# Thomas–Fermi screening

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Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid. [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. [1] It is named after Llewellyn Thomas and Enrico Fermi.

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.

Llewellyn Hilleth Thomas was a British physicist and applied mathematician. He is best known for his contributions to atomic physics and solid-state physics, in particular:

Enrico Fermi was an Italian and naturalized-American physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi held several patents related to the use of nuclear power, and was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. He made significant contributions to the development of statistical mechanics, quantum theory, and nuclear and particle physics.

## Contents

The Thomas–Fermi wavevector (in Gaussian-cgs units) is [1]

Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units. The term "cgs units" is ambiguous and therefore to be avoided if possible: cgs contains within it several conflicting sets of electromagnetism units, not just Gaussian units, as described below.

${\displaystyle k_{0}^{2}=4\pi e^{2}{\frac {\partial n}{\partial \mu }}}$,

where μ is the chemical potential (fermi level), n is the electron concentration and e is the elementary charge.

In thermodynamics, chemical potential of a species is energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. Thus, it is the partial derivative of the free energy with respect to the amount of the species, all other species' concentrations in the mixture remaining constant. The molar chemical potential is also known as partial molar free energy. When both temperature and pressure are held constant, chemical potential is the partial molar Gibbs free energy. At chemical equilibrium or in phase equilibrium the total sum of the product of chemical potentials and stoichiometric coefficients is zero, as the free energy is at a minimum.

The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by µ or EF for brevity. The Fermi level does not include the work required to remove the electron from wherever it came from. A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties, how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics.

The elementary charge, usually denoted by e or sometimes qe, is the electric charge carried by a single proton or, equivalently, the magnitude of the electric charge carried by a single electron, which has charge −1 e. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called the elementary positive charge.

Under many circumstances, including semiconductors that are not too heavily doped, neμ/kBT, where kB is Boltzmann constant and T is temperature. In this case,

${\displaystyle k_{0}^{2}=4\pi e^{2}n/(k_{\rm {B}}T)}$,

i.e. 1/k0 is given by the familiar formula for Debye length. 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

In plasmas and electrolytes, the Debye length, named after Peter Debye, is a measure of a charge carrier's net electrostatic effect in a 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 , the electric potential will decrease in magnitude by 1/e. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector for particles of density , charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures 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.

${\displaystyle k_{\rm {TF}}^{2}=4\left({\frac {3n}{\pi }}\right)^{1/3}}$.

If we restore the electron mass ${\displaystyle m_{e}}$ and the Planck constant ${\displaystyle \hbar }$, the screening wavevector in Gaussian units is ${\displaystyle k_{0}^{2}=k_{\rm {TF}}^{2}(m_{e}/\hbar ^{2})}$.

The electron rest mass is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about 9.109×10−31 kilograms or about 5.486×10−4 atomic mass units, equivalent to an energy of about 8.187×10−14 joules or about 0.5110 MeV.

The Planck constant, or Planck's constant, denoted is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

For more details and discussion, including the one-dimensional and two-dimensional cases, see the article on Lindhard theory.

## Derivation

### Relation between electron density and internal chemical potential

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. As the number of electrons in the system increases (with fixed temperature and volume), the internal chemical potential increases. This consequence is largely because electrons satisfy the Pauli exclusion principle: only one electron may occupy an energy level and lower-energy electron states are already full, so the new electrons must occupy higher and higher energy states.

The relationship is described by the electron number density ${\displaystyle n(\mu )}$ as a function of μ, the internal chemical potential. The exact functional form depends on the system. For example, for a three-dimensional Fermi gas, a noninteracting electron gas, at absolute zero temperature, the relation is ${\displaystyle n(\mu )\propto \mu ^{3/2}}$.

Proof: Including spin degeneracy,

${\displaystyle n=2{\frac {1}{(2\pi )^{3}}}{\frac {4}{3}}\pi k_{\rm {F}}^{3}\quad ,\quad \mu ={\frac {\hbar ^{2}k_{\rm {F}}^{2}}{2m}}.}$

(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, ${\displaystyle n(\mu )\propto e^{\mu /k_{\rm {B}}T}}$.

### Local approximation

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. Therefore it cannot be possible to define a chemical potential at a single point, independent of the electron density at nearby points.

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. This length usually corresponds to a few atoms in metals.

### Electrons in equilibrium, nonlinear equation

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. This gives rise to the "basic equation of nonlinear Thomas–Fermi theory": [1]

${\displaystyle \rho ^{\text{induced}}(\mathbf {r} )=-e[n(\mu _{0}+e\phi (\mathbf {r} ))-n(\mu _{0})]}$

where n(μ) is the function discussed above (electron density as a function of internal chemical potential), e is the elementary charge, r is the position, and ${\displaystyle \rho ^{\text{induced}}(\mathbf {r} )}$ is the induced charge at r. The electric potential ${\displaystyle \phi }$ is defined in such a way that ${\displaystyle \phi (\mathbf {r} )=0}$ 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.

### Linearization, dielectric function

If the chemical potential does not vary too much, the above equation can be linearized:

${\displaystyle \rho ^{\text{induced}}(\mathbf {r} )\approx -e^{2}{\frac {\partial n}{\partial \mu }}\phi (\mathbf {r} )}$

where ${\displaystyle \partial n/\partial \mu }$ is evaluated at μ0 and treated as a constant.

This relation can be converted into a wavevector-dependent dielectric function: [1]

${\displaystyle \epsilon (\mathbf {q} )=1+{\frac {k_{0}^{2}}{q^{2}}}}$ (cgs-Gaussian)

where

${\displaystyle k_{0}={\sqrt {4\pi e^{2}{\frac {\partial n}{\partial \mu }}}}}$

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.

## Example: A point charge

If a point charge Q is placed at r=0 in a solid, what field will it produce, taking electron screening into account?

One seeks a self-consistent solution to two equations:

• The Thomas–Fermi screening formula gives the charge density at each point r as a function of the potential ${\displaystyle \phi (\mathbf {r} )}$ at that point.
• The Poisson equation (derived from Gauss's law) relates the second derivative of the potential to the charge density.

For the nonlinear Thomas–Fermi formula, solving these simultaneously can be difficult, and usually there is no analytical solution. However, the linearized formula has a simple solution:

${\displaystyle \phi (\mathbf {r} )={\frac {Q}{r}}e^{-k_{0}r}}$ (cgs-Gaussian)

With k0=0 (no screening), this becomes the familiar Coulomb's law.

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.

## Fermi gas at arbitrary temperature

For a three-dimensional Fermi gas (noninteracting electron gas), the screening wavevector ${\displaystyle k_{0}}$ can be expressed as a function of both temperature and Fermi energy ${\displaystyle E_{\rm {F}}}$. The first step is calculating the internal chemical potential ${\displaystyle \mu }$, which involves the inverse of a Fermi–Dirac integral,

${\displaystyle {\frac {\mu }{k_{\rm {B}}T}}=F_{1/2}^{-1}\left[{2 \over {3\Gamma (3/2)}}\left({E_{\rm {F}} \over T}\right)^{3/2}\right]}$.

We can express ${\displaystyle k_{0}}$ in terms of an effective temperature ${\displaystyle T_{\rm {eff}}}$: ${\displaystyle k_{0}^{2}=4\pi e^{2}n/k_{\rm {B}}T_{\rm {eff}}}$, or ${\displaystyle k_{\rm {B}}T_{\rm {eff}}=n\partial \mu /\partial n}$. The general result for ${\displaystyle T_{\rm {eff}}}$ is

${\displaystyle {T_{\rm {eff}} \over T}={4 \over 3\Gamma (1/2)}{(E_{F}/k_{\rm {B}}T)^{3/2} \over F_{-1/2}(\mu /k_{\rm {B}}T)}}$.

In the classical limit ${\displaystyle k_{\rm {B}}T\gg E_{\rm {F}}}$, we find ${\displaystyle T_{\rm {eff}}=T}$, while in the degenerate limit ${\displaystyle k_{\rm {B}}T\ll E_{\rm {F}}}$ we find

${\displaystyle k_{\rm {B}}T_{\rm {eff}}=(2/3)E_{\rm {F}}}$.

A simple approximate form that recovers both limits correctly is

${\displaystyle k_{\rm {B}}T_{\rm {eff}}=\left[(k_{\rm {B}}T)^{p}+(2E_{\rm {F}}/3)^{p}\right]^{1/p}}$,

for any power ${\displaystyle p}$. A value that gives decent agreement with the exact result for all ${\displaystyle k_{\rm {B}}T/E_{\rm {F}}}$ is ${\displaystyle p=1.8}$, which has a maximum relative error of < 2.3%.

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## References

1. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, Toronto, 1976)