In condensed matter physics, Lindhard theory is a method of calculating the effects of electric field screening by electrons in a solid. It is based on quantum mechanics (first-order perturbation theory) and the random phase approximation. It is named after Danish physicist Jens Lindhard, who first developed the theory in 1954.
ThomasâÂÂFermi screening, plasma oscillations and Friedel oscillations can be derived as a special case of the more general Lindhard formula. In particular, ThomasâÂÂFermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the Fermi wavevector, i.e. the long-distance limit. The LorentzâÂÂDrude expression for the plasma oscillations are recovered in the dynamic case (long wavelengths, finite frequency).
This article uses cgs-Gaussian units.
The term
is a response function known as Lindhard function. Here is the FermiâÂÂDirac distribution function for electrons in thermodynamic equilibrium, is the kinetic energy with wave vector , is the electron mass and is a positive infinitesimal constant,.
At zero temperature, is a Heaviside step function, where is the Fermi wave vector associated to the Fermi energy . The sum can be carried out in the continuous limit using analytic continuation, resulting in , where in 3 dimensions can be written as
In the static limit, when , we have that , where
is the (static) Lindhard function. This function derivative diverges at . Also . Note that also appears in the calculation of the ground state energy of jellium when using HartreeâÂÂFock method.
The electron-electron Coulomb potential , where e is the elementary charge, can be written in Fourier space as (where is the wave vector), then the effective single particle potential is
The Lindhard formula for the longitudinal dielectric function is then given by
where is Lindhard function.
This Lindhard formula is valid also for nonequilibrium distribution functions. It can be obtained by first-order perturbation theory and the random phase approximation (RPA).
To understand the Lindhard formula, consider some limiting cases in 2 and 3 dimensions. The 1-dimensional case is also considered in other ways.
In the long wavelength limit (), Lindhard function reduces to
where is the three-dimensional plasma frequency (in SI units, replace the factor by .) For two-dimensional systems,
This result recovers the plasma oscillations from the classical dielectric function from Drude model and from quantum mechanical free electron model.
First, consider the long wavelength limit ().
For the denominator of the Lindhard formula,
and for the numerator,
Inserting these into the Lindhard formula and taking the limit of , we obtain
where we used , and .
Consider the static limit ().
The Lindhard formula becomes
Inserting the above equalities for the denominator and numerator, we obtain
Assuming a thermal equilibrium FermiâÂÂDirac carrier distribution, we get
here, we used and .
Therefore,
Here, is the 3D screening wave number (3D inverse screening length) defined as <blockquote>.</blockquote>Then, the 3D statically screened Coulomb potential is given by
And the inverse Fourier transformation of this result gives
known as the Yukawa potential. Note that in this Fourier transformation, which is basically a sum over all , we used the expression for small for every value of which is not correct.
For a degenerated Fermi gas (T=0), the Fermi energy is given by
So the density is
At T=0, , so .
Inserting this into the above 3D screening wave number equation, we obtain
This result recovers the 3D wave number from ThomasâÂÂFermi screening.
For reference, DebyeâÂÂHückel screening describes the non-degenerate limit case. The result is , known as the 3D DebyeâÂÂHückel screening wave number.
In two dimensions, the screening wave number is
Note that this result is independent of n.
Consider the static limit (). The Lindhard formula becomes
Inserting the above equalities for the denominator and numerator, we obtain
Assuming a thermal equilibrium FermiâÂÂDirac carrier distribution, we get
Therefore,
is 2D screening wave number(2D inverse screening length) defined as<blockquote>.</blockquote>Then, the 2D statically screened Coulomb potential is given by
It is known that the chemical potential of the 2-dimensional Fermi gas is given by
and .
As with the dielectric function, the magnetic susceptibility for an electron gas can be calculated as
where is the Bohr magneton.
In the static limit,
which corresponds to the spin susceptibility of Pauli paramagnetism.
This time, consider some generalized case for lowering the dimension. The lower the dimension is, the weaker the screening effect. In lower dimension, some of the field lines pass through the barrier material wherein the screening has no effect. For the 1-dimensional case, we can guess that the screening affects only the field lines which are very close to the wire axis.
In real experiment, we should also take the 3D bulk screening effect into account even though we deal with 1D case like the single filament. The ThomasâÂÂFermi screening has been applied to an electron gas confined to a filament and a coaxial cylinder. For a K<sub>2</sub>Pt(CN)<sub>4</sub>Cl<sub>0.32</sub>÷2.6H<sub>2</sub>0 filament, it was found that the potential within the region between the filament and cylinder varies as and its effective screening length is about 10 times that of metallic platinum.