6 Fermi-Dirac integral. For the Fermi distribution function the following identity holds, So Eq. (13) can be written as An integration by parts yields (a) 0 100 200 300 −100 −80 −60 −40 −20 0 20 T (K) E 0 (meV) (b) 0 5 10 15 20 −200 −150 −100 −50 0 50 n$_{s}$ (10$^{11}$cm −2 ) E 0 (meV) Figure 5. Chemical potential µ relative to the ground state energy E 0 . a) Curves with constant n s , bottom up: n s = 1 , 2 , 5 , 10 , 20 × 10 $^{11}$cm $^{$_{−}$11}$. b) Curves for constant T , top down: T = 0 , 50 , 100 , 150 , 200 , 250 , 300 K . f = 1 exp ( E − µ k$_{B}$T ) + 1 = 1 exp ( ε − µ$_{E}$$_{0}$ k$_{B}$T ) + 1 (35) − ∂f ( ε ) ∂ε = 1 4 k$_{B}$T cosh − 2 ( ε − µ$_{E}$$_{0}$ 2 k$_{B}$T ) . (36) ¯ ε = ε$_{F}$$_{0}$ [ ∫ ∞ 0 ( ε ε$_{F}$$_{0}$ ) 5 / 6 ( − ∂f ( ε ) ∂ε ) dε ] − 6 . (37) ¯ ε = ε 6 F 0 [ 5 6 ∫ ∞ 0 ε − 1 / $^{6}$f ( ε ) dε ] − 6 , ¯ ε = ε$_{F}$$_{0}$ ( ε$_{F}$$_{0}$ k$_{B}$T ) 5 ×   $^{5}$ 6 ∫ ∞ 0 ( ε k$_{B}$T ) − 1 / 6 exp ε − µ$_{E}$$_{0}$ k$_{B}$T ) + 1 d ( ε k$_{B}$T )   $^{−}$$^{6}$ , ¯ ε = ε$_{F}$$_{0}$ ( ε$_{F}$$_{0}$ k$_{B}$T ) 5 [ Γ ( 11 6 F − 1 / 6 ( µ$_{E}$$_{0}$ k$_{B}$T )] − 6 . (38) d $^{n}$µ$_{E}$$_{0}$ ( T ) dT n ∣ T → 0 = 0 for n ≥ 1 , (39) d $^{n}$¯ ε ( T ) dT n ∣ T → 0 = 0 for n ≥ 1 . (40) µ$_{E}$$_{0}$ ( T ) ≃ ε$_{F}$$_{0}$, (41) ¯ ε ( T ) ε$_{F}$$_{0}$. (42) F j ( η ) ≃ η j +1 Γ ( j + 2) for η ≫ 0 , (43) F j ( η ) ≃ exp η for η ≪ 0 . (44) For low temperatures we can take advantage of the behavior of µ$_{E}$$_{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 g s = 2 g v2D = 2 g v3D = 6 m e = 9 . 1094 × 10 $^{$_{−}$31}$kg m$_{d2D}$ = 0 . 1905 m e m z = 0 . 9163 m e m$_{de3D}$ = 0 . 322 m e m$_{dh3D}$ = 0 . 59 m e ϵ sc = 11 . 9 ϵ ins = 3 . 9 → ϵ ∗ = 7 . 9 D = 200 nm C = 16 → ε D = 0 . 4615 meV N A = 2 × 10 $^{15}$cm $_{−}$3 N D = 0 ε g0Si = 1 . 17 eV α Si = 4 . 73 × 10 $^{$_{−}$4}$eVK$_{−}$1 β Si = 636 K Table I. Values used for calculations.