Anomalous Quasiparticle Reflection from the Surface of a ^3He-^4He Dilute Solution
Hiroki Ikegami, Kitak Kim, Daisuke Sato, Kimitoshi Kono, Hyoungsoon Choi, Yuriy P. Monarkha
aa r X i v : . [ c ond - m a t . o t h e r] N ov Anomalous Quasiparticle Reflection from the Surface of a He- He Dilute Solution
Hiroki Ikegami, ∗ Kitak Kim, Daisuke Sato, Kimitoshi Kono, Hyoungsoon Choi, † and Yuriy P. Monarkha The Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan Department of Physics, KAIST, Daejeon 34141, Republic of Korea Institute for Low Temperature Physics and Engineering, 47 Nauky Avenue, Kharkov 61103, Ukraine (Dated: July 18, 2018)A free surface of a dilute He- He liquid mixture is a unique system where two Fermi liquidswith distinct dimensions coexist: a three-dimensional (3D) He Fermi liquid in bulk and a two-dimensional (2D) He Fermi liquid at the surface. To investigate a novel effect generated by theinteraction between the two Fermi liquids, mobility of a Wigner crystal of electrons formed on thefree surface of the mixture is studied. An anomalous enhancement of the mobility, compared with thecase where the 3D and 2D systems do not interact with each other, is observed. The enhancementis explained by non-trivial reflection of 3D quasiparticles from the surface covered with the 2D Hesystem.
Dimensionality is one of the defining characteristicsthat govern physical properties of systems. Althougha lot of experimental and theoretical investigations haveclarified static and dynamic properties of quantum many-body problems, the studies so far have been mostlylimited to the cases with a well-defined single dimen-sion. Mixed-dimensional systems, in which two quantummany-body systems with distinct dimensions coexist, areexpected to exhibit nontrivial dynamics that never occurin either subsystem by itself. Such systems are found invarious fields, for example, electrons in noble metals [1]and unconventional materials [2, 3] (electrons inside andon the surface), and ultracold atoms [4–6], but in manycases, the two subsystems are not well understood or welldistinguished.The free surface of a dilute He- He mixture [7]is unique in this regard since well-characterized two-dimensional (2D) and three-dimensional (3D) fermionicsystems of identical particles, i.e., He, coexist [8, 9].When a He quasiparticle (QP) in the 3D system ap-proaches the 2D He layer formed at the free surface andinteracts with it, the interaction is expected to affect thereflection process and surface dynamics. In this letter,we demonstrate that novel QP reflection from the sur-face manifests itself as anomalous enhancement of themobility of a Wigner crystal (WC) of electrons trappedon the surface of the mixture.We consider the mixture with a free surface. The 3D He system is formed in bulk; He atoms, which are sol-uble in liquid He at concentrations of He x up to ∼ He [7]. At the surface, He atoms arebound to the surface to form a dense 2D layer [Fig. 1(a)][11, 12], showing the 2D Fermi liquid behavior [8, 9].With increasing x , the thickness of the 2D He layerincreases in the range of several atomic layers except at x ∼ ∼ Heand is almost unchanged with x . Therefore, the Fermitemperature T F is high ( ∼ x [Fig. 1(c)]. This is in contrast to the lowFermi temperature of the 3D He T F ( < He systems have been investigated as the prototypi-cal Landau’s Fermi liquid in each dimension [7–9] owingto their cleanliness, the mixed-dimensional system pre-sented in this work will serve as an ideal model systemto study the interplay between the two subsystems.Electrons on a free surface of liquid helium [14, 15] areoften used as a sensitive microscopic probe for investi-gating fundamental properties of elementary excitationsin quantum fluids. The electrons undergo a transitionto a WC at a certain low temperature, where the crys-tallization generates a commensurate deformation of thehelium surface called a dimple lattice (DL) [Fig. 1(a)][14, 15]. The emergence of the DL strengthens the cou-pling of the WC to the liquid, making the transport ofthe WC sensitive to properties of elementary excitationsin the liquid [16–19]. This feature has been utilized toreveal, particularly, the specular nature of QP reflectionfrom the surface in normal and superfluid He [17–19].In this study, we similarly measure the mobility of a WCover the He- He liquid mixture down to ∼
10 mK overa wide range of x (up to 6.1%) to investigate the natureof QP dynamics near the surface. (So far there has beenan experimental study of WC mobility only at x = 0.5%[20]).The mobility is measured with the Sommer − Tannertechnique [22] in the Corbino geometry at a vertical mag-netic field B of 380–1100 Gauss. The Corbino disk con-sists of two concentric electrodes 18.0 and 11.9 mm indiameter and is attached to the ceiling of the samplecell. A bottom circular electrode, which is located 3.0mm below the Corbino disk, is used to provide a verticalpressing electric field E ⊥ . The free surface is set at a mid-way between the Corbino disk and the bottom electrode,and electrons are deposited on it. The longitudinal con-ductivity σ xx is measured by applying an ac voltage V ac of frequency f = 214 kHz to the inner electrode of theCorbino disk and recording the induced current I out on He electron He(a)(b) (c) x He concentration (%) n s ( a t o m i c l a y e r) ~1 m m ~0.1 Å T F T F x He concentration (%) F e r m i ene r g y ( K ) FIG. 1. (a) He- He mixture and WC formed on a free sur-face. A 2D He layer is formed at the free surface of themixture as a result of the larger zero-point motion of Hethan that of He [11, 12]. Electrons are trapped about 10 nmabove the surface. At low temperatures, the electrons forma WC dressed with a DL. The lattice constant of the WCis a = 0.93 µ m for our electron density of n e ∼ × m − , while the depth of the DL is only δ ∼ He layer n s (in the unit of atomic layerswith monolayer density 6.4 × m − ) as a function of x at T = 0. This graph is based on the analysis by Guo et al. oftheir surface tension data [8] (see Supplementary Material fordetails [21]). The surface is covered by a monolayer of Heeven at a very small x ( ∼
200 ppb). n s diverges approachingthe saturation concentration ( x s = 6.7%). (c) T F and T F as a function of x . (see Supplementary Material [21]). the outer electrode. A small V ac (= 1 mV rms ) is used toavoid nonlinear effects. The mobility µ is deduced fromthe Drude relation σ xx = en e µ/ [1 + ( µB ) ], where n e isthe electron density. Mobility at each x is measured ata certain value of n e and E ⊥ in the range of n e = (1.33–1.40) × m − and E ⊥ = (2.02–2.08) × V/m. Weuse a rather high E ⊥ to avoid the WC from decouplingfrom the DL easily by the ac drive. The magnitudes ofthe mobility are calibrated by multiplying by a factor ofabout unity (0.94–1.14) so that the mobility agrees withthe theoretical mobility of highly correlated electrons inthe ripplon scattering regime [15] above T m (see Supple-mentary Material [21]). The mixture is cooled to ∼ He x is determined fromthe amounts of He and He introduced into the cell. Wealso monitor x via the dielectric constant of the mixtureusing a parallel plate capacitor immersed in the liquid.Figure 2 shows the mobility µ as a function of temper-ature T at different x . With decreasing T , µ exhibits
10 100 M ob ili t y ( m / V s ) Temperature (mK) 300 T m FIG. 2. Mobility of the WC as a function of T for different He concentrations. T m indicates the transition temperatureto the WC phase. a sudden drop at the transition temperature to the WC, T m ∼
260 mK, due to the formation of a DL, followed bya further reduction at lower T . In the WC phase, two fea-tures are found: the mobility at each x asymptoticallyapproaches a constant value below several tens of mK,and µ is significantly suppressed when x is increased.Right below T m , µ is limited by the viscosity of thebulk mixture η . In this regime, µ is determined by theviscous drag force acting on the DL moving together withthe WC [15, 19, 23]. We evaluate the theoretical mobil-ity at our measurement frequency (214 kHz) using ex-perimentally known values of viscosity η [24–26], density ρ [27], and surface tension α of the mixture [8] (for thederivation of the theoretical mobility at a finite frequency,see Supplementary Material [21]; in the theoretical mo-bility, the contribution of the electron scattering by ther-mally excited ripplons, which is not negligible at a highmobility, is also included). As shown in Fig. 3, the theo-retical mobility is in excellent agreement with the experi-mental data at high x and high T (except for x = 1.0%,where the mean-free-path of bulk QPs is larger than theperiod of the DL) without any adjustable parameters.This agreement suggests that there is no contributionfrom the 2D He layer in the viscous regime.The experimental mobility deviates from the viscosity-limited one and approaches a temperature-independentvalue at low temperatures. These observations suggest acrossover from the viscous regime to the ballistic regimewith decreasing T as the mean free path l q of a He QPin the bulk mixture becomes longer than the lattice con-stant of the WC which is about 1 µ m (in the Fermi liquid, l q increases with decreasing T as l q ∝ T − ). The mobilityis then limited by friction caused by bulk QP reflectionfrom the moving DL. In the case of pure He, a He QP isdemonstrated to be reflected specularly [17–19]. For the
10 100 30010
10 100 30010
10 100 30010
10 100 30010 (a) 1.0% (b) 2.1% M ob ili t y ( m / V s ) Temperature (mK) (c) 4.9% (d) 6.1% T m T m T m T m r a =0.29 VSAPA R gas R gas VSAPA r a =0.43 r a =0.45 r a =0.59VSAPA R gas VSAPA R gas
FIG. 3. Experimental mobility of the WC compared withthe theoretical calculations. The experimental data are thesame as those shown in Fig. 2. Theoretical mobilities limitedby the viscosity (V), specular reflection (S), accommodationprocess (A), partial accommodation process (PA), and ripplonscattering in the electron gas regime (R gas ) are shown for n e = 1.35 × m − and E ⊥ = 2.05 × V/m. (a) x =1.0, (b) 2.1, (c) 4.9, and (d) 6.1%. An error in the theoreticalcurves associated with the deviation of n e and E ⊥ used inthe calculation from the actual values is estimated to be 3%at most. At x = 1.0%, the system is in the ballistic regimeat temperatures just below T m because of the long mean freepath, and therefore the viscous regime is not observed. mixture, a similar process is shown in Fig. 4(a); however,it is not trivial how a He QP is reflected from a surfaceelement dS in the presence of the 2D He layer.As a reflection law, specular and diffusive reflectionshave been conventionally considered. For the specularreflection of a He QP, the incident and reflection anglesare equal [Fig. 4(a)]. In this case, there is no momentumtransfer in the tangential direction to the surface element dS . For the conventional diffusive reflection [Fig. 4(b)],which generally occurs at a solid surface with microscopicirregularities, reflected He QPs are in thermal equilib-rium with the moving surface, and therefore their mo-mentum distribution is significantly different from thatof bulk QPs, resulting in a large average momentum ex-change in the direction of motion. Drag forces dF D actingon a surface element dS thus differ by orders of magni-tude for the two cases: dF D ∼ n p F V ( δ/a ) dS (1)for the specular reflection and dF D ∼ n p F V dS (2) (a) Specular Reflection (b)
Conventional (c)
Accommodated V V V
2D Surface He He- He Mixture He QuasiparticleDiffusive ReflectionDiffusive Reflection
FIG. 4. Schematic pictures of microscopic reflection processesof a He QP at the surface, seen in the reference frame mov-ing horizontally together with the DL. (a) Specular reflection.The He QP is reflected with an angle equal to the incidentangle. (b) Conventional diffusive reflection. Small arrowsindicate the probability of QP reflection in a given direction.Red arrow indicates the averaged direction of reflection. Afterreflection, He QPs are in thermal equilibrium with the mov-ing surface. (c) Accommodated diffusive reflection. ReflectedQPs are in thermal equilibrium with the 2D He layer whichis not moving horizontally with the DL. The drag force in thisprocess is by a factor less than that of specular reflection (seetext). for the conventional diffusive reflection [28], where V isthe horizontal velocity of the surface profile, n and p F are the number density and the Fermi momentum of Hein the mixture, δ ∼ a (= 0.93 µ m) is the lattice constant of the WC. Thesesuggest that (i) for specular reflection, dF D is by ( δ/a ) ( ∼ − ) smaller than the case of conventional diffusivereflection and (ii) when δ → dF D → dF D is independent of the dimple depthfor conventional diffusive reflection, which means that itis not applicable for the description of the drag force ofthe DL [28].As shown in Fig. 3, the theoretical mobility of thespecular reflection model evaluated at our measurementfrequency (black dashed line) is in qualitative agreementwith the experimental mobility (see Supplementary Ma-terial for the derivation of the mobility at a nonzero fre-quency [21]). However, the experimental mobility is stillhigher than that given by this model (as noted above,the conventional diffusive reflection results in even muchlower WC mobility). This noteworthy result means thatconventional specular and diffusive reflection laws can-not explain observed mobility data of the long mean-free-path regime.As an unusual QP reflection model, Monarkha andKono proposed a process involving the accommodation ofan incoming He QP with the surface layer of He atoms[28], which is shown schematically in Fig. 4(c). Notethat Fig. 4 is drawn for an observer moving horizontallywith the DL. The key features of this process are (1) themomentum distribution of reflected QPs is described bythe Fermi function f ( ε β,p ) in the reference frame boundto the element dS ′ of the 2D He layer (here ε β,p is theenergy of a QP with a momentum p and spin β ), i.e.reflected QPs are in thermal equilibrium with the Helayer, and (2) the 2D He layer does not move horizon-tally together with the DL but just oscillates verticallywith the amplitude of δ (this is the reason for a primesymbol in dS ′ ). Because of (1), this process representsdiffusive reflection; when the DL is stationary the mo-mentum distribution of reflected QPs is the same as thatof the conventional diffusive model. However, because of(2), the momentum distribution of reflected QPs is de-scribed by the equilibrium function in the frame which isnot moving horizontally with a surface relief. Therefore,in the reference frame fixed to the DL, as qualitativelydrawn in Fig. 4(c), the momentum distribution functionof reflected QPs f out ( p ) = f ( ε β,p + pV + p z ∇ ξ V ) (3)is close to the distribution function of incoming QPs f in ( p ) = f ( ε β,p + pV ), where ξ ( r ) describes the sur-face relief of the DL. Thus, this reflection process reducesthe average in-plane momentum exchange at the surfacesignificantly. Only a small drag force is caused by thelast term in the argument of the distribution function inEq. (3) associated with the oscillating vertical motion ofthe He layer with a small velocity of ∇ ξ V ≃ ( δ/a ) V .Obviously, for a flat surface ( ∇ ξ = 0), the drag force F D = 0.Remarkably, such a simple modification of the diffu-sive reflection model leads to a giant increase in the WCmobility − the drag sforce acting on the moving DL be-comes smaller by a factor ( δ/a ) as compared to thatgiven by conventional diffusive reflection. Detailed theo-retical analysis predicts that dc mobility is by a factor offour larger than that found for the case of specular reflec-tion [28] and by a smaller factor at a finite frequency asshown with blue dashed lines in Fig. 3 (see Supplemen-tary Material for the finite-frequency effect [21]). How-ever, not all incoming QPs are reflected by this accom-modated diffusive process; thus, we fit the data usingthe partial accommodation model (green solid lines inFig. 3), where a fraction r a among QPs are scattered bythe accommodated diffusive process and the others arereflected specularly. In this case, the dimple drag forceis defined as F D = (cid:18) − r a (cid:19) F (spec) D , (4)where F (spec) D is the drag force for the specular reflection. r a He concentration (%) x FIG. 5. Accommodation ratio r a as a function of He con-centration.
As can be seen in Fig. 5, r a increases with increasing x . This increase could be associated with a momentummismatch between He in the surface layer and He in thebulk mixture caused by the large difference in the Fermienergies of the two systems: T F < T F ∼ x , making the accommodation processmore favorable. Another possibility for the increase of r a with x is associated with the increase of the thicknessof the He layer according to Fig. 1(b).Our work can be extended to lower temperatures wherethe 2D He system is expected to undergo a superfluidtransition, potentially to a topological superfluid state.The results obtained in this work indicate that the WCmobility could be useful for detecting the superfluid stateof the surface layer because the transition should affectmicroscopic details of the accommodation process.In conclusion, we have demonstrated that the interplayof the 2D and 3D Fermi systems generates a new kindof QP reflection from an uneven surface relief movingalong the interface, which reveals itself as an anomalousenhancement in the WC mobility on the surface of He- He mixture.This work was partly supported by JSPS KAK-ENHI Grant Numbers JP24000007, JP26287084,and JP17H01145 and by National Research Foun-dation (NRF) of Korea Grant Numbers NRF-2015R1C1A1A01055813 and 2016R1A5A1008184. ∗ [email protected] † [email protected] [1] A. Zangwill, Physics at Surfaces (Cambridge UniversityPress, Cambridge, 1988).[2] N. C. Plumb, M. Salluzzo, E. Razzoli, M. M˚ansson,M. Falub, J. Krempasky, C. E. Matt, J. Chang,M. Schulte, J. Braun, H. Ebert, J. Min´ar, B. Delley, K.-J. Zhou, T. Schmitt, M. Shi, J. Mesot, L. Patthey, andM. Radovi´c, Phys. Rev. Lett. , 086801 (2014).[3] Y. H. Wang, D. Hsieh, E. J. Sie, H. Steinberg, D. R.Gardner, Y. S. Lee, P. Jarillo-Herrero, and N. Gedik,Phys. Rev. Lett. , 127401 (2012).[4] Y. Nishida and S. Tan, Phys. Rev. Lett. , 170401(2008).[5] M. Iskin and A. L. Suba¸sı, Phys. Rev. A , 063628(2010).[6] J. Okamoto, L. Mathey, and W.-M. Huang, Phys. Rev.A , 053633 (2017).[7] E. R. Dobbs, Helium Three (Oxford University Press,Oxford, 2000).[8] H. M. Guo, D. O. Edwards, R. E. Sarwinski, and J. T.Tough, Phys. Rev. Lett. , 1259 (1971).[9] D. O. Edwards, S. Y. Shen, J. R. Eckardt, P. P. Fatouros,and F. M. Gasparini, Phys. Rev. B , 892 (1975).[10] S. Yorozu, M. Hiroi, H. Fukuyama, H. Akimoto, H. Ishi-moto, and S. Ogawa, Phys. Rev. B , 12942 (1992).[11] A. F. Andreev, Sov. Phys. JETP , 939 (1966).[12] W. F. Saam, Phys. Rev. A , 1278 (1971).[13] D. O. Edwards and W. F. Saam, The Free Surface of Liq-uid Helium. Progress in Low Temperature Physics, Vol.7, Part A , Chap. 4, edited by Brewer, D (North-Holland,1978) .[14] E. Y. Andrei, ed.,
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Hiroki Ikegami, ∗ Daisuke Sato, and Kimitoshi Kono
RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
Kitak Kim and Hyoungsoon Choi † 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 (Dated: July 18, 2018)
INTERPOLATION OF PHYSICAL QUANTITIESOF HE- HE MIXTURE
In the main text, we use interpolated values of phys-ical quantities of a He- He mixture to analyze the ex-perimental data. In this section, we describe how weinterpolate these quantities.
Molar Volume
As the molar volume of the dilute mixture V m , weadopt the results of Edwards et al. [1]: V m = V [1 + α ( x , T ) x ] (1)with α = (0 . ± . − [(0 . ± . T. (2)Here V = 27.579 cc/mol is the molar volume of pure He [2, 3], x is the concentration of He, and T is thetemperature in the unit of K. This equation is valid for He concentrations less than 6.6 % and temperatures be-low 0.6 K. Our estimates of the dielectric constant andthe density of the mixture are also based on the molarvolume obtained from this equation.For the calculations of the theoretical mobility de-scribed in the main text, we neglect the temperature de-pendence of V m because V m changes by less than 0.06%from 0 to 0.3 K at a fixed x . Viscosity
The viscosity of the dilute mixture has been measuredby Kuenhold et al. (for x = 0.5, 1.3, 2.7, 5.0, and 7.0%)[4] and K¨onig and Pobell ( x = 0.98 and 6.1%) [5]. Weinterpolate these experimental data in the temperaturerange of 10 −
300 mK as a function of T and x accordingto the following procedure. First, we divide the temper-ature range into two regions: above and below T , which varies between 50 and 65 mK depending on x . Withineach temperature region, we fit the viscosity η to thefollowing function of T and x :ln( ηT ) = X m =0 1 X n =0 A mn (ln T ) m (ln x ) n , (3)yielding two sets of fitting parameters, A Hmn and A Lmn ,for the high- and low-temperature regions, respectively.The two fitted curves in the two temperature regionsare smoothly connected by imposing the relation:ln( ηT ) = f ( T − T ) ln( η H T )+ [1 − f ( T − T )] ln( η L T ) , (4)where η H and η L are the viscosity above and below T ,respectively. We use the function f ( T ) given by f ( T ) = 11 + exp( − T /α ) (5)with α = 0.003 K. Figure S1(a) shows the results of ourfitting. The interpolated values of viscosity for the valuesof x used in our measurements are shown in Fig. S1(b).The error in our estimation of the viscosity is less than7%. Surface Tension
The available experimental data on the surface ten-sion of the dilute mixture are very limited, particularlyin the range of x = 0.5 − et al. ( x =3.2 × − , 5.42 × − , 0.56, 6.16, 9.64, and 22%) [6]. Weinterpolate their data using the following phenomenolog-ical model. We assume that the surface layer of He is inthermal equilibrium with He atoms in the bulk having achemical potential µ . According to the thermodynamicrelation, the surface entropy S per unit area is related tothe surface tension α m as [7] S = − ( ∂α m /∂T ) µ . (6) V i sc o s i t y ( P ) Temperature (K) (a) (b)7.0%6.12%5.0%2.7%1.3%0.98%0.5% 6.1%5.5%4.9%3.1%2.1%1.0%
FIG. S1. (a) Fitting of the viscosity data of Kuenhold et al. [4] and K¨onig and Pobell [5] by the procedure described inthe supplemental text. (b) Interpolated viscosity at the Heconcentrations used in our experiment.FIG. S2. Results of fitting of a and b with Eqs. (9) and (10).The values of a and b are taken from Ref. [6]. Here the surface entropy is composed of two parts, S = S + S , where S is the entropy of He on the surfaceand S is that of pure He. S and the surface tension ofpure He α are related by S = − dα /dT . If He in thesurface layer is Fermi degenerate, the entropy should belinear in temperature, S = bT , leading to S = − ( dα /dT ) + bT. (7)From Eqs. (6) and (7), we derive∆ α ( T ) ≡ α ( T ) − α m ( T ) = a + bT / , (8) Pure HePure He1.0%2.1%3.1%4.9%5.5%6.1%
Temperature (K) S u r f a c e t e n s i o n ( d y n e / c m ) FIG. S3. Interpolated surface tension. The open symbolsrepresent the experimental data of the surface tension for each He concentration, taken from Ref. [6]: △
32 ppm, + 542ppm, ▽ (cid:3) × ⊗ where a and b are functions of µ . We fit the experimentalvalues of a and b obtained by Guo et al. [6] as a functionof µ to the following empirical forms: √ a = − B r ( µ − α ) A − β (9)with fitting parameters A , B , α , and β , and b = C + D √ F − µ (10)with fitting parameters C , D , and F . The results of thefitting are shown in Fig. S2.Next, we evaluate the dependence of µ on x and T .The dependence is obtained by approximating µ as µ ( T, x ) ≃ µ (0 , x ) + µ F ( T, x ) , (11)where µ F ( T, x ) is the chemical potential for a 3D non-interacting Fermi gas with effective mass m = 2.28 m [8] ( m is the mass of a He atom). For µ (0 , x ), weuse the experimental values obtained by Seligmann etal. [9]. From Eqs. (8) − (11) and using the temperaturedependence of the experimental surface tension of pure He [6, 10], α ( T ) = 0 . − . T / (dyne / cm) , (12)( T is in the unit of K), we obtain the interpolated α m ( T )as a function of x and T . The results of the interpolated α m ( T ) at several fixed x are shown in Fig. S3. The errorin α m ( T ) is estimated to be less than 2%. THICKNESS AND FERMI ENERGY OFSURFACE HE LAYER
Here we consider how the areal density of the 2D Helayer n s evolves with increasing He concentration in thebulk x . According to the thermodynamic relation, n s isdescribed in terms of the surface tension α m by n s = − ∂α m ∂µ (cid:12)(cid:12)(cid:12)(cid:12) T , (13)where µ is the chemical potential of He. (Here weassume that the interaction between He in the surfacelayer and He in the bulk mixture does not depend on µ .) Using the dependence of α m on µ obtained by Guo et al. [Eq. (8)] [6], we calculate n s ( µ ) from Eq. (13).The chemical potential µ is approximately related to x and T by Eq. (11). Therefore, we obtain n s as a func-tion of x and T . We show n s at T = 0 as a functionof x in Fig. 1b of the main text. It increases with in-creasing x and diverges as it approaches the saturationconcentration of He.Next, we consider how the Fermi energy of the 2D Helayer T F evolves with increasing x . We consider that apotential well present near the surface generates boundstates of He with eigenenergies of E n ⊥ in the directionnormal to the surface ( n = 0, 1, 2, · · · , l ). The bound He can move freely with a kinetic energy E nk in the planeparallel to the surface. In this situation, the energy of thelowest bound state E ⊥ corresponds to the binding energy E b . Because the surface He layer is in equilibrium withbulk He in the mixture, the bound states with energiesless than the Fermi energy of bulk He E F are occupiedat T = 0. This leads to the relation E F + E b = E F ,where the Fermi energy of the 2D He layer E F corre-sponds to the highest kinetic energy of the bound statesof n = 0. For calculating E F and E F shown in Fig. 1cin the main text, we use the experimental value of Ref.[9] for E F and the experimental value of E b = − MOBILITY ON PURE HE IN RIPPLONSCATTERING REGIME
The magnitude of the mobility presented in the maintext is calibrated by multiplying by a factor about unity(0.94 − T m agrees with the theoretical mobility of the highly cor-related electrons in the ripplon scattering regime. Thejustification for this calibration process is based on thefact that our mobility measured for pure He at temper-atures above T m agrees with the theoretical mobility inboth the ripplon [Eq. (3.70) in Ref. [12]] and gas scatter-ing [13] regimes without correction of the magnitude ofthe mobility as shown in Fig. S4. Note that the mobil-ity for pure He measured by Mehrotra et al. at similar electron densities deviates from the theoretical mobilityin the ripplon scattering regime [14]. The origin of thediscrepancy between our and their data is unknown. gas+ripplon scattering gas scattering M ob ili t y ( m / V s ) Temperature (mK) ripplon scattering
FIG. S4. Mobility of electrons on the surface of pure liquid He measured at an electron density of n e = 1.29 × m − and a pressing field of E ⊥ = 2.09 × V/m. The blue lineis the theoretical mobility limited by He gas scattering [13].The green line is the ripplon-limited mobility with the many-electron effect [Eq. (3.70) in Ref. [12]]. The dash-dotted lineis the theoretical mobility including both the gas scatteringand ripplon scattering.
THEORETICAL CALCULATION OF MOBILITY
Here we describe in detail the theoretical mobility ofthe WC at a nonzero frequency in the viscous and bal-listic regimes presented in the main text. To calculatethe mobility, we consider the motion of a WC in a spa-tially uniform ac electric field E k e − iωt by noting that thetransport properties of the WC dressed with a dimplelattice (DL) are significantly modified from those of bareelectrons by the reaction of the helium surface. The acelectric field induces uniform displacement of the elec-trons u e − iωt from their lattice sites, and the displacedelectrons experience a reactive force F D e − iωt from thehelium surface. In this situation, the equation of motionof an electron is described as F D + im e ων r u − e E k = − m e ω u , (14)where direct scattering by thermal ripplons with a rate of ν r is included. ( e is the elementary charge and m e is theelectron mass.) For a small displacement, the reactiveforce is linear in the velocity ˙ u and therefore generallyexpressed in the form of F D = m e ω [ w ( ω ) + iν ( ω )] u . (15)Using the dimensionless response function defined as [12,15, 16] Z ( ω ) = (cid:20) w ( ω ) ω (cid:21) + i ν ( ω ) ω , (16)the equation of motion is rewritten as − e E k = − m e ω h Re Z ( ω ) + i (cid:16) Im Z ( ω ) + ν r ω (cid:17)i u . (17)This equation indicates that the real part Re Z ( ω ) rep-resents the change in the inertia of an electron and theimaginary part ω Im Z ( ω ) describes the momentum relax-ation of the electron due to the coupling to the DL. FromEq. (17), the mobility of a WC is expressed as [12, 15, 16] µ = em e ω Im Z + ν r ( ω Re Z ) + ( ω Im Z + ν r ) . (18)This equation suggests that the problem of calculating µ reduces to evaluating Z ( ω ). Note that Re Z ( ω ) = 0 atzero frequency, and thus µ = em e ω Im Z + ν r .The reactive force from the helium surface arises fromthe coupling of the electrons to the surface and the dy-namics of the surface: F D = − i X g g ˜ U g ξ g e i g · u ( t ) . (19)Here ξ g is the Fourier component of the surface pro-file ξ ( r ) at the reciprocal lattice vector g , g = | g | ,and ˜ U g = U g exp( − W g ) is the electron-ripplon couplingstrength, where U g = e ( E ⊥ + E g ) and exp( − W g ) is theDebye − Waller factor. ( E ⊥ is the pressing field and E g is the polarization field defined in Refs. [12, 17].) TheDebye − Waller factor exp( − W g ) can be evaluated in themanner described in Refs. [12, 18].The dynamics of the free surface (i.e., the dynamics of ξ g ) is different in the viscous and ballistic regimes. Be-low, we describe the theoretical mobility in both regimes.We also give details of the direct scattering by thermallyexcited ripplons. Mobility in Viscous Regime
In this regime, the dynamics of the surface is deter-mined by the Navier − Stokes equation for an incompress-ible fluid ( ∇ · v = 0) with viscosity η : ρ ∂ v ∂t + ∇ p = η ∆ v , (20)( ρ , v , and p are the density, the velocity field, and thepressure of the fluid, respectively) with the followingboundary conditions at the free surface associated with the stress tensor σ ij : σ xz = η (cid:18) ∂v x ∂z + ∂v z ∂x (cid:19) = 0 ,σ yz = η (cid:18) ∂v y ∂z + ∂v z ∂y (cid:19) = 0 , (21) σ zz = − p + 2 η ∂v z ∂z = α ∆ ξ − P e , where α is the surface tension and P e is the pressure fromthe electrons. (The z -axis is taken normal to the surface.)For P e , we take the Fourier component and use the linearapproximation P e, g = n e ˜ U g e − i g · u ( t ) ≃ n e ˜ U g [1 − i g · u ( t )] ≡ P (0) e, g + P (1) e, g . (22)Solving Eq. (20) with the boundary conditions of Eq.(21), we obtain ξ for P (1) e, g as ξ (1) g = − gP (1) e, g ρ ∆ g , (23)where ∆ g = ω r,g − ω − δ g − iωγ g ,γ g = ηρ g φ ( κ ) ,δ g = ω χ ( κ ) ,φ ( κ ) = 2 − √ κ hp κ − i / ,χ ( κ ) = 4 κ (cid:26) √ hp κ + 1 i / − (cid:27) ,κ = ωρ/ ( ηg ), and ω r,g = p α/ρg / is the ripplon fre-quency with wave number g . (Note that the Fourier com-ponents of ξ ( r ) in the static state ξ (0) g = − gn e ˜ U g ρω r,g are ob-tained from P (0) e, g .) Substituting Eq. (23) into Eq. (19)and using Eqs. (15) and (16), we obtain [12, 15, 16] Z ( ω ) = 1 + ρ m e n e X g g (cid:12)(cid:12) ξ g (cid:12)(cid:12) (cid:16) ω r,g ω (cid:17) ω + δ g + i ωγ g ∆ g , (24)where n e is the density of electrons. To calculate thetheoretical mobility in the viscous regime described inthe main text, we first evaluate Z ( ω ) [Eq. (24)] usingthe interpolated values of ρ , α , and η of the mixturepresented in Sec. of this Supplementary Information,and then evaluate µ using Eq. (18). Note that at thezero-frequency limit, ω Re Z ( ω ) = 0 and ω Im Z = ν d = ηm e n e P g g (cid:12)(cid:12) ξ g (cid:12)(cid:12) , indicating µ to be proportional to η − . Mobility in Ballistic Regime
In this regime, the mobility of the WC is caused by thereflection of QPs. The reflection of QPs generates a dragforce on the moving DL, and the drag force is transferredto the WC. At a nonzero frequency, we must also includethe inertial term in the calculation of the mobility, whichcan be performed by using the response function Z ( ω ).The response function can be obtained from the dynam-ics of the free surface by noting that microscopic infor-mation of the reflection of QPs is incorporated in thedamping of a capillary wave. We consider the dynamicsof the free surface under the damping of a capillary wave,which is described by¨ ξ g + 2 γ g ˙ ξ g + ω r,g ξ g = − ˜ U g n e gρ e − i g · u ( t ) , (25)where γ g is the damping rate of a capillary wave withwave number g . Solving Eq. (25) for a small displace-ment u and using Eqs. (15), (16), and (19), we obtain[12, 16] Z ( ω ) = 1+ X g n e ˜ U g gm e ρω r,g g x (cid:0) ω r,g − ω − γ g (cid:1) + 2 iγ g ω r,g /ω (cid:0) ω r,g − ω (cid:1) + 4 γ g ω , (26)where g x is the component of g parallel to v = ˙ u . Notethat γ g includes information on the microscopic natureof the reflection of a QP.In the case of specular reflection, the damping rate γ ( s ) g caused by the reflection of QPs is given by [12, 15, 19] γ ( s ) g = − g π ~ ρ ∞ Z p df ( ε ) dp dp, (27)where ~ is Planck’s constant and f ( ε ) is the Fermi distri-bution function of He QPs with energy ε = p / (2 m ∗ )( p is the momentum and m ∗ is the effective mass of a He QP). Note that at T ≪ T F , the mobility is inde-pendent of temperature T [ γ ( s ) g = gp F / (8 π ~ ρ )] ( T F isthe Fermi temperature and p F is the Fermi momentum).From Eq. (27), we obtain Z ( ω ) using Eq. (26), withwhich we calculate the mobility using Eq. (18). Notethat at ω → ω Re Z → ω Im Z → κ s ( T ) m e n e X g g x (cid:12)(cid:12) ξ g (cid:12)(cid:12) , (28)reproducing the zero-frequency mobility given in Ref.[15], where κ s ( T ) = − π ~ Z ∞ p df ( ε ) dp dp. (29)In the case of the reflection associated with the com-plete accommodation of a QP into the surface layer, thedamping rate is given by [20] γ ( a ) g ( T ) = 14 γ ( s ) g ( T ) . (30) The mobility in the accommodation process presented inthe main text is calculated using Eqs. (26) and (30).Note that γ ( a ) g ( T ) and γ ( s ) g ( T ) have the same function of T but are different by a factor of four. Thus, at zero fre-quency, the mobility limited by the accommodation pro-cess is four times higher than that in the case of specularreflection. At a nonzero frequency, the term ω Re Z ( = 0)contributes to the mobility, resulting in the difference inmobilities between the specular and accommodation pro-cesses being less than a factor of four.In the practical situation at the surface of the mix-ture, not all but a part of the He QPs with ratio r a arereflected by the accommodation process. In this partialaccommodation process, the damping rate is describedas γ g ( T ) = (cid:18) − r a (cid:19) γ ( s ) g ( T ) , (31)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 accommo-dation model with the fitting parameter r a . Contribution from Direct Scattering by ThermalRipplons
In the theoretical mobility presented in the main text,we also include the contribution of the direct scatteringby thermally excited ripplons. We evaluate the rate ofcollision with the ripplons ν r using the Born approxima-tion. In this approximation, ν r is described as a simpleform including the dynamical structure factor (DSF). Forthe DSF of the WC, we use the high-temperature approx-imation [Eq. (8.2) in Ref. [12]], where the form of theDSF is the same as that of a non-degenerate electron gaswith the temperature replaced by the kinetic energy ofelectrons in the WC. ∗ [email protected] † [email protected][1] D. O. Edwards, E. M. Ifft, and R. E. Sarwinski, Phys.Rev. , 380 (1969).[2] E. C. Kerr and R. Taylor, Ann. Phys. , 292 (1964).[3] E. Tanaka, K. Hatakeyama, S. Noma, and T. Satoh,Cryogenics , 365 (2000).[4] K. A. Kuenhold, D. B. Crum, and R. E. Sarwinski, Phys.Lett. A , 13 (1972).[5] R. K¨onig and F. Pobell, J. Low Temp. Phys. , 287(1994).[6] H. M. Guo, D. O. Edwards, R. E. Sarwinski, and J. T.Tough, Phys. Rev. Lett. , 1259 (1971).[7] L. D. Landau and E. M. Lifshitz, Statistical Physics,Part 1, 3rd Edition (Butterworth-Heinemann, 1980)Chap. XV. [8] R. A. Sherlock and D. O. Edwards,Phys. Rev. A , 2744 (1973).[9] P. Seligmann, D. O. Edwards, R. E. Sarwinski, and J. T.Tough, Phys. Rev. , 415 (1969).[10] K. R. Atkins, Can. J. Phys. , 1165 (1953).[11] J. R. Eckardt, D. O. Edwards, P. P. Fatouros, F. M. Gas-parini, and S. Y. Shen, Phys. Rev. Lett. , 706 (1974).[12] Y. P. Monarkha and K. Kono, Two-DimensionalCoulomb Liquids and Solids (Springer-Verlag, Berlin,2004).[13] M. Saitoh, J. Phys. Soc. Jpn. , 201 (1977). [14] R. Mehrotra, C. J. Guo, Y. Z. Ruan, D. B. Mast, andA. J. Dahm, Phys. Rev. B , 5239 (1984).[15] Y. P. Monarkha and K. Kono, J. Phys. Soc. Jpn. ,3901 (1997).[16] Y. P. Monarkha and V. E. Syvokon, Low Temp. Phys. , 1067 (2012).[17] Y. P. Monarkha, Sov. J. Low Temp. Phys. , 600 (1976).[18] Y. P. Monarkha and V. B. Shikin, Sov. J. Low Temp.Phys. , 471 (1983).[19] Y. P. Monarkha and K. Kono, J. Phys. Soc. Jpn. , 960(2005).[20] Y. P. Monarkha and K. Kono, J. Phys. Soc. Jpn.75