Debye length

In plasmas and electrolytes, the Debye length <math>\lambda_{\rm D}</math> (Debye radius or Debye–Hückel screening length), is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists.<ref>Debye, P.; Hückel, E. (2019) [1923]. Translated by Braus, Michael J. "Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen" [The theory of electrolytes. I. Freezing point depression and related phenomenon]. Physikalische Zeitschrift. 24 (9): 185–206.</ref> With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/e. A Debye sphere is a volume whose radius is the Debye length. Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). The corresponding Debye screening wave vector <math>k_{\rm D}=1/\lambda_{\rm D}</math> for particles of density <math>n</math>, charge <math>q</math> at a temperature <math>T</math> is given by <math> k_{\rm D}^2=4\pi n q^2/(k_{\rm B}T) </math> in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures (<math>T \to 0</math>) are known as the Thomas–Fermi length and the Thomas–Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.

The Debye length is named after the Dutch-American physicist and chemist Peter Debye (1884-1966), a Nobel laureate in Chemistry.

Physical origin

The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of <math>N</math> different species of charges, the <math>j</math>-th species carries charge <math>q_j</math> and has concentration <math>n_j(\mathbf{r})</math> at position <math>\mathbf{r}</math>. According to the so-called "primitive model", these charges are distributed in a continuous medium that is characterized only by its relative static permittivity, <math>\varepsilon_r</math>. This distribution of charges within this medium gives rise to an electric potential <math>\Phi(\mathbf{r})</math> that satisfies Poisson's equation: <math display="block"> \varepsilon \nabla^2 \Phi(\mathbf{r}) = -\, \sum_{j = 1}^N q_j \, n_j(\mathbf{r}) - \rho_{\rm ext}(\mathbf{r}),</math> where <math>\varepsilon \equiv \varepsilon_r \varepsilon_0</math>, <math>\varepsilon_0</math> is the electric constant, and <math>\rho_{\rm ext}</math> is a charge density external (logically, not spatially) to the medium.

The mobile charges not only contribute in establishing <math>\Phi(\mathbf{r})</math> but also move in response to the associated Coulomb force, <math>- q_j \, \nabla \Phi(\mathbf{r})</math>. If we further assume the system to be in thermodynamic equilibrium with a heat bath at absolute temperature <math>T</math>, then the concentrations of discrete charges, <math>n_j(\mathbf{r})</math>, may be considered to be thermodynamic (ensemble) averages and the associated electric potential to be a thermodynamic mean field. With these assumptions, the concentration of the <math>j</math>-th charge species is described by the Boltzmann distribution, <math display="block"> n_j(\mathbf{r}) = n_j^0 \, \exp\left( - \frac{q_j \, \Phi(\mathbf{r})}{k_{\rm B} T} \right),</math> where <math>k_{\rm B}</math> is the Boltzmann constant and where <math>n_j^0</math> is the mean concentration of charges of species <math>j</math>.

Identifying the instantaneous concentrations and potential in the Poisson equation with their mean-field counterparts in the Boltzmann distribution yields the Poisson–Boltzmann equation: <math display="block"> \varepsilon \nabla^2 \Phi(\mathbf{r}) = -\, \sum_{j = 1}^N q_j n_j^0 \, \exp\left(- \frac{q_j \, \Phi(\mathbf{r})}{k_{\rm B} T} \right) - \rho_{\rm ext}(\mathbf{r}) .</math>

Solutions to this nonlinear equation are known for some simple systems. Solutions for more general systems may be obtained in the high-temperature (weak coupling) limit, <math>q_j \, \Phi(\mathbf{r}) \ll k_{\rm B} T</math>, by Taylor expanding the exponential: <math display="block"> \exp\left(- \frac{q_j \, \Phi(\mathbf{r})}{k_{\rm B} T} \right) \approx 1 - \frac{q_j \, \Phi(\mathbf{r})}{k_{\rm B} T}.</math>

This approximation yields the linearized Poisson–Boltzmann equation <math display="block"> \varepsilon \nabla^2 \Phi(\mathbf{r}) = \left(\sum_{j = 1}^N \frac{n_j^0 \, q_j^2}{ k_{\rm B} T} \right)\, \Phi(\mathbf{r}) -\, \sum_{j = 1}^N n_j^0 q_j - \rho_{\rm ext}(\mathbf{r}) </math> which also is known as the Debye–Hückel equation:<ref name=Kirby>Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. New York: Cambridge University Press. ISBN 978-0-521-11903-0.</ref><ref name=DLi>Li, D. (2004). Electrokinetics in Microfluidics. Academic Press. ISBN 0-12-088444-5.</ref><ref name=Clemmow>PC Clemmow & JP Dougherty (1969). Electrodynamics of particles and plasmas. Redwood City CA: Addison-Wesley. pp. § 7.6.7, p. 236 ff. ISBN 978-0-201-47986-7.</ref><ref name=Robinson>RA Robinson &RH Stokes (2002). Electrolyte solutions. Mineola, NY: Dover Publications. p. 76. ISBN 978-0-486-42225-1.</ref><ref name=Brydges>See Brydges, David C.; Martin, Ph. A. (1999). "Coulomb Systems at Low Density: A Review". Journal of Statistical Physics. 96 (5/6): 1163–1330. arXiv:cond-mat/9904122. Bibcode:1999JSP....96.1163B. doi:10.1023/A:1004600603161. S2CID 54979869.</ref> The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses divided by <math>\varepsilon</math>, has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale <math display="block"> \lambda_{\rm D} = \left(\frac{\varepsilon \, k_{\rm B} T}{\sum_{j = 1}^N n_j^0 \, q_j^2}\right)^{1/2}</math> that commonly is referred to as the Debye–Hückel length. As the only characteristic length scale in the Debye–Hückel equation, <math>\lambda_D</math> sets the scale for variations in the potential and in the concentrations of charged species. All charged species contribute to the Debye–Hückel length in the same way, regardless of the sign of their charges. For an electrically neutral system, the Poisson equation becomes <math display="block"> \nabla^2 \Phi(\mathbf{r}) = \lambda_{\rm D}^{-2} \Phi(\mathbf{r}) - \frac{\rho_{\rm ext}(\mathbf{r})}{\varepsilon} </math> To illustrate Debye screening, the potential produced by an external point charge <math>\rho_{\rm ext} = Q\delta(\mathbf{r})</math> is <math display="block"> \Phi(\mathbf{r}) = \frac{Q}{4\pi\varepsilon r} e^{-r/\lambda_{\rm D}}</math> The bare Coulomb potential is exponentially screened by the medium, over a distance of the Debye length: this is called Debye screening or shielding (Screening effect).

The Debye–Hückel length may be expressed in terms of the Bjerrum length <math>\lambda_{\rm B}</math> as <math display="block"> \lambda_{\rm D} = \left(4 \pi \, \lambda_{\rm B} \, \sum_{j = 1}^N n_j^0 \, z_j^2\right)^{-1/2},</math> where <math>z_j = q_j/e</math> is the integer charge number that relates the charge on the <math>j</math>-th ionic species to the elementary charge <math>e</math>.

In a plasma

For a weakly collisional plasma, Debye shielding can be introduced in a very intuitive way by taking into account the granular character of such a plasma. Let us imagine a sphere about one of its electrons, and compare the number of electrons crossing this sphere with and without Coulomb repulsion. With repulsion, this number is smaller. Therefore, according to Gauss theorem, the apparent charge of the first electron is smaller than in the absence of repulsion. The larger the sphere radius, the larger is the number of deflected electrons, and the smaller the apparent charge: this is Debye shielding. Since the global deflection of particles includes the contributions of many other ones, the density of the electrons does not change, at variance with the shielding at work next to a Langmuir probe (Debye sheath). Ions bring a similar contribution to shielding, because of the attractive Coulombian deflection of charges with opposite signs.

This intuitive picture leads to an effective calculation of Debye shielding (see section II.A.2 of <ref> Meyer-Vernet N (1993) Aspects of Debye shielding. American journal of physics 61, 249-257</ref>). The assumption of a Boltzmann distribution is not necessary in this calculation: it works for whatever particle distribution function. The calculation also avoids approximating weakly collisional plasmas as continuous media. An N-body calculation reveals that the bare Coulomb acceleration of a particle by another one is modified by a contribution mediated by all other particles, a signature of Debye shielding (see section 8 of <ref> Escande, D. F., Bénisti, D., Elskens, Y., Zarzoso, D., & Doveil, F. (2018). Basic microscopic plasma physics from N-body mechanics, A tribute to Pierre-Simon de Laplace, Reviews of Modern Plasma Physics, 2, 1-68 </ref>). When starting from random particle positions, the typical time-scale for shielding to set in is the time for a thermal particle to cross a Debye length, i.e. the inverse of the plasma frequency. Therefore in a weakly collisional plasma, collisions play an essential role by bringing a cooperative self-organization process: Debye shielding. This shielding is important to get a finite diffusion coefficient in the calculation of Coulomb scattering (Coulomb collision).

In a non-isothermic plasma, the temperatures for electrons and heavy species may differ while the background medium may be treated as the vacuum {{{1}}} and the Debye length is <math display="block"> \lambda_{\rm D} = \sqrt{\frac{\varepsilon_0 k_{\rm B}/q_e^2}{n_e/T_e+\sum_j z_j^2n_j/T_j}}</math> where

• λ_D is the Debye length,
• ε₀ is the permittivity of free space,
• k_B is the Boltzmann constant,
• q_e is the charge of an electron,
• T_e and T_i are the temperatures of the electrons and ions, respectively,
• n_e is the density of electrons,
• n_j is the density of atomic species j, with positive ionic charge z_jq_e

Even in quasineutral cold plasma, where ion contribution virtually seems to be larger due to lower ion temperature, the ion term is actually often dropped, giving <math display="block"> \lambda_{\rm D} = \sqrt{\frac{\varepsilon_0 k_{\rm B} T_e}{n_e q_e^2}}</math> although this is only valid when the mobility of ions is negligible compared to the process's timescale.<ref>I. H. Hutchinson Principles of plasma diagnostics ISBN 0-521-38583-0</ref>

Typical values

In space plasmas where the electron density is relatively low, the Debye length may reach macroscopic values, such as in the magnetosphere, solar wind, interstellar medium and intergalactic medium. See the table here below:<ref>Kip Thorne (2012). "Chapter 20: The Particle Kinetics of Plasma" (PDF). Applications of Classical Physics. Retrieved September 7, 2017.</ref>

Plasma	Density ne(m−3)	Electron temperature T(K)	Magnetic field B(T)	Debye length λD(m)
Solar core	1032	107	—	10−11
Tokamak	1020	108	10	10−4
Gas discharge	1016	104	—	10−4
Ionosphere	1012	103	10−5	10−3
Magnetosphere	107	107	10−8	102
Solar wind	106	105	10−9	10
Interstellar medium	105	104	10−10	10
Intergalactic medium	1	106	—	105

In an electrolyte solution

In an electrolyte or a colloidal suspension, the Debye length<ref name="ISO">International Standard ISO 13099-1, 2012, "Colloidal systems – Methods for Zeta potential determination- Part 1: Electroacoustic and Electrokinetic phenomena"</ref><ref name="Dukhin">Dukhin, A. S.; Goetz, P. J. (2017). Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound. Elsevier. ISBN 978-0-444-63908-0.</ref><ref>Russel, W. B.; Saville, D. A.; Schowalter, W. R. (1989). Colloidal Dispersions. Cambridge University Press. ISBN 0-521-42600-6.</ref> for a monovalent electrolyte is usually denoted with symbol κ⁻¹ <math display="block"> \kappa^{-1} = \sqrt{\frac{\varepsilon_{\rm r} \varepsilon_0 k_{\rm B} T}{2 e^2 I}}</math>

where

• I is the ionic strength of the electrolyte in number/m³ units,
• ε₀ is the permittivity of free space,
• ε_r is the dielectric constant,
• k_B is the Boltzmann constant,
• T is the absolute temperature in kelvins,
• <math>e</math> is the elementary charge,

or, for a symmetric monovalent electrolyte, <math display="block"> \kappa^{-1} = \sqrt{\frac{\varepsilon_{\rm r} \varepsilon_0 R T}{2\times10^3 F^2 C_0}}</math> where

• R is the gas constant,
• F is the Faraday constant,
• C₀ is the electrolyte concentration in molar units (M or mol/L).

Alternatively, <math display="block"> \kappa^{-1} = \frac{1}{\sqrt{8\pi \lambda_{\rm B} N_{\rm A} \times 10^{-24} I}} </math> where <math>\lambda_{\rm B}</math> is the Bjerrum length of the medium in nm, and the factor <math> 10^{-24} </math> derives from transforming unit volume from cubic dm to cubic nm.

For deionized water at room temperature, at pH=7, λ_B ≈ 1μm.

At room temperature (20 °C or 70 °F), one can consider in water the relation:<ref>Israelachvili, J. (1985). Intermolecular and Surface Forces. Academic Press. ISBN 0-12-375181-0.</ref> <math display="block"> \kappa^{-1}(\mathrm{nm}) = \frac{0.304}{\sqrt{I(\mathrm{M})}}</math> where

• κ⁻¹ is expressed in nanometres (nm)
• I is the ionic strength expressed in molar (M or mol/L)

There is a method of estimating an approximate value of the Debye length in liquids using conductivity, which is described in ISO Standard,<ref name="ISO"/> and the book.<ref name="Dukhin" />

In semiconductors

The Debye length has become increasingly significant in the modeling of solid state devices as improvements in lithographic technologies have enabled smaller geometries.<ref>Stern, Eric; Robin Wagner; Fred J. Sigworth; Ronald Breaker; Tarek M. Fahmy; Mark A. Reed (2007-11-01). "Importance of the Debye Screening Length on Nanowire Field Effect Transistor Sensors". Nano Letters. 7 (11): 3405–3409. Bibcode:2007NanoL...7.3405S. doi:10.1021/nl071792z. PMC 2713684. PMID 17914853.</ref><ref>Guo, Lingjie; Effendi Leobandung; Stephen Y. Chou (199). "A room-temperature silicon single-electron metal–oxide–semiconductor memory with nanoscale floating-gate and ultranarrow channel". Applied Physics Letters. 70 (7): 850. Bibcode:1997ApPhL..70..850G. doi:10.1063/1.118236.</ref><ref>Tiwari, Sandip; Farhan Rana; Kevin Chan; Leathen Shi; Hussein Hanafi (1996). "Single charge and confinement effects in nano-crystal memories". Applied Physics Letters. 69 (9): 1232. Bibcode:1996ApPhL..69.1232T. doi:10.1063/1.117421.</ref>

The Debye length of semiconductors is given: <math display="block"> L_{\rm D} = \sqrt{\frac{\varepsilon k_{\rm B} T}{q^2 N_{\rm dop}}}</math> where

• ε is the dielectric constant,
• k_B is the Boltzmann constant,
• T is the absolute temperature in kelvins,
• q is the elementary charge, and
• N_dop is the net density of dopants (either donors or acceptors).

When doping profiles exceed the Debye length, majority carriers no longer behave according to the distribution of the dopants. Instead, a measure of the profile of the doping gradients provides an "effective" profile that better matches the profile of the majority carrier density.

In the context of solids, Thomas–Fermi screening length may be required instead of Debye length.

See also

• Bjerrum length
• Debye–Falkenhagen effect
• Plasma oscillation
• Shielding effect
• Screening effect

Further reading

• Goldston & Rutherford (1997). Introduction to Plasma Physics. Philadelphia: Institute of Physics Publishing.
• Lyklema (1993). Fundamentals of Interface and Colloid Science. NY: Academic Press.