Specific Heat and Superfluid Density of 4He near T λ of a 33.6 nm Film Formed Between Si Wafers

We report measurements of superfluid density and specific heat of a 33.6 nm film near the superfluid transition. The film is formed between two patterned and directly bonded silicon wafers. These measurements were undertaken with the primary purpose of understanding coupling and proximity effects in a situation when the film was in contact with helium in a larger confinement (Perron et al. in Nat. Phys. 6:499, 2010; Perron and Gasparini in Phys. Rev. Lett. 109:035302, 2012). However, these data are also relevant to issues of correlation-length finite-size scaling. This is the thinnest hard-wall confined film for which such scaling has been tested for the specific heat and superfluid density. One expects that at some small thickness such scaling should fail. We compare our results with previous data of helium in a similar confinement but at larger thickness. We find good agreement with scaling in regions where previous data scaled, and confirm the lack of scaling where previously reported. In our analysis we consider a native oxide growth between the etching and bonding steps of cell fabrication and its effect on our scaling analysis.