From the post "Band Gap? Just A Kink",
\(f(ln(T))=\cfrac{m_e}{4A_or_{ec}}\)
\(f(ln(T))=E.h(ln(T))+1=\cfrac{m_e}{4A_or_{ec}}\)
where from the post "Oops! Smooth Operator",
\(h(ln(T))|_{lim\,T\rightarrow 0}=0\)
\(h(ln(T)) =\cfrac{1}{E}(\cfrac{m_e}{4A_or_{ec}}-1)\)
where \(h(ln(T))\) is a function in \(ln(T)\), \(E\) is a constant, \(A_o\) is the drag factor of free space, and
\( r_{ ec }=\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }v^{ 2 } } \)
and \(v^2=2c^2\)
So,
\(h(ln(T)) =\cfrac{1}{E}(\cfrac{2\pi \varepsilon _{ o }m^2_ec^2}{A_oq^2}-1)\)
Fro the case of room temperature or higher,
\(h(ln(T≥T_{room})) =\cfrac{1}{E}(\cfrac{2\pi \varepsilon _{ o }m^2_ec^2}{A_oq^2}-1)\)
In case of a nucleus with higher positive charge, \(n\)
\(h(ln(T≥T_{room})) =\cfrac{1}{E}(\cfrac{2\pi \varepsilon _{ o }m^2_ec^2}{A_onq^2}-1)\)
Unless we know how space density varies with temperature, \(T\) from which \(h(ln(T)\) is part of, we can go no further. The simplest monotonously decreasing function in \(ln(T)\) is probably,
\(h(ln(T))=-ln(T)=\cfrac{1}{E}(\cfrac{2\pi \varepsilon _{ o }m^2_ec^2}{A_onq^2}-1)\)
\(T=e^{-\cfrac{1}{E}(\cfrac{2\pi \varepsilon _{ o }m^2_ec^2}{A_onq^2}-1)}\)
It is interesting to note the electron momentum squared term in the exponent. At first glance, it seems that there is no engineering element in this expression. The presence of \(n\) in the denominator of the exponent suggests that the pull of the nucleus act against conductivity. \(n\) should be kept small or the nucleus be relatively shielded. It also suggest that loosely held valence electrons further away from the nucleus contribute more to conductivity. So the choice of elements in a superconducting lattice should contain elements that are high in the reactivity series. The lack of engineering tweak in the expression suggest that these valence electrons in orbit are passive in the conduction of electricity. It is possible that together with other outer electrons in the structure of the material, they form passive conduit that channels free electrons through it. As the conduit narrows, free electrons are repelled from the walls of the enclosure and is sped along the applied potential. Deviations perpendicular to the length of such channel decreases as the channel narrows with increasing \(r_e\), results in higher conductivity as the speed component along the channel increases. Cooling the material will increase \(r_e\) and so narrows the conduit. Heating up the material causes \(r_e\) to decrease and widens the conduit and conductivity drops. This is consistent with experimental observations.
Already, they look like sheets of graphene. Passe.
It is expected that \(T\) at the kink point be a low number, such that at room temperature most of the electrons is on the lower part of the curve as most material are poor conductor at room temperature.
