View source for Debye length

{{Short description|Measure of electrostatic effect and how far it persists}}

In [[plasma (physics)|plasmas]] and [[electrolyte]]s, 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 (chemistry)|solution]] and how far its electrostatic effect persists.<ref>{{cite journal |url=http://digital.library.wisc.edu/1793/79225 |last1=Debye |first1=P. |last2=Hückel |first2=E. |orig-year=1923 |trans-title=The theory of electrolytes. I. Freezing point depression and related phenomenon |title=Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen |journal=[[Physikalische Zeitschrift]] |volume=24 |issue=9 |pages=185–206 |translator-first=Michael J. |translator-last=Braus |year=2019 }}</ref>  With each Debye length the charges are increasingly [[Electric-field screening|electrically screened]] and the electric potential decreases in magnitude by 1/[[E_(mathematical_constant)|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 screening|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's law|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 theory|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 expansion|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>{{cite book |last=Kirby |first=B. J. |title=Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices |location=New York |publisher=Cambridge University Press |year=2010 |isbn=978-0-521-11903-0 }}</ref><ref name=DLi>{{cite book |last=Li |first=D. | title=Electrokinetics in Microfluidics |publisher=Academic Press |isbn=0-12-088444-5 |year=2004 }}</ref><ref name=Clemmow>{{cite book |title=Electrodynamics of particles and plasmas |url=https://books.google.com/books?id=SBNNzUrTjecC&q=particles+plasmas+inauthor:Clemmow&pg=PP1 |author=PC Clemmow & JP Dougherty |isbn=978-0-201-47986-7 |year=1969 |publisher=[[Addison-Wesley]] |location=Redwood City CA |pages=§&nbsp;7.6.7, p. 236 ff }}{{Dead link|date=January 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref name=Robinson>{{cite book |title=Electrolyte solutions |page=76 |url=https://books.google.com/books?id=6ZVkqm-J9GkC&pg=PR3 |author=RA Robinson &RH Stokes| isbn=978-0-486-42225-1 |publisher=[[Dover Publications]] |location=Mineola, NY |year=2002}}</ref><ref name=Brydges>See {{cite journal| last1=Brydges|first1=David C.| last2=Martin|first2=Ph. A.|journal=Journal of Statistical Physics|volume=96|issue=5/6| year=1999|pages=1163–1330|doi=10.1023/A:1004600603161|title=Coulomb Systems at Low Density: A Review|arxiv = cond-mat/9904122 |bibcode = 1999JSP....96.1163B |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 ([[Electric-field screening|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 {{nowrap|(<math>\varepsilon_r = 1</math>),}} 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''<sub>D</sub> is the Debye length,
* ''ε''<sub>0</sub> is the [[permittivity of free space]],
* ''k''<sub>B</sub> is the [[Boltzmann constant]],
* ''q''<sub>''e''</sub> is the [[elementary charge|charge of an electron]],
* ''T<sub>e</sub>'' and ''T<sub>i</sub>'' are the temperatures of the electrons and ions, respectively,
* ''n<sub>e</sub>'' is the density of electrons,
* ''n<sub>j</sub>'' is the density of atomic species ''j'', with positive [[ion]]ic charge ''z<sub>j</sub>q<sub>e</sub>''

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>{{cite book | chapter=Chapter 20: The Particle Kinetics of Plasma |title=Applications of Classical Physics |author=Kip Thorne |date=2012 |url=http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/ |chapter-url=http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1220.1.K.pdf |access-date=September 7, 2017}}</ref>

{| class="wikitable"
|-
! Plasma
! Density<br />{{nobold|''n''<sub>e</sub>(m<sup>−3</sup>)}}
! Electron temperature<br />{{nobold|''T''(K)}}
! Magnetic field<br />{{nobold|''B''(T)}}
! Debye length<br />{{nobold|''λ''<sub>D</sub>(m)}}
|- align=center
! style="text-align:left" | Solar core
| 10<sup>32</sup>
| 10<sup>7</sup>
| {{--}}
| 10<sup>−11</sup>
|- align=center
! style="text-align:left" | [[Tokamak]]
| 10<sup>20</sup>
| 10<sup>8</sup>
| 10
| 10<sup>−4</sup>
|- align=center
! style="text-align:left" | Gas discharge
| 10<sup>16</sup>
| 10<sup>4</sup>
| {{--}}
| 10<sup>−4</sup>
|- align=center
! style="text-align:left" | Ionosphere
| 10<sup>12</sup>
| 10<sup>3</sup>
| 10<sup>−5</sup>
| 10<sup>−3</sup>
|- align=center
! style="text-align:left" | Magnetosphere
| 10<sup>7</sup>
| 10<sup>7</sup>
| 10<sup>−8</sup>
| 10<sup>2</sup>
|- align=center
! style="text-align:left" | Solar wind
| 10<sup>6</sup>
| 10<sup>5</sup>
| 10<sup>−9</sup>
| 10
|- align=center
! style="text-align:left" | Interstellar medium
| 10<sup>5</sup>
| 10<sup>4</sup>
| 10<sup>−10</sup>
| 10
|- align=center
! style="text-align:left" | Intergalactic medium
| 1
| 10<sup>6</sup>
| {{--}}
| 10<sup>5</sup>
|-
|+ align="bottom" style="style="caption-side: bottom" | 
|}

== In an electrolyte solution ==
In an [[electrolyte]] or a [[colloids|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">{{cite book |last1=Dukhin |first1=A. S. |last2=Goetz |first2=P. J. |title=Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound |publisher=Elsevier |year=2017 |isbn=978-0-444-63908-0 }}</ref><ref>{{cite book |last1=Russel |first1=W. B. |last2=Saville |first2=D. A. |last3=Schowalter |first3=W. R. |title=Colloidal Dispersions |publisher=Cambridge University Press |year=1989 |isbn=0-521-42600-6 }}</ref> for a monovalent electrolyte is usually denoted with symbol ''κ''<sup>−1</sup>
<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/m<sup>3</sup> units,
* ''ε''<sub>0</sub> is the [[Vacuum permittivity|permittivity of free space]],
* ''ε''<sub>r</sub> is the [[Relative static permittivity|dielectric constant]],
* ''k''<sub>B</sub> is the [[Boltzmann constant]],
* ''T'' is the absolute temperature in [[kelvin]]s,
* <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]],
* ''C''<sub>0</sub> is the electrolyte concentration in [[molar concentration|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''<sub>B</sub> ≈ 1μm.

At room temperature ({{convert|20|°C|-1|disp=or}}), one can consider in water the relation:<ref>{{cite book |last=Israelachvili |first=J. |title=Intermolecular and Surface Forces |publisher=Academic Press |year=1985 |isbn=0-12-375181-0 }}</ref>
<math display="block"> \kappa^{-1}(\mathrm{nm}) = \frac{0.304}{\sqrt{I(\mathrm{M})}}</math>
where
* ''κ''<sup>−1</sup> is expressed in [[nanometre]]s (nm)
* ''I'' is the [[ionic strength]] expressed in [[molar concentration|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>{{cite journal| doi = 10.1021/nl071792z| volume = 7| issue = 11| pages = 3405–3409| last = Stern| first = Eric|author2=Robin Wagner |author3=Fred J. Sigworth |author4=Ronald Breaker |author5=Tarek M. Fahmy |author6=Mark A. Reed | title = Importance of the Debye Screening Length on Nanowire Field Effect Transistor Sensors| journal = Nano Letters| date = 2007-11-01|bibcode = 2007NanoL...7.3405S | pmid=17914853 | pmc=2713684}}</ref><ref>{{cite journal|last=Guo|first=Lingjie|author2=Effendi Leobandung|author3=Stephen Y. Chou|year=199|title=A room-temperature silicon single-electron metal–oxide–semiconductor memory with nanoscale floating-gate and ultranarrow channel|journal=Applied Physics Letters|volume=70|issue=7|pages=850|bibcode=1997ApPhL..70..850G|doi=10.1063/1.118236}}<!--| access-date = 2010-10-25--></ref><ref>{{cite journal |last=Tiwari| first=Sandip| author2=Farhan Rana| author3=Kevin Chan|author4=Leathen Shi| author5=Hussein Hanafi|year=1996 |title=Single charge and confinement effects in nano-crystal memories| journal=Applied Physics Letters |volume=69 |issue=9|pages=1232|bibcode=1996ApPhL..69.1232T|doi=10.1063/1.117421}}<!--| access-date = 2010-10-25--></ref>

The Debye length of [[semiconductor]]s 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''<sub>B</sub> is the Boltzmann constant,
* ''T'' is the absolute temperature in kelvins,
* ''q'' is the elementary charge, and
* ''N''<sub>dop</sub> 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]]
* [[Electric-field screening|Screening effect]]

== References ==
{{Reflist}}

== Further reading ==
* {{cite book | author=Goldston & Rutherford | title=Introduction to Plasma Physics | publisher=[[Institute of Physics Publishing]] |  location=Philadelphia | year=1997 }}
* {{cite book | author=Lyklema | title=Fundamentals of Interface and Colloid Science | publisher=[[Academic Press]] | location= NY | year=1993  }}

{{Authority control}}

[[Category:Electricity]]
[[Category:Electronics concepts]]
[[Category:Colloidal chemistry]]
[[Category:Plasma parameters]]
[[Category:Electrochemistry]]
[[Category:Length]]
[[Category:Peter Debye]]