Three-dimensional critical behavior with 2D, 1D, and 0D dimensionality crossover: surface and edge specific heats

The critical behavior at a second order phase transition is characterized by the divergence of the correlation length xi. We have studied the superfluid transition of 4He in a series of experimental cells in which this divergence of xi is modified due to finite-size confinement. In particular, the design of these cells is such that the smallest dimension is kept the same, 1 microm, but the geometry is such that one obtains crossover to dimensionality of 2, 1, and 0. This corresponds to films, channels, and boxes filled with helium. We measure the specific heat and compare these results with theoretical expectations. We identify surface and line specific heat contributions by analyzing the deviation of the specific heat from its behavior in the thermodynamic limit. The design of these cells is made possible by a combination of silicon lithography and direct wafer bonding.