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Let $e>0$ denote the elementary charge, so an electron's charge is $-e$. In the relaxation-time Drude model with positive transport effective mass,The steady drift velocity is ${\mathbf{v}}_d=-e\tau \mathbf{E}/m^{\ast }=-\mu \mathbf{E}$, defining positive electron mobility $\mu =e\tau /m^{\ast }$. The current density is opposite to the electron drift:By the definition of electrical conductivity, $\mathbf{J}=\sigma \mathbf{E}$, soIts units are $({\mathrm{m}}^{-3})(\mathrm{C})({\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1})=\mathrm{S}{\mathrm{m}}^{-1}$. The mobility definition alone establishes the result in the scalar linear-response regime; the relaxation-time equation supplies a microscopic approximation for $\mu$.

Doping changes the mobile carrier population by introducing controlled impurity levels. In an intrinsic semiconductor, the filled valence band is separated from the nearly empty conduction band by a band gap $E_g$. Thermal excitation produces an electron-hole pair, so the electron and hole number densities obey $n=p=n_i$. A electron hole represents a missing valence-band electron and behaves as a mobile positive charge carrier. For nondegenerate bands in thermal equilibrium,Here $N_C,N_V$ are the effective density of states, $E_C,E_V$ the band edges, and $E_F$ denotes the chemical potential, conventionally called the Fermi level. The semiconductor mass-action law holds in this nondegenerate equilibrium regime.

In a tetrahedrally bonded elemental semiconductor such as silicon, a substitutional group-V atom has an extra outer electron relative to the host. It is a donor impurity: a shallow level just below $E_C$ can ionise, releasing an electron to the conduction band and leaving a fixed positively charged donor. The resulting n-type semiconductor has electrons as majority carriers and holes as minority carriers. Conversely a group-III substitution supplies one fewer bonding electron. An acceptor impurity can take an electron from the valence band, leaving a mobile electron hole and a fixed negative acceptor; the resulting p-type semiconductor has holes as majority carriers.

The impurity state is shallow because the electron or hole responds to a screened Coulomb potential and an effective mass, rather than the bare atomic potential. A simple hydrogenic donor estimate iswhere ${ϵ}_r$ is the relative permittivity. This is a useful scaling model, with real band anisotropy and central-cell effects omitted. Dopants are fixed in the lattice; their ionisation changes mobile carrier numbers without making a uniformly doped bulk sample carry a large net electric charge.

The charge neutrality condition isWhen shallow dopants are almost fully ionised and $N_D-N_A\gg n_i$, an n-type region has $n\simeq N_D-N_A$ and $p\simeq n_i^2/n$. A p-type region similarly has $p\simeq N_A-N_D$ and $n\simeq n_i^2/p$. Thus compensation by the opposite impurity type reduces the majority-carrier density; the total dopant concentration is not generally the mobile carrier density. At sufficiently high impurity density the degenerate semiconductor regime requires the full Fermi-Dirac distribution and these exponential approximations fail.

There are three useful temperature regimes. At low temperature, dopant freeze-out leaves many shallow levels un-ionised. In the extrinsic regime, the dopants are largely ionised and set the majority-carrier density. At high enough temperature, intrinsic pair creation dominates and $n,p$ approach $n_i$. The Fermi level moves towards the conduction band with donor doping and towards the valence band with acceptor doping. The electrical conductivity isDoping therefore controls conductivity and its carrier type, but ionised impurities also scatter carriers and can reduce charge carrier mobility. Carrier density and mobility must both be considered.

A p-n junction forms when p-type and n-type regions are joined within a semiconductor. Initially the concentration gradients drive diffusion current: electrons diffuse from the n region to the p region and holes in the opposite direction. Recombination removes mobile majority carriers near the join, exposing fixed positive donors on the n side and fixed negative acceptors on the p side. This forms a depletion region and an electric field directed from n to p. Its drift current opposes diffusion. At equilibrium the net current vanishes and the Fermi level is uniform, although the band edges vary spatially through band bending.

For an abrupt nondegenerate junction with fully ionised dopants, its built-in potential isUnder the depletion approximation, the space-charge densities are $-e{N}_A$ on a p-side width $w_p$ and $+e{N}_D$ on an n-side width $w_n$. Integrating Poisson equation and imposing zero field outside gives $N_A{w}_p=N_D{w}_n$ andMost of the depletion width therefore extends into the more lightly doped side. The schematic shows the impurity levels before contact and the resulting space charge and potential after contact; there is no externally extractable equilibrium battery voltage from the built-in field alone.

A forward bias, with p positive relative to n, reduces the barrier from $V_{\mathrm{bi}}$ towards $V_{\mathrm{bi}}-V$ and injects minority carriers across the junction. In the ideal low-injection, diffusion-controlled regime this produces the Shockley diode equation, $I=I_s[e^{eV/(k_{B}T)}-1]$. Reverse bias increases the barrier and the depletion region; only a small thermally generated minority-carrier current flows before breakdown. Reverse-biased depletion regions also give a voltage-dependent junction capacitance. Real devices have recombination, leakage, series resistance and breakdown mechanisms beyond the ideal law. Controlled doping creates majority and minority carriers, and their junction produces a depletion barrier that makes rectification possible.

The Hall effect is a transverse electric response to a magnetic force on moving charge carriers. Take conventional current density $\mathbf{J}=J_x\hat{\mathbf{x}}$ and magnetic field $\mathbf{B}=B_z\hat{\mathbf{z}}$ in a conducting slab. For one carrier type of number density $n$ and signed electric charge $q$, the drift velocity is ${\mathbf{v}}_d=\mathbf{J}/(nq)$. Carriers initially deflect sideways until a Hall electric field balances the magnetic Lorentz force:The Hall coefficient is thereforeFor electrons it is $-1/(ne)$, whereas a simple hole-dominated conductor gives $+1/(pe)$. The sign reveals the effective carrier sign; the magnitude estimates their density in the one-carrier approximation.

For width $w$ along $y$ and thickness $t$ along $z$, $I=J_{x}wt$. Define the signed Hall voltage by $V_H=E_{y}w$, namely the electric potential at the lower-$y$ edge minus that at the higher-$y$ edge. ThenThis explicit contact convention fixes the voltage sign. A Hall bar with longitudinal and transverse contacts measures both electrical resistivity and the Hall response. Reverse the field and take the part of the transverse voltage odd in $B_z$ to suppress longitudinal-voltage pickup from misaligned contacts; reversing current provides another useful check.

A Hall measurement is useful for carrier identification, semiconductor doping measurements and magnetic-field sensing. Combining a one-carrier $R_H$ with electrical conductivity gives a mobility magnitude $\mu =\sigma |R_H|$. However, mixed electron and hole conduction is not described by a single density: in weak fields,The response weights mobilities differently from ordinary conductivity. Complicated energy bands, anisotropic Fermi surfaces and multiple bands can likewise change the simple interpretation in metals; a positive Hall coefficient does not mean that the material contains positively charged fundamental electrons.

Semiconductor doping introduces controlled impurities to alter equilibrium carrier populations and electrical conductivity. In an intrinsic semiconductor, thermally excited electrons in the conduction band and electron holes in the valence band have equal densities $n=p=n_i$. A shallow donor impurity, such as phosphorus in silicon, supplies an electron from an impurity level just below the conduction-band edge. Once ionized it is positively charged, and the material is n type. An acceptor impurity, such as boron in silicon, accepts a valence electron into a level just above the valence-band edge, leaving a mobile hole and a negatively charged ionized impurity; the material is p type.

With dopant densities $N_D,N_A$, charge neutrality requiresFor fully ionized, nondegenerate dopants at equilibrium, the semiconductor mass-action law gives $np=n_i^2$. In a substantially n-type sample, $N_D-N_A\gg n_i$, soThe corresponding p-type result is $p\simeq N_A-N_D$ and $n\simeq n_i^2/(N_A-N_D)$. The majority population grows while the minority population decreases; adding donors does not simply add equal numbers of free electrons and holes.

The Fermi energy shifts toward the conduction band for n-type material and toward the valence band for p-type material. In the nondegenerate approximation,where $N_C,N_V$ are effective band-edge state densities. At very low temperature, dopants may not be ionized and carrier freeze-out occurs. At intermediate temperatures, ionized dopants dominate the carrier density; at sufficiently high temperature, intrinsic excitation can dominate. Heavy doping can produce degenerate carrier statistics, invalidating these Boltzmann expressions.

The electrical conductivity is $\boxed{\sigma =e(n{\mu }_e+p{\mu }_h)}$. Doping usually raises the majority carrier density, but ionized impurity scattering can lower mobility, so conductivity need not scale indefinitely with dopant concentration. Joining p-type and n-type regions creates a p-n junction, with a depletion region and built-in electric field established by carrier diffusion and compensating drift. The fixed ionized dopants, not an absence of all charge, account for the depleted region's net charge density.

In a metal, a partially filled energy band or overlapping energy bands supplies electrons with nearby unoccupied states. At thermal equilibrium, oppositely directed group velocities are equally populated and the net electric current vanishes. An applied electric field distorts the occupation distribution, producing a small mean drift velocity. A filled energy band alone has no such low-field intraband response because its velocity contributions cancel over the entire Brillouin zone.

In the Drude model, let $n$ be the mobile electron number density, let $e>0$ be the elementary charge, and let $\tau$ be the momentum relaxation time. The mean motion obeysThe steady state is ${\mathbf{v}}_d=-e\tau \mathbf{E}/m$. The current density is opposite to the electron drift velocity:This gives microscopic Ohm's law, with electrical conductivity $\sigma$ in $\mathrm{S}{\mathrm{m}}^{-1}$ and electrical resistivity ${\rho }_{\mathrm{el}}$ in $\Omega \mathrm{m}$. For a uniform wire of length $\ell$ and cross-sectional area $A$, its electrical resistance is $R={\rho }_{\mathrm{el}}\ell /A$. The dissipated power per volume is $\mathbf{J}\cdot \mathbf{E}=\sigma E^2$, the Joule heating supplied by the external source.

The free electron model with Fermi-Dirac statistics improves the microscopic picture. The Pauli exclusion principle fills states up to the Fermi energy at low temperature. For a spherical, parabolic energy band with two spin states,Only states near the Fermi surface change their occupation in a weak-field response; the large equilibrium Fermi velocity is not the small drift velocity. The same $n{e}^2\tau /m$ formula results for this isotropic parabolic model when $\tau$ is evaluated for transport near the Fermi surface. In a crystal, the energy band group velocity, effective mass and possibly direction-dependent scattering replace the simplest bare-particle picture. The mean free path is approximately ${\ell }_{\mathrm{mfp}}=v_F\tau$.

A perfectly periodic static lattice is already included in Bloch states and does not by itself give a finite electrical resistivity. Momentum relaxation comes from defects, impurities, boundaries, and electron-phonon scattering. Independent scattering mechanisms approximately add their rates, giving Matthiessen's rule,The approximately temperature-independent ${\rho }_0$ is the residual resistivity. In ordinary simple metals, the phonon contribution is approximately proportional to temperature at high temperature; a three-dimensional acoustic-phonon transport model gives a $T^5$ contribution well below the relevant Debye temperature. Other mechanisms can dominate in particular materials or regimes, so neither power law is universal.

The classical Drude model incorrectly treats the equilibrium carrier statistics. Fermi-Dirac statistics explains the small electronic heat capacity, while electrons carrying both charge and heat give the approximate Wiedemann-Franz law ${\kappa }_e/(\sigma T)={\pi }^2{k}_B^2/(3e^2)$ for degenerate carriers with appropriate nearly elastic scattering. Both the simple electrical conductivity formula and this thermal relation depend on the assumptions about energy bands and scattering, rather than being exact for all metals.

Capacity convention. The printed equation requires $c$ to be volumetric heat capacity, in $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. If “specific heat capacity” means per unit mass instead, the storage term is ${\rho }_{m}c{\partial }_{t}T$ and both denominators in the printed equation must contain ${\rho }_{m}c$. This distinction is fixed by dimensions.

For a fixed material region $V$, conservation of thermal energy gives the heat balance with volumetric heat generation,Use the divergence theorem, $\mathbf{J}=-\kappa \nabla T$ and constant coefficients. Since the region is arbitrary,The heat source is power per volume, while $\kappa$ is thermal conductivity rather than thermal diffusivity.

For uniform electrical conductivity and imposed current, current density is $j=I/(\pi a^2)$ and Joule heating is $H=j^2/\sigma$. A long, steady, cylindrically symmetric wire therefore satisfiesIntegration gives $T^{\prime }=-Hs/(2\kappa )+C/s$. Regularity at $s=0$ makes $C=0$, and the surface condition fixes the remaining constant. Thus the Joule-heated cylindrical wire hasHere $s$ denotes the printed radial coordinate $\rho$ and avoids confusing it with mass density. The centre is hotter by $I^2/(4{\pi }^2{a}^2\kappa \sigma )$.

For the air boundary, the internal flux at $s=a$ is $-\kappa T^{\prime }(a)=Ha/2$. Equate this to $\alpha [T(a)-T_{\mathrm{air}}]$. The Newton cooling of a Joule-heated wire givesMultiplication by circumference $2\pi a$ verifies that surface loss per length equals $I^2/(\pi a^2\sigma )$, the electrical power per length. The full air-cooled profile uses this surface temperature plus the same parabolic interior excess, assuming constant material properties and negligible end losses.

Use peak phasors with time dependence $e^{-i\omega t}$, real positive electrical conductivity $\sigma$, and absolute permeability ${\mu }_a=\mu {\mu }_0$. Ampère-Maxwell equation and Faraday equations giveso ${ϵ}^{\prime }=ϵ+i\sigma /(\omega {ϵ}_0)$. The sign of the imaginary term follows from the stated time convention. For a transverse plane wave in a homogeneous conductor,In the good-conductor limit, choose the root that decays into $z>0$:The electric phasor and physical field are consequently ${\tilde{E}}_x=E_0{e}^{(i-1)z/\delta }$ and $E_x=\mathrm{Re}[E_0{e}^{i(z/\delta -\omega t)-z/\delta }]$, as required. Tangential boundary matching sets $E_0$; it need not equal the incident amplitude.

Faraday's law fixes the magnetic field, including its phase:For real $E_0$ and $\psi =z/\delta -\omega t$, this means $H_y=E_0{e}^{-z/\delta }(\cos \psi -\sin \psi )/(\omega {\mu }_a\delta )$, with other components zero. The time-averaged Poynting vector isThe energy entering a thin slab exceeds that leaving it by $-d⟨N_z⟩/dz$ per unit volume, givingDirectly, $\mathbf{J}=\sigma \mathbf{E}$ and $\mathbf{J}\cdot \mathbf{E}=J^2/\sigma$ at each time. Therefore $⟨J^2/\sigma ⟩=\sigma |{\tilde{E}}_x{|}^2/2$ is exactly this loss. The factors one half use peak rather than root mean square phasors; amplitude decays over $\delta$, while intensity decays over $\delta /2$.

For a cold collisionless unmagnetized plasma, electron inertia gives $\tilde{\sigma }=i{ϵ}_0{\omega }_p^2/\omega$ and real ${ϵ}^{\prime }=1-{\omega }_p^2/{\omega }^2$. Above the plasma frequency the wave propagates without this resistive attenuation. Below it, a semi-infinite plasma supports an evanescent field with electric and magnetic components in quadrature and zero mean inward energy flux: the incident energy is reflected, not absorbed as Joule heating. electron kinetic energy is reversibly stored, so the nondispersive dielectric field-energy formula alone is incomplete. Collisions introduce a real dissipative electrical conductivity and can produce absorption.^[ref]

Effective permittivity. Use peak phasors with time dependence $e^{-i\omega t}$. The Ampere-Maxwell law isThe conductive contribution to complex permittivity is thereforeHere ${ϵ}^{\prime }$ is the source's effective relative permittivity, not just the real part of a complex response. The positive imaginary term is the loss sign appropriate to this time convention.

The penetrating electric wave. Combine this equation with Faraday's law of induction. In a uniform source-free medium the transverse phasor satisfiesFor a good conductor of electrical conductivity $\sigma$, $\sigma \gg \omega ϵ{\epsilon }_0$, so $k^2\simeq i{\mu }_0\sigma \omega$. The root which decays into $z>0$ isChoose the surface phasor amplitude $E_0$ real. Taking the real part of ${\tilde{E}}_x=E_0{e}^{ikz}$ givesThe skin depth $\delta$ is an amplitude-decay distance in metres.

The magnetic wave and phase. The curl of an $x$-polarized field varying in $z$ is in the $y$ direction. Since $\nabla \times \tilde{\mathbf{E}}=i\omega {\mu }_0\tilde{\mathbf{H}}$,ThusThe phase sign follows the specified convention. Equivalently the surface impedance of a good conductor is $E_x/H_y=(1-i)/(\sigma \delta )$.

How the fields differ from those in vacuum. The two metal fields have the same exponential envelope and spatial period $2\pi \delta$, but their phase arguments differ by $\pi /4$.

An individual travelling wave in the lossless exterior has no attenuation, real wave impedance and in-phase electric and magnetic fields. The exterior total field also includes the reflected wave and may show interference, so it need not share that simple phase relation everywhere. At the interface the tangential fields match; $E_0$ is the field just inside the metal, not necessarily the incident amplitude. A good conductor reflects strongly and its wave impedance is much smaller than the vacuum impedance.

Average inward energy flux. For peak phasors, the Poynting vector averaged over a cycle is $⟨\mathbf{S}⟩=\mathrm{Re}(\tilde{\mathbf{E}}\times {\tilde{\mathbf{H}}}^{\ast })/2$. The fields giveConsequently the constant in the question isThe flux is in the $+z$ direction and has units $\mathrm{W}{\mathrm{m}}^{-2}$.

Local energy conservation. The Poynting flux and local heating in a skin layer satisfyThe decrease becomes Joule heating in each layer. The Poynting theorem includes this transfer from field energy to the metal; averaged electromagnetic energy storage has zero time derivative in the periodic state, so no energy is lost from the complete balance.

Distinguish electrical current density ${\mathbf{J}}_e$ from heat flux ${\mathbf{J}}_Q$. Uniform current across the circular section givesby Ohm's law. The local Joule heating rate is electrical work per unit volume and time:For a segment of length $\ell$, integration gives $H\pi a^2\ell =I^2\ell /(\sigma \pi a^2)=I^{2}R$, confirming the usual resistor power. The model takes electrical conductivity constant and ignores temperature-induced current redistribution.

Replacing the fixed surface temperature by Newton's law of cooling changes the boundary condition but not the radial derivative in steady state. At the surface,HenceHere $\gamma$ is a surface heat transfer coefficient in $\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, unrelated to the surface tension symbol used in C10. An independent energy-balance check uses a length $\ell$: the electrical input $I^2\ell /(\sigma \pi a^2)$ must equal the surface loss $2\pi a\ell \gamma [T(a)-T_{\mathrm{air}}]$, which gives the same answer.

The complete temperature profile isThermal conductivity controls the internal gradient, but cancels from the surface-temperature rise because all generated power must eventually cross the prescribed surface. This assumes positive constant $\gamma$, uniform electrical conductivity and negligible end or additional radiative losses; temperature-dependent material properties could make the heating and boundary problem nonlinear.

The asthenosphere exists because the mantle's ability to deform is strongly dependent on temperature, pressure, composition, water and deformation mechanism. Cooling from the surface creates a strong lithosphere. Beneath it, hotter mantle deforms by material creep at much lower stress on geological timescales. The asthenosphere is predominantly solid weak rock, not a global ocean of magma. On the much shorter timescale of a seismic wave, it still supports S waves.

The hot shallow mantle can be close to its solidus. Water weakening of rocks and, in some settings, a small connected melt fraction can further lower effective viscosity. At greater depths, changes in pressure, homologous temperature, mineral structure and water storage modify the deformation laws; increasing depth does not imply indefinitely decreasing viscosity. Explaining a weak upper-mantle interval does not require every location to contain the same amount of melt.

There are several physically distinct boundaries to test. The thermal lithosphere-asthenosphere boundary describes a thermal transition, the mechanical boundary a contrast in long-term strength or viscosity, and the seismic lithosphere-asthenosphere boundary a change in seismic velocity or other wave properties. They need not have identical depths. A smooth geothermal gradient can explain broad thermal weakening, but a very sharp decrease of seismic velocity may require changes in melt fraction, hydration, composition or seismic anisotropy. A sharp shallow reflector within a craton might instead be an internal lithospheric boundary.^[ref]

Begin with a thermal null model. A half-space cooling model givesIf a thermal boundary corresponds to a fixed temperature $T_{\ast }$, its depth isso it deepens as the square root of plate age. Test that prediction against independently determined seafloor ages, bathymetry and heat flux. An oceanic plate cooling model adds finite thickness and basal heat supply, allowing the old-age depth to approach a finite value. For continents, include radiogenic heating, crustal thickness, the history of continental rifting or tectonic thickening, and advective heating by magma or mantle flow. A cold, dry craton and a recently extended volcanic region should not share the same thermal profile.

Then calculate the observable consequences of competing mechanisms rather than merely assigning a boundary to a map:

Compare models against the observed boundary depth and its sharpness, including uncertainty and resolution. An age–depth trend with matching heat flux favours thermal control; a sharp discontinuity that fails the thermal prediction but coincides with an electrical anomaly can support a melt- or hydration-related transition. Neither observation alone proves a unique cause. The spatially varying boundary is best understood as the result of thermal evolution interacting with composition, fluids and deformation history.

Probably yes in connected fractures or transiently opened pore space, especially where a transform fault admits circulating seawater; a uniform water-filled porous layer several kilometres thick is not guaranteed. Porosity and permeability of a porous medium generally decrease as cracks close or seal, while ongoing faulting can reopen pathways. Fluid can occur at high pressure and temperature, including a supercritical state; it need not resemble cold surface water.

Several complementary tests address presence and origin:

No one anomaly uniquely identifies water. Seismic velocity can also reflect cracks, composition and temperature; electrical conductivity can reflect melt or conductive minerals. Serpentinization demonstrates incorporation of water into minerals, which must be distinguished from fluid currently occupying pore space. At an oceanic transform fault, an external seawater supply is the natural first hypothesis, but isotope ratios and chemical budgets can test contributions from deep fluids rather than presuming the source. A research example of coupled deformation and hydrothermal alteration is Mass transfer and fluid flow during detachment faulting and development of an oceanic core complex, Atlantis Massif by Chiara Boschi and others; a detachment system illustrates the methods without proving identical flow at every transform fault.

At a convergent margin, the incoming oceanic lithosphere is a chemically and mechanically layered package: sediments, oceanic crust and lithospheric mantle. Subduction does not deliver this whole package unchanged to one destination. Solids can be accreted, underplated, exhumed or recycled deeply, while released fluids and melts carry selected components to the overriding plate. The outcome depends on incoming composition and hydration, slab age, convergence rate, thermal structure, interface coupling and subsequent tectonic history.

The incoming material already contains a history. Sediments include terrigenous debris, biogenic carbonates, siliceous deposits, clay minerals, pore water and organic matter. Oceanic crust consists mainly of basalt, sheeted dikes and gabbro; hydrothermal alteration can add water, carbonates and mobile elements before subduction. Beneath it, residual peridotite forms the lithospheric mantle. Faulting at a spreading environment, a transform boundary or an outer-rise bending region can let seawater enter and cause serpentinization. The lithospheric mantle is not uniformly hydrated to its full thickness. Consequently, sediment thickness alone does not measure the incoming volatile inventory.

Sediments: separation starts near the trench. Compaction and increasing effective pressure expel pore fluids, which can escape through the forearc, transport dissolved constituents and influence pore pressure on the plate interface. Some sediments are scraped off into an accretionary prism, shortened by thrusts and retained in the upper plate. Others pass beneath the leading edge, then detach and undergo tectonic underplating beneath the forearc. Deeper underplated material may return to the surface during tectonic exhumation, preserving a record of burial and deformation. At a sediment-starved erosive margin, little prism may form, and tectonic erosion can entrain material from the overriding plate as well. Thus net continental gain and net continental loss are both possible; they can even alternate at one margin.

Sediments that remain attached to the slab undergo metamorphism, dehydration, and potentially decarbonation or carbonate dissolution. The released fluids can transport boron, barium, strontium, lead and other soluble constituents. Carbon can leave as dissolved species, remain in carbonate, or occur as reduced carbon depending on redox conditions and reaction history. Along sufficiently hot paths, sediment-derived melts or mixtures of sediment and altered rock can transfer material into the mantle wedge; buoyant mixed material may rise as a mélange diapir. Cold paths can instead retain more solid material and some volatile-bearing phases to greater depths. These are alternative reaction and transport pathways, not a single compulsory sequence. A carbon-budget study explicitly includes fluid dissolution, sediment diapirism and storage in the upper plate, showing why volcanic carbon dioxide alone cannot close the whole subduction carbon budget.^[ref]

Oceanic crust: metamorphic transformation and chemical filtration. Altered basalt and gabbro are progressively buried. Depending on their pressure-temperature paths and reaction kinetics, they can pass through blueschist facies or warmer assemblages and eventually form eclogite, dominated by garnet and omphacite. Eclogitization increases density and can strengthen negative slab buoyancy, but a kinetic delay can leave metastable rock. It is incorrect to assign every slab the same sequence or the same fixed depths for these changes.

During burial, breakdown of hydrous phases releases water-rich fluids; different mineral assemblages retain different quantities of water. The fluids migrate through or along the interface and into the mantle wedge, where they cause metasomatism and lower the solidus of peridotite. Flux melting of the mantle wedge generates much of the primary magma feeding a volcanic arc. This is different from saying that the entire descending oceanic crust melts beneath the arc. Most of its refractory mass can remain solid. Partial melting of oceanic crust is more plausible for unusually hot or young slabs, or other favorable thermal conditions, and produces a distinct crustal contribution.

The depth of dehydration is controlled by thermal structure and composition. Global thermal-petrological calculations, for example, predict more efficient deep water transport in old, fast, cold slabs than in hot ones, with substantial differences between volcanic crust, gabbro and hydrated peridotite. Such a calculation supports conditional deep retention; its modeled proportions should not be imposed on every margin.^[ref]

Mechanical detachment adds more branches. Upper-crustal slices, seamounts or oceanic plateaus may be accreted or underplated instead of continuing down the slab. A complete or partial ophiolite can be emplaced onto another plate by obduction in suitable convergence histories. Most intact oceanic crust is not saved this way. High-pressure fragments may be entrained in an interface channel or mélange and exhumed; their small preserved volume should not be mistaken for the dominant global mass flux. Magma generated after slab-derived transfer may also pond and crystallize beneath the upper crust by magmatic underplating, which is a different process from mechanical tectonic underplating of solid slices.

Lithospheric mantle: residual rock and a variable water carrier. Most incoming lithospheric mantle is depleted peridotite left after extraction of ridge melts. Its cold, dense, mechanically coherent portion contributes to slab pull and generally sinks with the plate. Hydrated portions contain serpentine and possibly other hydrous minerals. Breakdown of these phases releases water at conditions that depend on the thermal path, feeding further metasomatism of the mantle wedge and sometimes dehydration embrittlement. The latter is a proposed contributor to intermediate-depth earthquakes, not an exclusive explanation for all such events. Very cold paths may transfer some water onward in high-pressure hydrous phases or as hydrogen incorporated in nominally anhydrous minerals.

Do not confuse the incoming hydrated lithospheric mantle with newly serpentinized forearc mantle in the overriding plate. The latter receives slab-derived fluids and can temporarily store their water. Hydrated, relatively buoyant interface material can facilitate return flow and tectonic exhumation, while the colder deeper bulk of the lithospheric mantle continues down. Some peridotite may also be accreted or obducted with an ophiolite. These exchanges mean that even an exposed serpentinite requires structural and chemical context before it can be assigned to its original plate.

Deep recycling and eventual return. The residual eclogite, remaining sediments and lithospheric mantle can enter the deeper mantle. A slab can temporarily stagnate around the mantle transition zone or penetrate into the lower mantle, depending on buoyancy, phase transformations, geometry and mantle viscosity. Crustal material can remain as compositionally distinct domains, mix, or react with surrounding peridotite; it need not retain the near-surface eclogite assemblage indefinitely as pressure rises. Subsequent mantle convection can return recycled components toward spreading regions or mantle plumes, where partial melting incorporates them into new basalt. A primary study of Hawaiian basalt illustrates a specific model in which melts derived from recycled crust react with peridotite to form a pyroxenite source. It is an interpretation of combined chemical evidence, not a unique label attached to every enriched basalt.^[ref]

How the pathways can be observed. Seismic reflection profiles, drilling, wedge geometry and balanced structural sections constrain accretion, fluid escape and tectonic erosion. High-pressure metamorphism, mineral zoning, radiometric dating and pressure-temperature-time paths in exhumed rocks reveal burial, reaction and return. Their reconstructable histories are especially useful when deformation and fluid inclusions are linked to the dated mineral growth. Earthquake distributions, seismic tomography, seismic attenuation and electrical conductivity constrain the geometry and physical state of the slab and mantle wedge, although fluids, melts and temperature can produce nonunique signals.

Arc magmas carry complementary chemical evidence: relatively mobile barium, boron, lead or strontium can trace fluid-mediated transfer, whereas sediment-sensitive thorium and isotope ratios can help identify other contributions. Element mobility changes with fluid composition and pressure-temperature conditions, so no one element is an infallible tracer. Boron isotope ratios, radiogenic strontium isotope ratios and lead isotope ratios should be interpreted with measured incoming sediments and altered oceanic crust. Arc gases and forearc fluid discharge constrain volatile return; ocean-island basalts, mantle xenoliths and diamond inclusions provide possible evidence of deeper recycling. Each observation samples only part of the system.

For any element $X$, a useful control-volume statement isHere permanent accretion is counted outside the chosen reactive control volume, return includes fluid, melt and exhumed-solid exports, and transient underplating or reservoir growth contributes to storage until exported. For a different boundary choice, the terms must be regrouped consistently. The three incoming layers can have very different transfer efficiencies for bulk rock, water, carbon and a particular trace element. A small mass of escaping fluid can dominate an element's return while most of the solid slab continues into the deep mantle.

Charge carrier mobility measures weak-field drift speed per electric field. For electrons and electron holes,Both mobilities are positive, with SI units ${\mathrm{m}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$; the opposite drift signs reflect the opposite carrier charges. In a simple isotropic relaxation-time model, electron mobility is ${\mu }_e=e{\tau }_e/m_e^{\ast }$ and hole mobility is ${\mu }_h=e{\tau }_h/m_h^{\ast }$. These are momentum relaxation times, not the minority carrier lifetime for electron-hole recombination.

Scattering by phonons, ionized impurities and defects limits mobility. Increasing phonon populations commonly lowers mobility at high temperature; ionized-impurity scattering can dominate at low temperature and high semiconductor doping. Approximate addition of independent inverse relaxation times is the Matthiessen's rule. At sufficiently strong field, velocity saturation invalidates the constant-mobility drift law.

The Hall effect provides a measurement: a current-carrying bar in a perpendicular magnetic field develops a transverse voltage. In a one-carrier model $|R_H|=1/(en)$ and $\mu =\sigma |R_H|$, combining the transverse voltage with longitudinal electrical conductivity. Multiple carrier types and scattering-dependent Hall factors require a more detailed interpretation.

Generation increases the mobile electron or hole populations; interband excitation creates an electron-hole pair. Thermal excitation and absorbed photons are important sources, and impact ionization can multiply carriers in a strong field. Ordinary single-photon interband absorption needs enough photon energy to bridge the band gap, with the required momentum balance.

Electron-hole recombination removes a pair. Radiative electron-hole recombination emits a photon, with simple rate $Bnp$; trap-assisted electron-hole recombination uses localized defect levels and transfers energy to the lattice; Auger recombination transfers the electron-hole recombination energy to another carrier. The last process becomes especially important at high injection because its rate contains products such as $n^{2}p$ and $n{p}^2$. The detailed band structure and available defects determine which channel dominates.

At thermal thermodynamic equilibrium, generation and electron-hole recombination continue at equal rates. For the simple radiative model the net electron-hole recombination rate is $B(np-n_i^2)$, enforcing the semiconductor mass-action law. Under weak excitation with the majority density nearly fixed, an excess minority density often decays as $\Delta n(t)=\Delta n(0)e^{-t/\tau }$, defining its minority carrier lifetime.

A short light pulse followed by time-resolved electrical conductivity or luminescence measures the relaxation. Spatial diffusion, trapping and instrumental response must be separated from electron-hole recombination before identifying a single fitted decay constant with the intrinsic lifetime. These processes control photodetector response and light-emitting-device efficiency.

A covalent bond is stabilized by shared electronic states extending across neighboring atomic nuclei. In a simple two-centre description, combining comparable atomic orbitals produces a bonding state with enhanced electron density between the nuclei and an antibonding state with a node there. For equivalent normalized orbitals ${\varphi }_A,{\varphi }_B$ with real overlap $S$,Their energy ordering follows from the molecular Hamiltonian; constructive interference and reduced gradients help explain why the bonding combination can be lower. Opposite-spin electrons can occupy the bonding orbital together. The familiar molecular-hydrogen bond is a basic example, while extended covalent networks form solids such as diamond. Orbital shape and hybridization make covalent bonding strongly directional and constrain bond angles. Occupying antibonding states reduces the net bond strength.

Ionic bonding instead emphasizes electric charge transfer and the electrostatic attraction of the resulting oppositely charged ions. Removing an electron from one atom costs ionization energy and adding it to another gains electron affinity; these isolated-atom terms alone need not favor transfer. In a crystal, the electrostatic lattice energy can stabilize the ionic arrangement. For a schematic lattice of singly charged ions at nearest-neighbor distance $r$,where the geometrical Madelung constant $A_M$ sums attractions and repulsions throughout the lattice. Short-range overlap repulsion prevents collapse and gives an equilibrium separation. Sodium chloride is an example of an extended ionic crystal, rather than isolated NaCl molecules in the solid.

Ideal ionic bonding is comparatively nondirectional, so packing, ion sizes and electric charge balance strongly influence coordination. ions cannot freely migrate in the ordinary solid; melting or dissolution can make ionic electric charge transport possible. Covalent materials do not obey one universal electrical conductivity rule: their electronic bands and filling determine whether mobile electronic carriers are available.

The two descriptions are limiting models, not a sharp classification of every bond. Unequal electronegativities can produce polar covalent bonds with partial electric charge transfer, and a material may contain both covalent groups and ionic interactions between those groups. Chemical bonding reflects a balance of electronic kinetic energy, electron-nucleus attraction, electron-electron repulsion and nuclear repulsion; it cannot be reduced to a force between two assigned point charges in every material.

The Drude model treats mobile electric charge carriers as independent particles with density $n$, signed electric charge $q$, isotropic mass $m$, and momentum relaxation time $\tau$. In uniform steady fields,At zero magnetic field the drift velocity is $q\tau \mathbf{E}/m$, giving electrical conductivity and electrical resistivityThe electrical conductivity is positive for either carrier sign. Scattering dissipates directed drift energy into the environment; a finite steady electric current is compatible with a short-lived drift momentum. The independent-carrier relaxation-time approximation omits detailed band anisotropy, quantum interference and microscopic collision correlations.

For electric current along $x$ and magnetic field $B\hat{\mathbf{z}}$, the magnetic force deflects carriers toward one side of the sample. electric charge accumulates until a transverse Hall field cancels that force in an open transverse circuit. With $J_y=0$, $E_y=v_{x}B$, soElectrons have negative $R_H$ in this convention, whereas positive carriers have positive $R_H$. For a strip of thickness $d$ along $z$, the magnitude of the Hall electric potential difference across its width is $|V_H|=|IB/(nqd)|$. Thus Hall measurements determine carrier sign and, within this one-carrier model, density, while combining $R_H$ with ${\sigma }_0$ determines charge carrier mobility $\mu =|q|\tau /m={\sigma }_0|R_H|$.

Keeping both in-plane electric current components, the resistivity tensor isThis fixes the off-diagonal sign convention and shows that inversion is not componentwise reciprocation. Under the Hall boundary condition $J_y=0$, the ideal one-carrier isotropic model gives $E_x/J_x={\rho }_0$, independent of magnetic field: there is no ordinary longitudinal magnetoresistance in this minimal model. Multiple carrier types or anisotropic bands can change both this conclusion and the simple $R_H=1/(nq)$ inference. Their electrical conductivity must be added before inverting to obtain the total electrical resistivity.

Use physical fields proportional to the real part of $e^{-i\omega t}$. The Drude frequency response isThus the sketch is an even Lorentzian line shape, with maximum ${\sigma }_0^2$ at zero frequency, half that value at $|\omega |={\tau }^{-1}$, and an ${\omega }^{-2}$ tail. The positive-frequency half is shown in the preceding figure. Squaring the modulus differs from plotting either $|\sigma |$ or the real part of $\sigma$.

For a single carrier species, the plasma frequency and electrical conductivity satisfyThereforeThe corresponding relaxation time is $\tau \simeq 3.57\times 10^{-14}\mathrm{s}$. This is an angular frequency; the ordinary frequency is $\boxed{f_{1/2}={\omega }_{1/2}/(2\pi )\simeq 4.46\times 10^{12}\mathrm{Hz}}$. The units of electrical conductivity used in the calculation are ${\mathrm{\Omega }}^{-1}{\mathrm{m}}^{-1}$, consistently with the SI plasma-frequency relation.

A magnetic field quantizes a charged carrier's cyclotron motion into Landau levels. For an isotropic parabolic band, neglecting spin splitting,The degeneracy of each orbital level per unit perpendicular area is $eB/h$ per spin. Increasing the field changes both the level spacing and the number of states in each level. Successive levels cross the Fermi energy, producing an oscillatory density of states and hence quantum oscillations in thermodynamic and transport quantities.

For a general closed semiclassical orbit, Onsager quantization relates its wavevector-space area to field:Setting this area to an extremal Fermi-surface cross-section gives successive inverse-field crossings separated byA typical oscillatory contribution is proportional to $\cos (2\pi F/B+\phi )$. The oscillations are periodic in $1/B$, not in $B$. In three dimensions, extremal cross-sections dominate because contributions from other longitudinal momenta cancel by phase averaging. Rotating the field therefore maps the shape of the Fermi surface.

Oscillations in magnetization are the de Haas-van Alphen effect; oscillations in longitudinal electrical resistivity are the Shubnikov-de Haas effect. They are most readily observed in clean samples at low temperature. The thermal damping factor isand a common first-harmonic disorder factor is $R_D=\exp [-\pi /({\omega }_c{\tau }_q)]$. Thus the Landau spacing must be sufficiently large compared with thermal broadening and disorder broadening, approximately $k_{B}T\lesssim \hslash {\omega }_c$ and ${\omega }_c{\tau }_q\gtrsim 1$, with the displayed factors giving the quantitative suppression.

The temperature dependence measures the cyclotron effective mass, and the field-dependent damping probes the quantum lifetime ${\tau }_q$. That lifetime need not equal the momentum relaxation time controlling electrical conductivity: small-angle scattering can broaden Landau levels while relaxing electric current inefficiently. Spin splitting can generate further modulation or resolved spin structure. Oscillations are not the same as the ordinary smooth magnetoresistance or cyclotron resonance; they probe quantized orbital state counting near the Fermi energy.

Relaxation-time model. The Drude model treats mobile charge carriers as independent particles accelerated by the Lorentz force between scattering events. Collisions randomize the drift momentum toward local equilibrium and are represented by a uniform, field-independent momentum relaxation time. For the present calculation the carrier density, isotropic mass and relaxation time are fixed, the fields are uniform and stationary, and quantum orbital quantization and intercarrier correlations are neglected. The steady drift velocity obeysHere $n$ is the number of carriers of that species per unit volume, $q$ their signed charge, $m>0$ their inertial or isotropic band mass, $\tau$ the momentum relaxation time, and ${\sigma }_0$ their single-species electrical conductivity at zero field.

One-carrier tensor. With $\mathbf{B}=B\hat{\mathbf{z}}$ and $\mathbf{J}=nq\mathbf{v}$, solving for the electric field givesSince $(\mathbf{J}\times \mathbf{B}{)}_x=B{J}_y$ and $(\mathbf{J}\times \mathbf{B}{)}_y=-B{J}_x$,Thus the resistivity tensor, defined by $\mathbf{E}=\mathbf{\rho }\mathbf{J}$, isIts off-diagonal signs follow from the stated axes and signed $q$. For zero transverse electric current, $E_y/J_x=B/(nq)$ is the Hall electrical resistivity; reversing charge reverses it. The single-species longitudinal electrical resistivity is independent of $B$ in this ideal isotropic model.

Adding electrons and electron holes. Let both species have density $n$ and charges $q=\pm e$, with $e>0$. The stipulated equal zero-field electrical conductivity means equal positive charge carrier mobility $\mu ={\sigma }_0/(ne)$, even if their masses and momentum relaxation times separately differ. Put $r=1/{\sigma }_0$ and $h=B/(ne)$. Inverting each electrical resistivity matrix givesBoth species experience the same electric field, so their electric current densities add. Therefore it is their conductivity tensors, rather than their resistivity tensors, that must be summed. The Hall parts cancel andThis is compensated two-carrier magnetoresistance. Although the net Hall field vanishes, each carrier species is bent in its own direction; maintaining the total longitudinal electric current requires an increasing longitudinal field. Equal carrier numbers alone do not ensure zero Hall response if charge carrier mobilities differ; both equalities given here are used.

Large-field limit and quantum oscillations. Expanding the exact diagonal entry,For $\mu B\gg 1$,Here the last equality applies to either species with its own mass and transport time; the stipulated charge carrier mobilities give the same threshold. This is the classical strong-field condition ${\omega }_c{\tau }_{\mathrm{tr}}\gg 1$: carriers turn substantially before their drift momentum relaxes.