For Show

In the following graph, 1/x*(1-(1-1/x)^(1/2)) in blue is plotted together with 1/(2*x)*(1-(1-1/(2*x))^(1/2)) in black where x has been replaced with 2x.  This means, the black curve represents higher temperature; each value of x corresponds to 2x for the black curve.  Also plotted in the same corresponding color are the derivatives of the curves.  (The black curve should be above the blue curve if not manipulated so.)

The drop in band gap is illustrated on the derivative curves.  The band gap is from the top of the discontinuity all the way down to where its value approaches infinity.  It is noticed that the top of the discontinuity in the higher temperature curve dropped resulting in a drop in  \(E_{BG}\).  This corresponds to a steeper gradient above the kink; in value, a lower value, more negative.

These curves are illustrative only.