Fluidity in Domain Walls in Dilute 3 He- 4 He Films on Graphite: Possible 1D Fermi Fluid and Dirac Fermions in Helium Film
aa r X i v : . [ c ond - m a t . o t h e r] A ug Fluidity in Domain Walls in Dilute He- He Films on Graphite:Possible 1D Fermi Fluid and Dirac Fermions in Helium Film
Masashi Morishita ∗ Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (Dated: August 7, 2019)The heat capacity of a small amount of He atoms dissolved in submonolayer He film has beenmeasured. The measured heat capacity is finite and suggests that He atoms are mobile at an arealdensity regime higher than that of the √ × √ He films are believed to be solid.Moreover, at higher areal densities, the measured heat capacity is proportional to T and dependson the amounts of He atoms. These behaviors are anomalous to that of a 2D Fermi fluid, andcannot be explained by uniform melting. One possible explanation for these anomalous behaviorsis that helium atoms exhibit fluidity inside the domain walls of the adsorption structure, and thedissolved He atoms behave as a one-dimensional Fermi fluid or as Dirac fermions, depending onthe structure of the domain walls. The behaviors of the measured heat capacity strongly suggestthis possibility.
The quantum properties of low dimensional matterhave attracted much attentions in condensed matterphysics. Graphene is one of the most fascinating and pe-culiar examples that can be treated as two-dimensional(2D) [1], because it exhibits novel and unique fea-tures, and studies on its properties and applications haveevolved explosively within the last decade. A heliumfilm adsorbed onto a graphite surface provides an almostideal 2D system, and exhibits a well-defined layer-by-layer structure. Each layer is independent from eachother and exhibits high flatness and uniformity. The He atom has a nuclear spin of 1/2, and He solid filmprovides a 2D quantum spin system and has been in-vestigated vigorously [2–4]. With increase in areal den-sity, their magnetism exhibit rather complicated change,which has been discussed with the evolution of the ad-sorption structure [5, 6]. On the other hand, informationon the properties and adsoprtion structures of helium-4( He) films is limited due to the lack of an appropriatemethod for their observation.In this letter, I report the results of heat capacity mea-surements of a small amount of He atoms dissolved intosubmonolayer He films on graphite. Results stronglysuggest that He atoms are mobile at an areal densityregime higher than that of the √ × √ He films have been believed to be solid. At higher arealdensities the measured heat capacity is proportional to T . This anomalous temperature variation cannot beexplained if He atoms move around the entire surfaceof the graphite. At these areal densities, He films areexpected to have domain wall superstructures. A possi-ble explanation for these anomalous observations is that He atoms in domain walls exhibit fluidity, and that Heatoms move around only inside domain walls. Fluidityinside domain walls provides regular confined geometrywith the width of atomic size for He atoms. He atomscan be expected to behave as a one-dimensional Fermi ∗ [email protected] fluid or as Dirac fermions, depending on the structure ofdomain walls (striped or honeycomb).The heat capacity measurement of dilute He- He mix-ture films can be utilized to clarify the nature of Hefilms, and this approach was first adopted by Ziouzia etal. [7]. The heat capacity of He is very small [8, 9],and gives very little information about the nature of Hefilms. A small amount of He atoms dissolve only intothe top layer of He thin film. When the top layer of He film is a fluid, the He atoms behave as a Fermi fluidand exhibit finite heat capacity, giving information onthe He film. On the other hand, when the He film issolid, the dissolved He are almost localized and exhibitsalmost no heat capacity contribution.The heat capacity is measured by the usual adiabaticheat-pulse method. The graphite substrate used in thiswork is Grafoil. The total surface area of the substrateis approximately 390 m . To ensure uniformity of He- He film, the following procedures are adopted in samplepreparation. At first, a sufficient amount of the sample He is introduced into the sample cell to cover the het-erogeneous surface of the Grafoil substrate. After the He film is annealed by raising the temperature once,a designated amount of He gas is introduced, and thesample film is annealed again. Typically, in a series ofmeasurements, the amount of He is fixed at some valuewhich corresponds to the areal densities ( ρ ) of 0.1 nm − or 0.2 nm − , while the amount of He is gradually in-creased. Annealing is performed after the introductionof each sample over 6-8 h at a high temperature withthe sample vapor pressure at around 500 Pa. After theannealing, the temperature is slowly decreased over 8-10h until the vapor pressure becomes much less than 1 Pa.The vapor pressure is measured by an in-situ pressuregauge. Other experimental details are similar to those ofour previous works [10, 11].The measured heat capacity of a small amount of Hedissolved in submonolayer He films at select areal den-sities are shown in Fig. 1. As reported elsewhere [12],with increasing areal density from the fluid phase andapproaching the areal density of 6.3 nm − , the measuredheat capacity approaches zero. This value of areal den-sity corresponds to that of the √ × √ He- He film into the √ × √ He atoms are mobile,although at this areal density regime, the He film is be-lieved to be solid. Furthermore, at the higher areal densi-ties than 7.2 nm − the measured heat capacity is propor-tional to T (as shown in Fig. 1(b)) and the magnitudeof the measured heat capacity is almost proportional tothe amount of He atoms (as shown below in Fig. 4.).The heat capacity of a Fermi fluid is proportional to T at low temperatures, and its slope is independent of thenumber of particles. Therefore, He atoms dissolved insubmonolayer He films at these areal densities cannotbe considered a Fermi fluid, and uniform melting of Hefilms cannot explain the observations.There are some candidates for the possible origins ofthe observed anomalous heat capacity. The T variation -2 -2 -2 -2 -2 -2 -2 -2 -2 C ( m J / K ) T (mK) r = 0.1 nm -2 (a) -2 -2 -2 -2 -2 -2 -2 -2 C ( m J / K ) T (mK ) r = 0.1 nm -2 (b) FIG. 1. (color online). Measured heat capacity of dilute He- He mixture films are plotted at select areal densities (a)below 7.2 nm − as functions of T , and (b) above 7.2 nm − asfunctions of T (b). Numbers in the figure indicate the totalareal density of He and He, and here the areal density of He is 0.1 nm − . The dotted lines indicate the expected heatcapacity of ideal 2D Fermi gas of 0.1 nm − , and the brokenlines indicate the origin of the vertical axes. The solid linesare guides for the eye. reminded us of 2D phonon contribution. However, theheat capacities of pure He films, whose origins can beattributed to phonons, are far smaller than the measuredones here [8, 9, 13]. In other words, the magnitude ofthe observed heat capacity can be explained only by thenon-realistic Debye temperature of the order of 1 mK. He nuclear spin contribution can also be excluded, be-cause the interactions between He nuclear spins shouldbe extremely weak in the context of this experiment, andfurthermore, entropy changes calculated from measuredheat capacities are much larger than the expected change, N k B ln 2, where N is the number of He atoms. A filmconsisting of a He- He mixture can exhibit phase sepa-ration into He-rich and He -rich phases, and the mixingof these phases with increasing temperature is also a can-didate for the origin of the observed heat capacity. How-ever, heat capacity contribution from the mixing shouldbe independent of the amount of He. The He amountdependence of the observed heat capacity can excludethis possibility.Helium films are thought to solidify with the impor-tant contribution of hardcore repulsion between heliumatoms [14, 15]. At low areal densities, corrugation of theadsorption potential helps to solidify the film into com-mensurate structures. The √ ×√ He monolayer filmon graphite [16] and is discussed in He monolayer film[5, 6]. In the DWs, the role of the corrugation would beless important, and He could exhibit fluidity. The situ-ation is somewhat similar to the possible fluidity insidedislocations and grain boundaries in hcp He concerningits observed “supersolid”-like behavior [17]. If DWs ex-hibit fluidity, He atoms should crowd onto the DWs toreduce their zero-point energies, and move about in theDWs. Therefore, confined geometries with the width ofatomic size are provided for He atoms. Although thestructures of dislocations and grain boundaries in hcp He are irregular, the DWs are arranged regularly, andthe behaviors of He atoms dissolved in them can be ex-pected to reflect the regular structure. DWs exhibit twodifferent structures, namely the striped (SDW) and hon-eycomb domain wall (HDW) structures.In the case of SDW structures (which appear in a lowerareal density regime), He atoms should travel in one-dimensional (1D) space and behave as a 1D Fermi fluid.Hence, He atoms dissolved in SDWs in He film are apossible candidate for a Tomonaga-Luttinger liquid, al-though evidence for this is yet to be obtained in heatcapacity measurements.In the case of HDW structures (which appear in ahigher areal density regime), He atoms should travel inhoneycomb lattices. Their degree of freedom is similar tothat of electrons in graphene. In graphene, electrons be-have as massless Dirac fermions and their dispersion nearthe Dirac points is linear [1]. Similar behaviors have beenobserved in an ultracold gas of potassium atoms in hon-eycomb lattices [18], and in carbon monoxide moleculesin a hexagonal pattern [19]. He atoms in the HDWsof He films are similarly expected to have linear disper-sion. In this case, their heat capacity is expected to beproportional to T , and observed anomalous T variationat high areal densities can be explained. An almost T dependence has been reported in the heat capacity mea-surement of a multilayered organic material, in whichmassless Dirac fermions are expected [20].The exponent ( α ) of the measured heat capacity, whichis obtained by fitting the measured values with C ∝ T α in the low temperature regime, where the second deriva-tive of the smoothed values is not negative, is plottedin Fig. 2. The rather sudden change from T -linear to T behavior at around 7.0 nm − can be attributed tothe structural phase transition between the SDW andHDW structures. In the case of submonolayer pure Hefilm, the structural phase transition between SDW andHDW structures is predicted to occur around 6.8 nm − [6]. This value is similar to that of the areal densitywhere the exponent of the measured heat capacity sud-denly changes, although the masses and the quantumstatistics are different between He and He.Next, let us pay attention to the behaviors in a hightemperature regime. The heat capacity of a 1D Fermifluid approaches N k B / N k B . In Fig. 1(a),the measured heat capacities tend to saturate to N k B / He in domainwalls. That is, some fraction of He atoms dissolves in √ × √ − , the mea-sured heat capacities tend to saturate once to N k B / He: 0.1 nm -2 He: 0.2 nm -2 E x ponen t a r total (nm -2 ) FIG. 2. (color online). Exponent of measured heat capacityof dilute He- He mixture films at a low temperature regime.The colored rectangles are guides for the eye. He: 0.1 nm -23
He: 0.2 nm -2 g (J/K ) r total (nm -2 ) FIG. 3. (color online). Areal density variation of the slope ofmeasured heat capacity. The broken line and the solid line areexpected behavior for the striped domain wall structure with ρ = 0.1 and ρ = 0.2 nm − , respectively, with assumptionsdescribed in the text.
2D Fermi gas with linear dispersion overshoots once andthen decreases and saturates to 2
N k B , which is twice theexpected value for an ordinary 2D Fermi gas. The mea-sured heat capacities appear to approach N k B , and not2 N k B , with a rather large distribution. The thermal deBroglie length of He atoms at 10 mK is about 10 nm,which is similar to the platelet size of graphite; at 100 mKit is several nm, which is similar to the lattice constantsof the honeycomb domain wall structures. Therefore, in asufficiently low temperature regime, He atoms can be af-fected by the honeycomb structures. However, at highertemperatures, He atoms should behave as ordinary 2Dfermions, and their heat capacity should approach N k B .The excess observed at around 7.6 nm − can be explainedby that the heat capacity exceeds N k B before the Heatoms loose the nature of Dirac fermions, or the lineardispersion, with increasing temperature. Conversely, theobservation of the excess supports the peculiarity of thissystem. Indeed, at these areal densities, the measuredheat capacities tend to decrease and seemingly approach N k B at high temperatures.The slope of the heat capacity of 1D Fermi gas atlow temperatures is γ = g k B mL / ~ N , where g is thenumber of degrees of spin freedom, m is the mass, and L is the length. Unfortunately, the number of He atoms inSDWs depends on the (total) areal density due to finitesolubility as mentioned above. However, the solubilitycan be thought of as almost proportional to the length ofthe DWs, and the total length of the SDWs increaseslinearly with the (total) areal density. The expectedchanges in γ , assuming the solubility of He reaches avalue corresponding to 0.2 nm − at the total areal den-sity of 6.8 nm − , are shown in Fig. 3 for cases with ρ = 0.1 and 0.2 nm − . The slopes of the measured heatcapacity, which are obtained by linear fitting of the mea-sured heat capacity at low temperatures (typically below20 mK), are also shown in Fig. 3. The agreement withthe expected areal density variation is good at low arealdensities. The decreases in the slopes of the measuredheat capacity at high areal densities can be attributed tothe coexistence of HDWs. He: 0.1 nm -23
He: 0.2 nm -2 g ( J / K ) g ( J / K ) r total (nm -2 ) FIG. 4. (color online). Areal density variation of the coef-ficient of the T term of the measured heat capacities. Thescale of vertical axis for ρ = 0.1 nm − is shown on the left,and for ρ = 0.2 nm − on the right. These scales differ fromeach other according to the amount of He. The arrows in-dicate the areal densities where the honeycomb domain wallstructure with displayed periodicity has a regular hexagonalstructure. He: 0.1 nm -23
He: 0.2 nm -2 H e v e l o c i t y ( m / s ) r total (nm -2 ) FIG. 5. (color online). Areal density variation of the speedof He atoms estimated from the measured heat capacity, as-suming He atoms behave as Dirac fermions. The increaseobserved at areal densities above 8.6 nm − may be incorrectdue to the coexistence of domain wall structure and incom-mensurate solid phase. The coefficients of the T term, γ is obtained by fittingthe measured values with C = γ T at a low temperatureregime (typically below 30 mK). As shown in Fig. 4, the T term disappears around 9.1 nm − . This means thatup to this areal density, He- He films have HDW struc-tures, although the DW structure of He film is thought to collapse into an incommensurate solid at 7.9 nm − from heat capacity measurements [8] or 8.4 nm − fromtheoretical simulations [16].If He atoms behave as Dirac fermions, their speedsare the same, because they exhibit linear dispersion. Thespeed of He atoms, v , can be estimated from γ with thefollowing formula, v = (cid:0) gζ (3) k B A/ π ~ γ (cid:1) / , assum-ing that interactions between He atoms are weak, andwhere g = 4 is the number of degrees of freedom, ζ (3) isthe Riemann zeta function, and A is the surface area [21].The speeds are shown in Fig. 5 as functions of total arealdensity. The estimated speed has maxima at around 8.4nm − , and the velocity is much higher than the Fermivelocity in He films behaving as 2D Fermi fluid. At 8.4nm − , the HDW structure is expected to have a regularstructure with the periodicity of 4 × v at this areal densitycan be attributed to this reason.The obtained magnitude of v appears to saturate atapproximately 160 m/s. This behavior suggests the ex-istence of some critical velocity. Although one possibleorigin is the critical velocity of the 2D superfluid He,measurements with smaller amounts of He are desirable.In summary, the heat capacity of a small amountof He atoms dissolved in submonolayer He film ongraphite was measured. The observed behaviors suggestthe nature of He atoms to be that of 1D fermions in thelow areal density regime, and that of Dirac fermions inthe higher areal density regime. These results stronglysuggest that the films exhibit fluidity in the domain walls.The origin of the fluidity and natures of He and also Heatoms in domain walls must be understood further withsuccessive research.
ACKNOWLEDGMENTS
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