0$. For h-BN the inequivalent sublattice energies therefore leave the direct two-band gapGraphene's equivalent carbon sites instead have ${ϵ}_B={ϵ}_N$ in this model, so the two bands touch at the valleys. Near a valley, $|\gamma (K+\mathbf{q})|\simeq (\sqrt{3}a/2)|\mathbf{q}|$: graphene has approximately linear, gapless Dirac cones while h-BN has gapped band edges. The model panel in the original figure in solution displays this distinction.

First, undoped h-BN is an electrical insulator with thermally suppressed carriers, whereas neutral graphene is a zero-gap semimetal whose transport is readily changed by gating. Disorder, temperature and contacts still affect its measured conductivity. Second, h-BN suppresses interband absorption below its large gap and is transparent over much of the visible range, whereas graphene permits low-energy interband absorption unless occupations block it. A quantitative optical threshold in real h-BN also involves additional bands and strong excitonic effects; a two-orbital model should not be presented as a numerical prediction of its complete optical spectrum.

For this radially layered coaxial cable, defineUse the solution regular at the axis with bounded exterior electric potential, so that the empty interior and exterior have the constant electric potentials fixed by their adjacent conducting cylinders.

There is no free charge in either electrical insulator or on their interface. Gauss's law and continuity of normal electric displacement field give $D_{\rho }=C/\rho$ in both layers. The electric potential difference fixes the constant becauseThe electric potential, including the empty regions bounded by the conducting cylinders, is thereforeBoth inner and outer Dirichlet boundary conditions hold, and the electric potential is continuous at the material interface, whereTaking the negative gradient of the electric potential yieldsThe radial electric field generally jumps at $\rho =c$, but $ϵE_{\rho }(c^{-})={ϵ}^{\prime }{E}_{\rho }(c^{+})$, as required by the absence of free surface charge density there. The conducting cylinders carry opposite linear charge densities $\lambda =2\pi C$ and $-\lambda$.

For an Amperian loop of radius $\rho$, the enclosed electric current is zero, $I$, or zero according as $\rho <a$, $a<\rho <b$, or $\rho >b$. The magnetostatic Ampère law therefore gives $H_{\phi }=I/(2\pi \rho )$ in the gap, and zero outside. Using the two magnetic permeabilities,The tangential magnetic field strength is continuous at the material interface, while the tangential magnetic field need not be. At each ideal conducting surface and at the material interface, the displayed electric field and magnetic field formulas specify one-sided limits; a single interface value is unnecessary when the limiting values differ.

In an electrical insulator, lattice vibrations usually carry most heat. A small temperature gradient produces an imbalance between phonons moving in opposite directions. In an isotropic kinetic approximation,where $c_V$ is heat capacity per unit volume, $v_s$ an appropriate acoustic speed and ${\ell }_{\mathrm{tr}}$ the transport mean free path. The factor one third is the angular average of the squared direction cosine. With several branches and frequency-dependent scattering, replace this estimate by the corresponding weighted mode integral. A short lifetime matters for conductivity only insofar as it relaxes heat current.

For a clean three-dimensional crystal at $T\ll {\Theta }_D$, the Debye heat capacity varies as $T^3$. If diffuse boundary scattering fixes ${\ell }_{\mathrm{tr}}$ near a specimen dimension, then ${\kappa }_{\mathrm{ph}}\propto T^3$. The size dependence is a test of that boundary-limited regime. Specular surfaces, defects and small specimens can change the simple law; at ballistic scales a local conductivity description itself needs care.

As temperature rises, more phonons are excited, but phonon-phonon scattering shortens their transport path. Normal processes conserve total crystal momentum, whereas an Umklapp process satisfies ${\mathbf{q}}_1+{\mathbf{q}}_2-{\mathbf{q}}_3=\mathbf{G}\ne 0$ with reciprocal vector $\mathbf{G}$ and can relax directed heat flow. Low-temperature Umklapp scattering is strongly suppressed; its increasing importance often produces a conductivity maximum followed by a decline. Isotope and impurity scattering lower or broaden that maximum.

At sufficiently high temperature, $c_V$ approaches its classical constant while a typical anharmonic Umklapp rate grows approximately as $T$. Under that further approximation,Thus a clean crystalline insulator commonly shows a low-temperature rise, a peak, then a high-temperature fall. This is not a universal curve for every insulator: glasses, strong disorder, multiple scattering mechanisms, radiative heat transfer and structural transitions can give different behavior.

A steady-state bar experiment supplies a concrete test. Inject known heating power, measure the small temperature difference between two separated sensors and infer $\kappa =PL/(A\Delta T)$ after correcting for heat leaks. Keep the gradient small enough that a single representative temperature is meaningful, calibrate sensor contact effects and subtract radiative/background losses. Repeat with different widths or isotope/defect content to distinguish boundary limitation from intrinsic scattering. A power-dependent result may reflect a changing temperature profile rather than a new microscopic law.

In an ordinary three-dimensional crystalline electrical insulator, heat is carried mainly by phonons. Kinetic theory gives the estimatewhere $c_V$ is heat capacity per unit volume, $v_s$ is a typical sound speed, and $\ell$ is the phonon mean free path. The thermal conductivity $\kappa$ has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$.

At low temperature, $T\ll {\Theta }_D$, the Debye model gives $c_V\propto T^3$. With phonon-phonon collisions frozen out, diffuse scattering at the sample boundaries makes $\ell$ approximately constant, and $v_s$ is nearly constant. HenceAt high temperature, of order or above the Debye temperature, the Dulong-Petit law makes $c_V$ approximately constant. Anharmonic phonon scattering, especially resistive Umklapp scattering, has a collision rate approximately proportional to $T$. Thus $\ell \propto T^{-1}$ andThese two regimes usually give an intermediate maximum of thermal conductivity. The assumptions matter: glassy materials, strong defects or specular boundary reflection can change the low-temperature law; these are not universal laws for every electrical insulator.

An electrical insulator has no freely conducting bulk charge in the ideal static description, but an applied electric field can displace bound positive and negative charge relative to one another. The resulting electric dipole moments per unit volume define the polarization density $\mathbf{P}$. Distortion of electronic charge clouds, relative displacement of ions, and partial alignment of permanent molecular electric dipoles can all contribute. Thermal agitation opposes the last mechanism.

For a weak field in a linear isotropic dielectric, the constitutive law isHere ${\chi }_e$ is the electric susceptibility, $ϵ$ is the permittivity, and $\mathbf{D}$ is the electric displacement field. In an anisotropic material, ${\chi }_e$ is a tensor and $\mathbf{P}$ need not be parallel to $\mathbf{E}$. These response laws describe the macroscopic field, not necessarily the microscopic field acting on one molecule.

Spatial variation and surface termination of the polarization density give bound charge and bound current through the bound charge densitieswhere $\hat{\mathbf{n}}$ is outward from the material. Thus uniform polarization density has no bulk bound charge, but generally has opposite bound charges on opposing surfaces. Gauss's law can be rewritten asWith no free surface charge, the normal component of $\mathbf{D}$ is continuous across an interface; the tangential electric field is continuous in electrostatics.

For example, fill a parallel-plate capacitor of plate area $A$ and gap $d$ with a uniform dielectric, neglecting fringing. At fixed free plate charge $Q$, $D=Q/A$, soThe induced bound surface charges oppose the original field, reducing the electric field and potential difference at fixed $Q$. At fixed applied potential difference, the field remains $V/d$ and the battery instead supplies more free charge. The electric displacement field separates these free-charge and bound-charge roles.

A positive electric potential difference between gate and substrate attracts electrons towards the insulating interface and repels the mobile electron holes of the p-type semiconductor substrate. The exposed, negatively charged acceptor impurities form a depletion region; the thin mobile electron layer at its surface is the semiconductor inversion layer. The conduction band and valence band bend downwards towards the positively biased gate, because the electron potential energy is $-e\psi$.

To specify the conventional MOS inversion threshold, let $N_A$ be the ionized acceptor density, $n_i$ the intrinsic carrier concentration, andIn the neutral bulk the intrinsic level lies $e{\varphi }_F$ above the semiconductor chemical potential. A positive surface potential ${\psi }_s$ lowers both band edges by $e{\psi }_s$, giving, in the nondegenerate approximation,At ${\psi }_s={\varphi }_F$, the surface is intrinsic: $n_s=p_s=n_i$. The usual onset of strong inversion instead meansThus the surface electron density has grown to the bulk majority-carrier density. The chemical potential need not actually enter the conduction band. The phrase “just forms” can also refer to the earlier weak-inversion crossover; the diagram uses the conventional strong-inversion threshold and labels it explicitly.

Within the depletion approximation the negative space charge is approximately $-e{N}_A$ over a widthwhere ${ϵ}_s$ is the absolute semiconductor permittivity. Its outer edge is where the bands become flat and mobile electron holes recover their bulk density. The electrical insulator blocks steady carrier transport while sustaining an electric field; the metal gate supplies positive surface charge balancing depletion and inversion charge. Under gate bias, the gate and substrate electrochemical potentials are different, so a single equilibrium Fermi level should not be drawn through the whole biased stack.

The large effective field primarily represents exchange interactions, which arise from Coulomb interactions, electronic hopping and the Pauli exclusion principle. It is not usually the much smaller classical magnetic dipole field of neighbouring ions. Direct exchange involves overlapping magnetic-ion orbitals. Superexchange instead uses virtual charge transfer through a ligand while real charge transport remains gapped, allowing substantial magnetic order in an electrical insulator.

Use dimensionless spin operators ${\mathbf{S}}_i$ with eigenvalue $S=1/2$, and count each bond once. One consistent Heisenberg model convention isThen $J_{ij}<0$ favours a Heisenberg ferromagnet, while $J_{ij}>0$ favours a Heisenberg antiferromagnet. Replacing neighbouring spins by their mean values gives the effective fieldFor $z$ equivalent ferromagnetic neighbours with $J_{ij}=-J_F<0$ and uniform order, $M=n{g}_J{\mu }_B⟨S_z⟩$, soThe last equality is the same molecular-field result derived above, not an exact transition temperature for every lattice.

Microscopically, a half-filled orbital pair with virtual hopping $t_{ij}$ and a large on-site repulsion $U$ permits an energy lowering of the singlet but not the parallel-spin triplet, yielding antiferromagnetic $J\simeq 4|t_{ij}{|}^2/U$ in this convention. Ligand-mediated hopping is the analogous mechanism in many ionic insulators. Different orbital occupancies and nearly orthogonal ligand paths can suppress this antiferromagnetic term and permit ferromagnetic exchange to dominate. Thus the sign depends on electronic structure and bonding geometry, not simply whether the sample conducts.

For antiferromagnetic coupling on a bipartite lattice, the field on each sublattice is proportional to the opposite sublattice magnetization, with opposite alignment preferred. Two coupled self-consistency equations then permit $M_A=-M_B\ne 0$ and vanishing total magnetization: the order parameter is staggered. The single positive-$\lambda$ uniform equation describes ferromagnetism and must be extended to these sublattices to describe an antiferromagnet.

An Umklapp process conserves reduced crystal momentum modulo a nonzero reciprocal lattice vector. It can occur in electron-phonon scattering as well as phonon-phonon scattering; the lattice takes up the reciprocal-lattice momentum.

In a three-phonon Umklapp scattering process, the first-zone wavevectors obey the displayed equation for a nonzero reciprocal lattice vector $\mathbf{G}$, together with conservation of energy. The crystal momentum transferred to the lattice allows these processes to relax a phonon heat current. They are suppressed at sufficiently low temperature and often determine the approximate $1/T$ high-temperature thermal conductivity of a crystalline electrical insulator.