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D Integration of Fermi Functions

    The Fermi function and its antiderivative is given by
\begin{align}f(E) &= \frac{1}{1+e^{\frac{E-E_F}{k_BT}}}\\
F(E) &= \int f(E)\,dE = -k_BT\ln\left(1+e^{-\frac{E-E_F}{k_BT}}\right)
The product $f(E)(1-f(E+\Delta E))$ can be written as
\begin{gather}f(E)(1-f(E+\Delta E)) = \frac{f(E)-f(E+\Delta E)}
{1-e^{-\frac{\Delta E}{k_BT}}},
and the corresponding antiderivative is
\begin{gather}\int f(E)(1-f(E+\Delta E))\,dE = -\frac{k_BT}{1-e^{-\frac{\Delta E...
{1+e^{-\frac{E+\Delta E-E_F}{k_BT}}}\right).
A usual approximation for   metal tunnel junctions, where the Fermi energy is much bigger than the conduction band edge, $E_F \gg E_c$, is to set $E_c = -\infty$, thus
\int_{E_c}^{\infty} f(E)(1-f(E+\Delta E))\,dE \approx
...elta E))\,dE
\approx\frac{\Delta E}{1-e^{-\frac{\Delta E}{k_BT}}}.

Christoph Wasshuber