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Analytic approximations for the Fermi energy of an ideal Fermi gas
1.The inversion of Eq. (1) (as would be required, for example, if the density of states f were to be inferred from electrical measurements of F and η) and other useful properties of Eq. (1) are immediately obtained through the observation that the change of variable and converts Eq. (1) into the standard form of the well‐studied Stieltjes transform. See D. V. Widder, Transform Theory (Academic, New York, 1971), particularly Secs. 5.14, 6.4 and 7.9 and Chap. 9.
2.J. S. Blakemore, Semiconductor Statistics (Pergamon, New York, 1962), Appendices B and C. Further tables are cited here.
3.A. Sommerfeld, Z. Phys. 47, 1 (1928).
4.I. Adawi, J. Stat. Phys. 12, 263 (1975).
5.Important exceptions are the accurate empirical approximations in N. G. Nilsson, Phys. Status Solidi A 19, 75 (1973).
6.J. McDougall and E. C. Stoner, Philos. Trans. R. Soc. London A 237, 67 (1938).
7.W. Shockley, Electrons and Holes in Semiconductors (Van Nostrand, Princeton, 1950), p. 242.
8.R. W. Dixon, Bell Syst. Tech. J. 55, 973 (1976).
9.P. A. Barnes and T. L. Paoli, IEEE J. Quantum Electron. QE‐12, 633 (1976).
10.See Ref. 1, Sec. 5.13.
11.Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U.S. GPO, Washington, D.C., 1964), Secs. 3.6.25, 3.6.24 (as corrected in later printings), 6.1.1, and 6.1.9.
12.F. Stern, J. Appl. Phys. 47, 5382 (1976), Fig. 6 at optical loss and
13.L. D. Landau and E. M. Lifshitz, Statistical Physics, 2nd ed. (Addison‐Wesley, Reading, Mass., 1969), Sec. 55;
13.E. Fermi, Z. Phys. 36, 902 (1926).
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