Thomas-Fermi screening of electrostatic fields in a type-I superconductor
The empirical London equations make distinct predictions for a superconductor's longitudinal and transverse electrodynamic response. Conventionally, this framework attributes the Meissner effect to the transverse component, with the longitudinal response predicted to remain unchanged on entering the superconducting state, i.e. a static electric field is screened over the Thomas-Fermi length ($\lambda_{\mathrm{TF}} \sim 1$~\AA), just as in the nonsupercondcuting metal. J.~E.~Hirsch~[\textit{Phys.~ Rev.~B}~\textbf{69},~214515~(2004)] has developed an alternative formalism, which predicts that the longitudinal screening length should instead be governed by the London penetration depth as the system is cooled below the superconducting transition temperature ($T_{\mathrm{c}}$). Here we use a combination of qPlus atomic force microscopy and scanning tunneling microscopy/spectroscopy of single-crystal Pb(111) at $\sim$~340~mK~ \mbox{($T/T_{\mathrm{c}}\sim0.05$)} in an attempt to detect any modifications in electrostatic screening of the longitudinal tip field upon entering the superconducting phase. We quench superconductivity using a magnetic field of 200 mT normal to the Pb(111) surface. By measuring force-distance curves and local field-emission resonance spectra at the same sample position in the presence and absence of the magnetic field, we constrain the change in screening between the normal and superconducting state to less than $\sim 1 \%$. This is between two to three orders of magnitude smaller than that predicted by Hirsch's theory, and entirely consistent with unmodified Thomas-Fermi screening across the superconducting transition.