Fluid and registered phases in the second layer of 3He on graphite
aa r X i v : . [ c ond - m a t . o t h e r] J un Fluid and registered phases in the second layer of He on graphite
M.C. Gordillo
Departamento de Sistemas F´ısicos, Qu´ımicos y Naturales,Universidad Pablo de Olavide. E-41013 Seville, Spain
J. Boronat
Departament de F´ısica, Universitat Polit`ecnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
A quantum Monte Carlo approach, considering all the corrugation effects, was used to calculatethe complete phase diagram of the second He layer adsorbed on graphite. We found that a first-layer triangular solid was in equilibrium with a gas in the second layer. At a surface density0 . ± .
001 ˚A − , this fluid changes into two first-layer registered phases: 4/7 and 7/12 solids.The 7/12 arrangement transforms into an incommensurate triangular structure of ρ = 0 . ± . − upon further helium loading. A recently proposed hexatic phase was found to be unstable withrespect to those commensurate solids. Helium adsorbed on graphite at temperatures close tozero is a standard setup to study the properties of stablequasi-two-dimensional quantum fluids and solids . Theinterplay between the additional third dimension, the ef-fects of the corrugation of the different substrates, andthe quantum statistics of the adsorbed isotope ( He isa boson and He a fermion) produce very rich phasediagrams. In particular, there is a wealth of experi-mental data on the behavior of He atoms adsorbed ongraphite, both on the first (clean or preplated) and sec-ond layers . Recent experimental and theo-retical work suggests that the low-temperature phasediagram of the first layer includes a liquid-gas coexis-tence, followed by a solidification at high He densities(first to a √ × √ . It is worth notic-ing that the transition from gas to liquid is rather uniquein quantum fluids and can be properly modeled only ifthe corrugation of the substrate is fully taken into ac-count. Once promoted to the second layer, He atomsappear to be in a fluid-like phase, that eventually turnsinto a solid upon increase in the amount of helium ad-sorbed .The theoretical knowledge of that second- He layer islimited to He-preplated graphite . A full calculationof that phase diagram produced a set of results thatcompared favorably with the experimental data, predict-ing the existence of a very dilute liquid that, at higherdensities, is in equilibrium with a registered 7/12 solidthat progresses to the formation of an incommensuratetriangular phase close to third layer promotion. The den-sity of that commensurate structure compares very favor-able to the one found experimentally . However,some caution has to be exercised in comparing the arealdensities in a theoretical calculation (simply the num-ber or atoms divided by the surface) to the same experi-mental magnitude. Calorimetric measurements are typi-cally given in terms of coverage, i.e., as a certain amountin excess of the density corresponding to the first-layercommensurate √ × √ ) with its neutron scat-tering counterparts, and therefore some uncertainties inthe comparison to our simulations, further complicatedwith the presence of defects that vary from a sample toanother. On another quarter, those theoretical resultsalso indicate that an accurate description of the phasediagram demands the consideration of both the corruga-tion of the substrates and the relaxation of the first-layeratoms from their crystallographic positions.In this Rapid Communication, we will be concernedwith the quantum Monte Carlo description of a Helayer on top of an incommensurate He solid adsorbedon graphite. As in Ref. 19, the description of the systemwill be as realistic as possible, including corrugation andrelaxation effects. In addition, we also analyzed differ-ent first-layer densities to allow for a compression uponhelium loading. A recent suggestion about the observa-tion of a stable hexatic He phase will be also considered.The starting point of our microscopic approach is theHamiltonian for a system with two He layers adsorbedon graphite, H = − ¯ h m N X i =1 ∇ i + N X i =1 V ext ( r i ) + N X i 38 ˚A, and r b = 1 . 89 ˚A . Weconsidered unpolarized systems, i.e., N d = N u = N/ u u ( r ) is the numerical solution of the one-body Schr¨odinger equation that describes a single Heatom on top of a triangular lattice formed by first-layer He atoms located in the crystallographic positions of anincommensurate triangular phase, neglecting the influ-ence of the graphite structure . In the present work, weused three first layer triangular lattices of densities takenfrom different experimental works: 0.109 (from Ref. 3),0.113 (intermediate from those of Ref. 12 and 29) and0.116 ˚A − (from Ref. 16). This was done in order to takeinto account a possible compression of the bottom layerupon increasing of the overall helium density. The valueof b was optimized variationally ( b = 2.96 ˚A).The bottom-layer trial wave function wasΨ d ( r N u +1 , r N u +2 , . . . , r N ) = N d Y i u d ( r i ) × −130−125−120−115−110−105−100−95 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 E n e r gy p e r H e a t o m ( K ) Surface area (Å ) FIG. 1. (Color online) Energy per He atom as a functionof the inverse of the total He density. Full squares, a singlelayer incommensurate solid; full circles, a second layer systemincluding a first layer solid of density 0.109 ˚A − ; open squares,second layer arrangement on top of a 0.113 ˚A − incommensu-rate triangular phase. Dotted line, third order polynomial fitto the single layer energy values, intended as a guide-to-the-eye; dashed line, double-tangent Maxwell line to determinethe coexistence between phases. N d Y i 109 ˚A − ; open squares, ρ = 0 . − ; open circles, ρ = 0 . 116 ˚A − . In order to obtain the stability ranges of the differentphases, we performed double-tangent Maxwell construc-tions using our FN-DMC results. This means that the x -axis in Fig. 1 represents the inverse of the total (first+ second layer) density. In that figure, full squares weretaken from Ref. 18 and correspond to the incommen-surate solid phase of the (single) first layer. Full circlesstand by the results from a simulation including 16 × . × . 74 ˚A ; 288 He atoms)in the first layer plus the necessary atoms in the sec-ond one to account for the displayed surface per atom.This corresponds to a bottom layer of density ρ = 0 . − , in line with experimental results of Ref. 3. Opensquares are simulation data for a first layer comprising14 × − (44 . × . 34 ˚A ; 224 atoms). It can be seenthat the open squares are consistently above the opencircles in the inverse density range displayed. Therefore,we should draw the double-tangent Maxwell line (dashedline in Fig. 1) between the 0.109 ˚A − data and the resultsfor a single layer solid. From that line, we can establishthat a single-layer structure of density 0.106 ± − is in equilibrium with a two-layer system with total den-sity 0.111 ± − . This means that from 0.106 ˚A − up, one would have a mixture of clean first-layer zoneswith very dilute second layer systems of 0.111-0.109 =0.002 ˚A − in the adequate proportions to produce totalintermediate densities in the range from 0.106 ˚A − to0.111 ˚A − . This is in excellent agreement with the ex-perimental values given in Refs. 2 ( ∼ − ), 8 ( ∼ − ) and 12 ( ∼ − ), obtained with differenttechniques. −90−85−80−75−70−65 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 E n e r gy p e r H e a t o m ( K ) Surface area (Å ) FIG. 3. (Color online) Same as in the previous figure, butincluding a double-layer incommensurate solid (upper full cir-cles), an hexatic phase (full squares), a 4 / / 12 (down triangle) commensurate phases on top of a first-layer solid of density 0.113 ˚A − . Full and dotted lines corre-spond to Maxwell constructions between the different stablephases. The energy when we increase the density (or decreasethe surface per atom), is displayed in Fig. 2. The sym-bols are the same as in Fig. 1, but we included a thirdset of calculations (open circles) in which the underly-ing incommensurate solid density was 0.116 ˚A − , follow-ing the experimental findings of Ref. 16. Two thingsare immediately apparent: first, this last setup is alwaysmetastable with respect to the first two arrangements,and second, on increasing the helium density, the ener-gies corresponding to the open squares start to go belowthe ones represented by full circles. This means that thefirst layer solid undergoes a compression upon heliumloading. This is in line with previous results for a double He layer on graphene. Third-order polynomial fits tothe data in Fig. 2, not shown for simplicity, indicate thatthe crossing is produced at a density ρ = 0 . ± . − (6 . ± . 01 ˚A in Fig. 2). This corresponds to asecond-layer density of 0 . ± . 002 ˚A − .Fig. 3 reports the coexistence between the second-layerfluid (open squares), and the 4/7 and 7/12 registeredstructures with the first layer (full triangles). The dot-ted line is a double-tangent Maxwell line between the 4/7solid and a fluid of density 0 . ± . 001 ˚A − (6 . ± . in Fig. 3). In both cases, the underlying first-layerdensity was 0.113 ˚A − , since a more compressed trian-gular solid increases the overall energy per particle. Thisresult is in good agreement with the experimental 0.111˚A − value provided in Ref. 12. Since the line can beprolonged to higher densities to include the 7/12 struc-ture, we conclude that our results support the coexis- −23−22.8−22.6−22.4−22.2−22−21.8−21.6−21.4−21.2−21−20.8 0 0.01 0.02 0.03 0.04 0.05 E n e r gy p e r H e a t o m on t h e s ec ond l a y e r ( K ) Density (Å −2 ) FIG. 4. (Color online) Energy per He atom on the secondlayer as a function of that layer density. The density of thefirst layer was ρ = 0 . 109 ˚A − . Full line, third-order poly-nomial fit to the simulation data, intended exclusively as aguide-to-the-eye. tence between both registered solids and the 0.166 ˚A − fluid. This is not surprising given their very close densi-ties ( ρ / = 0 . 177 ˚A − , ρ / = 0 . 179 ˚A − ). In the samefigure, full circles stand for the energy results for thehexatic phase recently proposed in Ref. 16 to accountfor the experimental data. As we can see, those dataare above both the results of the commensurate struc-tures, and the second-layer incommensurate triangularsolid represented by the full circles. This means that thisphase is unstable with respect to any of those solids, atleast in the limit T = 0.The full line in Fig. 3 is another double-tangentMaxwell line, this time between the 7/12 structure and asecond-layer incommensurate solid of ρ = 0 . ± . − (5 . ± . 01 ˚A in Fig. 3). This density is ratherclose to the one corresponding to the third-layer promo-tion observed in different experiments ( ρ = 0 . 184 ˚A − Ref. 3; ρ = 0 . 186 ˚A − , Ref. 7; ρ = 0 . 187 ˚A − , Ref.32; ρ = 0 . 19 ˚A − , Ref. 11). Those values are smalleror compatible with the one deduced from our data inFig. 3. This implies that experimentally we should havean equilibrium between a clean second-layer 7/12 struc- ture of ρ = 0 . 179 ˚A − and a setup with a low-densityfluid on top of a second layer solid . The nature ofthe transformations undergone by the second layer uponfurther helium loading is beyond the scope of the presentwork.At this point, a remaining question is that of the natureof the second-layer fluid before solidification. To solvethat, we plotted the energies per He atom on the secondlayer versus the He density on that layer alone. Thisis done in Fig. 4. As one can see, our FN-DMC resultscorrespond to a gas phase, since the energy per atomincreases monotonically as a function of the He density,with no discernible plateau that would be the tell-talesignal of a liquid-gas transition .In summary, we have undertaken the calculation of therather complicated equation of state of the second layerof He on graphite. The comparison between previoustheoretical descriptions of the same system on preplatedgraphite and the experimental data suggested thenecessity of including in the calculations all corrugationand dynamic effects. That implies the consideration ofdifferent first-layer densities to take into account a pos-sible compression, in line with what happened for Heon graphene. With all that, our quantum Monte Carloresults compare very favorably with the available experi-mental data. This is true for the second-layer promotiondensity ρ = 0 . 106 ˚A − and the upper density limit fora fluid (0.053 ˚A − versus the 0.055 ˚A − of Ref. 3, andthe 0.050-0.060 ˚A − interval proposed on Ref. 6). Thesolidification into the registered structures is also wellpredicted ( ρ ∼ . 178 ˚A − , the same value that experi-ment ). This also validates our value for the first layerdensity upon compression (0.113 ˚A − ), and differs fromthe one proposed in Ref. 16 (0.116 ˚A − ).Our data also suggest that the registered 4/7 and 7/12solids are in equilibrium with a three-layer system of ρ ∼ . 19 ˚A − . This means that from ρ = 0 . 179 ˚A − up thereis a mixture of a second-layer 4/7 and 7/12 structuresand a third-layer fluid. 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