Full range analytic approximations for Fermi energy and Fermi–Dirac integral ℱ−1/2 in terms of ℱ1/2

New full range analytic approximation formulas for efficient calculation of the reduced Fermi energy η and the Fermi–Dirac integral ℱ−1/2 in terms of u=ℱ1/2, which is simply related to the free-carrier concentration and temperature, are presented. The maximum fractional error is 0.033% for η and 0.05% for ℱ−1/2, while the maximum absolute error for η is 0.000 41. These formulas facilitate analyses of carrier transport and device characteristics in structures involving smaller band-gap semiconductors.