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## Table I. Values used for calculations. 

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Fermi-Dirac integral. For the Fermi distribution function 

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the following identity holds, 

So Eq. (13) can be written as 

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An integration by parts yields 

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For low temperatures we can take advantage of the behavior of µ$_{E}$$_{0}$ ( T ) and ¯ ε ( T ) for T → 0 . It can easily be shown that for constant n$_{s}$ and therefore constant ε$_{F}$$_{0}$ all derivatives vanish at this point. 

So µ$_{E}$$_{0}$ ( T ) and ¯ ε ( T ) are very flat functions at T → 0 , for k$_{B}$T ≪ ε$_{F}$$_{0}$ they can be approximated by 

See appendix A for details. The Fermi-Dirac integral F j ( η ) can be approximated by 32 

which is the first term of an asymptotic series derived with help of a Sommerfeld expansion, 33 and by 

For T → 0 the argument of the Fermi-Dirac integral in Eq. (29) increases beyond any border. Which approxima- tion for the Fermi-Dirac integral is applicable depends 

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