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The hydrophobic core of a lipid bilayer makes transfer of a hydrated ion into bare lipid energetically costly. An ion channel supplies a water-accessible route across that barrier. Its narrow selectivity filter presents coordinating chemical groups that compensate for loss or rearrangement of the ion's hydration shell. Selectivity depends on the free-energy balance of hydration, coordination, electrostatics and geometry, rather than a sieve that passes all smaller ions.

A potassium-channel filter presents backbone carbonyl oxygens in a geometry favourable for potassium; smaller sodium does not automatically gain the same energetic compensation. Sodium and calcium channels use different filter architectures: charged side chains and their coordination environments favour particular ions, with ion occupancy and competing species also affecting permeation. In canonical vertebrate voltage-gated sodium channels, the DEKA selectivity ring illustrates chemical discrimination distinct from potassium-channel carbonyl coordination. The precise mechanism therefore depends on the channel family.

Rapid flux requires low successive free-energy barriers, not permanent binding of the preferred ion. Several occupied sites, nearby water and ion–ion interactions can facilitate passage without requiring the ion to cross the lipid. For scale, a $100\mathrm{pS}$ open channel driven by $100\mathrm{mV}$ carries $10\mathrm{pA}$, approximately $6\times 10^7$ monovalent ions per second. A nanometre-scale route with favourable coordination can support such rapid, thermally driven motion. An ion channel is not an ATP-driven pump: net flux follows the electrochemical potential difference,which vanishes at the species' Nernst equilibrium potential. ATP-consuming pumps maintain the gradients that make sustained channel currents possible.

Permeation selectivity and opening probability are separate properties. Voltage-sensor movement, ligand binding or mechanical tension can regulate access to the pore, while inactivation can terminate conduction during a maintained stimulus. Patch-clamp unitary currents establish discrete conducting states; reversal potentials under changed ionic compositions quantify selectivity. Structure determination locates the filter and gates, and mutations or specific blockers test their functional roles. Agreement between these measurements is stronger evidence than structure or current shape alone.

The passive upper limiting membrane potential for a purely sodium-selective pathway is its Nernst equilibrium potential. Equal inside and outside electrochemical potentials, using ideal concentrations, requireThis is the maximum absolute inside-minus-outside voltage under the stated passive assumptions. If the reference resting membrane potential is $-60\mathrm{mV}$, the corresponding increase is about $121\mathrm{mV}$.

Actual action potentials usually peak below $E_{\mathrm{Na}}$ because voltage-gated sodium channels undergo ion-channel inactivation, while outward potassium and leak electric currents oppose inward sodium electric current. Finite membrane capacitance and finite available time also prevent instantaneous equilibration. At $V=E_{\mathrm{Na}}$, the sodium driving force itself is zero.

Use the supplied $-50\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ as a standard transformed Gibbs free energy of ATP hydrolysis, rather than replacing it with a different commonly quoted standard value. Approximate thermodynamic activities by concentrations divided by $c^{\circ }=1\mathrm{M}$, at an explicitly assumed $T=300\mathrm{K}$. The dimensionless reaction quotient isConsequentlyand the reverse synthesis cost is $g_{\mathrm{ATP}}=63.22\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Substituting bare millimolar numbers in this quotient without the standard state would give the wrong logarithm.

For outside-to-inside proton movement, the membrane potential is $\Delta \psi =-0.125\mathrm{V}$ and ${\mathrm{pH}}_{\mathrm{in}}-{\mathrm{pH}}_{\mathrm{out}}=0.5$, so $[H^{+}{]}_{\mathrm{in}}/[H^{+}{]}_{\mathrm{out}}=10^{-0.5}$. The molar electrochemical potential change isThus the available proton-motive force magnitude is $0.15476\mathrm{V}$, supplying $14.93\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ per transported proton. Assuming one proton per c-subunit per revolution, three ATP per revolution and no leakage, slippage or transport overhead,The smallest integer permitting synthesis is therefore $\boxed{n_c=13}$ in this ideal budget. A ring of thirteen gives only a small driving margin; inefficiency or sustained finite output could require a larger ring. The result is an energetic lower bound and near-reversible estimate, not a unique structural determination from these data. Reasonable temperatures near $300$–$310\mathrm{K}$ leave the minimum integer thirteen. Additional membrane transport costs have been excluded explicitly.

An ion channel supplies a water-accessible pathway through the otherwise strongly insulating, hydrophobic lipid bilayer. Moving a hydrated ion directly into that low-permittivity region is energetically expensive. The channel's polar pore and coordinating groups reduce the barrier while retaining selectivity. Transport is passive: the electrochemical potential difference drives net movement, and the channel does not itself pump an ion uphill. For an ideal species of charge number $z$,The net electric current of a pathway permeable only to that species reverses at its Nernst equilibrium potential. An open channel can carry zero net electric current at that voltage even while individual ions cross both ways.

The selectivity filter balances loss of the hydration shell against favorable coordination within the pore. In a typical potassium channel, backbone carbonyl oxygen atoms create appropriately arranged binding sites for a substantially dehydrated potassium ion. The smaller sodium ion is not automatically preferred simply because it fits geometrically: its hydration is stronger and the filter does not compensate its dehydration in the same way. Geometry, flexibility, coordination chemistry and ion occupancy all matter. Sodium-selective and calcium-selective channels use different arrangements, often involving charged side chains; a single rigid-sieve explanation is inadequate.

Rapid passage coexists with transient binding because several nearby sites have comparable free energies and relatively low barriers between them. Multiple ions occupying a pore influence one another through electrostatic interactions, enabling correlated exchange and displacement rather than irreversible trapping at one deep site. High throughput therefore requires favorable passage kinetics as well as strong discrimination against unwanted species. A more tightly bound species can even block a pathway instead of being its fastest permeant.

Gating and selectivity are distinct. A voltage-sensing domain couples membrane potential to activation-gate conformation; ligand-gated ion channels couple binding to opening; mechanosensitive ion channels couple membrane stress to opening. Ion-channel inactivation introduces additional unavailable states. None of these mechanisms alone identifies which ion the open pore prefers.

For $N$ similar independent channels, a local Ohmic approximation with outward-positive electric current isHere open probability controls availability, single-channel electrical conductance controls current per open pore, and $V-E_{\mathrm{rev}}$ is the electrical driving factor in this approximation. Nonlinear current-voltage relations and mixed-ion permeation may require a fuller model. Single-channel recording resolves discrete openings; reversal shifts when ionic concentrations change test selectivity; gating electric currents test sensor motion; structural evidence identifies the pore and its coordinating groups. Together these distinguish rapid selective permeation from the separate process of opening the gate.

An ionic atmosphere. A negatively charged molecule attracts positive counterions and repels negative co-ions; signs reverse for a positive molecule. The resulting electric double layer partially screens the bare electric field. Far from an isolated molecule, the electrolyte recovers its neutral bulk composition. At thermodynamic equilibrium, spatial uniformity of each species' electrochemical potential requiresThe logarithm implicitly uses a common reference density. This result for Boltzmann ion densities in a symmetric electrolyte neglects ion correlations, finite size and nonideal activities, but shows immediately why equal concentrations of the two signs do not persist close to a charged surface.

Self-consistent screening. Insert these densities into Poisson's equation, with solvent permittivity $ϵ$ and fixed charge density ${\rho }_f$:This is the Poisson-Boltzmann equation. For a symmetric monovalent salt with bulk density $n_s$ of each sign, outside the fixed charge it becomesFor $|z_{i}e\psi |\ll k_{B}T$, linearizing and using bulk neutrality gives the Debye–Hückel approximationThus added salt shortens the Debye length: electrostatic screening becomes stronger over a shorter distance. The factor of two in a $1:1$ salt comes from two ionic species, not from double-counting the density of salt pairs.

For example, a charged sphere of radius $a$ and total charge $Q$ has the exterior linearized solutionassuming fixed uniform surface charge and no ions inside. It follows from the decaying spherical solution and $-ϵ{\psi }^{\prime }(a)=Q/(4\pi a^2)$. At zero salt it reduces to the Coulomb potential; at finite salt it decays exponentially as well as geometrically. The excess mobile charge integrates to $-Q$ in an infinite neutralizing reservoir, although a finite sphere surrounding the molecule generally contains only part of that ionic atmosphere.

Biophysical scales and limitations. In water at room temperature, the monovalent formula is approximately ${\lambda }_D=0.304\mathrm{nm}/\sqrt{I/(\mathrm{mol}{\mathrm{L}}^{-1})}$, with $I$ the ionic strength. It gives about $0.79\mathrm{nm}$ at $0.15\mathrm{mol}{\mathrm{L}}^{-1}$ and $9.6\mathrm{nm}$ at $1\mathrm{mmol}{\mathrm{L}}^{-1}$. Consequently the same charged protein can exert very different long-range interactions mediated by electrostatics at different salt concentrations. Close to strongly charged DNA, the linear approximation fails; nonlinear screening, counterion condensation, ion-specific adsorption and changes of surface ionization through charge regulation may matter. Mean-field theory also misses strong multivalent-ion correlations and finite-size saturation: the exponential law cannot be extrapolated to arbitrarily large near-surface concentrations.

A larger source–drain bias opens an energy window between reservoir electrochemical potentials. When a charge-addition/removal transition enters that window, sequential tunnelling becomes allowed even away from a zero-bias degeneracy. The low-conductance regions in a gate/bias map are Coulomb diamonds.

For an illustrative symmetric device with reservoir potentials $\pm e{V}_{\mathrm{SD}}/2$, let $q=Q_{\mathrm{ext}}/e$ and $E_{\mathrm{ch}}=e^2/C$. In the charge-$N$ interval, the classical blockade domain isIt shrinks to zero at $q=N\pm 1/2$. Unequal capacitances change edge slopes and voltage division; the figure's symmetry is a chosen model, not a universal device property.

Quantum confinement adds discrete single-particle levels and spin structure. In a constant-interaction approximation, successive addition energies contain $e^2/C$ plus the relevant orbital spacing, so diamond sizes need not all match. Excited-state tunnelling can produce extra conductance lines. At stronger coupling, cotunnelling and charge fluctuations can give leakage beyond a simple sequential classical model. Thus temperature and coupling must be small enough to resolve the charging and level structure separately.

Use the one-way density per unit length $g_{+}/L=2/(h|v|)$. The electron-flux magnitude in an interval is charge times velocity times line density:If $g$ is instead defined for a finite length $L$, the traversal frequency is $|v|/L$; the same cancellation follows. The printed $I=Nev$ therefore requires $N$ to be a line density, not an unscaled total particle number.

Subtract the opposite reservoir's incoming flux. With the stated electron electrochemical potentials, conventional positive current is opposite to electron-number flux, soAt low temperature and small bias, each fully transmitted occupied spin-degenerate subband contributes $2e^2/h$. Hence $\boxed{G=N_{b}2e^2/h}$. More generally, the Landauer formula weights each mode by its transmission probability; imperfect transmission removes exact integer plateaux.

Spin-polarized injection produces different spin-resolved electrochemical potentials in Cu, ${\mu }_s={\mu }_{\uparrow }-{\mu }_{\downarrow }$. Their imbalance diffuses beyond the charge-current path. Oppositely directed spin-resolved currents can cancel in net charge while transporting spin.

The second Py contact is a spin-selective voltage probe. In a weakly invasive model,Its open-circuit potential adjusts until the total detector charge current vanishes. That potential changes when the detector magnetization reverses relative to the injected spins. Zero detector charge current does not imply zero spin imbalance or zero measured voltage.

Reverse injector current and each electrode's alignment to test the predicted signal symmetries. Remove spin-independent thermoelectric offsets and stray-field Hall contributions. A nonzero voltage alone would not prove spin accumulation.^[ref]

The map shows Coulomb blockade with discrete single-electron charging: low-current diamonds separate charge-stable regions from finite-bias transport windows. A small conducting island or quantum dot must be tunnel-coupled to source and drain, with a gate shifting its electrochemical potential. Charging prevents sequential transport until an island addition/removal energy enters the bias voltage window.

The tunnel resistances should substantially exceed the resistance quantum $h/e^2\simeq 25.8\mathrm{k\Omega }$, preventing charge fluctuations from washing out integer island occupancy, while still permitting a measurable current. Temperature and tunnel broadening must be small compared with the addition-energy scale. Here the smaller diamonds imply $e^2/C_{\Sigma }\simeq 2.5\mathrm{meV}$, so $k_{B}T\ll 2.5\mathrm{meV}$, or $T\ll 29\mathrm{K}$; resolving the inferred 1 meV orbital structure requires the stricter scale $T\ll 12\mathrm{K}$. Stray capacitance and unintended multiple islands can complicate this simple interpretation.

This is transverse magnetic focusing. The depleted gates form reflecting boundaries and two constrictions. A narrow quantum point contact collimates carriers into cyclotron trajectories with cyclotron radius $r_c=m^{\ast }{v}_F/(e|B|)=\hslash k_F/(e|B|)$. When an integer number of semicircular skipping arcs links the two openings, the collector couples particularly strongly to the injector population. A floating voltage contact then changes its electrochemical potential to satisfy zero net current, producing a peak in the measured nonlocal voltage. Sweeping the field changes the arc diameter, moving the focused population into or away from the collector.

The connection condition is $L=2j{r}_c$ for integer $j\ge 1$, with the higher orders requiring specular reflections. The effect needs ballistic electron transport, sufficiently narrow openings and a smooth boundary. Electron phase coherence can add interference structure, but the basic geometrical focusing condition is semiclassical.^[ref]

There is a polarity qualification in the printed geometry. The electron charge is negative: for a field out of the page, an electron launched upward through the left opening bends left, not toward the right opening. A direct left-opening-to-right-opening focusing path in the upper region therefore requires the field into the page; the printed outward field supports the reverse connection. Reversing conventional current reverses the occupation perturbation and voltage sign, but does not reverse the curvature of an individual electron. Thus the standard injector-to-collector explanation describes the focusing mechanism, while the asserted signed voltage increase requires a consistent field/lead convention. The calculated results below are peak-field magnitudes; taking outward field positive gives negative fields for direct left-to-right electron injection.

Integer Hall transport and filling. A clean two-dimensional electron gas at low temperature in a strong perpendicular magnetic field forms Landau levels with orbital spacing $\hslash {\omega }_c$, where ${\omega }_c=e|B|/m^{\ast }$. Examples include semiconductor heterojunctions and inversion layers. On integer plateaus,with integer plateau index $\nu$. The density of states in one resolved Landau branch is $e|B|/h$ per unit area, so the actual filling factor isIt varies continuously with density and field. Within a disorder-broadened plateau, localized states can change occupation without changing the integer number of transporting edge branches; the filling need not equal the plateau integer at every point in that interval.

At a boundary, the confining potential bends the Landau energies through the Fermi energy, giving chiral edge states, interpreted semiclassically as skipping orbits. Opposite edges carry opposite propagation directions. An ideal spin-resolved transmitted edge channel contributes $e^2/h$; $\nu$ such channels give $I=\nu (e^2/h)(V_S-V_D)$. Chirality and suppressed backscattering between well-separated edges explain vanishing longitudinal voltage. Ideal reservoirs and appropriate probe equilibration are part of this edge-state transport model.

A gated antidot. A local repulsive gate potential depletes an island inside the Hall bar. Edge states circulate around its boundary as closed antidot orbits. When a localized orbit is weakly coupled to extended channels on opposite sides, resonance at the reservoir electrochemical potential permits backscattering and produces conductance dips on an otherwise quantized transmitting plateau. If the path instead adds a transmission route between otherwise disconnected channels, resonant peaks can occur: the sign depends on the device's tunnel connectivity and what conductance is measured. An isolated antidot too far from every conducting edge has negligible transport signature.

For the simplest opposite-edge backscattering picture with one active branch,where $f$ branches would otherwise transmit. A gate moves the resonance through the Fermi energy by changing both confinement and orbit area. Stronger depletion of the constrictions also changes $f$, producing broad conductance plateaus or pinch-off, with narrow resonances superposed where coupling is finite. Low temperature resolves individual resonances; thermal and tunnel broadening can merge them.

For a fixed-area single-electron orbit, a phase change of $2\pi$ corresponds toSweeping gate voltage at fixed field shifts enclosed area and level energy, giving oscillatory resonance tracks. This period is not automatically set by the lithographic gate area. Charging energy, self-consistent screening and spin branches modify spacings; in particular coupled spin branches in a charging model can generate two resonances per $h/e$ cycle without changing the electron flux quantum.^[ref]

No antidot dimensions, capacitances, lever arms or measured spectrum are supplied, so absolute periods and resonance voltages cannot be extracted. The sketch illustrates the mechanisms and regimes requested, rather than inventing a numerical device data set.

Improving the bare edge potential. A picture of noninteracting Landau levels simply intersecting a prescribed smooth confinement ignores the redistribution and screening of electron charge. A self-consistent electrostatic calculation combines gate and donor potentials, electron density and Coulomb interactions. Where a partially occupied Landau level has a large density response, electrons screen strongly and create a compressible strip with nearly pinned level energy. Regions with the chemical potential in a gap are incompressible and support much of the potential change. Consequently the screened edge alternates between compressible and incompressible strips, rather than preserving an arbitrary externally drawn slope. Finite temperature, disorder, exchange and edge reconstruction can require refinements beyond the simplest electrostatic model.^[ref]

Fractional states and composite fermions. Very clean two-dimensional systems show plateaus at selected fractional fillings, with small longitudinal resistance and a many-electron excitation gap. Electron-electron correlations are essential; partial filling of a noninteracting Landau level is not enough. The simplest composite fermion binds two statistical flux quanta to each electron. This is a representation of correlated electron motion, not a physical solenoid attached to a particle. With $B>0$, ${\Phi }_0=h/e$, and density $n_s$, the mean effective field isDefine a signed effective filling ${\nu }^{\ast }=n_s{\Phi }_0/B^{\ast }$. ThenFor ${\nu }^{\ast }=p>0$, filled composite-fermion levels produce $\nu =p/(2p+1)=1/3,2/5,3/7,\dots$. For reversed effective field write ${\nu }^{\ast }=-p$, giving $\nu =p/(2p-1)=1,2/3,3/5,\dots$; the nontrivial fractional upper sequence starts with $p=2$. If effective filling is defined as nonnegative using $|B^{\ast }|$, retain these as separate field-direction branches instead of using the signed expression blindly. At $\nu =1/2$ the mean $B^{\ast }$ vanishes and the effective filling diverges, so it is not an integer composite-fermion Hall plateau in this simplest construction. Attaching two flux quanta preserves fermionic statistics; the emergent integer-like gaps organize a prominent sequence of fractional electronic states.^[ref]

A gated quantum dot is an electron island confined in all spatial directions, weakly coupled by tunnel barriers to source and drain and tuned by a plunger gate. Small capacitance makes charge addition costly, while confinement resolves orbital levels. In the constant-interaction model,The charging contribution to successive addition potentials is $e^2/C_{\Sigma }$; the often used charging-energy scale $E_C=e^2/(2C_{\Sigma })$ differs by a factor of two. Orbital spacing and spin filling add further structure.

For weak coupling, $k_{B}T$ and lifetime broadening must be smaller than the relevant addition energies to see Coulomb blockade. Sequential tunnelling proceeds when an addition level is inside the source-drain bias window and the appropriate charge states are accessible. A gate sweep gives conductance peaks near charge degeneracies and low conductance between them. A bias-versus-gate map shows Coulomb diamonds, with excited-state lines when additional levels enter the window. Their slopes and sizes depend on source, drain and gate capacitances and the bias convention. Cotunnelling gives higher-order transport inside nominally blocked regions; stronger coupling and many-body effects can invalidate the simple sequential picture.

Coherent dot states as a qubit. A single-electron spin has basis $|0⟩=|\uparrow ⟩$, $|1⟩=|\downarrow ⟩$ in a magnetic field, split by $E_Z=|g^{\ast }|{\mu }_{B}B$. A qubit is a controllable coherent state $a|\uparrow ⟩+b|\downarrow ⟩$, not just a pair of classical current levels. Initialize through spin-selective loading or relaxation; resonant magnetic or spin-orbit-assisted electrical driving produces rotations. Adjacent occupied dots can couple through a gate-controlled exchange interaction. In the weak-tunnel symmetric Hubbard limit,Local spin rotations combined with a suitable exchange entangling operation provide universal control in the ideal spin-qubit proposal. Gate pulses must avoid unwanted orbital excitation and leakage, and complete coherent operations before dephasing destroys the information.^[ref]

For measurement, place the reservoir electrochemical potential between Zeeman levels so that only one spin state can tunnel out, or use Pauli spin blockade in a double dot. A nearby quantum point contact or single-electron transistor detects the resulting charge change. Such spin-to-charge conversion makes transport useful for reading a spin even when the stored information is not itself a net current.^[ref]

A double dot can alternatively encode a charge qubit in left/right occupation, with Hamiltonian $H=(\epsilon {\sigma }_z+2t_c{\sigma }_x)/2$. It allows electrical control and charge sensing, but couples strongly to charge noise. Spin qubits also suffer hyperfine, electrical and spin-orbit-related noise, as well as relaxation and readout errors. An observed Coulomb diamond alone establishes charging physics, not initialization, coherent control or a functioning qubit.

Constancy of each ionic electrochemical potential gives the Boltzmann distribution in the local electric potential:Positive potential repels positive ions and attracts negative counterions. The coordinate $s$ measures distance from a locally planar surface; the same local relations hold for a spherical potential with its appropriate radial coordinate.

A direct band gap has its conduction minimum and valence maximum at the same crystal momentum. A photon can then connect the band-edge states with negligible momentum change. An indirect band gap requires an additional phonon to supply or remove momentum, making radiative recombination much less probable. Direct-gap materials such as gallium arsenide or gallium nitride are therefore usually better for a light-emitting diode than silicon.

For the graded junction, let $q=|e|>0$, ${\epsilon }_s={\epsilon }_0{\epsilon }_r$, and use electrostatic potential $\phi$ with $E_x=-{\phi }^{\prime }$. Fully ionized dopants have charge density $\rho =q(N_D-N_A)=q\alpha x$, so Poisson's equation is ${\phi }^{\prime \prime }=-q\alpha x/{\epsilon }_s$. The paper's displayed plus sign corresponds to using the negative potential $V=-\phi$, or a signed electron charge in that formula. The positive barrier magnitude and charge magnitude must be used consistently in the width below.

The intended compensation point is $x=0$, where $N_D=N_A$ and the net doping changes sign. A locally neutral model with comparable electron and hole mobilities has its lowest carrier conductivity near there. The depletion approximation neglects mobile-carrier conductivity throughout the layer, so it does not by itself locate an exact conductivity minimum; unequal mobilities or asymmetric carrier profiles can shift it.

Let the depletion edges be $x=\pm d$. With zero exterior field, integrating $E_x^{\prime }=\rho /{\epsilon }_s$ and imposing $E_x(\pm d)=0$ gives the linearly graded p-n junction fieldIts integral gives $\phi (x)=\phi (0)+q\alpha (d^{2}x-x^3/3)/(2{\epsilon }_s)$, so the electric potential increases toward the n-type side while the electron band edges $E_c,E_v$ bend downward together. Their separation is the local band gap, and the equilibrium electrochemical potential is flat. The built-in potential arises when carrier diffusion exposes ionized dopants until the opposing field balances further diffusion.

Finally the positive barrier isWith total width $w=2d$, this yields the linearly graded junction depletion widthHere $\alpha >0$ labels p-to-n progression with increasing $x$; otherwise use its magnitude in the width formula.

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.

A positive gate voltage attracts electrons towards the oxide interface of the p-type semiconductor and repels mobile electron holes. The uncovered ionized acceptors form a negatively charged depletion region. Increasing the voltage further creates a thin electron-rich semiconductor inversion layer at the surface. The conduction band and valence band both bend downwards towards the positively biased gate, because an electron's electrostatic energy is $-e\psi$.

In an ideal nondegenerate substrate with acceptor density $N_A$, write ${\varphi }_F=(k_{B}T/e)\ln (N_A/n_i)>0$ and surface potential ${\psi }_s$ relative to the neutral bulk. The electron and hole surface densities areThe first change of majority-carrier type occurs at ${\psi }_s={\varphi }_F$, where $n_s=p_s=n_i$. The conventional MOS inversion threshold for appreciable strong inversion isAt threshold the inversion sheet is just developing, while a much thicker depletion region lies between it and the neutral p-type bulk. Inversion does not require the conduction band to cross the Fermi level: the surface can remain nondegenerate.

The metal and semiconductor Fermi levels need not be equal. The ideal oxide blocks particle exchange, so the gate and substrate can maintain different electrochemical potentials. In a biased MOSFET, the gate-to-substrate Fermi-level difference is ${\tilde{\mu }}_g-{\tilde{\mu }}_s=-e{V}_g$, when the applied voltage is defined in the electron-reservoir convention. Each region can separately be in internal equilibrium. If an external zero-voltage connection instead equilibrates the reservoirs, their Fermi levels coincide, but different work functions can still cause band bending. Here the chemical potential shown in a band diagram is the electron electrochemical potential; the purely chemical contribution alone is not the equilibrium criterion for charged particles.

At equilibrium, majority electrons undergo diffusion from n to p, while majority electron holes diffuse from p to n. Their departure exposes positive donors on the n side and negative acceptors on the p side. The resulting electric field points from n to p and opposes further diffusion. Equilibrium is reached when the chemical potential is constant across the junction. The built-in potential is this electrostatic adjustment, not a discontinuity of the material's band gap.

Since electronic band energies shift by $-eV(x)$, both the conduction band and valence band undergo band bending down from p to n by $e{V}_b$ at zero applied bias. Their separation remains the same gap. In the nondegenerate thermal equilibrium limit the built-in voltage can also be expressed using neutral-edge densities as $e{V}_b=k_{B}T\log (n_n/n_p)=k_{B}T\log (n_n{p}_p/n_i^2)$.

Take the n-contact potential and its electrochemical potential as reference. Applying $V_{\mathrm{ext}}$ to the p side changes the total depletion-layer drop toThus $V_{\mathrm{ext}}<0$ is reverse bias: it increases the barrier and widens the depletion region. The bands have horizontal tangents at the neutral edges in this approximation and a smooth cubic bend within the linearly graded layer.

A biased junction is not in global thermal equilibrium. Its contact electron electrochemical potentials obeyUnder reverse bias the p-contact level is higher by $e|V_{\mathrm{ext}}|$. A single horizontal equilibrium chemical potential cannot describe this state; electron and hole quasi-Fermi levels describe the nonequilibrium populations. The sketch marks the contact levels. Detailed spatial quasi-Fermi profiles require transport and recombination information beyond the electrostatic doping profile.

ATP synthase couples downhill proton motion across a cell membrane to synthesis of ATP from ADP and inorganic phosphate. Its membrane-embedded $F_o$ sector contains an $a$ subunit with offset proton-access half-channels and a rotating $c$-subunit ring. Reversible protonation of sites on the ring permits rotation relative to the stationary $a$ subunit. The central rotor transmits this motion through the $\gamma$ shaft to the soluble $F_1$ head. A peripheral stator holds the head relative to the membrane sector. The common bacterial schematic has an ${\alpha }_3{\beta }_3$ head, catalytic sites mainly on the three $\beta$ subunits, and a central $\gamma /ϵ$ rotor; accessory stator components depend on the organism.

The binding-change mechanism cycles the catalytic sites between open, loose-binding and tight-binding conformations. Rotor orientation changes their affinities for ADP, phosphate and ATP; the sites operate out of phase. Tight binding favors formation of bound ATP, and mechanical rotation changes the conformation to permit product release. In the ideal tightly coupled cycle, one revolution gives three catalytic turnovers:The proton count is not universally fixed: an ideal ring with $n_c$ proton-carrying subunits gives $n_c$ protons per revolution and $n_c/3$ protons per synthesized ATP, before additional transport costs.

For a proton moving from side 1 to side 2, its molar electrochemical potential change isThe chemical gradient and membrane electric potential difference together supply the proton-motive force. Synthesis requires the proton free-energy release to meet the positive Gibbs free energy of ATP synthesis. The coupling is reversible: isolated $F_1$ supplied with ATP can hydrolyse it and rotate its shaft, while the intact enzyme can use hydrolysis-driven rotation to pump protons against their gradient. The sketch is structural and schematic; it does not reconstruct the absent stepping trace.

Use number concentrations and write the electrochemical potential of a sodium ion as $\tilde{\mu }={\mu }^{\circ }+k_{\mathrm{B}}T\ln c+e\psi$. Define the inward sodium-motive force as available work per positive charge for transport from outside to inside. With $V_m={\psi }_i-{\psi }_o$,Its negative is the signed inward free-energy cost per charge; stating this direction removes a common sign ambiguity.

Long-open sodium channels establish electrochemical equilibrium, $k_{\mathrm{B}}T\ln (c_{i0}/c_o)+e{V}_{m0}=0$. This does not generally require equal concentrations: impermeant counterions can support a nonzero equilibrium membrane potential. Let $n_0=c_{i0}V$. For a large fixed-concentration external reservoir, after pumping $n$ ions out, $c_i(n)=(n_0-n)/V$ and $V_m(n)=V_{m0}-ne/C$. The sodium export cost from a capacitive vesicle for exporting another ion isThe initial-equilibrium relation was used in the last step. The first term is the incremental electrical cost and the second is the change in chemical cost from their initial balanced values.

If the intended initial reference is uncharged with $V_{m0}=0$ and $c_{i0}=c_o$, then $n_0=c_{o}V$ and the requested form in the listed variables isThat equal-concentration reference is an additional assumption. For example, $c_{i0}=2c_o$ and $V_{m0}=-(k_{\mathrm{B}}T/e)\ln 2$ also give zero initial sodium flux, but a different ion inventory. Thus the open-channel condition alone cannot determine $n_0$ from $c_{o}V$. If concentrations are molar rather than number densities, use the corresponding Avogadro constant conversion and $RT$ for molar free energies. Treating the capacitor charge discretely gives a charging-energy increment $(n+1/2)e^2/C$ rather than the differential $n{e}^2/C$; the distinction is negligible for the thousands of ions used next.

ATP synthase combines a membrane $F_o$ proton-conducting rotor with an $F_1$ catalytic head. The c-ring and central gamma shaft rotate relative to a stator and the alpha/beta head. Proton flow down its electrochemical potential can drive conformational changes in three beta catalytic sites to bind substrates, form and release ATP. Reverse operation uses ATP hydrolysis to turn the shaft and pump protons. The immobilized $F_1$ assay isolates this latter rotation; it does not require the membrane proton motor to be present.