Thomas–Fermi screening
[1] It is a special case of the more general Lindhard theory; in particular, Thomas–Fermi 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.
where μ is the chemical potential (Fermi level), n is the electron concentration and e is the elementary charge.
For the example of semiconductors that are not too heavily doped, the charge density n ∝ eμ / kBT, where kB is Boltzmann constant and T is temperature.
In the opposite extreme, in the low-temperature limit T = 0, electrons behave as quantum particles (fermions).
Such an approximation is valid for metals at room temperature, and the Thomas–Fermi screening wavevector kTF given in atomic units is
For more details and discussion, including the one-dimensional and two-dimensional cases, see the article on Lindhard theory.
The internal chemical potential (closely related to Fermi level, see below) of a system of electrons describes how much energy is required to put an extra electron into the system, neglecting electrical potential energy.
Then the required relationship is described by the electron number density
(in this context—i.e., absolute zero—the internal chemical potential is more commonly called the Fermi energy).
As another example, for an n-type semiconductor at low to moderate electron concentration,
The main assumption in the Thomas–Fermi model is that there is an internal chemical potential at each point r that depends only on the electron concentration at the same point r. This behaviour cannot be exactly true because of the Heisenberg uncertainty principle.
No electron can exist at a single point; each is spread out into a wavepacket of size ≈ 1 / kF, where kF is the Fermi wavenumber, i.e. a typical wavenumber for the states at the Fermi surface.
Nevertheless, the Thomas–Fermi model is likely to be a reasonably accurate approximation as long as the potential does not vary much over lengths comparable or smaller than 1 / kF.
Finally, the Thomas–Fermi model assumes that the electrons are in equilibrium, meaning that the total chemical potential is the same at all points.
(In electrochemistry terminology, "the electrochemical potential of electrons is the same at all points".
In semiconductor physics terminology, "the Fermi level is flat".)
This balance requires that the variations in internal chemical potential are matched by equal and opposite variations in the electric potential energy.
at the points where the material is charge-neutral (the number of electrons is exactly equal to the number of ions), and similarly μ0 is defined as the internal chemical potential at the points where the material is charge-neutral.
This relation can be converted into a wavevector-dependent dielectric function:[1] (in cgs-Gaussian units)
At long distances (q → 0), the dielectric constant approaches infinity, reflecting the fact that charges get closer and closer to perfectly screened as you observe them from further away.
If a point charge Q is placed at r = 0 in a solid, what field will it produce, taking electron screening into account?
However, the linearized formula has a simple solution (in cgs-Gaussian units):
Note that there may be dielectric permittivity in addition to the screening discussed here; for example due to the polarization of immobile core electrons.
In that case, replace Q by Q/ε, where ε is the relative permittivity due to these other contributions.
A simple approximate form that recovers both limits correctly is
However, this form of the effective temperature does not correctly recover the specific heat and most other properties of the finite-
A simple form for an effective temperature which correctly recovers all the density-functional properties of even the interacting electron gas, including the pair-distribution functions at finite
, has been given using the classical map hyper-netted-chain (CHNC) model of the electron fluid.
is the Wigner–Seitz radius corresponding to a sphere in atomic units containing one electron.
or less, electron-electron interactions become negligible compared to the Fermi energy, then, using a value of
