Superconductors — Question 7

Question 7

The density of superconducting carriers controls the screening length

Use the supplied London relation $\lambda=\sqrt{m_s/(\mu_0n_sq_s^2)}$. A model has $n_s(T)=n_s(0)[1-(T/T_c)^4]$, with carrier mass and charge fixed.

1. Find the penetration-depth ratio as a function of temperature.

2. Find the temperature where penetration depth doubles and the remaining fraction of superconducting carrier density.

Question 7 - Solution

The density of superconducting carriers controls the screening length

(a) Method and calculation

$$\frac{\lambda(T)}{\lambda(0)}=\sqrt{\frac{n_s(0)}{n_s(T)}}.$$$$\lambda(T)/\lambda(0)=[1-(T/T_c)^4]^{-1/2}.$$

(b) Interpretation and further calculation

$$2=[1-(T/T_c)^4]^{-1/2}\Rightarrow T/T_c=(3/4)^{1/4}=.930605.$$$$n_s(T)/n_s(0)=1/4.$$

Physical check

The divergence near Tc signals weakening screening within the model; microscopic details are not derived from this phenomenological law.