Inphysics,screening is the damping ofelectric fields caused by the presence of mobilecharge carriers. It is an important part of the behavior of charge-carrying mediums, such as ionized gases (classicalplasmas),electrolytes, and electronic conductors (semiconductors,metals).In a fluid, with a givenpermittivityε, composed of electrically charged constituent particles, each pair of particles (with chargesq₁ andq₂) interact through theCoulomb force as$${\displaystyle \mathbf{F}=\frac{q_1{q}_2}{4\pi \epsilon {\left|\mathbf{r}\right|}^2}\hat{\mathbf{r}},}$$where the vectorr is the relative position between the charges. This interaction complicates the theoretical treatment of the fluid. For example, a naive quantum mechanical calculation of the ground-state energy density yields infinity, which is unreasonable. The difficulty lies in the fact that even though the Coulomb force diminishes with distance as1/r², the average number of particles at each distancer is proportional tor², assuming the fluid is fairlyisotropic. As a result, a charge fluctuation at any one point has non-negligible effects at large distances.

In reality, these long-range effects are suppressed by the flow of particles in response to electric fields. This flow reduces theeffective interaction between particles to a short-range "screened" Coulomb interaction. This system corresponds to the simplest example of a renormalized interaction.^[1]

Insolid-state physics, especially formetals andsemiconductors, thescreening effect describes theelectrostatic field and Coulomb potential of anion inside the solid. Like the electric field of thenucleus is reduced inside an atom or ion due to theshielding effect, the electric fields of ions in conducting solids are further reduced by the cloud ofconduction electrons.

Consider a fluid composed of electrons moving in a uniform background of positive charge (one-component plasma). Each electron possesses a negative charge. According to Coulomb's interaction, negative charges repel each other. Consequently, this electron will repel other electrons creating a small region around itself in which there are fewer electrons. This region can be treated as a positively charged "screening hole". Viewed from a large distance, this screening hole has the effect of an overlaid positive charge which cancels the electric field produced by the electron. Only at short distances, inside the hole region, can the electron's field be detected. For a plasma, this effect can be made explicit by an${\displaystyle N}$-body calculation.^[2]: §5 If the background is made up of positive ions, their attraction by the electron of interest reinforces the above screening mechanism. In atomic physics, a germane effect exists for atoms with more than one electron shell: theshielding effect. In plasma physics, electric-field screening is also called Debye screening or shielding. It manifests itself on macroscopic scales by a sheath (Debye sheath) next to a material with which the plasma is in contact.

The screened potential determines the inter atomic force and thephonondispersion relation in metals. The screened potential is used to calculate theelectronic band structure of a large variety of materials, often in combination withpseudopotential models. The screening effect leads to theindependent electron approximation, which explains the predictive power of introductory models of solids like theDrude model, thefree electron model and thenearly free electron model.

The first theoretical treatment ofelectrostatic screening, due toPeter Debye andErich Hückel,^[3] dealt with a stationary point charge embedded in a fluid.

Consider a fluid of electrons in a background of heavy, positively charged ions. For simplicity, we ignore the motion and spatial distribution of the ions, approximating them as a uniform background charge. This simplification is permissible since the electrons are lighter and more mobile than the ions, provided we consider distances much larger than the ionic separation. Incondensed matter physics, this model is referred to asjellium.

Letρ denote thenumber density of electrons, andφ theelectric potential. At first, the electrons are evenly distributed so that there is zero net charge at every point. Therefore,φ is initially a constant as well.

We now introduce a fixed point chargeQ at the origin. The associatedcharge density isQδ(r), whereδ(r) is theDirac delta function. After the system has returned to equilibrium, let the change in the electron density and electric potential be Δρ(r) and Δφ(r) respectively. The charge density and electric potential are related byPoisson's equation, which gives$${\displaystyle -{\mathrm{\nabla }}^2[\mathrm{\Delta }\varphi (r)]=\frac{1}{{\epsilon }_0}[Q\delta (r)-e\mathrm{\Delta }\rho (r)],}$$whereε₀ is thevacuum permittivity.

To proceed, we must find a second independent equation relatingΔρ andΔφ. We consider two possible approximations, under which the two quantities are proportional: the Debye–Hückel approximation, valid at high temperatures (e.g. classical plasmas), and the Thomas–Fermi approximation, valid at low temperatures (e.g. electrons in metals).

In the Debye–Hückel approximation,^[3] we maintain the system in thermodynamic equilibrium, at a temperatureT high enough that the fluid particles obeyMaxwell–Boltzmann statistics. At each point in space, the density of electrons with energyj has the form$${\displaystyle {\rho }_j(r)={\rho }_j^{(0)}(r)\exp \left[\frac{e\varphi (r)}{k_{\mathrm{B}}T}\right]}$$wherek_B is theBoltzmann constant. Perturbing inφ and expanding the exponential to first order, we obtain$${\displaystyle e\mathrm{\Delta }\rho \simeq {\epsilon }_0{k}_0^2\mathrm{\Delta }\varphi }$$where$${\displaystyle k_0\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\frac{\rho e^2}{{\epsilon }_0{k}_{\mathrm{B}}T}}}$$

The associated lengthλ_D ≡ 1/k₀ is called theDebye length. The Debye length is the fundamental length scale of a classical plasma.

In the Thomas–Fermi approximation,^[4] named afterLlewellyn Thomas andEnrico Fermi, the system is maintained at a constant electronchemical potential (Fermi level) and at low temperature. The former condition corresponds, in a real experiment, to keeping the metal/fluid in electrical contact with a fixedpotential difference withground. The chemical potentialμ is, by definition, the energy of adding an extra electron to the fluid. This energy may be decomposed into a kinetic energyT part and the potential energy −eφ part. Since the chemical potential is kept constant,$${\displaystyle \mathrm{\Delta }\mu =\mathrm{\Delta }T-e\mathrm{\Delta }\varphi =0.}$$

If the temperature is extremely low, the behavior of the electrons comes close to thequantum mechanical model of aFermi gas. We thus approximateT by the kinetic energy of an additional electron in the Fermi gas model, which is simply theFermi energyE_F. The Fermi energy for a 3D system is related to the density of electrons (including spin degeneracy) by$${\displaystyle \rho =2\frac{1}{(2\pi {)}^3}\left(\frac{4}{3}\pi k_{\mathrm{F}}^3\right),E_{\mathrm{F}}=\frac{{\hslash }^2{k}_F^2}{2m},}$$wherek_F is the Fermi wavevector. Perturbing to first order, we find that$${\displaystyle \mathrm{\Delta }\rho \simeq \frac{3\rho }{2E_{\mathrm{F}}}\mathrm{\Delta }{E}_{\mathrm{F}}.}$$

Inserting this into the above equation for Δμ yields$${\displaystyle e\mathrm{\Delta }\rho \simeq {\epsilon }_0{k}_0^2\mathrm{\Delta }\varphi }$$where$${\displaystyle k_0\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\sqrt{\frac{3e^2\rho }{2{\epsilon }_0{E}_{\mathrm{F}}}}=\sqrt{\frac{m{e}^2{k}_{\mathrm{F}}}{{\epsilon }_0{\pi }^2{\hslash }^2}}}$$is called the Thomas–Fermi screening wave vector.

This result follows from the equations of a Fermi gas, which is a model of non-interacting electrons, whereas the fluid, which we are studying, contains the Coulomb interaction. Therefore, the Thomas–Fermi approximation is only valid when the electron density is low, so that the particle interactions are relatively weak.

Our results from the Debye–Hückel or Thomas–Fermi approximation may now be inserted into Poisson's equation. The result is$${\displaystyle \left[{\mathrm{\nabla }}^2-k_0^2\right]\varphi (r)=-\frac{Q}{{\epsilon }_0}\delta (r),}$$which is known as thescreened Poisson equation. The solution is$${\displaystyle \varphi (r)=\frac{Q}{4\pi {\epsilon }_{0}r}{e}^{-k_{0}r},}$$which is called a screened Coulomb potential. It is a Coulomb potential multiplied by an exponential damping term, with the strength of the damping factor given by the magnitude ofk₀, the Debye or Thomas–Fermi wave vector. Note that this potential has the same form as theYukawa potential. This screening yields adielectric function${\displaystyle \epsilon (r)={\epsilon }_0{e}^{k_{0}r}}$.

A mechanical${\displaystyle N}$-body approach provides together the derivation of screening effect and ofLandau damping.^[2][5] It deals with a single realization of a one-component plasma whose electrons have a velocity dispersion (for a thermal plasma, there must be many particles in a Debye sphere, a volume whose radius is the Debye length). On using the linearized motion of the electrons in their own electric field, it yields an equation of the type$${\displaystyle \mathcal{E}\mathrm{\Phi }=S,}$$

where${\displaystyle \mathcal{E}}$ is a linear operator,${\displaystyle S}$ is a source term due to the particles, and${\displaystyle \mathrm{\Phi }}$ is the Fourier-Laplace transform of the electrostatic potential. When substituting an integral over a smooth distribution function for the discrete sum over the particles in${\displaystyle \mathcal{E}}$, one gets$${\displaystyle ϵ(\mathbf{k},\omega )\mathrm{\Phi }(\mathbf{k},\omega )=S(\mathbf{k},\omega ),}$$where${\displaystyle ϵ(\mathbf{k},\omega )}$ is the plasma permittivity, or dielectric function, classically obtained by a linearizedVlasov-Poisson equation,^[6]: §6.4${\displaystyle \mathbf{k}}$ is the wave vector,${\displaystyle \omega }$ is the frequency, and${\displaystyle S(\mathbf{k},\omega )}$ is the sum of${\displaystyle N}$ source terms due to the particles.^{[2]: Equation 20}

By inverse Fourier-Laplace transform, the potential due to each particle is the sum of two parts^[2]: §4.1 One corresponds to the excitation ofLangmuir waves by the particle, and the other one is its screened potential, as classically obtained by a linearized Vlasovian calculation involving a test particle.^[6]: §9.2 The screened potential is the above screened Coulomb potential for a thermal plasma and a thermal particle. For a faster particle, the potential is modified.^[6]: §9.2 Substituting an integral over a smooth distribution function for the discrete sum over the particles in${\displaystyle S(\mathbf{k},\omega )}$, yields the Vlasovian expression enabling the calculation of Landau damping.^[6]: §6.4

In real metals, the screening effect is more complex than described above in the Thomas–Fermi theory. The assumption that the charge carriers (electrons) can respond at any wavevector is just an approximation. However, it is not energetically possible for an electron within or on aFermi surface to respond at wavevectors shorter than the Fermi wavevector. This constraint is related to theGibbs phenomenon, whereFourier series for functions that vary rapidly in space are not good approximations unless a very large number of terms in the series are retained. In physics, this phenomenon is known asFriedel oscillations, and applies both to surface and bulk screening. In each case the net electric field does not fall off exponentially in space, but rather as an inverse power law multiplied by an oscillatory term. Theoretical calculations can be obtained fromquantum hydrodynamics anddensity functional theory (DFT).

• Bjerrum length
• Debye length

• Fitzpatrick, Richard (2011-03-31)."Debye Shielding".The University of Texas at Austin. Retrieved2018-07-12.

• Condensed matter physics
• Electromagnetism concepts
• Plasma phenomena

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