Debye length

Physical origin

The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of N different species of charges, the j -th species carries charge q j and has concentration n j ( r ) at position 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, ε r . This distribution of charges within this medium gives rise to an electric potential Φ ( r ) that satisfies Poisson's equation:

ε ∇ 2 Φ ( r ) = − ∑ j = 1 N q j n j ( r ) − ρ E ( r ) ,

where ε ≡ ε r ε 0 , ε 0 is the electric constant, and ρ E is a charge density external (logically, not spatially) to the medium.

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

n j ( r ) = n j 0 exp ⁡ ( − q j Φ ( r ) k B T ) ,

where k B is Boltzmann's constant and where n j 0 is the mean concentration of charges of species 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 ∇ 2 Φ ( r ) = − ∑ j = 1 N q j n j 0 exp ⁡ ( − q j Φ ( r ) k B T ) − ρ E ( 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, q j Φ ( r ) ≪ k B T , by Taylor expanding the exponential:

exp ⁡ ( − q j Φ ( r ) k B T ) ≈ 1 − q j Φ ( r ) k B T .

This approximation yields the linearized Poisson-Boltzmann equation

ε ∇ 2 Φ ( r ) = ( ∑ j = 1 N n j 0 q j 2 k B T ) Φ ( r ) − ∑ j = 1 N n j 0 q j − ρ E ( r )

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

λ D = ( ε k B T ∑ j = 1 N n j 0 q j 2 ) 1 / 2

that commonly is referred to as the Debye–Hückel length. As the only characteristic length scale in the Debye–Hückel equation, λ 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

∇ 2 Φ ( r ) = λ D − 2 Φ ( r ) − ρ E ( r ) ε

To illustrate Debye screening, the potential produced by an external point charge ρ E = Q δ ( r ) is

Φ ( r ) = Q 4 π ε r e − r / λ 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 λ B as

λ D = ( 4 π λ B ∑ j = 1 N n j 0 z j 2 ) − 1 / 2 ,

where z j = q j / e is the integer charge number that relates the charge on the j -th ionic species to the elementary charge 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):

Hannes Alfvén pointed out that: "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."

In a plasma

In a plasma, the background medium may be treated as the vacuum ( ε r = 1 ), and the Debye length is

λ D = ε 0 k B / q e 2 n e / T e + ∑ j z j 2 n j / T j

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

The ion term is often dropped, giving

λ D = ε 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.

In an electrolyte solution

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

κ − 1 = ε r ε 0 k B T 2 N A e 2 I

where

I is the ionic strength of the electrolyte, and here the unit should be mole/m³, 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. e is the elementary charge,

or, for a symmetric monovalent electrolyte,

κ − 1 = ε r ε 0 R T 2 F 2 C 0

where

R is the gas constant, F is the Faraday constant, C₀ is the molar concentration of the electrolyte.

Alternatively,

κ − 1 = 1 8 π λ B N A I

where

λ B is the Bjerrum length of the medium.

For water at room temperature, λ_B ≈ 0.7 nm.

At room temperature (25 °C), one can consider in water for 1:1 electrolytes the relation:

κ − 1 ( n m ) = 0.304 I ( 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.

The Debye length of semiconductors is given:

L D = ε k B T q 2 N d

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_d is the 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.