Finite-Size Scaling of the Superfluid Density of HELIUM-4 Confined Between Two Silicon Surfaces

Scaling of finite-size effects are tested by measuring the superfluid density of He confined between two Si surfaces. This is done using a torsional oscillator technique with cells of 4 different confinement spacings, that is, 3.9 mu m, 2.8mu m, 5190A and 1060A. These are the first measurements of He confined in a sufficiently well-defined planar geometry to show cross-over behavior from three dimensional-like to finite -size to two dimensional. The spacing between two Si surfaces are achieved by a photolithographic technique and a high temperature vacuum bonding. The height of SiO _2 steps which determines the confinement spacing is measured by an ellipsometer before bonding process. The spacings are also measured and estimated by an interferometer technique and superfluid density (mass loading of He in the cell on the oscillator). The results with different techniques are proven to be consistent with one another. The reductions of superfluid density near T_ lambda due to the finite-size effects are observed for above four cells. In addition, the dimensionality crossover from 3D to 2D are observed for 3.9mu m and 5190A cells. This cross-over effect had never been reported for He system. The reasons for our success might be that the confinement spacings of the cells are well-defined and uniform, and also the quality factors of the torsional oscillators for those cells are high enough. The data for above four cells are analyzed for scaling with the exponent of the bulk correlation length, nu . We find, contrary to theoretical expectations, that this scaling does not work. We discuss these results in the light of finite-size scaling predictions and earlier measurements of confined helium.