Re-evaluation of the derivatives of the half order Fermi integrals

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Abstract
The Fermi integrals of half orders are important in the simulation of semiconductor transport processes. Several of these integrals (−1/2, 1/2, 3/2, 5/2) have been recently retabulated since the 1938 study by McDougall and Stoner [Phil. Trans. Roy. Soc. A 237, 67 (1938)], but the derivatives were not re-evaluated. The original integral values were calculated without the aid of high speed computers by using approximate series evaluation and tabulations of exponentials and zeta functions. In addition, a discrepancy was found in the literature since the original study in 1938. The second derivative of F1/2 has been mistakenly represented as being proportional to a Fermi integral of another order. This article tabulates the half order Fermi integrals from −1/2 to 5/2 over the reduced energy range −5 to 20 in 0.25 increments. The first two derivatives of F−1/2 are also calculated by numerical integration and tabulated to aid in interpolation. It is shown that the second derivative of F1/2 is not proportional to another Fermi integral. A suitable interpolation scheme is proposed to calculate the values of the Fermi integrals of various order to high accuracy over the total reduced energy range.