Anomalous Quasiparticle Reflection from the Surface of a $^3$He-$^4$He Dilute Solution

TL;DR

This study investigates how interactions between 3D and 2D $^3$He Fermi liquids at the surface of a dilute $^3$He-$^4$He mixture lead to anomalous quasiparticle reflection, affecting electron mobility on the surface.

Contribution

It reveals a novel quasiparticle reflection mechanism caused by the interaction between 3D and 2D $^3$He Fermi liquids at the liquid surface.

Findings

Observed anomalous enhancement of electron mobility.

Attributed the enhancement to non-trivial quasiparticle reflection.

Demonstrated interaction effects between 3D and 2D Fermi liquids.

Abstract

A free surface of a dilute $^3$He-$^4$He liquid mixture is a unique system where two Fermi liquids with distinct dimensions coexist: a three-dimensional (3D) $^3$He Fermi liquid in bulk and a two-dimensional (2D) $^3$He Fermi liquid at the surface. To investigate a novel effect generated by the interaction between the two Fermi liquids, 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 non-trivial reflection of 3D quasiparticles from the surface covered with the 2D $^3$He system.

Equations

$$V_m=V_4\left[1+\alpha \left(x_3,T\right)x_3\right]$$

$$\alpha =(0.284\pm 0.005)-\left[(0.032\pm 0.003)\right]T.$$

$$\ln (\eta T^2)=\sum _{m=0}^4\sum _{n=0}^1{A}_{mn}(\ln T{)}^m(\ln x_3{)}^n,$$

$$\ln (\eta T^2)=f(T-T_0)\ln ({\eta }^H{T}^2)+[1-f(T-T_0)]\ln ({\eta }^L{T}^2),$$

$$f(T)=\frac{1}{1+\exp (-T/\alpha )}$$

$$S=-{\left(\partial {\alpha }_m/\partial T\right)}_{{\mu }_3}.$$

$$S=-\left(d{\alpha }_4/dT\right)+bT.$$

$$\Delta \alpha (T)\equiv {\alpha }_4(T)-{\alpha }_m(T)=a+b{T}^2/2,$$

$$\sqrt{a}=-B\sqrt{\frac{({\mu }_3-\alpha {)}^2}{A^2}-1}+\beta$$

$$b=C+\frac{D}{\sqrt{F-{\mu }_3}}$$

$${\mu }_3(T,x_3)\simeq {\mu }_3(0,x_3)+{\mu }_F(T,x_3),$$

$${\alpha }_4(T)=0.387-0.0065T^{7/3}(dyne/cm),$$

$$n_s=-{\frac{\partial {\alpha }_m}{\partial {\mu }_3}|}_T,$$

$${\mathbf{F}}_{\mathrm{D}}+i{m}_e\omega {\nu }_r\mathbf{u}-e{\mathbf{E}}_{\parallel }=-m_e{\omega }^2\mathbf{u},$$

$${\mathbf{F}}_{\mathrm{D}}=m_e\omega [w(\omega )+i\nu (\omega )]\mathbf{u}.$$

$$Z(\omega )=\left[1+\frac{w(\omega )}{\omega }\right]+i\frac{\nu (\omega )}{\omega },$$

$$-e{\mathbf{E}}_{\parallel }=-m_e{\omega }^2\left[\mathrm{Re}Z(\omega )+i\left(\mathrm{Im}Z(\omega )+\frac{{\nu }_r}{\omega }\right)\right]\mathbf{u}.$$

$$\mu =\frac{e}{m_e}\frac{\omega \mathrm{Im}Z+{\nu }_r}{{\left(\omega \mathrm{Re}Z\right)}^2+{\left(\omega \mathrm{Im}Z+{\nu }_r\right)}^2}.$$

$${\mathbf{F}}_{\mathrm{D}}=-i\sum _{\mathbf{g}}\mathbf{g}{\tilde{U}}_g{\xi }_{\mathbf{g}}{e}^{i\mathbf{g}\cdot \mathbf{u}(t)}.$$

$$\rho \frac{\partial \mathbf{v}}{\partial t}+\nabla p=\eta \Delta \mathbf{v},$$

$${\sigma }_{xz}$$

$${\sigma }_{yz}$$

$${\sigma }_{zz}$$

$$P_{e,\mathbf{g}}$$

$${\xi }_{\mathbf{g}}^{(1)}=-\frac{g{P}_{e,\mathbf{g}}^{(1)}}{\rho {\Delta }_g},$$

$${\Delta }_g$$

$${\gamma }_g$$

$${\delta }_g$$

$$\varphi (\kappa )$$

$$\chi (\kappa )$$

$$Z(\omega )=1+\frac{\rho }{2m_e{n}_e}\sum _{\mathbf{g}}g{\left|{\xi }_g^0\right|}^2{\left(\frac{{\omega }_{r,g}}{\omega }\right)}^2\frac{{\omega }^2+{\delta }_g^2+i2\omega {\gamma }_g}{{\Delta }_g},$$

$${\ddot{\xi }}_{\mathbf{g}}+2{\gamma }_g{\dot{\xi }}_{\mathbf{g}}+{\omega }_{r,g}^2{\xi }_{\mathbf{g}}=-\frac{{\tilde{U}}_g{n}_{e}g}{\rho }{e}^{-i\mathbf{g}\cdot \mathbf{u}(t)},$$

$$Z(\omega )=1+\sum _{\mathbf{g}}\frac{n_e{\tilde{U}}_g^{2}g}{m_e\rho {\omega }_{r,g}^2}{g}_x^2\frac{\left({\omega }_{r,g}^2-{\omega }^2-4{\gamma }_g^2\right)+2i{\gamma }_g{\omega }_{r,g}^2/\omega }{{\left({\omega }_{r,g}^2-{\omega }^2\right)}^2+4{\gamma }_g^2{\omega }^2},$$

Full text

Supplementary Material:

Anomalous Quasiparticle Reflection

from the Surface of a 3He-4He Dilute Solution

Hiroki Ikegami

hikegami@riken.jp

RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

  

Kitak Kim

Department of Physics, KAIST, Daejeon 34141, Republic of Korea

  

Daisuke Sato

RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

  

Kimitoshi Kono

RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

  

Hyoungsoon Choi

h.choi@kaist.ac.kr

Department of Physics, KAIST, Daejeon 34141, Republic of Korea

  

Yuriy P. Monarkha

Institute for Low Temperature Physics and Engineering, 47 Nauky Avenue, Kharkov 61103, Ukraine

I Interpolation of Physical Quantities of 3He-4He Mixture

In the main text, we use interpolated values of physical quantities of a 3He-4He mixture to analyze the experimental data. In this section, we describe how we interpolate these quantities.

I.1 Molar Volume

As the molar volume of the dilute mixture $V_{m}$, we adopt the results of Edwards et al. Edwards et al. (1969):

[TABLE]

with

[TABLE]

Here $V_{4}=$ 27.579 cc/mol is the molar volume of pure 4He Kerr and Taylor (1964); Tanaka et al. (2000), $x_{3}$ is the concentration of 3He, and $T$ is the temperature in the unit of K. This equation is valid for 3He concentrations less than 6.6 $\%$ and temperatures below 0.6 K. Our estimates of the dielectric constant and the density of the mixture are also based on the molar volume obtained from this equation.

For the calculations of the theoretical mobility described in the main text, we neglect the temperature dependence of $V_{m}$ because $V_{m}$ changes by less than 0.06% from 0 to 0.3 K at a fixed $x_{3}$.

I.2 Viscosity

The viscosity of the dilute mixture has been measured by Kuenhold et al. (for $x_{3}=$ 0.5, 1.3, 2.7, 5.0, and 7.0%) Kuenhold et al. (1972) and König and Pobell ($x_{3}=$ 0.98 and 6.1%) König and Pobell (1994). We interpolate these experimental data in the temperature range of 10$-$300 mK as a function of $T$ and $x_{3}$ according to the following procedure. First, we divide the temperature range into two regions: above and below $T_{0}$, which varies between 50 and 65 mK depending on $x_{3}$. Within each temperature region, we fit the viscosity $\eta$ to the following function of $T$ and $x_{3}$:

[TABLE]

yielding two sets of fitting parameters, $A_{mn}^{H}$ and $A_{mn}^{L}$, for the high- and low-temperature regions, respectively.

The two fitted curves in the two temperature regions are smoothly connected by imposing the relation:

[TABLE]

where $\eta^{H}$ and $\eta^{L}$ are the viscosity above and below $T_{0}$, respectively. We use the function $f(T)$ given by

[TABLE]

with $\alpha=$ 0.003 K. Figure S1(a) shows the results of our fitting. The interpolated values of viscosity for the values of $x_{3}$ used in our measurements are shown in Fig. S1(b). The error in our estimation of the viscosity is less than 7%.

I.3 Surface Tension

The available experimental data on the surface tension of the dilute mixture are very limited, particularly in the range of $x_{3}=$ 0.5$-$6.1%. The data measured with the finest steps are those reported by Guo et al. ($x_{3}=$ 3.2$\times$10*-3*, 5.42$\times$10*-2*, 0.56, 6.16, 9.64, and 22%) Guo et al. (1971). We interpolate their data using the following phenomenological model. We assume that the surface layer of 3He is in thermal equilibrium with 3He atoms in the bulk having a chemical potential $\mu_{3}$. According to the thermodynamic relation, the surface entropy $S$ per unit area is related to the surface tension $\alpha_{m}$ as Landau and Lifshitz (1980)

[TABLE]

Here the surface entropy is composed of two parts, $S=S_{3}+S_{4}$, where $S_{3}$ is the entropy of 3He on the surface and $S_{4}$ is that of pure 4He. $S_{4}$ and the surface tension of pure 4He $\alpha_{4}$ are related by $S_{4}=-d\alpha_{4}/dT$. If 3He in the surface layer is Fermi degenerate, the entropy should be linear in temperature, $S_{3}=bT$, leading to

[TABLE]

From Eqs. (6) and (7), we derive

[TABLE]

where $a$ and $b$ are functions of $\mu_{3}$. We fit the experimental values of $a$ and $b$ obtained by Guo et al. Guo et al. (1971) as a function of $\mu_{3}$ to the following empirical forms:

[TABLE]

with fitting parameters $A$, $B$, $\alpha$, and $\beta$, and

[TABLE]

with fitting parameters $C$, $D$, and $F$. The results of the fitting are shown in Fig. S2.

Next, we evaluate the dependence of $\mu_{3}$ on $x_{3}$ and $T$. The dependence is obtained by approximating $\mu_{3}$ as

[TABLE]

where $\mu_{F}(T,x_{3})$ is the chemical potential for a 3D non-interacting Fermi gas with effective mass $m_{\rm 3D}=$ 2.28$m_{3}$ Sherlock and Edwards (1973) ($m_{3}$ is the mass of a 3He atom). For $\mu_{3}(0,x_{3})$, we use the experimental values obtained by Seligmann et al. Seligmann et al. (1969). From Eqs. (8)$-$(11) and using the temperature dependence of the experimental surface tension of pure 4He Guo et al. (1971); Atkins (1953),

[TABLE]

($T$ is in the unit of K), we obtain the interpolated $\alpha_{m}(T)$ as a function of $x_{3}$ and $T$. The results of the interpolated $\alpha_{m}(T)$ at several fixed $x_{3}$ are shown in Fig. S3. The error in $\alpha_{m}(T)$ is estimated to be less than 2%.

II Thickness and Fermi Energy of Surface 3He Layer

Here we consider how the areal density of the 2D 3He layer $n_{s}$ evolves with increasing 3He concentration in the bulk $x_{3}$. According to the thermodynamic relation, $n_{s}$ is described in terms of the surface tension $\alpha_{m}$ by

[TABLE]

where $\mu_{3}$ is the chemical potential of 3He. (Here we assume that the interaction between 3He in the surface layer and 3He in the bulk mixture does not depend on $\mu_{3}$.) Using the dependence of $\alpha_{m}$ on $\mu_{3}$ obtained by Guo et al. [Eq. (8)] Guo et al. (1971), we calculate $n_{s}(\mu_{3})$ from Eq. (13). The chemical potential $\mu_{3}$ is approximately related to $x_{3}$ and $T$ by Eq. (11). Therefore, we obtain $n_{s}$ as a function of $x_{3}$ and $T$. We show $n_{s}$ at $T=0$ as a function of $x_{3}$ in Fig. 1b of the main text. It increases with increasing $x_{3}$ and diverges as it approaches the saturation concentration of 3He.

Next, we consider how the Fermi energy of the 2D 3He layer $T_{F}^{\rm 2D}$ evolves with increasing $x_{3}$. We consider that a potential well present near the surface generates bound states of 3He with eigenenergies of $E_{\bot}^{n}$ in the direction normal to the surface ($n=$ 0, 1, 2, $\cdots$, $l$). The bound 3He can move freely with a kinetic energy $E_{k}^{n}$ in the plane parallel to the surface. In this situation, the energy of the lowest bound state $E_{\bot}^{0}$ corresponds to the binding energy $E_{b}$. Because the surface 3He layer is in equilibrium with bulk 3He in the mixture, the bound states with energies less than the Fermi energy of bulk 3He $E_{F}^{\rm 3D}$ are occupied at $T=$ 0. This leads to the relation $E_{F}^{\rm 2D}+E_{b}=E_{F}^{\rm 3D}$, where the Fermi energy of the 2D 3He layer $E_{F}^{\rm 2D}$ corresponds to the highest kinetic energy of the bound states of $n=$ 0. For calculating $E_{F}^{\rm 2D}$ and $E_{F}^{\rm 3D}$ shown in Fig. 1c in the main text, we use the experimental value of Ref. Seligmann et al. (1969) for $E_{F}^{\rm 3D}$ and the experimental value of $E_{b}=-$ 2.28 K Eckardt et al. (1974).

III Mobility on pure 4He in Ripplon Scattering Regime

The magnitude of the mobility presented in the main text is calibrated by multiplying by a factor about unity (0.94$-$1.14) so that the mobility at temperatures above $T_{m}$ agrees with the theoretical mobility of the highly correlated electrons in the ripplon scattering regime. The justification for this calibration process is based on the fact that our mobility measured for pure 4He at temperatures above $T_{m}$ agrees with the theoretical mobility in both the ripplon [Eq. (3.70) in Ref. Monarkha and Kono (2004)] and gas scattering Saitoh (1977) regimes without correction of the magnitude of the mobility as shown in Fig. S4. Note that the mobility for pure 4He measured by Mehrotra et al. at similar electron densities deviates from the theoretical mobility in the ripplon scattering regime Mehrotra et al. (1984). The origin of the discrepancy between our and their data is unknown.

IV Theoretical Calculation of Mobility

Here we describe in detail the theoretical mobility of the WC at a nonzero frequency in the viscous and ballistic regimes presented in the main text. To calculate the mobility, we consider the motion of a WC in a spatially uniform ac electric field ${\bf{E}}_{\parallel}e^{-i\omega t}$ by noting that the transport properties of the WC dressed with a dimple lattice (DL) are significantly modified from those of bare electrons by the reaction of the helium surface. The ac electric field induces uniform displacement of the electrons ${\bf u}e^{-i\omega t}$ from their lattice sites, and the displaced electrons experience a reactive force ${\bf{F}}_{\rm D}e^{-i\omega t}$ from the helium surface. In this situation, the equation of motion of an electron is described as

[TABLE]

where direct scattering by thermal ripplons with a rate of $\nu_{r}$ is included. ($e$ is the elementary charge and $m_{e}$ is the electron mass.) For a small displacement, the reactive force is linear in the velocity $\dot{\bf u}$ and therefore generally expressed in the form of

[TABLE]

Using the dimensionless response function defined as Monarkha and Kono (1997); Monarkha and Syvokon (2012); Monarkha and Kono (2004)

[TABLE]

the equation of motion is rewritten as

[TABLE]

This equation indicates that the real part ${\mathop{\rm Re}\nolimits}Z(\omega)$ represents the change in the inertia of an electron and the imaginary part $\omega{\mathop{\rm Im}\nolimits}Z(\omega)$ describes the momentum relaxation of the electron due to the coupling to the DL. From Eq. (17), the mobility of a WC is expressed as Monarkha and Kono (2004, 1997); Monarkha and Syvokon (2012)

[TABLE]

This equation suggests that the problem of calculating $\mu$ reduces to evaluating $Z(\omega)$. Note that ${\mathop{\rm Re}\nolimits}Z(\omega)=0$ at zero frequency, and thus $\mu=\frac{e}{m_{e}}\frac{1}{{\omega{\mathop{\rm Im}\nolimits}Z+\nu_{r}}}$.

The reactive force from the helium surface arises from the coupling of the electrons to the surface and the dynamics of the surface:

[TABLE]

Here $\xi_{\bf{g}}$ is the Fourier component of the surface profile $\xi({\bf r})$ at the reciprocal lattice vector ${\bf{g}}$, $g=\left|\bf{g}\right|$, and $\tilde{U}_{g}=U_{g}\exp(-W_{g})$ is the electron-ripplon coupling strength, where $U_{g}=e(E_{\bot}+E_{g})$ and $\exp(-W_{g})$ is the Debye$-$Waller factor. ($E_{\bot}$ is the pressing field and $E_{g}$ is the polarization field defined in Refs. Monarkha (1976); Monarkha and Kono (2004).) The Debye$-$Waller factor $\exp(-W_{g})$ can be evaluated in the manner described in Refs. Monarkha and Shikin (1983); Monarkha and Kono (2004).

The dynamics of the free surface (i.e., the dynamics of $\xi_{\bf{g}}$) is different in the viscous and ballistic regimes. Below, we describe the theoretical mobility in both regimes. We also give details of the direct scattering by thermally excited ripplons.

IV.1 Mobility in Viscous Regime

In this regime, the dynamics of the surface is determined by the Navier$-$Stokes equation for an incompressible fluid ($\nabla\cdot{\bf{v}}=0$) with viscosity $\eta$:

[TABLE]

($\rho$, ${\bf{v}}$, and $p$ are the density, the velocity field, and the pressure of the fluid, respectively) with the following boundary conditions at the free surface associated with the stress tensor $\sigma_{ij}$:

[TABLE]

where $\alpha$ is the surface tension and $P_{e}$ is the pressure from the electrons. (The $z$-axis is taken normal to the surface.) For $P_{e}$, we take the Fourier component and use the linear approximation

[TABLE]

Solving Eq. (20) with the boundary conditions of Eq. (21), we obtain $\xi$ for $P^{(1)}_{{e,\bf g}}$ as

[TABLE]

where

[TABLE]

$\kappa=\omega\rho/(\eta g^{2})$, and $\omega_{r,g}=\sqrt{\alpha/\rho}g^{3/2}$ is the ripplon frequency with wave number $g$. (Note that the Fourier components of $\xi({\bf r})$ in the static state $\xi_{g}^{(0)}=-\frac{gn_{e}\tilde{U}_{g}}{\rho\omega_{r,g}^{2}}$ are obtained from $P^{(0)}_{{e,\bf g}}$.) Substituting Eq. (23) into Eq. (19) and using Eqs. (15) and (16), we obtain Monarkha and Kono (1997); Monarkha and Syvokon (2012); Monarkha and Kono (2004)

[TABLE]

where $n_{e}$ is the density of electrons. To calculate the theoretical mobility in the viscous regime described in the main text, we first evaluate $Z(\omega)$ [Eq. (24)] using the interpolated values of $\rho$, $\alpha$, and $\eta$ of the mixture presented in Sec. I of this Supplementary Information, and then evaluate $\mu$ using Eq. (18). Note that at the zero-frequency limit, $\omega{\mathop{\rm Re}\nolimits}Z(\omega)=0$ and $\omega{\mathop{\rm Im}\nolimits}Z=\nu_{d}=\frac{\eta}{{m_{e}n_{e}}}\sum\limits_{\bf{g}}{g^{3}}\left|{\xi_{g}^{0}}\right|^{2}$, indicating $\mu$ to be proportional to $\eta^{-1}$.

IV.2 Mobility in Ballistic Regime

In this regime, the mobility of the WC is caused by the reflection of QPs. The reflection of QPs generates a drag force on the moving DL, and the drag force is transferred to the WC. At a nonzero frequency, we must also include the inertial term in the calculation of the mobility, which can be performed by using the response function $Z(\omega)$. The response function can be obtained from the dynamics of the free surface by noting that microscopic information of the reflection of QPs is incorporated in the damping of a capillary wave. We consider the dynamics of the free surface under the damping of a capillary wave, which is described by

[TABLE]

where $\gamma_{g}$ is the damping rate of a capillary wave with wave number $g$. Solving Eq. (25) for a small displacement $\bf{u}$ and using Eqs. (15), (16), and (19), we obtain Monarkha and Syvokon (2012); Monarkha and Kono (2004)

[TABLE]

where $g_{x}$ is the component of $\bf{g}$ parallel to ${\bf v}=\dot{\bf u}$. Note that $\gamma_{g}$ includes information on the microscopic nature of the reflection of a QP.

In the case of specular reflection, the damping rate $\gamma_{g}^{(s)}$ caused by the reflection of QPs is given by Monarkha and Kono (1997, 2005, 2004)

[TABLE]

where $\hbar$ is Planck’s constant and $f(\varepsilon)$ is the Fermi distribution function of 3He QPs with energy $\varepsilon=p^{2}/(2m_{\rm 3D}^{*})$ ($p$ is the momentum and $m_{\rm 3D}^{*}$ is the effective mass of a 3He QP). Note that at $T\ll T_{F}$, the mobility is independent of temperature $T$ [$\gamma_{g}^{(s)}=gp_{F}^{4}/(8\pi^{2}\hbar^{3}\rho)$] ($T_{F}$ is the Fermi temperature and $p_{F}$ is the Fermi momentum). From Eq. (27), we obtain $Z(\omega)$ using Eq. (26), with which we calculate the mobility using Eq. (18). Note that at $\omega\to 0$, $\omega{\mathop{\rm Re}}Z\to 0$ and

[TABLE]

reproducing the zero-frequency mobility given in Ref. Monarkha and Kono (1997), where

[TABLE]

In the case of the reflection associated with the complete accommodation of a QP into the surface layer, the damping rate is given by Monarkha and Kono (2006)

[TABLE]

The mobility in the accommodation process presented in the main text is calculated using Eqs. (26) and (30). Note that $\gamma_{g}^{(a)}(T)$ and $\gamma_{g}^{(s)}(T)$ have the same function of $T$ but are different by a factor of four. Thus, at zero frequency, the mobility limited by the accommodation process is four times higher than that in the case of specular reflection. At a nonzero frequency, the term $\omega{\mathop{\rm Re}}Z(\neq 0)$ contributes to the mobility, resulting in the difference in mobilities between the specular and accommodation processes being less than a factor of four.

In the practical situation at the surface of the mixture, not all but a part of the 3He QPs with ratio $r_{a}$ are reflected by the accommodation process. In this partial accommodation process, the damping rate is described as

[TABLE]

where $r_{a}=0$ corresponds to specular reflection and $r_{a}=1$ to complete accommodation. In the main text, the experimental data are fitted to this partial accommodation model with the fitting parameter $r_{a}$.

IV.3 Contribution from Direct Scattering by Thermal Ripplons

In the theoretical mobility presented in the main text, we also include the contribution of the direct scattering by thermally excited ripplons. We evaluate the rate of collision with the ripplons $\nu_{r}$ using the Born approximation. In this approximation, $\nu_{r}$ is described as a simple form including the dynamical structure factor (DSF). For the DSF of the WC, we use the high-temperature approximation [Eq. (8.2) in Ref. Monarkha and Kono (2004)], where the form of the DSF is the same as that of a non-degenerate electron gas with the temperature replaced by the kinetic energy of electrons in the WC.