As \(T\) increases, \(r_c\) decreases, since the \(B\) field generated by the spinning electron increases.
\(B=\cfrac{\mu_o}{4\pi}\cfrac{qv\times \hat{r}}{r_c^2}\)
Why should this decrease electric conductivity? The Lorentz force that acts on passing charges due to this \(B\) field is perpendicular to both the \(B\) field and the velocity of the charge. A high \(B\) field sends the charge clashing into the structure lattice and reduces its drift velocity.
For high conductivity, \(B\) field should be minimum but non zero. For a given material, this occurs when,
\(r_c=\cfrac { m_{ e }c^{ 2 } }{ 2AT }\sqrt { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } -1}\)
is a maximum, ie
\(\cfrac{d\,r_c}{d\,T}=0\)
or
\(\cfrac { q^{ 2 } }{ 4\pi T\varepsilon _{ o }m_{ e }c^{ 2 } } \frac { 1 }{ \sqrt { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } -1 } } -\cfrac { m_{ e }c^{ 2 } }{ 2AT^{ 2 } } \sqrt { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } -1 } =0\)
\(\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }c^{ 2 } } =\cfrac { m_{ e }c^{ 2 } }{ 2AT } \left( { AT\cfrac { q^{ 2 } }{ \pi (m_{ e }c^{ 2 })^{ 2 }\varepsilon _{ o } } -1 } \right) \)
\( \cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }c^{ 2 } } =\cfrac { q^{ 2 } }{ 2\pi \varepsilon _{ o }m_{ e }c^{ 2 } } -\cfrac { m_{ e }c^{ 2 } }{ 2AT } \)
\( \cfrac { m_{ e }c^{ 2 } }{ 2AT } =\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }m_{ e }c^{ 2 } } \)
\( T=\cfrac { Aq^{ 2 } }{ 2\pi \varepsilon _{ o }(m_{ e }c^{ 2 })^{ 2 } } \)
where
\(A=\cfrac{\tau_o}{2}\cfrac{qv_{\small{T}}}{\{r^2_{\small{T}}+r_{or}^2\}^{3/2}}.{r_{or}}\)
A particular temperature exists for which the material is most conductive. Beyond the maximum value, \(r_c\) decreases monotonously, \(B\) increases monotonously, drift velocity decreases and conductivity decreases. As temperature decreases, the complex roots changes to real roots. What was a conductor can change rapidly into a semiconductor after attaining maximum conductivity with decreasing temperature.
And superconductors are all stuck at minimum \(B\), at a specific temperature.
Good night.
