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For mobile thermal electrons with a fixed neutralizing background, the Debye length is the characteristic distance over which plasma screening suppresses an electrostatic perturbation. Linearizing the Boltzmann density response and using Poisson's equation gives the displayed scale. Other mobile species contribute additively to ${\lambda }_D^{-2}$ through $n_s{q}_s^2/({ϵ}_0{k}_B{T}_s)$.

Connecting a conductor to a large reference reservoir fixes its electric potential, conventionally at zero, while allowing electric charge to enter or leave. Grounding does not fix the conductor's net electric charge. In electrostatics, its surface is then a prescribed-potential boundary for Poisson's equation.

For a uniformly doped p-n junction whose depleted thickness $d$ lies almost entirely on the donor side, Poisson's equation gives $dE/dx=e{N}_D/ϵ$. The minimum field obeys $E(d)=0$. Integrating the triangular electric field profile gives $V_{\mathrm{dep}}=e{N}_D{d}^2/(2ϵ)$. The external reverse bias is this total voltage minus the built-in junction potential.

Write $e=q>0$ for the elementary charge and $ϵ={ϵ}_0{ϵ}_r$ for the permittivity. Taking the bulk electric potential as zero, the Boltzmann distribution of point ions gives $c_i(x)=c_i^{\infty }\exp [-e{z}_i\varphi (x)/(k_{B}T)]$. Here $c_i$ is a number density, not a molar concentration. Combining this with Poisson's equation gives the planar Poisson-Boltzmann equationMultiplying by ${\varphi }^{\prime }$ and integrating from the neutral bulk, where $\varphi ={\varphi }^{\prime }=0$, yieldsFor the solution occupying $x>0$, with the field on the other side neglected, Gauss's law gives $\sigma =-ϵ{\varphi }^{\prime }(0)$. On the branch that decays to zero without crossing it,Source sign qualification: the unsigned square root printed in the paper gives $|\sigma |$. A negative wall electric potential, such as the next part's $-73\mathrm{mV}$, has ${\varphi }^{\prime }(0)>0$ and hence $\sigma <0$, providing a concrete counterexample to interpreting that printed expression as signed surface charge density. This is the planar Grahame equation, with an explicit sign convention.

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.

For any closed surface enclosing a region $V$, Gauss's law for gravity states that the outward flux of the gravitational field isHere $G$ is the gravitational constant and $M_V$ is the enclosed mass; the negative sign expresses attractive gravity.

Use the gravitational potential sign conventions specified by this paper: $\mathbf{g}=\nabla \phi$. Thus $\phi$ is the negative of the usual Newtonian gravitational potential $\Phi$ defined by $\mathbf{g}=-\nabla \Phi$. For example, setting the value at infinity to zero gives $\phi =GM/r$ for a point mass. The divergence theorem now givesSince $V$ is arbitrary, continuity of the integrands implies the local Poisson's equationConversely, integrating this Poisson's equation over any $V$ and applying the divergence theorem recovers Gauss's law for gravity. At a discontinuity of the mass density, the local equation is interpreted piecewise, or globally in the sense of distributions.

For a vacuum electric potential, put $\Phi =R(r)Y(\theta )$ in Laplace's equation in spherical coordinates. Multiplication by $r^2/(RY)$ separates the radial and angular terms. With separation constant $\ell (\ell +1)$ and $x=\cos \theta$, the angular Legendre differential equation isRegularity at both polar axes selects nonnegative integers $\ell$ and the Legendre polynomials. The radial equation is $(r^2{R}^{\prime }{)}^{\prime }-\ell (\ell +1)R=0$, whose independent solutions are $r^{\ell }$ and $r^{-\ell -1}$. Hence the angularly regular axisymmetric vacuum solution isThe radial domain and its boundary conditions determine which terms are admissible.

For the given charge density, only $P_1(\cos \theta )=\cos \theta$ is forced. Write $\Phi =f(r)\cos \theta$. The interior radial Poisson equation is $f^{\prime \prime }+2f^{\prime }/r-2f/r^2=-4$. The identity for the Poisson equation with a spherical harmonic source gives the particular solution $f_p=-r^2$, since the radial operator takes $r^2$ to four. Finiteness at zero and decay at infinity therefore giveContinuity of the electric potential and its radial derivative at $a$ requires $Aa-a^2=B/a^2$ and $A-2a=-2B/a^3$. Solving gives $A=4a/3$, $B=a^4/3$, soAll other angular modes have zero coefficients after imposing regularity, matching and decay. At the origin the interior expression is $(4a/3)z-rz$, which extends with a continuous gradient and value zero. The angular charge density has no unique value exactly at that point; Poisson's equation holds classically for $r>0$ and as a weak solution through the origin. There is no additional point charge.

Reflect the source across the equatorial plane and subtract the two ball kernels. This gives the Dirichlet Green function inside a hemisphere $G_{\mathrm{hemi}}(\mathbf{x};{\mathbf{x}}_0)=G_B(\mathbf{x};{\mathbf{x}}_0)-G_B(\mathbf{x};{\mathbf{x}}_0^{-})$. It is zero on the spherical boundary because each ball kernel is zero there, and on the plane because reflection makes their values equal.

For a point charge $e$, Poisson's equation in SI units is $\Delta V=-e\delta (\mathbf{x}-{\mathbf{x}}_0)/{\epsilon }_0$, so $V=-e{G}_{\mathrm{hemi}}/{\epsilon }_0$. Writing ${\mathbf{x}}_0^{\ast -}$ for the reflected inverse point givesThe construction uses the real charge, its negative reflected image, a negative inversion image of magnitude $ea/r_0$, and its positive reflected image. Reflection and inversion commute. All three images lie outside the open upper half-ball, and the prescribed zero boundary values determine the electrostatic potential uniquely.

For $s=|\mathbf{x}-{\mathbf{x}}_0|>0$, the radial Laplacian gives ${\nabla }^2(1/s)=s^{-2}{\partial }_s(s^2{\partial }_s(1/s))=0$. The outward flux of $\nabla [-1/(4\pi s)]$ through a small sphere centred at the source is $[1/(4\pi s^2)]4\pi s^2=1$. Applying Green's second identity to a smooth test function outside that small sphere and shrinking its radius therefore gives the distributional identityThe kernel tends to zero at infinity. Thus it is the fundamental solution with the sign convention $\Delta G=+\delta$. The sign will reverse when converting it into the electrostatic potential of a positive point charge through Poisson's equation.

A charged colloid in a polar electrolyte is surrounded by an electric double layer. For a negatively charged surface, counterions are positive and co-ions negative; their distributions reverse for a positive surface. A compact Stern layer contains specifically adsorbed or closely approaching ions, followed by a diffuse layer where electrostatic attraction competes with thermal motion. Mobile charge integrated over the surroundings balances the particle charge in an overall neutral system.

For a mean-field dilute electrolyte, $n_i(r)=n_{i,\infty }\exp [-z_{i}e\psi (r)/(k_{B}T)]$ with Poisson's equation relates charge density and potential. At weak potential, the diffuse potential outside a sphere decays approximately as $\psi (r)\propto e^{-\kappa (r-a)}/r$. The inverse screening length obeys ${\kappa }^2=e^2{\sum }_i{z}_i^2{n}_{i,\infty }/(ϵk_{B}T)$: added salt shortens the Debye length. The zeta potential is the potential at the hydrodynamic slip plane, not necessarily the surface potential. The simple model neglects ion size, correlations and chemical charge regulation; a polar solvent with almost no dissociated ions need not provide a short screening length.

Let $\Phi (x)$ be the electrostatic potential in volts, relative to the bulk value, and $x$ the distance normal to the surface. The positive quantity $e$ is the elementary charge; the monovalent ions carry $\pm e$. The bulk number density $n_0$ refers to each ionic species in a symmetric electrolyte, so the total bulk ion density is $2n_0$. ${ϵ}_0$ is vacuum permittivity, ${ϵ}_r$ relative permittivity, $k_B$ the Boltzmann constant and $T$ absolute temperature. The hyperbolic sine acts on the dimensionless potential $u=e\Phi /(k_{B}T)$.

The printed sign is inconsistent with physical Poisson–Boltzmann screening under these usual definitions. Indeed,and Poisson's equation gives ${\Phi }^{\prime \prime }=-\rho /({ϵ}_0{ϵ}_r)=+2e{n}_0\sinh u/({ϵ}_0{ϵ}_r)$, with a plus sign. The literal printed equation and the physical screened solution must therefore be distinguished.

Gauss's law states that the outward electric flux through a closed surface is the enclosed electric charge divided by the vacuum permittivity:In electrostatics, the electric field is the negative gradient of the electrostatic potential, $\mathbf{E}=-\nabla V$. Substitution into the differential equation form of Gauss's law gives Poisson's equation ${\nabla }^{2}V=-\rho /{ϵ}_0$. Consequently, in a region containing no charge density,which is Laplace's equation. Charges on the bounding conductors enter through the boundary conditions; they need not vanish for Laplace's equation to hold inside the empty gap.

For this space-charge-filled parallel-plate capacitor, place the first plate at $x=0$ and the second at $x=d=x_o$, ignoring fringing field of a capacitor. With vacuum permittivity between the plates and fixed charge density $\rho$, Poisson's equation and the prescribed boundary electric potentials giveIntegrating twice and imposing both values yieldsThe electric field is minus the gradient of the electric potential, henceThus the electric potential is a parabola, concave downward for $\rho >0$ and upward for $\rho <0$. The signed electric field is a straight line. Its magnitude has a V-shaped corner if its zerolies inside the gap; otherwise its magnitude is a single straight segment. An interior zero requires $|V_1-V_2|<|\rho |d^2/(2{ϵ}_0)$. At that point the electric potential has its interior extremum. For $\rho =0$, these reduce to a linear electric potential and a uniform electric field.

Donor doping supplies conduction electrons to n-type semiconductor; acceptor doping supplies holes to p-type semiconductor. In the nondegenerate, fully ionized limit, majority densities are approximately $n_n=N_D$ and $p_p=N_A$, while minority densities follow $np=n_i^2$. On contact, majority carriers diffuse across the junction, exposing positive donor ions on the n side and negative acceptor ions on the p side. The depletion-region electric field points from n to p and opposes further diffusion. At thermal equilibrium the Fermi level is uniform and drift balances diffusion, with no net current.

For an abrupt junction with uniform doping and permittivity ${\epsilon }_s$, charge neutrality gives $N_D{x}_n=N_A{x}_p$. Poisson's equation then givesPositive applied $V$ means forward bias: it lowers the barrier and narrows the depletion region. Most depletion width lies in the more lightly doped side. The formula assumes the depletion approximation and fails once its operating assumptions break down.

forward bias increases minority densities at the depletion edges by the factor $e^{qV/k_{B}T}$. In the neutral material, the steady diffusion–recombination equation gives exponentially decaying excess minorities, with lengths $L_n=\sqrt{D_n{\tau }_n}$ and $L_p=\sqrt{D_p{\tau }_p}$. Taking the carrier gradients at the junction yieldsThis ideal diode law assumes low injection, nondegenerate statistics, negligible depletion-region recombination and series resistance, and sufficiently long neutral regions. Its exponential comes from the bias-dependent injection barrier; there is no universal abrupt “turn-on” voltage. reverse bias suppresses injection, giving an approximate $-I_S$ current until additional leakage or breakdown matters.^[ref]

Rectification uses the asymmetric current-voltage characteristic; controlled reverse breakdown enables voltage references, with current limiting. Bias-dependent depletion capacitance enables varactors. photodiodes and solar cells collect optically generated carriers through the junction field and diffusion toward it; efficient light-emitting diodes use radiative electron–hole recombination, preferably in a direct-gap material. Series resistance, high injection, trap-assisted recombination, heating and finite carrier storage explain departures from the ideal steady-state law.

Substitution into Poisson's equation, ${\nabla }^2\psi =-\rho /\epsilon$, gives the Poisson-Boltzmann equationFor a plane, ${\nabla }^2\psi =d^2\psi /d{s}^2$; for a sphere, it is $r^{-2}d(r^2{\psi }^{\prime })/dr$. The bulk conditions are $\psi \to 0$ and $n_{\pm }\to n_0$. At a planar wall with negligible internal field and normal pointing into the liquid, Gauss's law imposes $-\epsilon {\psi }^{\prime }(0)={\sigma }_s$; alternatively, a prescribed surface potential gives $\psi (0)={\psi }_0$.

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.

In a conductor, mobile electrons redistribute in response to an electric potential, producing an induced charge density that opposes its source. Finite electronic compressibility means a finite density change requires a change of local kinetic Fermi energy. Screening is therefore not perfect at every microscopic distance. On scales comparable to atomic structure or the Fermi wavelength, a continuum local-density model itself fails.

Thomas-Fermi screening treats electrons as a locally uniform degenerate free electron gas, replaces the ionic background by a smooth neutralizing charge, and neglects nonlocal response, exchange and correlation corrections. The electrochemical chemical potential is uniform in equilibrium. Because an electron has charge $-e$, its electrostatic energy is $-eV$, givingAt zero potential $\mu =E_{F0}$. Linearizing at fixed $\mu$, with $|eV|\ll E_{F0}$, givesInsert this in Poisson's equation to obtain $({\nabla }^2-{\xi }^{-2})V=-{\rho }_{\mathrm{free}}/{ϵ}_0$, where the Thomas-Fermi screening length isAway from a point source, direct radial differentiation shows ${\nabla }^2(e^{-r/\xi }/r)={\xi }^{-2}{e}^{-r/\xi }/r$. The decaying solution is therefore $V=C{e}^{-r/\xi }/r$. Integrating Poisson's equation over a small sphere containing charge $Q$ fixes $C=Q/(4\pi {ϵ}_0)$. For a sufficiently large source, the linearized formula is only a far-field approximation, not a justified description of its singular core.

For monovalent sodium, the electron number density is its atomic number density,This gives $E_{F0}=3.15\mathrm{eV}$ andRoom temperature is far below the corresponding Fermi temperature, so degeneracy is well justified, and sodium's simple conduction band makes a free-electron estimate useful. However $k_F\xi =0.615$ and the mean atomic spacing is $n_0^{-1/3}=0.340\mathrm{nm}$. The potential varies on a scale shorter than $k_F^{-1}$ and the atomic spacing, so the slowly varying local-density and continuum assumptions are not quantitatively controlled at the screening length itself. A nonlocal treatment includes Friedel oscillations, absent from the pure exponential.

Nearly unchanged interlayer force constants in two-layer and twenty-layer films are consistent with very local screening and bonding forces: reducing the distant bulk need not strongly change the force between adjacent layers. They also indicate that metallic electronic response can survive in the thin film. The observation does not by itself validate the numerical Thomas-Fermi approximation, establish an identical full phonon spectrum, or rule out surface-induced electronic changes; it directly constrains the measured local force constant.

Linearize the electron Boltzmann distribution, $n_e(\varphi )\simeq n_{e0}[1+e\varphi /(k_{B}T)]$, about a uniform fixed-ion background. Poisson's equation gives ${\nabla }^2\varphi =\varphi /{\lambda }_D^2$, where the electron Debye length isDefine the plasma parameter as the number of electrons in a Debye sphere:Some authors omit $4\pi /3$ in this definition. Mobile ions alter the total screening length by contributing their own susceptibility; the electron-only convention is used consistently here.

For a second-order scalar partial differential equation, classification refers to its principal second-derivative form. An elliptic partial differential equation has a definite principal form, exemplified by Poisson's equation; it typically determines a spatial field from boundary data. A parabolic partial differential equation has a distinguished first-order evolution direction and diffusive second spatial derivatives, exemplified by $u_t=D{u}_{xx}$. Its initial data relax or spread with time. Thus the pseudo-time equation is parabolic, while its steady-state spatial equation is elliptic.

Let $r=\Delta \tau /h^2$ and impose periodic indices modulo $J$. With ${\mathbf{\Phi }}^n=({\Phi }_0^n,\dots ,{\Phi }_{J-1}^n{)}^T$, the linear system iswhere $S$ shifts one grid cell cyclically. For $J\ge 3$, $A$ has diagonal $1+2r$, adjacent and corner entries $-r$, and all other entries zero. It is a cyclic tridiagonal matrix, rather than a nonperiodic tridiagonal one.

The periodic Laplacian annihilates the constant vector, so its steady equation requires $\boxed{\sum _j{\rho }_j=0}$. A strictly positive gravitational density violates this condition: summing the update gives ${\bar{\Phi }}^{n+1}={\bar{\Phi }}^n-4\pi G\Delta \tau \bar{\rho }$, an indefinitely drifting mean. Periodic gravitational calculations normally solve for $\rho -\bar{\rho }$ and fix the arbitrary mean gravitational potential. The per-step matrix $A$ itself remains invertible; the issue is existence of a periodic steady solution of Poisson's equation.