Lindhard theory


Lindhard theory, named after Danish professor Jens Lindhard, is a method of calculating the effects of electric field screening by electrons in a solid. It is based on quantum mechanics and the random phase approximation.
Thomas–Fermi screening can be derived as a special case of the more general Lindhard formula. In particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector is much smaller than the Fermi wavevector, i.e. the long-distance limit.
This article uses cgs-Gaussian units.

Formula

The Lindhard formula for the longitudinal dielectric function is given by
Here, is a positive infinitesimal constant, is and is the carrier distribution function which is the Fermi–Dirac distribution function for electrons in thermodynamic equilibrium.
However this Lindhard formula is valid also for nonequilibrium distribution functions.

Analysis of the Lindhard formula

To understand the Lindhard formula, consider some limiting cases in 2 and 3 dimensions. The 1-dimensional case is also considered in other ways.

Three dimensions

Long wavelength limit

First, consider the long wavelength limit.
For the denominator of the Lindhard formula, we get
and for the numerator of the Lindhard formula, we get
Inserting these into the Lindhard formula and taking the limit, we obtain
where we used, and.
This result is the same as the classical dielectric function.

Static limit

Second, consider the static limit.
The Lindhard formula becomes
Inserting the above equalities for the denominator and numerator, we obtain
Assuming a thermal equilibrium Fermi–Dirac carrier distribution, we get
here, we used and.
Therefore,
Here, is the 3D screening wave number defined as
Then, the 3D statically screened Coulomb potential is given by
And the 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, 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 is the 3D Thomas–Fermi screening wave number.
For reference, Debye–Hückel screening describes the nondegenerate limit case. The result is, the 3D Debye–Hückel screening wave number.

Two dimensions

Long wavelength limit

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.

Static limit

Second, consider the static limit.
The Lindhard formula becomes
Inserting the above equalities for the denominator and numerator, we obtain
Assuming a thermal equilibrium Fermi–Dirac carrier distribution, we get
here, we used and.
Therefore,
is 2D screening wave number defined as
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.
So, the 2D screening wave number is
Note that this result is independent of n.

One dimension

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.

Experiment

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 Thomas–Fermi screening has been applied to an electron gas confined to a filament and a coaxial cylinder. For a K2Pt4Cl0.32·2.6H20 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.

General