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 solution and how far its electrostatic effect persists. A Debye sphere is a volume whose radius is the Debye length. With each Debye length, charges are increasingly electrically screened. Every Debye‐length λD{displaystyle lambda _{D}}lambda _{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 kD=1/λD{displaystyle k_{D}=1/lambda _{D}}{displaystyle k_{D}=1/lambda _{D}}
for particles of density n{displaystyle n}n, charge q{displaystyle q}q at a temperature T{displaystyle T}T is given by :kD2=4πnq2/kBT{displaystyle :k_{D}^{2}=4pi nq^{2}/k_{B}T}{displaystyle :k_{D}^{2}=4pi nq^{2}/k_{B}T} in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures (T→0{displaystyle Tto 0}{displaystyle Tto 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.




Contents






  • 1 Physical origin


  • 2 Typical values


  • 3 In a plasma


  • 4 In an electrolyte solution


  • 5 In semiconductors


  • 6 See also


  • 7 References


  • 8 Further reading





Physical origin


The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of N{displaystyle N}N different species of charges, the j{displaystyle j}j-th species carries charge qj{displaystyle q_{j}}q_{j} and has concentration nj(r){displaystyle n_{j}(mathbf {r} )}n_{j}(mathbf {r} ) at position r{displaystyle mathbf {r} }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, εr{displaystyle varepsilon _{r}}varepsilon _{r}.
This distribution of charges within this medium gives rise to an electric potential Φ(r){displaystyle Phi (mathbf {r} )}Phi (mathbf {r} ) that satisfies Poisson's equation:



ε∇2Φ(r)=−∑j=1Nqjnj(r)−ρE(r){displaystyle varepsilon nabla ^{2}Phi (mathbf {r} )=-,sum _{j=1}^{N}q_{j},n_{j}(mathbf {r} )-rho _{E}(mathbf {r} )}varepsilon nabla ^{2}Phi (mathbf {r} )=-,sum _{j=1}^{N}q_{j},n_{j}(mathbf {r} )-rho _{E}(mathbf {r} ),

where ε≡εrε0{displaystyle varepsilon equiv varepsilon _{r}varepsilon _{0}}varepsilon equiv varepsilon _{r}varepsilon _{0}, ε0{displaystyle varepsilon _{0}}varepsilon _{0} is the electric constant, and ρE{displaystyle rho _{E}}rho _{E} is a charge density external (logically, not spatially) to the medium.


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



nj(r)=nj0exp⁡(−qjΦ(r)kBT){displaystyle n_{j}(mathbf {r} )=n_{j}^{0},exp left(-{frac {q_{j},Phi (mathbf {r} )}{k_{B}T}}right)}n_{j}(mathbf {r} )=n_{j}^{0},exp left(-{frac {q_{j},Phi (mathbf {r} )}{k_{B}T}}right),

where kB{displaystyle k_{B}}k_{B} is Boltzmann's constant and where nj0{displaystyle n_{j}^{0}}n_{j}^{0} is the mean
concentration of charges of species j{displaystyle j}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:



ε∇2Φ(r)=−∑j=1Nqjnj0exp⁡(−qjΦ(r)kBT)−ρE(r){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_{B}T}}right)-rho _{E}(mathbf {r} )}varepsilon nabla ^{2}Phi (mathbf {r} )=-,sum _{j=1}^{N}q_{j}n_{j}^{0},exp left(-{frac {q_{j},Phi (mathbf {r} )}{k_{B}T}}right)-rho _{E}(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, qjΦ(r)≪kBT{displaystyle q_{j},Phi (mathbf {r} )ll k_{B}T}q_{j},Phi (mathbf {r} )ll k_{B}T, by Taylor expanding the exponential:



exp⁡(−qjΦ(r)kBT)≈1−qjΦ(r)kBT{displaystyle exp left(-{frac {q_{j},Phi (mathbf {r} )}{k_{B}T}}right)approx 1-{frac {q_{j},Phi (mathbf {r} )}{k_{B}T}}}exp left(-{frac {q_{j},Phi (mathbf {r} )}{k_{B}T}}right)approx 1-{frac {q_{j},Phi (mathbf {r} )}{k_{B}T}}.

This approximation yields the linearized Poisson-Boltzmann equation


ε∇2Φ(r)=(∑j=1Nnj0qj2kBT)Φ(r)−∑j=1Nnj0qj−ρE(r){displaystyle varepsilon nabla ^{2}Phi (mathbf {r} )=left(sum _{j=1}^{N}{frac {n_{j}^{0},q_{j}^{2}}{k_{B}T}}right),Phi (mathbf {r} )-,sum _{j=1}^{N}n_{j}^{0}q_{j}-rho _{E}(mathbf {r} )}varepsilon nabla ^{2}Phi (mathbf {r} )=left(sum _{j=1}^{N}{frac {n_{j}^{0},q_{j}^{2}}{k_{B}T}}right),Phi (mathbf {r} )-,sum _{j=1}^{N}n_{j}^{0}q_{j}-rho _{E}(mathbf {r} )

which also is known as the Debye–Hückel equation:[1][2][3][4][5]
The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses divided by ε{displaystyle varepsilon }varepsilon , has the units of an inverse length squared and by
dimensional analysis leads to the definition of the characteristic length scale


λD=(εkBT∑j=1Nnj0qj2)1/2{displaystyle lambda _{D}=left({frac {varepsilon ,k_{B}T}{sum _{j=1}^{N}n_{j}^{0},q_{j}^{2}}}right)^{1/2}}lambda _{D}=left({frac {varepsilon ,k_{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, λD{displaystyle lambda _{D}}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


∇2Φ(r)=λD−2Φ(r)−ρE(r)ε{displaystyle nabla ^{2}Phi (mathbf {r} )=lambda _{D}^{-2}Phi (mathbf {r} )-{frac {rho _{E}(mathbf {r} )}{varepsilon }}}nabla ^{2}Phi (mathbf {r} )=lambda _{D}^{-2}Phi (mathbf {r} )-{frac {rho _{E}(mathbf {r} )}{varepsilon }}

To illustrate Debye screening, the potential produced by an external point charge ρE=Qδ(r){displaystyle rho _{E}=Qdelta (mathbf {r} )}rho _{E}=Qdelta (mathbf {r} ) is


Φ(r)=Q4πεre−r/λD{displaystyle Phi (mathbf {r} )={frac {Q}{4pi varepsilon r}}e^{-r/lambda _{D}}}Phi (mathbf {r} )={frac {Q}{4pi varepsilon r}}e^{-r/lambda _{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 λB{displaystyle lambda _{B}}lambda _{B} as



λD=(4πλB∑j=1Nnj0zj2)−1/2{displaystyle lambda _{D}=left(4pi ,lambda _{B},sum _{j=1}^{N}n_{j}^{0},z_{j}^{2}right)^{-1/2}}lambda _{D}=left(4pi ,lambda _{B},sum _{j=1}^{N}n_{j}^{0},z_{j}^{2}right)^{-1/2},

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



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 table):





































































Plasma
 Density
.mw-parser-output .nobold{font-weight:normal}
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


[6]

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In a low density plasma, localized space charge regions may build up large potential
drops over distances of the order of some tens of the Debye lengths. Such regions have been called electric double layers. An electric double layer is the simplest space charge distribution that gives a potential drop in the layer and a vanishing electric field on each side of the layer. In the laboratory, double layers have been studied for half a century, but their importance in cosmic plasmas has not been generally recognized.


— Hannes Alfvén, [7]



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 (εr=1{displaystyle varepsilon _{r}=1}varepsilon _{r}=1), and the Debye length is


λD=ε0kB/qe2ne/Te+∑jzj2nj/Ti{displaystyle lambda _{D}={sqrt {frac {varepsilon _{0}k_{B}/q_{e}^{2}}{n_{e}/T_{e}+sum _{j}z_{j}^{2}n_{j}/T_{i}}}}}{displaystyle lambda _{D}={sqrt {frac {varepsilon _{0}k_{B}/q_{e}^{2}}{n_{e}/T_{e}+sum _{j}z_{j}^{2}n_{j}/T_{i}}}}}

where



λD is the Debye length,

ε0 is the permittivity of free space,


kB is the Boltzmann constant,


qe is the charge of an electron,


Te and Ti are the temperatures of the electrons and ions, respectively,


ne is the density of electrons,


nj is the density of atomic species j, with positive ionic charge zjqe


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


λD=ε0kBTeneqe2{displaystyle lambda _{D}={sqrt {frac {varepsilon _{0}k_{B}T_{e}}{n_{e}q_{e}^{2}}}}}lambda _{D}={sqrt {frac {varepsilon _{0}k_{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.[8]



In an electrolyte solution


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


κ−1=εrε0kBT2×103NAe2I{displaystyle kappa ^{-1}={sqrt {frac {varepsilon _{r}varepsilon _{0}k_{B}T}{2times 10^{3}N_{A}e^{2}I}}}}{displaystyle kappa ^{-1}={sqrt {frac {varepsilon _{r}varepsilon _{0}k_{B}T}{2times 10^{3}N_{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,

εr is the dielectric constant,


kB is the Boltzmann constant,


T is the absolute temperature in kelvins,


NA is the Avogadro number.


e{displaystyle e}e is the elementary charge,


or, for a symmetric monovalent electrolyte,


κ−1=εrε0RT2×103F2C0{displaystyle kappa ^{-1}={sqrt {frac {varepsilon _{r}varepsilon _{0}RT}{2times 10^{3}F^{2}C_{0}}}}}{displaystyle kappa ^{-1}={sqrt {frac {varepsilon _{r}varepsilon _{0}RT}{2times 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,


κ−1=18πλBNA×103I{displaystyle kappa ^{-1}={frac {1}{sqrt {8pi lambda _{B}N_{A}times 10^{3}I}}}}{displaystyle kappa ^{-1}={frac {1}{sqrt {8pi lambda _{B}N_{A}times 10^{3}I}}}}

where



λB{displaystyle lambda _{B}}lambda _{B} is the Bjerrum length of the medium.

For water at room temperature, λB ≈ 0.7 nm.


At room temperature (25 °C), one can consider in water the relation:[10]


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



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.[11][12][13]


The Debye length of semiconductors is given:


LD=εkBTq2ND{displaystyle {mathit {L}}_{D}={sqrt {frac {varepsilon k_{B}T}{q^{2}N_{D}}}}}{displaystyle {mathit {L}}_{D}={sqrt {frac {varepsilon k_{B}T}{q^{2}N_{D}}}}}

where




ε is the dielectric constant,


kB is the Boltzmann's constant,


T is the absolute temperature in kelvins,


q is the elementary charge, and


ND 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


  • Debye–Falkenhagen effect


References




  1. ^ Kirby BJ. Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ Li D (2004). Electrokinetics in Microfluidics.


  3. ^ PC Clemmow & JP Dougherty (1969). Electrodynamics of particles and plasmas. Redwood City CA: Addison-Wesley. pp. §&nbsp, 7.6.7, p. 236 ff. ISBN 978-0-201-47986-7.


  4. ^ RA Robinson &RH Stokes (2002). Electrolyte solutions. Mineola, NY: Dover Publications. p. 76. ISBN 978-0-486-42225-1.


  5. ^ 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.


  6. ^ Kip Thorne (2012). "Chapter 20: The Particle Kinetics of Plasma" (PDF). APPLICATIONS OF CLASSICAL PHYSICS. Retrieved September 7, 2017.


  7. ^ Hannes Alfvèn (2012) [1981]. "II.6. Electric Double Layers, II.6.1. General Properties of Electric Double Layers". Cosmic Plasma. 82. D. Reidel Publishing Company. p. 29.


  8. ^ I. H. Hutchinson Principles of plasma diagnostics
    ISBN 0-521-38583-0


  9. ^ Russel, W.B., Saville, D.A. and Schowalter, W. R. Colloidal Dispersions, Cambridge University Press, 1989


  10. ^ Israelachvili, J., Intermolecular and Surface Forces, Academic Press Inc., 1985,
    ISBN 0-12-375181-0


  11. ^ 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.


  12. ^ 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.


  13. ^ 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.




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