# Fermi-Dirac and Bose-Einstein Integrals

C.1 Functional Expressions  Fermi-Dirac integrals of the form

occur widely in semiconductor transport theory and carrier density calculations, where E is the energy above the edge of the respective band normalised by the thermal energy kT and n is a similarly normalised value of the carrier fermi – energy or electrochemical potential. Г(п) is the Gamma function equal to (n = 1)!

for positive integral values of n and 4n (2m – 1)!!/2m for n = (m + Ч2), where the!! sign represents the product 1.3 … (2m – 1).  In a similar way, the Bose-Einstein integral is defined:

his integral has similar application to the case of bosons and its properties have been explored elsewhere, although not as fully as the Fermi-Dirac integrals. In the theory of optoelectronic devices such as light emitting diodes and solar cells, it has been pointed out that a non-zero electrochemical potential can be assigned to photons generated in a biased device, making the properties for non-zero argu­ments of increased interest. These Bose-Einstein integrals share many properties in common with Fermi-Dirac integrals explored in the present Appendix.  Generally,

where Ij represents the Fermi-Dirac integral and I j represents the Bose – Einstein.

C.2 General Properties

Both integrals share the differentiation formula: — Ij(n) = Ij-1(n) dn  For n negative or zero, the expansion for (1 ± x)-1 can be used to derive the following series expression:

This results in the “classical” or completely non-degenerate expression: I j(n)e exp(n), П << 0

Equation (C.5) can be used to derive the following relationship that is also valid for positive n:  I+(n) = I-(n)-1-(2n)/2j

This leads to a recurrence relationship:

I-(n) = X2~jr I+(2rn)

More complex relationships exist between integrals of the same order but with positive and negative arguments. For example (Blackmore xxxx),

n j +1

I+(n) = cos( jn )I+(-n) +’ [1+Rj(n )],n > 0 (C.9)

j J T(j + 2) J

where Rj(n) is a generally infinite series in negative powers of n. When j is an integer, this expression is simplified.

C.3 Special Cases

C.3.1 ц = 0

Integrals involving zero argument can be expressed in terms of the Reimann zeta function E,(n) since:

as may be seen from Eq. (C.5). Invoking Eq. (C.8), it follows: I+(0) = %(j +1)(1 – 2-1)

Since E,(n) equals п 26, п 490, n 6I945, n 8I9450 and so on for even values of n (2, 4, 6 and 8, respectively), this expression is simplified for odd integral values of j.

C.3.2 Integral j

In this case, relationships such as given by Eq. (C.9) is limited to a finite number of terms. Other simplifications occur as discussed later. The case for j = 0 is the highest order given analytically: I ± (П) = ± /n/Т ± exp(n)]