Effect of Fluxoid Quantization on Transitions of Superconducting Films

This paper presents an elementary theory for the transition of a superconducting film in the presence of a perpendicular magnetic field. The theory is based on the Ginzburg-Landau theory, with emphasis on the qualitatively important consequences of fluxoid quantization. The theory predicts that the transition occurs at a field $H_T\mathrm{}\left(T\right)=\frac{4\pi \lambda _e^{}{}_{}{}^2\mathrm{}\left(T\right)\mathrm{}H_{\mathrm{cb}}^{}{}_{}{}^2\mathrm{}\left(T\right)\mathrm{}}{{\varphi }_0}$, where ${\lambda }_e\mathrm{}\left(T\right)\mathrm{}$ and $H_{\mathrm{cb}}\mathrm{}\left(T\right)\mathrm{}$ are the usual penetration depth and bulk critical fields, respectively, and ${\varphi }_0$ is the flux quantum $\frac{\mathrm{hc}}{2e}$. Experimental data agree very well with this result if the transition is determined by measuring thermal conductivity or flux penetration. The resistive critical field for full normal resistance appears to be about twice this value, probably because of a residual filamentary structure. The theory also predicts that the angular dependence of the transition field should be given by $\left(\frac{H_{T}sin\theta }{H_{T\perp }}\right)+{\left(\frac{H_{T}cos\theta }{H_{T\parallel }}\right)}^2=1\mathrm{}$. This unusual form agrees with the thermal conductivity measurements of Morris.

The same theory leads in an elementary way to a quantitative interpretation of the periodic variation of $T_c$ with flux through a cylinder in the experiments of Little and Parks. The result for the maximum change is $\frac{\Delta T_c}{T_c}=\frac{\varphi _0^{}{}_{}{}^2}{\left[32\mathrm{}{\pi }^2{R}^2\lambda _e^{}{}_{}{}^2\mathrm{}\right(0\left)\mathrm{}H_{\mathrm{cb}}^{}{}_{}{}^2\mathrm{}\right(0\left)\right]}$. This agrees with their experimental value if ${\lambda }_e\mathrm{}\left(0\right)=4700\mathrm{}$ Å, whereas from the limited mean free path in the sample one would estimate ${\lambda }_e\mathrm{}\left(0\right)\mathrm{}\approx 2000$ Å. The agreement is probably within the uncertainties as to the details of the transition region. It should be noted that the present theory predicts a change larger by a factor of order $\left(\frac{T_F}{T_c}\right)\left(\frac{l}{{\xi }_0}\right)$ than the theory given by Little and Parks. The present theory also gives a semiquantitative account of the parabolic background effect observed by Little and Parks.