Finite-size scaling of ${}^4\mathrm{He}$ at the superfluid transition

Experimental results for confined ${}^4\mathrm{He}$ are reviewed that are relevant to correlation-length scaling near the superfluid transition. Data are discussed for which the uniform confinement represents dimensionality crossover from three dimensions (3D) to 2D, 1D, and 0D. In addition, data for the onset of superfluidity are discussed representing 2D to 1D crossover. Collectively, these data for the specific heat, superfluid density, and thermal conductivity yield, in some cases, excellent agreement with expectations of correlation-length scaling and, in others, surprising disagreement. This is especially true in the case of 3D to 2D crossover where data are most plentiful. Here there is a clear distinction between scaling when the confined helium is normal and the lack of scaling when helium becomes superfluid. By far the most problematic result is the lack of scaling for the superfluid density for 3D to 2D crossover and, to some extent, for 3D to 1D crossover. Connectivity and proximity effects can be identified with some data. These might explain some experimental results and present opportunities for further studies of weakly coupled superfluid regions. Measurements to test the universality of finite-size effects along the superfluid transition lines as function of pressure and ${}^3\mathrm{He}$ concentration are also discussed. In the case of the specific heat, data indicate that the nonuniversal behavior of the critical exponent $\alpha$, obtained from bulk measurements, is responsible for the observation of a distinct scaling locus for confined pure ${}^4\mathrm{He}$ versus that of the confined mixtures.

Figure 1

(Color online) Sketch showing the qualitative difference between the specific heat at constant pressure of bulk $C(t,\infty )$ and confined helium $C(t,L)$.

Figure 2

(Color online) Sketch showing the qualitative difference between the superfluid density of bulk ${\rho }_{sb}\left(t\right)$ and confined helium ${\rho }_s(t,L)$.

Figure 3

(Color online) Sketch showing the qualitative difference in the thermal resistivity between bulk $r(t,\infty )$ and confined helium $r(t,L)$. Adapted from 75.

Figure 4

Four independent measurements of bulk helium above the superfluid transition (5; 86; 35; 138) combined into one data set (open circles). The solid line represents a fit of the combined data to Eq. (44). The pluses are the difference between individual points and the solid line. From 158.

Figure 5

The equivalent plot to Fig. 4 for data below the bulk transition temperature. From 158.

Figure 6

An infrared transmission image of two 5-cm silicon wafers that are bonded together to form a closed structure used to confine liquid helium. These images are for a planar confinement where the light areas are bonded posts of silicon dioxide and the dark area is an open region to be filled with helium. The bright spot in the center of (a) is the hole in one of the wafers used to introduce helium into the cell. The only imperfections in the bonding occur in the outer border region and does not affect the area where helium will be confined.

Figure 7

(Color online) Measurement of the separation between two bonded wafers used for helium confinement as a function of position along the wafer surface. (a), (b), and (c) Cells designed for 2D, 1D, and 0D crossover, respectively. The wafers are separated with less than $\sim 1\%$ deviation from the average value.

Figure 8

The temperature shift of the specific-heat maximum of ${}^4\mathrm{He}$ from $T_{\lambda }$ for thin films. The shift is predicted to fall on a locus having a slope of $-\nu$. The dash dotted line is drawn with this slope. A fit to the open circles, the solid line, demonstrates a slope less than this. The dashed line is drawn as a guide to the eye and shows the behavior of the thinnest films. From 87.

Figure 9

Semilogarithmic plot of the specific-heat maximum vs film thickness. The symbols are the same as those in Fig. 8. The data determine a value of $\nu$ less than expected. The data which collapsed upon a single locus in Fig. 8 do not do so here. This demonstrates that one feature may scale while another may not for the same data. From 87.

Figure 10

The shift in the superfluid onset or critical temperature $T_c$ for ${}^4\mathrm{He}$ on various substrates. The symbols refer to data found in Table 1 of 87. From 87.

Figure 11

Experimental setup for the ac measurement of small uniform samples of confined helium between two silicon wafers. Adapted from 158.

Figure 12

(Color online) Oxide pattern used to confine helium to a planar geometry. This is achieved when a second silicon wafer is bonded to the structure. The outer border and posts provide a uniform separation between the two silicon wafers. The structure in the center minimizes the amount of liquid in this region and shifts the local superfluid onset well below the specific maximum.

Figure 13

(Color online) Data of specific heat for eight different planar confinements for temperatures above the bulk transition temperature $T_{\lambda }$.

Figure 14

(Color online) The data from Fig. 13 plotted in scaling form according to Eq. (5). The data show remarkable collapse onto a single locus. Also shown, as a dashed line, is the calculation of 184.

Figure 15

(Color online) The data of Fig. 14 plotted on a log-log scale to emphasise the surface specific heat region. The dashed line is the calculation of the surface region by 160. This agrees with the data remarkably well.

Figure 16

(Color online) Data from helium confined in a planar geometry plotted according to Eq. (50). Here a least-squares fit to Eq. (50) is shown as the solid line. This is done by varying $\nu$. (b) The deviation between the data and the best fit line shown in (a).

Figure 17

The value of the specific heat at $T_{\lambda }$ for helium confined to a planar geometry. Using Eq. (10) one can determine the critical exponents and the maximum value of the specific heat for bulk helium. The theoretical calculation of 200 is shown as the dashed line. This underestimates the effect of confinement but this is likely within the approximation used in the theory.

Figure 18

(Color online) Specific-heat data below $T_{\lambda }$for helium confined to the same planar confinements, as shown in Fig. 13.

Figure 19

(Color online) Scaling of the specific heat below $T_{\lambda }$ according to Eq. (5) but above the maximum. Here the data scale as well as they do above $T_{\lambda }$; see Fig. 15.

Figure 20

(Color online) Same scaling plot as for Fig. 19 for the region near the maximum value of their specific heat (shown here as a minimum in the scaling function). The vertical line at 15 in the scaling variable is the position of the superfluid transition of the confined helium. The data do not collapse in this region as they do in Figs. 14, 19.

Figure 21

(Color online) Data for ${}^4\mathrm{He}$ confined to a planar geometry obeys the equation predicting the shift of the heat-capacity maximum from $T_{\lambda }$ [see Eq. (9)]. This yields the expected correlation-length exponent $\nu$.

Figure 22

(Color online) The maximum of the single-point scaling function $f_1(t,L)$ is expected to be a constant value regardless of film thickness [see Eq. (11)]. The prediction of this value from 187 is shown by the dashed horizontal line. The uniform planar films measured do not support this result. This is the clearest indication of a problem with scaling at the heat-capacity maximum for planar films.

Figure 23

The maximum value of the specific heat for planar films vs $L^{\alpha ∕\nu }$ and fit to Eq. (12) to yield the maximum value of the bulk specific heat. This analysis produces a value of $517.4\mathrm{J}{\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$ which is significantly larger than other determinations (see Table ). The deviation of the data from this fit is not as obvious as in the higher resolution plot, Fig. 22. See also Fig. 49.

Figure 24

Temperature shift of the onset of superfluidity for thin films of ${}^4\mathrm{He}$ is consistent with a power law $\nu$, except for the thinnest films. The crosses are from data taken with the AFR technique while the open circles are from thermal conductivity measurements. The boxes represent a feature seen in the critical thinning data of helium and fall on the same locus as the superfluid onset data.

Figure 25

Superfluid density of helium confined between Mylar surfaces spaced by alumina powder where the average separation is $0.46\mathrm{\mu }\mathrm{m}$. The data are measured using a torsional oscillator. From 87.

Figure 26

Initial deviation from bulk behavior of the data shown in Fig. 25. Data in the region where Eq. (51) is valid should fall on a straight line and be described with an exponent of $\nu =0.671$. The data determine an exponent nearly double this expected value. From 87.

Figure 27

Superfluid fraction of ${}^4\mathrm{He}$ confined to a uniform planar geometry for six different confinements. The 0.0483- and $0.2113\text{−}\mathrm{\mu }\mathrm{m}$ data are obtained using the AFR technique while the others are measured using a torsional oscillator. The horizontal lines indicate the magnitude of the superfluid fraction where the discontinuous jump to zero is expected to occur (168).

Figure 28

The ratio of the superfluid density of ${}^4\mathrm{He}$ confined to a planar geometry to the bulk superfluid density. Data over a range of sizes that span more than 80 show a distinct lack of scaling. The data are measured using either a torsional oscillator (TO) or adiabatic fountain resonance (AFR). The solid line is the field theory calculation of the scaling locus from 184 and the dashed line is the $\psi$-theory calculation of 162.

Figure 29

The amplitude ${\widehat{R}}_{\lambda }$ of the thermal conductivity above $T_{\lambda }$. The open and solid diamonds are the data of 204 with $L=0.0025\mathrm{cm}$ for 22.3 bars and saturated vapor pressure (SVP), respectively. The open circles are the data of 140 at SVP for $L=0.22\mathrm{cm}$ and the solid circles are the data at SVP for $L=0.57\mathrm{cm}$ from 8. The dashed lines are from Eq. (3) of 8. The solid lines are the behavior of the bulk ${\widehat{R}}_{\lambda }$ based upon fits to the theory of 59. From 8.

Figure 30

Scaling of the thermal resistance of ${}^4\mathrm{He}$ confined to a planar geometry above $T_{\lambda }$ in the region where a bulk-plus-surface description applies. Data are taken from Fig. 29. These data do not collapse onto a single locus as expected. In addition, none of the data fall onto a line of slope $\nu =0.671$ as predicted by Eq. (38), but instead determine slopes greater than 1.

Figure 31

The thermal boundary resistance $R_K^{\mathrm{tot}}$ between ${}^4\mathrm{He}$ and a solid surface for $T<T_{\lambda }$. The data, open and closed circles, are from 61. The solid line is the renormalization-group result of 74 and the dashed line is a hydrodynamic result based upon a theory by 134. The dotted line is the Frank and Dohm renormalization group result for 28 bars. From 74.

Figure 32

Data for the critical thinning of helium films due to the thermodynamic Casimir force for three different film thicknesses. The thickness of each film is given in Å. From 77.

Figure 33

The data of Fig. 32 scaled using Eq. (54). The data collapse onto a single locus using the bulk correlation-length exponent $\nu$. From 77.

Figure 34

(Color online) Specific heat of helium confined to uniform, one-dimensional confinements. The 8- and $0.26\text{−}\mathrm{\mu }\mathrm{m}$ cylinder data are from 47, and 141, respectively. The $1\text{−}\mathrm{\mu }\mathrm{m}$ channel data are from 163.

Figure 35

(Color online) Data for 1D confinements scaled according to Eq. (5) for $T>T_{\lambda }$. The dashed line is the theoretical prediction for the surface region by 160. The $1.02\text{−}\mathrm{\mu }\mathrm{m}$ channel data show a unique initial deviation from bulk behavior at large values of the scaling variable due to edges in the confinement structure. The other confinements do not have such edges.

Figure 36

(Color online) The scaling of 1D data for $T<T_{\lambda }$. The Nuclepore data have a different behavior than the more uniform confinements. The dashed line is the calculation of 160 for the surface region. This underestimates the effects of confinement below $T_{\lambda }$ as was also seen in the planar data of Sec. 5A.

Figure 37

(Color online) The data of Fig. 34 plotted according to the single-point scaling function $f_1$. While the data do not collapse near the maximum, there is no systematic opening with size as was seen in the planar data.

Figure 38

(Color online) The temperature shift of superfluid onset for 1D confinements. All data except the open circles and crosses are from confinement in Nuclepore filters. The open circles are data from mica filters with pores of cross section closer to a diamond shape. The solid line is drawn with exponent $\nu$ and the data are consistent with this. The crosses are the temperature of the specific-heat maximum for confinement in Nuclepore.

Figure 39

The thermal resistivity of ${}^4\mathrm{He}$ confined to glass cylinder arrays measured by 166. The circles and squares represent data from cylinders with radii of 0.5 and $1.0\mathrm{\mu }\mathrm{m}$, respectively. The plus signs are the bulk resistivity measurement of 204, fit to a power-law shown as the solid line. From 166.

Figure 40

The scaling of the thermal conductivity data shown in Fig. 39 using Eq. (55). The data show the expected collapse onto a single locus above $T_{\lambda }$ but open up for temperatures below this. From 166.

Figure 41

The value of the thermal conductivity at $T_{\lambda }$from 117 and 166 vs $L^{-1}$ as the plus and boxes, respectively. The solid line is drawn with the expected slope of $\mu ∕\nu$ while the dashed line is the theoretical prediction of 212. From 166.

Figure 42

Scanning electron microscope images of the structures used to confine helium to 0D (164). (a) The $2\text{−}\mathrm{\mu }{\mathrm{m}}^3$ boxes made in ${\mathrm{SiO}}_2$. (b) The fill channels (darker regions) that span the majority of the cell and whose cross section is $10\mathrm{nm}\times 1\mathrm{\mu }\mathrm{m}$. The square oxide regions seen in (b) allow helium to flow from the cell fill hole to the channels.

Figure 43

(Color online) Comparison of the crossover from three dimensions to two, one, and zero dimensions where the smallest spatial length in each confinement is $1\mathrm{\mu }\mathrm{m}$. The data show the expected behavior: As more spatial dimensions are made finite, the greater the heat capacity is suppressed and the specific-heat maximum shifts to colder temperatures.

Figure 44

(Color online) The data above $T_{\lambda }$in Fig. 43 cast in scaling form according to Eq. (5). The solid lines are the expected slope and magnitude of the scaling locus in the surface region for all three lower dimensionality crossovers. The dashed lines are drawn with the edge specific-heat exponent and then have their magnitude adjusted to match the 1D and 0D data.

Figure 45

(Color online) The equivalent plot to Fig. 44 for $T<T_{\lambda }$. The solid and dashed lines are drawn on the 0D data with slopes appropriate for the surface and edge specific-heat regions.

Figure 46

(Color online) ${}^4\mathrm{He}$ confined to arrays of $(1\mathrm{\mu }\mathrm{m}{)}^3$ and $(2\mathrm{\mu }\mathrm{m}{)}^3$ boxes. The shift of the maximum in the specific heat from $T_{\lambda }$ is qualitatively as one expects; the smaller confinement has a larger shift. However, the overall magnitude of the data does not behave as expected since the two sets show little change as function of confinement.

Figure 47

(Color online) Scaling plot of the 0D data below $T_{\lambda }$ according to Eq. (5). The lack of scaling is not surprising since the specific-heat data overlap and show little change as function of confinement size.

Figure 48

Shift in the temperature of the specific-heat maximum as function of $L$ for three crossover dimensions. The data for a given dimensionality crossover fall on lines with the expected slope of $-\nu$.

Figure 49

The magnitude of $C_{\max }$ vs $L$ for three crossover dimensions. Fits of the 2D and 1D data to the power law $L^{\alpha ∕\nu }$ are shown as solid lines though those data (the 2D fit is from the wider range of data seen in Fig. 23). The 0D data are clearly inconsistent with this.

Figure 50

The cell design used to study 2D to 1D crossover by 50. Shallow channels of fixed width radially span the inner oxide ring that separates the two $0.3189\text{−}\mathrm{\mu }\mathrm{m}$ planar regions seen in (a). One can relate the resonant frequency of helium oscillated between the two planar regions to the superfluid fraction of the helium within the channels. An AFM image of a channel section is seen in (b). Multiple cells were built where the channel height was the same but the width was varied.

Figure 51

Superfluid density for helium confined to channels of two different widths, $19$ and $10\mathrm{\mu }\mathrm{m}$, measured by 50. The height and length of both channels are 10 and $2\mathrm{mm}$, respectively. This geometry represents a dimensionality crossover from two dimensions to one dimension.

Figure 52

The shift in the superfluid transition temperature of helium confined in finite-width channels (filled circles) relative to a channel of infinite width. Here the height and length of the channels are $10\mathrm{nm}$ and $2\mathrm{mm}$. Also shown (open circles) is the expected temperature shift based upon the 2D shift equation of the BKT theory [Eq. (58)] and the power-law shift of Sobnack-Kusmartsev [Eq. (59)], upper dashed line. The filled box is the measured shift of the superfluid transition temperature for filling channels used to fill 0D boxes with helium.

Figure 53

The measured specific-heat critical exponent $\alpha$ (a) and the ratio of the specific-heat amplitudes above and below the superfluid transition $A∕A^{\prime }$ (b) vs $T_{\lambda }(P_{\mathrm{sat}},x=0)-T_{\lambda }(P,x)$. The symbols are defined in the opening paragraph of Sec. 9A. The solid line is the theoretical result of 31.

Figure 54

(Color online) Data for the specific heat measured at constant pressure and concentration with two planar confinements that differ by a factor of 20 in height. From 123.

Figure 55

(Color online) The specific-heat data shown in Fig. 54 above $T_{\lambda }$ cast in scaling form according to Eq. (60). The mixtures with $x>0$ show good collapse amongst themselves but define a locus that is dramatically different from that defined by the $x=0$ data. The values of $\alpha$ and $A$ in the scaling function come from fitting bulk data at each concentration. From 123.

Figure 56

(Color online) The same data as Fig. 55 except the value of $\alpha$ used in the scaling function is taken to be that from the $x=0$ data. This produces a reasonable collapse of all the data but is a less consistent analysis. From 123.

Figure 57

(Color online) Scaling of the data from Fig. 54 below $T_{\lambda }$. This plot is analogous to Fig. 55. The mixture data with $x>0$ stand apart from the $x=0$ data.

Figure 58

Data above $T_{\lambda }$ for mixtures confined to 1D channels scaled according to Eq. (60) (161). As seen in the planar data, the mixtures with $x>0$ define a locus that differs from the one defined by the $x=0$ data.

Figure 59

The equivalent plot to Fig. 58 below $T_{\lambda }$. The mixtures data for $x>0$ define a separate locus from the $x=0$ data.

Figure 60

The superfluid fraction of both pure ${}^4\mathrm{He}$ and mixtures of ${}^2\mathrm{He}\text{−}{}^4\mathrm{He}$confined to a 48.3-nm-high and $3\text{−}\mathrm{\mu }\mathrm{m}$-wide channel. The solid line is the superfluid fraction for bulk pure ${}^4\mathrm{He}$. The bulk data for the $x>0$ mixtures have been omitted for clarity. From 121.

Figure 61

The scaling of the superfluid fraction of helium mixtures confined to a 48.3-nm planar geometry according to Eq. (62). The solid line is the field-theory calculation of 184. Adapted from 121.

Figure 62

The thermal resistivity near $T_{\lambda }\left(P\right)$ of helium confined to $1\text{−}\mathrm{\mu }\mathrm{m}$-diameter glass capillaries as function of pressure. The data are as follows: open circles, SVP; solid circles, 6.95 bars; open squares, 11.25 bars; solid squares, 14.73 bars; open triangles, 22.31 bars; solid triangles, 28.00 bars. From 166.

Figure 63

The thermal resistivity data shown in Fig. 62 below $T_{\lambda }$ scaled according to Eq. (63). For $X>-2$, the data show a collapse within their precision. Below this, they show a systematic opening with pressure. The trend with pressure is more easily seen in the inset. From 166.