Debye length

In plasmas and electrolytes, the Debye length (also called Debye radius), named after Peter Debye, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists.[1] A Debye sphere is a volume whose radius is the Debye length. With each Debye length, charges are increasingly electrically screened. Every Debye‐length ${\displaystyle \lambda _{\rm {D}}}$, the electric potential will decrease in magnitude by 1/e. Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). The corresponding Debye screening wave vector ${\displaystyle k_{\rm {D}}=1/\lambda _{\rm {D}}}$ for particles of density ${\displaystyle n}$, charge ${\displaystyle q}$ at a temperature ${\displaystyle T}$ is given by ${\displaystyle k_{\rm {D}}^{2}=4\pi nq^{2}/(k_{\rm {B}}T)}$ in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures (${\displaystyle T\to 0}$) 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.

Physical origin

The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of ${\displaystyle N}$ different species of charges, the ${\displaystyle j}$-th species carries charge ${\displaystyle q_{j}}$ and has concentration ${\displaystyle n_{j}(\mathbf {r} )}$ at position ${\displaystyle \mathbf {r} }$. According to the so-called "primitive model", these charges are distributed in a continuous medium that is characterized only by its relative static permittivity, ${\displaystyle \varepsilon _{r}}$. This distribution of charges within this medium gives rise to an electric potential ${\displaystyle \Phi (\mathbf {r} )}$ that satisfies Poisson's equation:

${\displaystyle \varepsilon \nabla ^{2}\Phi (\mathbf {r} )=-\,\sum _{j=1}^{N}q_{j}\,n_{j}(\mathbf {r} )-\rho _{\rm {ext}}(\mathbf {r} )}$,

where ${\displaystyle \varepsilon \equiv \varepsilon _{r}\varepsilon _{0}}$, ${\displaystyle \varepsilon _{0}}$ is the electric constant, and ${\displaystyle \rho _{\rm {ext}}}$ is a charge density external (logically, not spatially) to the medium.

The mobile charges not only contribute in establishing ${\displaystyle \Phi (\mathbf {r} )}$ but also move in response to the associated Coulomb force, ${\displaystyle -q_{j}\,\nabla \Phi (\mathbf {r} )}$. If we further assume the system to be in thermodynamic equilibrium with a heat bath at absolute temperature ${\displaystyle T}$, then the concentrations of discrete charges, ${\displaystyle n_{j}(\mathbf {r} )}$, 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 ${\displaystyle j}$-th charge species is described by the Boltzmann distribution,

${\displaystyle n_{j}(\mathbf {r} )=n_{j}^{0}\,\exp \left(-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}\right)}$,

where ${\displaystyle k_{\rm {B}}}$ is Boltzmann's constant and where ${\displaystyle n_{j}^{0}}$ is the mean concentration of charges of species ${\displaystyle j}$.

Identifying the instantaneous concentrations and potential in the Poisson equation with their mean-field counterparts in Boltzmann's distribution yields the Poisson–Boltzmann equation:

${\displaystyle \varepsilon \nabla ^{2}\Phi (\mathbf {r} )=-\,\sum _{j=1}^{N}q_{j}n_{j}^{0}\,\exp \left(-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}\right)-\rho _{\rm {ext}}(\mathbf {r} )}$.

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, ${\displaystyle q_{j}\,\Phi (\mathbf {r} )\ll k_{\rm {B}}T}$, by Taylor expanding the exponential:

${\displaystyle \exp \left(-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}\right)\approx 1-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}}$.

This approximation yields the linearized Poisson-Boltzmann equation

${\displaystyle \varepsilon \nabla ^{2}\Phi (\mathbf {r} )=\left(\sum _{j=1}^{N}{\frac {n_{j}^{0}\,q_{j}^{2}}{k_{\rm {B}}T}}\right)\,\Phi (\mathbf {r} )-\,\sum _{j=1}^{N}n_{j}^{0}q_{j}-\rho _{\rm {ext}}(\mathbf {r} )}$

which also is known as the Debye–Hückel equation:[2][3][4][5][6] The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses divided by ${\displaystyle \varepsilon }$, has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale

${\displaystyle \lambda _{\rm {D}}=\left({\frac {\varepsilon \,k_{\rm {B}}T}{\sum _{j=1}^{N}n_{j}^{0}\,q_{j}^{2}}}\right)^{1/2}}$

that commonly is referred to as the Debye–Hückel length. As the only characteristic length scale in the Debye–Hückel equation, ${\displaystyle \lambda _{D}}$ 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

${\displaystyle \nabla ^{2}\Phi (\mathbf {r} )=\lambda _{\rm {D}}^{-2}\Phi (\mathbf {r} )-{\frac {\rho _{\rm {ext}}(\mathbf {r} )}{\varepsilon }}}$

To illustrate Debye screening, the potential produced by an external point charge ${\displaystyle \rho _{\rm {ext}}=Q\delta (\mathbf {r} )}$ is

${\displaystyle \Phi (\mathbf {r} )={\frac {Q}{4\pi \varepsilon r}}e^{-r/\lambda _{\rm {D}}}}$

The bare Coulomb potential is exponentially screened by the medium, over a distance of the Debye length.

The Debye–Hückel length may be expressed in terms of the Bjerrum length ${\displaystyle \lambda _{\rm {B}}}$ as

${\displaystyle \lambda _{\rm {D}}=\left(4\pi \,\lambda _{\rm {B}}\,\sum _{j=1}^{N}n_{j}^{0}\,z_{j}^{2}\right)^{-1/2}}$,

where ${\displaystyle z_{j}=q_{j}/e}$ is the integer charge number that relates the charge on the ${\displaystyle j}$-th ionic species to the elementary charge ${\displaystyle e}$.

In a plasma

In a non-isothermic plasma, the temperatures for electrons and heavy species may differ while the background medium may be treated as the vacuum (${\displaystyle \varepsilon _{r}=1}$), and the Debye length is

${\displaystyle \lambda _{\rm {D}}={\sqrt {\frac {\varepsilon _{0}k_{\rm {B}}/q_{e}^{2}}{n_{e}/T_{e}+\sum _{j}z_{j}^{2}n_{j}/T_{i}}}}}$

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

${\displaystyle \lambda _{\rm {D}}={\sqrt {\frac {\varepsilon _{0}k_{\rm {B}}T_{e}}{n_{e}q_{e}^{2}}}}}$

although this is only valid when the mobility of ions is negligible compared to the process's timescale.[7]

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:[8]

Plasma	Density ne(m^−3)	Electron temperature T(K)	Magnetic field B(T)	Debye length λD(m)
Solar core	10^32	10^7	—	10^−11
Tokamak	10^20	10^8	10	10^−4
Gas discharge	10^16	10^4	—	10^−4
Ionosphere	10^12	10^3	10^−5	10^−3
Magnetosphere	10^7	10^7	10^−8	10^2
Solar wind	10^6	10^5	10^−9	10
Interstellar medium	10^5	10^4	10^−10	10
Intergalactic medium	1	10^6	—	10^5

In an electrolyte solution

In an electrolyte or a colloidal suspension, the Debye length[9][10][11] for a monovalent electrolyte is usually denoted with symbol κ⁻¹

${\displaystyle \kappa ^{-1}={\sqrt {\frac {\varepsilon _{\rm {r}}\varepsilon _{0}k_{\rm {B}}T}{2\times 10^{3}N_{\rm {A}}e^{2}I}}}}$

where

I is the ionic strength of the electrolyte in molar units (M or mol/L),

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,

N_A is the Avogadro number.

${\displaystyle e}$ is the elementary charge,

or, for a symmetric monovalent electrolyte,

${\displaystyle \kappa ^{-1}={\sqrt {\frac {\varepsilon _{\rm {r}}\varepsilon _{0}RT}{2\times 10^{3}F^{2}C_{0}}}}}$

where

R is the gas constant,

F is the Faraday constant,

C₀ is the electrolyte concentration in molar units (M or mol/L).

Alternatively,

${\displaystyle \kappa ^{-1}={\frac {1}{\sqrt {8\pi \lambda _{\rm {B}}N_{\rm {A}}\times 10^{3}I}}}}$

where

${\displaystyle \lambda _{\rm {B}}}$ is the Bjerrum length of the medium.

For water at room temperature, λ_B ≈ 0.7 nm.

At room temperature (20 °C or 70 °F), one can consider in water the relation:[12]

${\displaystyle \kappa ^{-1}(\mathrm {nm} )={\frac {0.304}{\sqrt {I(\mathrm {M} )}}}}$

where

κ⁻¹ is expressed in nanometers (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,[9] and the book.[10]

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.[13][14][15]

The Debye length of semiconductors is given:

${\displaystyle L_{\rm {D}}={\sqrt {\frac {\varepsilon k_{\rm {B}}T}{q^{2}N_{\rm {dop}}}}}}$

where

ε is the dielectric constant,

k_B is the Boltzmann's 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, the Debye length is also called the Thomas–Fermi screening length.

See also

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

References

1. 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.
2. Kirby, B. J. (2010). Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. New York: Cambridge University Press. ISBN 978-0-521-11903-0.
3. Li, D. (2004). Electrokinetics in Microfluidics. Academic Press. ISBN 0-12-088444-5.
4. 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.
5. RA Robinson &RH Stokes (2002). Electrolyte solutions. Mineola, NY: Dover Publications. p. 76. ISBN 978-0-486-42225-1.
6. 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.
7. I. H. Hutchinson Principles of plasma diagnostics ISBN 0-521-38583-0
8. Kip Thorne (2012). "Chapter 20: The Particle Kinetics of Plasma" (PDF). Applications of Classical Physics. Retrieved September 7, 2017.
9. International Standard ISO 13099-1, 2012, "Colloidal systems – Methods for Zeta potential determination- Part 1: Electroacoustic and Electrokinetic phenomena"
10. 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.
11. Russel, W. B.; Saville, D. A.; Schowalter, W. R. (1989). Colloidal Dispersions. Cambridge University Press. ISBN 0-521-42600-6.
12. Israelachvili, J. (1985). Intermolecular and Surface Forces. Academic Press. ISBN 0-12-375181-0.
13. 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.
14. 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.
15. 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.

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.