Long-distance correlation-length effects and hydrodynamics of ${}^4\mathrm{He}$ films in a Corbino geometry

Previous measurements of the superfluid density ${\rho }_s$ and specific heat for ${}^4\mathrm{He}$ have identified effects that are manifest at distances much larger than the correlation length ${\xi }_{3\text{D}}$ [Perron et al., Nat. Phys. 6, 499 (2010); Perron and Gasparini, Phys. Rev. Lett. 109, 035302 (2012); Perron et al., Phys. Rev. B 87, 094507 (2013)]. We report here measurements of the superfluid density which are designed to explore this phenomenon further. We determine the superfluid fraction ${\rho }_s/\rho$ from the resonance of 34-nm films of varying widths $4\le W\le 100\mu \text{m}$. The films are formed across a Corbino ring separating two chambers where a thicker 268-nm film is formed. This arrangement is realized using lithography and direct Si-wafer bonding. We identify two effects in the behavior of ${\rho }_s/\rho$: one is hydrodynamic, for which we present an analysis, and the other is a correlation-length effect which manifests as a shift in the transition temperature $T_c$ relative to that of a uniform 34-nm film uninfluenced by proximity effects. We find that one can collapse both ${\rho }_s/\rho$ and the quality factor of the resonance onto universal curves by shifting $T_c$ as $\mathrm{\Delta }{T}_c\sim W^{-\nu }$. This scaling is a surprising result on two counts: it involves a very large length scale $W$ relative to the magnitude of ${\xi }_{3\text{D}}$ and the dependence on $W$ is not what is expected from correlation-length finite-size scaling which would predict $\mathrm{\Delta }{T}_c\sim W^{-1/\nu }$.

Figure 1

A schematic rendering, not to scale, of the Corbino confinement. The cell is formed with two 50-mm-diameter silicon wafers. The support posts and outer border maintain wafer separation of 268 nm except over the Corbino ring of width $W$ which has a 34-nm film above it. The 268-nm dimension was chosen since it is large enough so that in the region where the 34-nm film has its transition the 268-nm film has a superfluid fraction which is within 10% of the bulk value. The square posts $({100\mu \mathrm{m})}^2$ are at 200-$\mu \mathrm{m}$ separation and define the 268-nm separation of the two wafers inside and outside the Corbino ring. The 34-nm height is defined by the oxide pattern in the upper wafer.

Figure 2

A sample resonance signal for the amplitude and phase at $t=0.06$ The data fit (solid lines) according to the adiabatic fountain resonance equations derived in [18]. The resonant frequency ${\omega }_0$ is indicated by the dashed vertical line. The solid vertical line indicates the magnitude of the temperature oscillation.

Figure 3

A sample resonance signal for the amplitude and phase at $t=0.004$. The data fit according to the adiabatic fountain resonance equations derived in [18]. Again, the resonant frequency is indicated by the dashed vertical line. The amplitude signal is somewhat distorted because of finite velocity effects, however the fit of the phase still yields the resonant frequency.

Figure 4

The superfluid fraction of the Corbino cell with $W=100$ $\mu \mathrm{m}$ is shown with the 33.6-nm planar data. A hydrodynamic effect causes the Corbino data to behave like a 268-nm film until near $t_c$ where there is a rapid decrease in the superfluid fraction.

Figure 5

The ratio ${\rho }_s/{\rho }_{sb}$ vs $t$ for the Corbino cell with $w=100$ $\mu \mathrm{m}$, and the 33.6-nm planar data. The difference in the behavior of these data can be understood via the hydrodynamics discussed below.

Figure 6

The ratio ${\rho }_s/{\rho }_{sb}$ vs $t$ for the Corbino cell with $w=100$ $\mu \mathrm{m}$ and the 33.6-nm planar data are plotted with the calculated values for the Corbino cell. The calculated values for this quantity are the pluses using the data from ${\omega }_{+}$ of the 40-$\mu \mathrm{m}$ ring, and the crosses using ${\rho }_{s1}$(268 nm) from the scaled value of ${\rho }_{s1}\left(0.2113\mu \mathrm{m}\right)$. The diamonds are calculated using an internal cell resonance, Eq. (20). See text.

Figure 7

The ratio ${\rho }_s/{\rho }_{sb}$ vs $t$ for the Corbino cell with $w=40$ $\mu \mathrm{m}$ obtained from ${\omega }_{+}$, and the data for 211-nm planar film scaled to 268 nm.

Figure 8

The superfluid fraction ${\rho }_s/\rho$ vs the reduced temperature $t$. The open circles are the 33-nm planar data to which the Corbino data are compared. The two horizontal lines are the expected Kosterlitz-Thouless jump for the 33.6- and 268-nm planar data.

Figure 9

The ratio ${\rho }_s/{\rho }_{sb}$ vs $t$ for all the Corbino cells and for the 33.6-nm planar data. The symbols are identified in Fig. 8.

Figure 10

A plot of planar cell thickness vs critical temperature $t_c=1-T_c/T_{\lambda }$. All data were obtained using AFR. The fit yields a critical exponent $\nu =0.66\pm 0.02$ which agrees with the value determined for the bulk superfluid fraction $\nu =0.6705\pm 0.0006$ [21].

Figure 11

The ratio ${\rho }_s/{\rho }_{sb}$ is plotted for the Corbino cell with $W=4$ $\mu \mathrm{m}$ as well as for the 33-nm planar cell and the cell with a 31.7-nm film in the presence of $\left(2\mu \mathrm{m}\right){}^3$ boxes spaced $4\mu \mathrm{m}$.

Figure 12

A plot of $\delta t_c$ vs $W^{-1}$. The straight line is given by $\delta t_c\cong {(W/{\xi }_0^{-})}^{-\nu }$.

Figure 13

Scaling of the ratio ${\rho }_s/{\rho }_{sb}$ when plotted vs $t$ shifted by $\delta t_c=[T_c\left(\infty \right)-T_c\left(W\right)]/T_{\lambda }$.

Figure 14

Scaling of the dissipation $1/Q$ when plotted vs $t+\delta t_c$. This is the same shift as in Fig. 13 for ${\rho }_s/{\rho }_{sb}$.