Thermal Conductivity of Liquid ${}^3\mathrm{H}\mathrm{e}$ in Aerogel: A Gapless Superfluid

We have measured the thermal conductivity of liquid ${}^3\mathrm{H}\mathrm{e}$ in 98% aerogel at ultralow temperatures. Aerogel introduces disorder on a scale comparable to the superfluid coherence length. At low pressures the liquid in the aerogel shows normal-state behavior with conductivity linear in temperature. At pressures above $\sim 6\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{s}$ the onset of superfluidity suppresses the conductivity and the thermal conductivity again tends towards linear behavior in the very low temperature limit, providing strong evidence that here the liquid ${}^3\mathrm{H}\mathrm{e}$ in the aerogel is behaving as a gapless superfluid.

Figure 1

A thermal conductance measurement at 7.6 bars. Temperatures $T_{\mathrm{b}\mathrm{o}\mathrm{x}}$ and $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ are plotted against time. As a long 2.14 pW pulse of heat is applied to the heater wire, $T_{\mathrm{b}\mathrm{o}\mathrm{x}}$ rises, while $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ remains unaffected. Inset A shows the experimental arrangement and inset B the assumed thermal resistance chain.

Figure 2

The measured thermal conductance for various pressures with the linear temperature dependence divided out. The dotted line represents the expected fluid conductance when the liquid in the aerogel is normal. The solid line corresponds to the dashed lines of Fig. 3 but includes the effect of the boundary resistance causing the fall at the lowest temperatures (see text). Above 7 bars the data fall below the normal state value as the liquid in the aerogel becomes superfluid.

Figure 3

The ratio, ${\kappa }_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}/{\kappa }_n$, representing the conductivity as a fraction of the normal state value, corrected for the boundary resistance. Dashed lines are guides for the eye. For $P<6\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{s}$, ${\kappa }_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}/{\kappa }_n$ is unity over the whole range, indicating normal state behavior. At higher pressures the data fall below the normal state value as the liquid in the aerogel becomes superfluid. At low temperatures the conductivity ratio is clearly tending to a positive ${\kappa }_{\mathrm{a}\mathrm{e}\mathrm{r}\mathrm{o}}/{\kappa }_n$ at $T=0$, indicating gapless behavior. The errors indicated on the lowest temperature points, representing 20% of the boundary resistance, are shown only to indicate the relative magnitude of this correction. The jump in the 7 bars data marks the A-B transition; see text.

Figure 4

The estimated low temperature limit for the conductivity ratio as a function of pressure.