Thomas–Fermi screening

Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid. 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. It is named after Llewellyn Thomas and Enrico Fermi. T e f f T = 4 3 Γ ( 1 / 2 ) ( E F / k B T ) 3 / 2 F − 1 / 2 ( μ / k B T ) {displaystyle {T_{ m {eff}} over T}={4 over 3Gamma (1/2)}{(E_{F}/k_{ m {B}}T)^{3/2} over F_{-1/2}(mu /k_{ m {B}}T)}} . Thomas–Fermi screening is a theoretical approach to calculate the effects of electric field screening by electrons in a solid. 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. It is named after Llewellyn Thomas and Enrico Fermi. The Thomas-Fermi wavevector (in Gaussian-cgs units) is where μ is the chemical potential (fermi level), n is the electron concentration and e is the elementary charge. Under many circumstances, including semiconductors that are not too heavily doped, n∝eμ/kBT, where kB is Boltzmann constant and T is temperature. In this case, 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 If we restore the electron mass m e {displaystyle m_{e}} and the Planck constant ℏ {displaystyle hbar } , the screening wavevector in Gaussian units is k 0 2 = k T F 2 ( m e / ℏ 2 ) {displaystyle k_{0}^{2}=k_{ m {TF}}^{2}(m_{e}/hbar ^{2})} . For more details and discussion, including the one-dimensional and two-dimensional cases, see the article: 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. 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 n ( μ ) {displaystyle n(mu )} , is 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 n ( μ ) ∝ μ 3 / 2 {displaystyle n(mu )propto mu ^{3/2}} .