Superconductors — Question 10

Question 10

A spatial distribution of transition temperatures broadens a resistance transition

A long wire of length L and area A has local transition temperature $T_c(x)=T_1+(T_2-T_1)(x/L)^2$, with $T_2>T_1$. At uniform temperature T, each slice has resistivity zero below its local Tc and constant $\rho_n$ above it. Slices carry the same current; ignore proximity effects.

1. Find total R(T) for $T_1<T<T_2$ and the limiting regimes outside that interval.

2. Find the temperature where resistance equals half its fully normal value and determine when the whole wire has zero resistance.

Question 10 - Solution

A spatial distribution of transition temperatures broadens a resistance transition

(a) Method and calculation

$$T>T_c(x)\Rightarrow x<L\sqrt{\frac{T-T_1}{T_2-T_1}}.$$$$R(T)=R_n\sqrt{\frac{T-T_1}{T_2-T_1}},\quad R_n=\rho_nL/A.$$

(b) Interpretation and further calculation

$$R=R_n/2\Rightarrow T=T_1+(T_2-T_1)/4.$$$$R=0\ (T\le T_1),\quad R=R_n\ (T\ge T_2).$$The lowest local transition temperature controls the onset of a completely zero-resistance series path.

Physical check

A broad measured transition can reflect inhomogeneous local critical temperatures rather than a finite resistance in every superconducting region.