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.
where Ij represents the Fermi-Dirac integral and I j represents the Bose – Einstein.
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.3.1 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)]