Anomalous Quasiparticle Reflection from the Surface of a ${}^3\mathrm{He}\text{−}{}^4\mathrm{He}$ Dilute Solution

A free surface of a dilute ${}^3\mathrm{He}\text{−}{}^4\mathrm{He}$ liquid mixture is a unique system where two Fermi liquids with distinct dimensions coexist: a three-dimensional (3D) ${}^3\mathrm{He}$ Fermi liquid in the bulk and a two-dimensional (2D) ${}^3\mathrm{He}$ Fermi liquid at the surface. To investigate a novel effect generated by the interaction between the two Fermi liquids, the mobility of a Wigner crystal of electrons formed on the free surface of the mixture is studied. An anomalous enhancement of the mobility, compared with the case where the 3D and 2D systems do not interact with each other, is observed. The enhancement is explained by the nontrivial reflection of 3D quasiparticles from the surface covered with the 2D ${}^3\mathrm{He}$ system.

Figure 1

(a) ${}^3\mathrm{He}\text{−}{}^4\mathrm{He}$ mixture and WC formed on a free surface. A 2D ${}^3\mathrm{He}$ layer is formed at the free surface of the mixture as a result of the larger zero-point motion of ${}^3\mathrm{He}$ than that of ${}^4\mathrm{He}$ [11, 12]. Electrons are trapped about 10 nm above the surface. At low temperatures, the electrons form a WC dressed with a DL. The lattice constant of the WC is $a=0.93\mu \mathrm{m}$ for our electron density of $n_e\sim 1.35\times {10}^{12}{\mathrm{m}}^{-2}$, while the depth of the DL is only $\delta \sim 0.1\AA$. (b) Areal density of the 2D ${}^3\mathrm{He}$ layer $n_s$ (in the unit of atomic layers with monolayer density $6.4\times {10}^{18}{\mathrm{m}}^{-2}$) as a function of $x_3$ at $T=0$. This graph is based on the analysis by Guo et al. of their surface tension data [8] (see Supplemental Material for details [14]). The surface is covered by a monolayer of ${}^3\mathrm{He}$ even at a very small $x_3$ ($\sim 200\mathrm{ppb}$). $n_s$ diverges approaching the saturation concentration ($x_3^s=6.7\%$). (c) $T_F^{2\mathrm{D}}$ and $T_F^{3\mathrm{D}}$ as a function of $x_3$ (see Supplemental Material [14]).

Figure 2

Mobility of the WC as a function of $T$ for different ${}^3\mathrm{He}$ concentrations. $T_m$ indicates the transition temperature to the WC phase.

Figure 3

Experimental mobility of the WC compared with the theoretical calculations. The experimental data are the same as those shown in Fig. 2. Theoretical mobilities limited by the viscosity ($V$), specular reflection ($S$), accommodation process ($A$), partial accommodation process (PA), and ripplon scattering in the electron gas regime ($R_{\mathrm{gas}}$) are shown for $n_e=1.35\times {10}^{12}{\mathrm{m}}^{-2}$ and $E_{\perp }=2.05\times {10}^4\mathrm{V}/\mathrm{m}$. (a) $x_3=1.0\%$, (b) 2.1%, (c) 4.9%, and (d) 6.1%. An error in the theoretical curves associated with the deviation of $n_e$ and $E_{\perp }$ used in the calculation from the actual values is estimated to be 3% at most. At $x_3=1.0\%$, the system is in the ballistic regime at temperatures just below $T_m$ because of the long mean-free path, and therefore the viscous regime is not observed.

Figure 4

Schematic pictures of microscopic reflection processes of a ${}^3\mathrm{He}$ QP at the surface, seen in the reference frame moving horizontally together with the DL. (a) Specular reflection. The ${}^3\mathrm{He}$ QP is reflected with an angle equal to the incident angle. (b) Conventional diffusive reflection. Small arrows indicate the probability of QP reflection in a given direction. Red arrow indicates the averaged direction of reflection. After reflection, ${}^3\mathrm{He}$ QPs are in thermal equilibrium with the moving surface. (c) Accommodated diffusive reflection. Reflected QPs are in thermal equilibrium with the 2D ${}^3\mathrm{He}$ layer which is not moving horizontally with the DL. The drag force in this process is by a factor less than that of specular reflection (see the text).

Figure 5

Accommodation ratio $r_a$ as a function of ${}^3\mathrm{He}$ concentration.