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.

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

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.

Under many circumstances, including semiconductors that are not too heavily doped, $n \propto e μ / k B T$ , where k_B is Boltzmann constant and T is temperature. In this case,

k_{0}^{2}={\frac {4\pi e^{2}n}{k_{\rm {B}}T}},

i.e. $1/ k 0$ 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 k_TF given in atomic units is

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

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

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

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