Surface-Dominated Finite-Size Effects in Nanoconfined Superfluid Helium

Superfluid ${}^4\mathrm{He}$ (He II) is a widely studied model system for exploring finite-size effects in strongly confined geometries. Here, we study He II confined in millimeter-scale channels of 25 and 50 nm height at high pressures using a nanofluidic Helmholtz resonator. We find that the superfluid density is measurably suppressed in the confined geometry from the transition temperature down to 0.6 K. Importantly, this suppression can be accounted for by rotonlike thermal excitations with an energy gap of 5 K. We show that the surface-bound excitations lead to the previously unexplained lack of finite-size scaling of suppression of the superfluid density.

Figure 1

(a) The confined volume of the nanofluidic Helmholtz resonator. The two circular basins are connected to a pressurized ${}^4\mathrm{He}$ bath via four inlet channels and are interconnected via the strongly confined central channel (see Ref. [32] for AFM images). The basins form parallel plate capacitors (electrodes not shown) used for both sensing and forcing the Helmholtz modes. (b) Pressure amplitude of the two Helmholtz modes used in the study. Note that the fundamental (symmetric) mode shows zero pressure gradient across the strongly confined channel making only the second (antisymmetric) mode sensitive to the superfluid density in the strong confinement. (c) Temperature dependence of the two modes for 50 nm central channel confinement and $P\approx 1$ bar, measured with the one-basin protocol. (d) The three measurement protocols (see Ref. [32] for more detailed description). The “$\delta C$” box represents a bridge circuit that measures differential capacitance between one of the basins and a reference capacitor [the one-basin (i) and crosstalk (iii)] or between the two basins [the antisymmetric (ii)]. The coloring of the basin indicates the applied electrostatic force driving the Helmholtz mode: white—not forced, red—forced, blue—forced with a 180° phase shift.

Figure 2

Superfluid density as a function of reduced temperature in the transition region. The confinement of the central channel of the resonator is shown in the legend. The confined superfluid density is calculated using (3) and the black points show the bulk superfluid density for the corresponding device calculated using (2). Thick dotted horizontal lines indicate the expected KT jump [19]. The error bars are calculated as the standard deviation of a set of measurements with different protocols (i)–(iii) and different drive amplitudes. Inset: ratio of confined and bulk superfluid densities as a function of the scaling variable according to (1). Note that our data do not follow a universal scaling, as has been previously observed [14].

Figure 3

(a) Normal fluid density enhancement for helium slabs with 25 and 50 nm confinements. (b) Temperature dependence of the normal fluid density enhancement as a function of pressure for 25 nm confinement. The black dashed lines in (a),(b) are linear fits to (5), the roton gap is given by the slope of the line. (c) Data from panels (a) and (b) scaled with $g(P,D)=D^{-1}(1+P/P^{*})[1+(D^{*}/D{)}^3]$ with $P^{*}=25$ bar and $D^{*}=25\mathrm{nm}$ (see text).