The Thomas–Fermi wavevector (in Gaussian-cgs units) is[1]
##{\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.
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,
##{\displaystyle k_{0}^{2}={\frac {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
##{\displaystyle k_{\rm {TF}}^{2}=4\left({\frac {3n}{\pi }}\right)^{1/3}}##.
If we restore the electron mass ##{\displaystyle m_{e}}m_{e} and the Planck constant {\displaystyle \hbar }\hbar##, the screening wavevector in Gaussian units is ##{\displaystyle k_{0}^{2}=k_{\rm {TF}}^{2}(m_{e}/\hbar ^{2})}##