Homes Scaling in Ionic Liquid Gated La$_{2}$CuO$_{4+x}$ Thin Films

Finding more efficient ways of exploring the doping phase diagrams of high temperature superconductors as well as probing the fundamental properties of these materials are essential ingredients for driving the discovery of new materials. We use a doping technique involving gating with ionic liquids to systematically and continuously tune the T$_{c}$ of superconducting La$_{2}$CuO$_{4+x}$ thin films. We probe both the transport properties and the penetration depth of these samples and find that Homes scaling $\lambda^{-2}\propto\sigma T_{c}$ is obeyed, consistent with these materials being in the dirty limit. This result is independent of the precise mechanism for the gating process as all of the parameters of the scaling relationship are determined by direct measurements on the films.