Studies of two-dimensional finite-size effects on the superfluid density of helium-4 confined between two silicon wafers
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Measurements of the superfluid fraction of 4 He in films of finite lateral extent are reported in this study. These are done by confining 4 He between two silicon wafers which are patterned with small channels 9.4 nm high, uniform width W and 2000 μm long. Five cells were studied with different widths varying from 3 to 19 μm. The superfluid density was obtained using the Adiabatic Fountain Resonance technique. These are the first measurements of helium confined in films of finite lateral extent. This work is relevant to finite-size 2D behavior of the superfluid density which is related to the recent theory of Sobnack and Kusmartsev (SK) which predicts a new class of transition for thin films that are laterally constrained. For this kind of geometry, Sobnack and Kusmartsev state the superfluid onset takes place at lower temperatures than expected from Berezinskii-Kosterlitz-Thouless (BKT) theory. This predicted shift in the critical temperature ( T c ) should behave as a power law such that T c ( L , W ) = T c ( L )[1 - (2/[Special characters omitted.] <math> <f> <g>x</g><sup>*</sup><inf>2D</inf></f> </math> W ) 1/2 ]. The present study supports the idea that there is a new class of transition for films of finite lateral extent. The data show a shift in T c to a lower temperature which is larger than the logarithmic dependence expected from finite-size scaling and BKT theory. However, this shift is different than the one proposed by Sobnack and Kusmartsev. When examining the behavior of the shift for confinement at several widths, we found that the shift favors a power law with a larger exponent (1.446±0.089) than predicted by SK theory. Also an attempt to measure the superfluid density for 1D crossover using a torsional oscillator technique is reported. This technique was improved by regulating the temperature of the cell in addition to the nearly isothermal reference stages S 1 and S 2 . Despite this improvement on the technique, it is clear from our results that the torsional oscillator in combination with the geometry of our cell is not a suitable technique for determining ρ s /ρ since only about 10% of the superfluid helium in the cell is free to move. This is consistent with the torsional oscillator imparting a velocity field perpendicular to 50% or less of the channels. Finally, the design of one of our cells allowed us to measure the specific heat of 4 He confined on a film between silicon wafers at a separation of L = 0.3189 μm. These new data are an addition to the six different confinement sizes measured previously in our laboratory. These data are analyzed to test scaling predictions for the specific heat near the superfluid transition. These most recent data were found to collapse on the earlier measurements for T > T λ . For T < T λ , the data confirmed the earlier observation that near the specific heat maximum and into the superfluid side there is lack of scaling.