## Notes on Fermi-Dirac Integrals (4th Edition)

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#### Abstract

Fermi-Dirac integrals appear frequently in semiconductor problems, so an understanding of their properties is essential. The purpose of these notes is to collect in one place, some basic information about Fermi-Dirac integrals and their properties.

We also present Matlab functions (in a zipped file) that calculate Fermi-Dirac integrals (the "script F" defined by Dingle (1957)) in three different ways.

The function, "FD_int_approx.m", evaluates Fermi-Dirac integrals using analytic approximations developed by Bednarczyk et al. (1978) and Aymerich-Humet et al. (1981, 1983).

The folder, "FD_int_Pulfrey", includes a set of Matlab files that calculate Fermi-Dirac integrals using the approximations proposed by Halen and Pulfrey (1985, 1986). The main function is "FDjx.m". This function gives a better accuracy, but the simulation time is a little longer.

Fermi-Dirac integrals can be evaluated accurately by numerical integration. The function, "FD_int_num.m", calculates Fermi-Dirac integrals using the composite trapezoidal rule. This approach provides very high accuracy, but the CPU time is considerably longer.

In the 4th edition, we also provide a new table-based Matlab script ("TBFD_integral.m", download available at https://github.com/wang159/FDIntegral_Table) that is less likely to give large errors in a wide range of input while still much faster than the rigorous numerical integration.

A set of Matlab scripts to benchmark the CPU time and accuracy of the four approaches ("FD_int_approx.m", "FDjx.m", "FD_int_num.m", "TBFD_integral.m") is also available as the zipped file ("FDIntegral_benchmark.zip").

#### References

D. Bednarczyk and J. Bednarczyk, Phys. Lett. A, 64, 409 (1978)

J. S. Blakemore, Solid-St. Electron, 25, 1067 (1982)

X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, Solid-St. Electron, 24, 981 (1981)

X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, J. Appl. Phys., 54, 2850 (1983)

P. V. Halen and D. L. Pulfrey, J. Appl. Phys., 57, 5271 (1985)

P. Van Halen and D. L. Pulfrey, J. Appl. Phys., 59, 2264 (1986)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipies: The art of scientific computing, 3rd Ed., Cambridge University Press, 2007.

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