Mobile solid 3 He on carbon nanotube promoted by topological frustration
Igor Todoshchenko, Masahiro Kamada, Jukka-Pekka Kaikkonen, Yongping Liao, Alexander Savin, Marco Will, Elena Sergeicheva, Thanniyil Sebastian Abhilash, Esko Kauppinen, Pertti Hakonen
MMobile solid He on carbon nanotube promoted bytopological frustration
Low-dimensional He on a nanotube displays dimerization and delocalizationof topological defects in its ground state.Igor Todoshchenko ∗ , Masahiro Kamada , , Jukka-Pekka Kaikkonen ,Yongping Liao , Alexander Savin , , Marco Will , , Elena Sergeicheva ,Thanniyil Sebastian Abhilash , Esko Kauppinen , Pertti Hakonen , ∗ Low Temperature Laboratory, Department of Applied Physics, Aalto University,P.O. Box 15100, FI-00076 Aalto, Finland QTF Centre of Excellence, Department of Applied Physics, Aalto University,P.O. Box 15100, FI-00076 Aalto, Finland Nano Materials Group, Department of Applied Physics, Aalto University,P.O. Box 13500, FI-00076 Aalto, Finland ∗ To whom correspondence should be addressed;E-mail: igor.todoshchenko@aalto.fi, pertti.hakonen@aalto.fi
Abstract
Low dimensional fermionic quantum systems are exceptionally interesting because theyreveal distinctive physical phenomena, including among others, topologically protected ex-citations, edge states, frustration, and fractionalization. Two-dimensional He has indeedshown a remarkable variety of phases of matter including the unusual quantum spin liquid.Our aim was to lower the dimension of the He system even more by confining it on a sus-pended carbon nanotube. We demonstrate that He on a nanotube merges both fermionicand bosonic phenomena, with a quantum phase transition between solid 1/3 phase and afluid-like solid. The bosonic dimer fluid-like solid contains topology-induced vacancieswhich are delocalized owing to large zero-point motion. We thus observe a quantum phasetransition from fermionic He crystal in to a bosonic one in quasi-1D geometry. a r X i v : . [ c ond - m a t . o t h e r] O c t hysics of quantum liquids and solids is eminently rich and often counterintuitive. Bosonicliquid He possesses superfluidity, e.g. ability to flow without resistance, due to Bose-Einsteincondensation of particles below 2.1 K [1, 2]. Its fermionic counterpart, liquid He displayssuperfluidity below 2.5 mK due to paring of atoms into bosonic dimers (so-called Cooper pairs)[3, 4]. Moreover, a notion has been put forward that bosonic quantum solid could behave likea fluid, due to mobile delocalized defects such as vacancies. Even more, those vacancies, ifpresent at low enough temperature, must Bose-condense, and the crystal must then behave asa superfluid, keeping at the same time its translational symmetry [5, 6, 7]. Search for thisspectacular supersolid state has been ongoing since 1980s, and Kim and Chan announced hintsof superflow in solid He in 2004 [8, 9]. However, later it was established that the equilibriumconcentration of vacancies is exponentially small at low temperatures [10, 11], and the observedsuperflow was most likely through superfluid inside cores of dislocations or on grain boundaries[12]. Thus, it is extremely hard to observe a superflow even in “the most quantum” crystal, solidhelium, in three dimensions.Lowering the dimensionality, however, opens up the possibility for topologically stabilizedvacancies. In particular, carbon nanotube (CNT), which can be viewed as a rolled-and-gluednarrow strip of graphene (a sheet of most symmetric c-plane of graphite), has generally such chi-rality that its symmetry differs from the regular commensurate-solid superlattice composed ofadsorbed helium atoms on graphite. This mismatch ensures a finite amount of defects/vacanciesin adsorbed solid helium even at zero temperature, i.e. so-called zero-point vacancies, whichgive their hallmark on the ground state properties, and which have not yet been ever foundin the bulk 3D helium crystals. In the planar geometry of graphite, where carbon atoms areassembled into a plain hexagonal lattice, only the famous 1/3 solid phase (every third carbonhexagon is occupied by helium atom), is well established [13, 14], although Greywall also re-ported a sign of commensurate dimer phases [15]. The dimer commensurate phases should be2ven more stable on a curved nanotubes due to a larger distance between adsorption cites. Thevacancies in such bosonic dimer crystal are also bosonic, and due to small mass and the associ-ated strong zero-point motion, they delocalize at low enough temperature to procure a mobile,fluid-like solid phase as we demonstrate in this work. The relevance of this kind of topologicalfrustration has recently been demonstrated also in nanomagnetism [16]. Our defect-rich systemdiffers distinctly from one-dimensional Bose-Einstein condensates in an optical lattice where asupersolid has recently been created using coupling to additional cavities [17].At low helium coverages the periodic CNT lattice potential is not able to localize the ad-sorbed atoms, hence fluid and gas phases are formed. On graphite, a uniform fluid He phasehas been observed to form at coverages ρ < ρ LS ≈ . nm − [13, 18]. In 2012 Sato et al. ob-served that at extremely small densities, ρ < . nm − , the liquid forms clusters while the restof substrate is covered with gas phase [19]. As our present experimental work demonstrates,such coexisting phases of liquid clusters and a rarefied gas also appear in adsorbed He on CNT.Extraordinary electrical and mechanical properties of suspended CNTs facilitate their useas ultra-sensitive detectors for force sensing [20] and for surface physics [21]. Helium atomsprovide additional mass δm and an additional spring constant δk for the nanotube (due to changein tension [22]). The relative frequency shift δF/F = δk/ k − δm/ m of the mechanicalresonance of the tube thus provides information of the involved energies and the nature of thephases formed by the adsorbed helium.In our experiments, low-coverage He layer on CNT displayed a ”clustered liquid + dilutegas” to ”uniform fluid” transition at temperatures 0.1 ... 0.5 K. This transition manifested itselfas an abrupt drop of the CNT resonant frequency F ( T ) with increasing temperature T , due toredistribution of helium mass from the ends to uniform coverage over the tube (see Fig. 1A).At larger coverages helium is distributed uniformly, and observed jumps in F ( T ) traces areidentified as changes in a stiffness of the oscillator due to structural phase transitions, as illus-3igure 1: Phase transitions at different areal densities ρρρ . ( A ) Resonant frequency F vs. T across the transition from ”clustered liquid - gas” coexistence to ”uniform fluid”. F dropsowing to redistribution of He atoms from clusters near ends of the nanotube to a uniformlayer. ( B ) Transition from normal solid to fluid. Resonant frequency is higher in the solid phasebecause localized He atoms increase the tension in the CNT (see text); the panel also illustratesthe determination of the jump δF . ( C ) Transition from ”gliding” solid to fluid. Absence ofjump of F across the transition indicates that He atoms are delocalized in this new solid phaseand do not produce additional tension of the tube. ( D ) Transition from gliding to normal solidmanifested by an abrupt increase of tension due to localization of atoms, which increases thetension of the tube. The second transition from normal solid to fluid is seen as a drop of theresonant frequency due to decrease of tension, as in (B). The linear F ( T ) at the lowest T (seealso Fig. 4A) can be extrapolated to the fluid phase, with no jump, as in the panel (C).4rated in Fig. 1, B–D. As seen in the F ( T ) traces in Fig. 1, there are abrupt frequency changesby δF , either positive or negative, depending on the stiffness modification across the criticaltemperature T c . Basic He phases on graphite are illustrated in the insets of Fig. 2 and in fig. S3.The phase diagram, constructed from F ( T ) sweeps, is displayed in Fig. 2. The transitiontemperature T c has a maximum around ρ = 1 − nm − , above which it decreases very rapidlywith increasing coverage. This can be explained by 2D van der Waals (vdW) approach forliquid-gas coexistence (see the supplementary materials, IV). A fit using the 2D vdW phase sep-aration equation to the measured transition temperatures is illustrated by the parabola-lookingcurve in Fig. 2 at densities below 4.2 nm − . The calculated magnitude of the jump δF (Eq. 4in the supplementary materials) is in perfect agreement with our experiment, as shown in Fig. 3.When the liquid state fully overtakes the gas phase at 4.2 nm − , the jump δF approacheszero, but with further increase of coverage there emerges again a large frequency drop (Figs. 1Band 3). This frequency decrease indicates a transition from solid to fluid, as localized solidhelium phase enhances tension in the tube (see the supplementary materials, V). We note that theorigin of the increase of tension is due to localized He whose zero point motion in anharmonicadsorption potential leads to additional force between carbon atoms. The coverage 6.4 nm − corresponds to the 1/3 commensurate phase on graphite (see the insert in Fig. 2). The mostsymmetric 1/3 phase evidently maximizes the additional tension and thereby the frequencyjump δF , as seen in Fig. 3. The melting temperature of the 1/3 phase on CNT, T c ≈ T c ≈ He atoms on a CNT, which smears the carbon latticepotential.The frequency change δF across the melting temperature disappears at coverages ρ ≈ . nm − (Fig. 1C), which means that the additional tension due to the solid helium phase van-ishes. We attribute such a ”soft” solid state to a mobile solid phase which is formed by a gas of5igure 2: Phase diagram of adsorbed He on a CNT.
The individual points on the phasediagram were obtained from F ( T ) traces illustrated in Fig. 1. Circles mark transitions from thephase-separated ”clustered liquid + gas” system to ”uniform fluid”; squares are transitions fromthe mixture ”1/3 solid plus liquid” to uniform fluid. Diamonds mark the transition from glidingsolid to fluid (below 8.2 nm − ) and to the high- T solid. Solid squares denote the transition fromsolid to liquid at high coverages. Different colors represent different mechanical resonances.The solid curve depicts the fit of the liquid-gas phase separation curve using the van der Waalsequation of state. The transition occurs at temperature T c ( ρ ) where the line ρ = const( T ) crosses the phase separation curve below critical temperature T cr as illustrated in the insert.Vertical arrows indicate the coverage of the principal commensurate solid phases of which 3/8and 2/5 phases are composed from dimers (see fig. S3). The insets illustrate, from left to right,liquid+gas coexistence, 1/3, and 3/8 phases. 6igure 3: Frequency drop across phase transitions.
Difference δF between the resonantfrequency in the low-temperature phases and in the high-temperature uniform fluid phase, de-termined as illustrated in Fig. 1B. Circles: transition from ”clustered liquid + gas” two-phasesystem to a ”uniform fluid”. The difference δF vs coverage dependence is described very wellby Eq. 4 of the supplementary materials (solid curve) assuming that liquid clusters are concen-trated symmetrically near ends of the tube. Squares: transition from solid to fluid. The jump isdue to the loss of the additional tension provided by solid helium. The dashed lines indicates theconcentration of the 1/3 commensurate phase. Diamonds: jump from the gliding solid to fluidphase (at ρ > . nm − extrapolated through normal solid phase). Absence of the change of theresonant frequency indicates that there is no additional tension provided by gliding solid phase.Insert: F ( T → vs coverage. The line shows general softening trend due to increase in themass. The bump at . < ρ < . nm − is due to additional stiffness caused by solidificationof helium. The abrupt softening of solid helium at 6.4 nm − implies quantum phase transitionfrom normal to soft liquid-like solid. 7elocalized vacancies. This solid with mobile vacancies is expected to produce only small addi-tional tension, like a liquid does, because in this phase helium atoms are delocalized, in contrastto regular solids. This interpretation is corroborated by the observation of two transitions athigher coverage (cid:38) . nm − shown in Fig. 1D: the solid-induced tension appears abruptly at T ≈ . K (”gliding” solid to ”high- T ” solid transition), and then disappears again with meltingof solid at T = 0 . ... 0.25 K depending on the coverage. The frequency jump δF between1/3 phase and the gliding solid is observed down to the lowest temperatures, as shown in theinsert in Fig. 3, which indicates a quantum phase transition in the adsorbed film as a functionof coverage.We interpret the gliding solid phase as a bosonic dimer solid with delocalized topologi-cally stabilized vacancies which provides fast mass transport. According to Andreev [23], in amagnetically-disordered fermionic solid He a vacancy, due to its magnetic moment, forms astrongly localized magnetic polaron. Indeed, the monomer fermionic 1/3 phase shows no signsof mobile vacancies through mismatch-induced defects. However, when the coverage increasesfurther, bosonic dimer commensurate solids such as 2/5 [15] and the here-introduced 3/8 phasesbecome possible (see fig. S3C and the insert in Fig. 2).The temperature dependence of the resonant frequency F ( T ) was also analysed to un-derstand the behavior across the quantum phase transition as a function of He coverage (seeFig. 4A). Change of F reflects the change in helium pressure P as explained in the supplemen-tary materials, VI. The power-law fitting of the data, F ( T ) − F (0) ∝ T β gave β = 2 . ± . inthe normal solid below . nm − , corresponding to 1D phonons. In the gliding solid phase, wefind the exponent β = 1 . ± . (fig. S5A). The observed linear P ( T ) dependence impliesideal gas type of behavior with a fixed number of particles which can be related to delocalizedzero-point vacancies [5]. Truly, the nanotube as a seamlessly rolled graphene sheet works asa template for a superlattice of He dimers. However, helium superlattice with a clearly larger8igure 4:
Quantum phase transition between monomer 1/3 phase and gliding dimer solid. ( A ) Temperature dependence of resonant frequency F at different coverages indicated by thevalues on the right within the frame; the numbers point to curves in the corresponding orderfrom top to bottom. We find F ( T ) − F (0) ∝ T β with β (cid:39) at coverages < . nm − ,which corresponds to a solid with 1D phonon excitations (see text). The quadratic behaviorchanges abruptly to linear at ρ ≈ . nm − , which persists till complete monolayer coverage.Linear temperature dependence can be obtained with constant number of weakly interactingexcitations, such as vacancies induced by topological mismatch. The quadratic fit for the highestdensity . nm − of normal solid, and the linear fit for the lowest density . nm − of glidingsolid are shown by black curves. ( B ) Illustration of high-coverage He superlattice on a CNTwith mismatch-induced defects. The defects are produced when, for example, the dimerized3/8 superlattice on graphene (see fig. S3B) is rolled around the horizontal axis to form a CNT.The red line marks the connection of top and bottom edges of the graphene ”stencil”.9eriod cannot generally be matched with the CNT lattice, and the dimer lattice will containplenty of defects (see Fig. 4B). The system of point defects is thus topologically protected dueto the mismatch, and no new defects appear with increasing temperature as the correspondingactivation energy is much higher.The observed quantum state of weakly interacting delocalized vacancies is truly extraordi-nary because, in contrast to Cooper pairs stabilized by phonons, pairing of fermionic He atomson CNT into bosonic dimers is promoted by carbon lattice. Owing to topological frustration,the dimer phase contains defects and acquires properties reminiscent of bosonic supersolid inits ground state. Equally intriguing would be to study the nuclear spin system of the dimer solidphase because this phase, combining the paired He and frustrated 2D lattice, would facilitatea combination of exotic magnetic phenomena of p -wave-paired superfluid and quantum spinliquid. References [1] P. Kapitza, Viscosity of liquid helium below the λ -point. Nature , 74 (1938).[2] J. F. Allen, A. D. Misener, Flow of liquid helium II.
Nature , 75 (1938).[3] D. D. Osheroff, R. C. Richardson, D. M. Lee, New magnetic phenomena in liquid He3below 3 mK.
Phys. Rev. Lett. , 885-–888 (1972).[4] A. J. Leggett, Interpretation of recent results on He3 below 3 mK: a new liquid phase? Phys. Rev. Lett. , 1227–123 (1972).[5] A. F. Andreev, I. M. Lifshitz, Quantum theory of defects in crystals. Sov. Phys. JETP ,1107–1113 (1969).[6] A. J. Leggett, Can a solid be ”superfluid”? Phys. Rev. Lett. , 1543–1546 (1970).107] G. V. Chester, Speculations on Bose-Einstein condensation and quantum crystals. Phys.Rev. A , 256–258 (1970).[8] E. Kim, M. H. W. Chan, Probable observation of a supersolid helium phase. Nature ,225–227 (2004).[9] E. Kim, M. H. W. Chan, Observation of Superflow in Solid Helium,
Science , 1941–1944 (2004).[10] J. Day and J. Beamish, Pressure-driven flow of solid helium.
Phys. Rev. Lett. , 105304(2006).[11] M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof’ev, B. V. Svistunov, M. Troyer, Thefate of vacancy-induced supersolidity in He.
Phys. Rev. Lett. , 080401 (2006).[12] S. Sasaki, R. Ishiguro, F. Caupin, H. J. Maris, S. Balibar, Superfluidity of Grain Bound-aries and Supersolid Behavior, Science , 1098–1100 (2006).[13] M. Bretz, J. G. Dash, D. C. Hickernell, E. O. McLean, O. E. Vilches, Phases of He and He monolayer films adsorbed on basal-plane oriented graphite.
Phys. Rev. A , 1589–1615 (1973).[14] H. Godfrin, H.-J. Lauter, ”Experimental properties of He adsorbed on graphite” in
Progress in Low Temperature Physics, Volume XIV , W. P. Halperin Ed. (Elsevier, 1995),pp. 213–320.[15] D. S. Greywall, Heat capacity and the commensurate-incommensurate transition of Headsorbed on graphite.
Phys. Rev. B , 309–318 (1993).1116] S. Mishra, D. Beyer, K. Eimre, S. Kezilebieke, R. Berger, O. Gr¨oning1, C. A. Pignedoli1,K. M¨ullen, P. Liljeroth, P. Ruffieux, X. Feng, R. Fasel, Topological frustration inducesunconventional magnetism in a nanographene. Nature Nanotechnology , 22–28 (2020).[17] J. L´eonard, A. Morales, P. Zupancic, T. Esslinger, T. Donner, Supersolid formation in aquantum gas breaking a continuous translational symmetry, Nature , 87-–90 (2017).[18] K.-D. Morhard, J. Bossy, H. Godfrin, NMR measurements on He adsorbed on Grafoil atsubmonolayer coverages and millikelvin temperatures.
Phys. Rev. B , 446–454 (1995).[19] D. Sato, K. Naruse, T. Matsui, H. Fukuyama, Observation of self-binding in monolayer He.
Phys. Rev. Lett. , 235306 (2012).[20] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, A. Bachtold, A nanomechanicalmass sensor with yoctogram resolution.
Nature Nanotechnology , 301–304 (2012).[21] Z. Wang, J. Wei, P. Morse, J. G. Dash, O. E. Vilches, D. H. Cobden, Phase transitions ofadsorbed atoms on the surface of a carbon nanotube. Science , 552–555 (2010).[22] G. Y. Chen, T. Thundat, E. A. Wachter, R. J. Warmack, Adsorption-induced surface stressand its effects on resonance frequency of microcantilevers.
J. Appl. Phys. , 3618-3622(1995).[23] A. F. Andreev, ”Quantum crystals” in Progress in Low Temperature Physics, Volume VIII ,D. F. Brewer Ed. (North Holland, 1982), pp. 67–132.[24] N. Wei, P. Laiho, A. T. Khan, A. Hussain, A. Lyuleeva, S. Ahmed, Q. Zhang, Y. Liao, Y.Tian, Er-X. Ding, Y. Ohno, E. I. Kauppinen, Fast and ultraclean approach for measuringthe transport properties of carbon nanotubes.
Adv. Funct. Mater. , 1907150 (2019).1225] J.-P. Kaikkonen, A. T. Sebastian, P. Laiho, N. Wei, M. Will, Y. Liao, E. I. Kauppinen, P. J.Hakonen, Suspended superconducting weak links from aerosol-synthesized single-walledcarbon nanotubes. Nano Research , 1–6 (2020).[26] V. Gouttenoire, T. Barois, S. Perisanu, J.-L. Leclercq, S. T. Purcell, P. Vincent, A. Ayari,Digital and FM demodulation of a doubly clamped single-walled carbon-nanotube oscil-lator: Towards a nanotube cell phone. Small , 1060–1065 (2010).[27] A. N. Cleland, Foundations of Nanomechanics (Springer, 2003), pp. 233–236 .[28] G. A. Steele, A. K. H¨uttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, H.S. J. van der Zant, Strong coupling between single-electron tunneling and nanomechanicalmotion.
Science , 1103–1107 (2009).[29] B. Lassagne, Y. Tarakanov, J. Kinaret, D. Garcia-Sanchez, A. Bachtold, Coupling mechan-ics to charge transport in carbon nanotube mechanical resonators.
Science , 1107–1110(2009).[30] B. Dzyubenko, H.-C. Lee, O. E. Vilches, D. H. Cobden, Surface electron perturbationsand the collective behaviour of atoms adsorbed on a cylinder.
Nature Physics , 398–402(2015).[31] J. Beyer, M. Schmidt, J. Engert, S. AliValiollahi, H. J. Barthelmess, Reference measure-ments of SQUID-based magnetic-field fluctuation thermometers. Supercond. Sci. Tech. ,065010 (2013).[32] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics, Volume III (Quantum Me-chanics) (Pergamon Press, 1969). 1333] W. E. Carlos, M. W. Cole, Band structure and thermodynamic properties of He atoms neara graphite surface.
Phys. Rev. B , 3713–3720 (1980).[34] B. Brami, F. Joly, C. Lhuillier, Is there a liquid phase in a (sub)monolayer of He adsorbedon graphite?
J. Low Temp. Phys. , 63–76 (1994).[35] P. A. Whitlock, G. V. Chester, B. Krishnamachari, Monte Carlo simulation of a heliumfilm on graphite. Phys. Rev. B , 8704–8715 (1998).[36] A. D. Lueking, M. W. Cole, Commensurate phases of gases adsorbed on carbon nanotubes. Phys. Rev. B , 195425 (2007).[37] G. Y. Gor, N. Bernstein, Revisiting Bangham’s law of adsorption-induced deformation:changes of surface energy and surface stress. Phys. Chem. Chem. Phys. , 9788-9798(2016).[38] F. Joly, C. Lhuillier, B. Brami, The helium-graphite interaction. Surf. Sci.264
J. Low Temp. Phys. , 63–76 (1994).[35] P. A. Whitlock, G. V. Chester, B. Krishnamachari, Monte Carlo simulation of a heliumfilm on graphite. Phys. Rev. B , 8704–8715 (1998).[36] A. D. Lueking, M. W. Cole, Commensurate phases of gases adsorbed on carbon nanotubes. Phys. Rev. B , 195425 (2007).[37] G. Y. Gor, N. Bernstein, Revisiting Bangham’s law of adsorption-induced deformation:changes of surface energy and surface stress. Phys. Chem. Chem. Phys. , 9788-9798(2016).[38] F. Joly, C. Lhuillier, B. Brami, The helium-graphite interaction. Surf. Sci.264 , 419–422(1992).
ACKNOWLEDGMENTS
We are grateful to Henri Godfrin, Ari Harju, Ville Havu, Andreas Huettel, Martti Puska, andErkki Thuneberg for discussions, and Petri Tonteri from Densiq Ltd. (90620 Oulu, Finland) forproviding us with the ultra-pure grafoil. We thank Miika Haataja for measuring the surface areaof the grafoil sample and Qiang Zhang for participation in the optimization of the CNT process.
Funding:
This work was supported by Academy of Finland projects No. 314448 (BOLOSE),No. 312295 (CoE, Quantum Technology Finland), and No. 316572 (CNTstress). This workwas also supported within the EU Horizon 2020 programme by ERC (QuDeT, No. 670743),and in part by Marie-Curie training network project (OMT, No. 722923). J.-P. K. is grateful14or the financial support from Vilho, Yrj¨o and Kalle V¨ais¨al¨a Foundation of the Finnish Academyof Science and Letters. This research project utilized the Aalto University OtaNano/LTL infras-tructure which is part of European Microkelvin Platform (EMP, No. 824109 EU Horizon 2020).
SUPPLEMENTARY MATERIALS
Materials and MetodsSupplementary TextFigs. S1 to S5References ( ) 15 upplementary Materials for
Mobile solid He phase on carbon nanotube
I. Materials and methodsFabrication of nanotube samples:
The CNTs were synthesized in the gas phase with thefloating catalyst chemical vapour deposition growth method (FC-CVD) followed by the directthermophoretic deposition onto prefabricated chips [24]. For contact material we employedmolybdenum-rhenium/palladium (MoRe/Pd) bilayer. The 150 nm MoRe layer was first co-sputtered on top of a strongly doped Si wafer with 285 nm thick SiO layer. In a second step a5 nm Pd layer was sputtered on top of the existing MoRe. Optical lithography and reactive ionetching (RIE) were used to pattern 119 source drain electrode pairs per chip. Clean FC-CVDsynthesized nanotubes with diameters of . ± . nm were deposited on the clean chip. In thelast fabrication step, the chip was annealed in vacuum at 220 ◦ C for 10 min to increase contacttransparency [25]. No superconductivity was observed in the measured samples. The Coulombenergy of CNTs amounted approximately to meV. The doped substrate acted as back gatewith capacitance C g (cid:39) . − . aF to the tube.All source-drain pairs were checked regarding room temperature resistance of the CNTusing an automatic probe station. Two devices with a room temperature resistance of ∼ k Ω were chosen and bonded to a PCB in a parallel configuration. Scanning electron microscopeimages confirmed L = 700 nm for the length of the suspended part of one nanotube device afterall measurements were completed, while the total length of the tubes varied around 2-3 µ m. Theother device had detached before imaging, but according to nearly equal mechanical frequency,we estimate L (cid:39) nm’s for its suspended part as well. Biasing:
The bias point of the nanotube for detection of mechanical motion was deter-16ined by measuring its differential conductance dI/dU bias and the so-called transconductance dI/dU gate . Both these values should be large enough to provide sufficient current for sensi-tive detection. In Fig. S1A we show the transconductance at different bias and gate voltages,while Fig. S1B displays DC I-V characteristics at different gate voltages. On the basis of thesemeasurements, the working point [ U gate , U bias ] was chosen to have large amplification for me-chanical motion and low Joule heating. Preparation of He sample:
After electrical characterization of the CNT sample along withits mechanical resonance properties in vacuum, He atoms were gradually added to the samplechamber. As a single nanotube can adsorb only very little amount of He, we used a grafoilsample of surface area 17.6 m as a ballast. Using grafoil ballast, we were able to control thecoverage of the adsorbed He with an accuracy of 0.05 nm − ; the calibration of our coveragescale is discussed below.After each addition we waited for 10-20 hours and subsequently performed temperaturesweeps up and down at a rate of 10 ... 30 mK / h. Typically, the T sweep at each coverage wasrepeated at least once, often several times. Most of the data were collected on two resonanceswith vacuum frequencies 325.35 and 348.55 MHz, see Fig. S2. Both lines also displayed quitesimilar coverages, which hints that these two signals must originate from very similar CNTs.The similarity of the two nanotubes also confirmed by equal resistance at room temperature andby nearly equal Coulomb energies (see Fig. S1A). Mechanical resonance detection:
When a RF voltage ∆ U = U RF cos Ω t is applied on topof the DC gate voltage U gate , the tube starts to oscillate because of the oscillating electrical field ∆ E . The oscillating electrical force F = Q · ∆ E will produce the mechanical motion of thetube; here Q = C g ( U gate + ∆ U ) denotes the total charge on the nanotube with capacitance C g tothe gate. The equation of motion for the vertical position z of a CNT with radius r at height H C g U gate U RF cos Ω t/ ( r log 2 H/r ) =
M ∂ z/∂t + γ∂z/∂t + kz , where M is the effective mass of the tube, k is a spring constant, and γ denotes dissipation.The conductivity σ ( E ) of the tube also oscillates at the same frequency, as it is field-dependent.As a result, the current through the tube I = ∆ U σ ∝ cos Ω t obtains an additional constantterm which is proportional to the amplitude of mechanical oscillations, I DC ∝ C g U gate U RF σ (cid:48) E M − Ω + 2 πi Γ Ω , (1)where σ (cid:48) E denotes the derivative of the conductance with respect to the electric field E , Ω =2 πF , and Γ are the central frequency and the width of the mechanical resonance of the tube,respectively. In logarithmic approximation, the tube capacitance per unit length can be writtenas C g /L = 2 πε / log (2 H/r ) .Most of our data was obtained using the so-called frequency modulation method [26]. Whenthe driving RF frequency Ω is modulated over Ω ± δ at an audio frequency ω ∼ kHz, Ω( t ) =Ω+ δ cos ωt , the nanotube will produce a mixing current component at frequency ω proportionalto the mechanical response: < I > ω ∝ dd Ω 1Ω − Ω + 2 πi Γ Ω . (2)This audio frequency signal can be recorded accurately using a sensitive low-frequency lock-inamplifier. Originally, after adjusting the detection phase, only the real part of the differentiatedLorentzian in Eq. (2) was required in the analysis [26]. In the general case, however, there ap-pears a weakly temperature-dependent phase shift between the voltage and the conductance thatneeds to be accounted for. Therefore, the phase of the Lorentzian becomes a fitting parameter,and the real and imaginary parts need to be considered when analysing the measured responsecurves, see examples of the fit in Fig. S2. σ (cid:48) E can also be written as d I/dU bias dU gate overage scale of adsorbed He atoms:
Owing to substantial tension F , the mechanicalresonant frequency of our samples can be approximated by F = (1 / L ) (cid:112) F /µ , where µ denotes the mass per unit length. The coverage ρ was calculated from the frequency shift in theuniform fluid phase where helium is distributed uniformly over the tube and does not contributeappreciably to the tension of the tube: ∆ FF = − M He M C = − ρ He ρ C (3)where M He is the total mass of the adsorbed He, M C and ρ C = 38 . nm − denote the massand the areal density of carbon atoms in the nanotube, respectively, and F is the resonantfrequency of the nanotube in vacuum. This yields the relationship ρ He = − . nm − ∆ F/F for the helium coverage. For calculations of frequency shifts with non-uniform mass distribution µ ( x ) along the nanotube ( x -axis), the mode displacement z of the oscillating nanotube wasapproximated with the cosine form z = z max (1 − cos 2 πx/L ) / which differs less than 10 %from the exact oscillating shape [27]. The modification of the kinetic energy due to helium masswas calculated by integrating µ ( x ) ˙ z ( x ) over the nanotube.Even though the frequency of the fitted peaks can be determined with very high accuracy,our coverage scale has an uncertainty due to frequency shifts with temperature. This shift influ-enced in particular the measurements of ”clustered liquid + gas” to ”uniform liquid” transitiontemperature. The coverage was derived from the frequency shift determined at 0.25 K accord-ing to Eq. (3). The errors come from the fact that 0.25 K is rather close to ”liquid + gas” to”uniform fluid” transition temperature (see Fig. 1A of the Main text), and the data were extrap-olated from higher temperatures. Small shifts due to the Coulomb spring effects and backgroundcharge variation [28, 29, 30] were eliminated by tuning the gate and bias voltages of the workingpoint. Temperature:
Noise thermometry was employed for measurement of temperature over the19hole experimental range. The noise thermometer was manufactured by Physical-TechnicalInstitute (Physikalisch-Technische Bundesanstalt) in Berlin [31].
II: Electrical conductivity of the nanotube (Fig. S1)
Owing to the Coulomb gap, the conductance σ of the nanotube varies over several orders ofmagnitude depending on the applied gate voltage U gate and on the bias voltage U bias . A usefulquantity of the differential conductance is the so-called transconductance dI/dU gate which canbe measured by applying a small AC voltage to the gate. The working point in our measure-ments [ U gate , U bias ] was located near one of the largest and widest peaks seen in Fig. S1A. Thebias voltage U bias should be kept small enough to avoid unnecessary Joule heating IU bias (seeFig. S1B for typical IV-curves). However, in our sample the heating was primarily depositedon the contacts on electrodes, owing to the ballistic nature of the conductance of the tubes.The working point was optimized by measuring dI/dU gate ( U gate , U bias ) and I ( U gate , U bias ) mani-folds and readjusting the values in order to compensate for the drift due to background chargevariation. III: Quantum diffusion of the adsorbed helium at low densities He atoms on the nanotube lattice are attracted to the centers of carbon hexagons. The energybarrier separating two neighboring hexagon centers located at the distance of d = 2 . ˚A (equi-librium distance in the planar graphite/grafoil geometry) is quite large, U = 2 . meV = 33 . K[14]. The hopping time t h to the neighboring hexagon is t h ∼ / ( f P ) where P is the proba-bility of tunneling through the barrier, and f is the attempt frequency. At temperatures as lowas 0.1 K, the probability of thermally activated tunneling P th ∼ exp [ − U/ ( k B T )] is negligible,and hopping is governed by the quantum tunneling which probability can be estimated in the20uasi-classical approximation as P q ∼ exp (cid:104) − (cid:112) m ( U − E ) d/ (cid:126) (cid:105) where E is the lowest energylevel in the potential well [32].The shape of the potential depends on the curvature of the carbon lattice. By neglecting thecurvature and taking the first term of the Fourier expansion U ( x ) = ( U/ − cos 2 πx/d ) forthe shape of the potential, we obtain U ( d ) ≈ ( π /d ) U x for its harmonic approximation. Thisyields for the oscillation frequency f = (cid:112) U/ (2 md ) ∼ / s and for the lowest energylevel E = π (cid:126) f ≈ K. Consequently, the width of the barrier decreases to d ∗ ≈ . ˚A, andthe quantum tunneling probability becomes as large as P q ∼ . . The corresponding diffusioncoefficient in the limit of small density can be written as D ∼ h f P ∼ − m / s. Hence, weobtain a characteristic time scale τ ∼ (1 / L /D ∼ µ s that governs the diffusion of a heliumatom across the tube length of L = 700 nm. Similar estimate can be made using the band modelfor helium atoms on graphite [33]. Using the group velocity v g = (cid:126) k/m for k = 2 π/L , weobtain t = v g /L (cid:39) µ s for the traversal time. Thus, at small densities, the atoms on the tube canarrange themselves to an equilibrium configuration on time scales much less than a few minutescorresponding to the duration of our single resonance curve measurement. On the other hand,the diffusion time is much longer than the mechanical oscillation period, which prevents massredistribution within one oscillation cycle. IV: 2D van der Waals approach at low densities
A pressure P in two dimensions is introduced as the force per unit length of the boundary, aris-ing from the kinetic energy of the particles confined to in-plane motion on the CNT. The equa-tion for ideal gas is identical to the 3D formula, P = m ( N/S ) (cid:16) v x + v y (cid:17) = ( N a /S m ) k B T = RT /S m , where the surface area S m of one mole plays the role of the molar volume in 3D,and v i describes the average of square velocity of particles in the direction i = { x, y } . ThevdW correction b to the molar surface area S m is a constant equal to the total area occupied21y the hard cores of one mole of particles. Particles feel an attraction to the rest of the par-ticles, which is proportional to the coverage ρ ∝ /S m . This leads to a change in the pres-sure at the boundary, but since the deficit of pressure is proportional to the number of parti-cles, the correction is quadratic in coverage, P → P + a/S m . The functional form of thevdW equation is therefore the same as in 3D, but with the replacement of molar volume withmolar surface area, ( P + a/S m )( S m − b ) = RT . The reduced form reads then universally, [ P/P cr + 3( ρ/ρ cr ) ] · (3 ρ cr /ρ −
1) = 8(
T /T cr ) , where T cr , ρ cr , P cr are the critical values fortemperature, density and pressure.The above vdW form can also be applied to 2D He, a fermionic quantum system. The idealgas law for fermions contains additional ρ terms, while the exchange interaction yields anadditional increase in pressure. However, both modifications can be included in the parametersof the vdW equation. A fit of the 2D quantum vdW equation to the transition temperature with T cr = 0 . K and ρ cr = 4 . nm − is shown in Fig. 2 of the Main text.The validity of the vdW approach was also confirmed by calculating the frequency drop atthe phase separation curve. We assume that liquid helium clusters start to form near the endsof the tube where the atomic binding energy is larger due to a surface roughness at the ends ofnanotube [19]. Redistribution of He atoms from liquid puddles near the ends to a uniform fluidlayer over the tube at the phase transition changes the resonant frequency according to δFF = (cid:20) − π sin π ρρ + 148 π sin 2 π ρρ (cid:21) ρ ρ C , (4)where ρ = 3 ρ cr , and ρ C = 38 . nm − is the areal density of carbon atoms. The frequencyjump calculated from Eq. 4, plotted in Fig. 3 of the Main text as a solid curve, fits the experi-mental data points very well. Note that there is no fitting parameter in Eq. 4, once the criticalcoverage ρ cr = 1 . nm − is determined from the phase diagram (Fig. 2 of the main text).The largest liquid–gas co-existence coverage 4.2 nm − is five times larger than the corre-22ponding coverage in He on graphite, 0.8 nm − [19]. This difference can be attributed totransverse, vertical motion of He on the nanotube. In the variational Monte-Carlo study on He monolayer on graphite there is no self-binding for a strict 2D system, but accounting forthe vertical motion of atoms shows that He film forms a clustered liquid with critical coverageof ≈ − [34]. Although this effect is small on graphite [35], it is expected to be largerin our case because the number of carbon atoms binding He is smaller on a CNT. Hence, theobserved quantum vdW equation of state on a CNT is influenced also by quantum mechanicalmotion in the third, radial direction.
V: Commensurate solid phases of helium on a hexagonal carbon lattice (Fig. S3)
As discussed in the Main text, stability of commensurate phases of He on graphite lattice isinfluenced by the curvature of the lattice in a CNT. Here we discuss briefly how the orientationof the He structure on the curved CNT influences its stability. Basic structures of He on agraphite lattice are illustrated in Fig. S3. For easier comparison of the structures, all frames aredrawn with the same arm-chair type of top and bottom edges.Among the possible commensurate solid helium phases on the carbon hexagonal lattice,the most symmetric and stable is the monomer 1/3 phase in which helium atoms occupy everythird hexagon. The 1/3 phase has been experimentally observed on graphite by Bretz et al. atthe coverage of 6.4 nm − with both He and He [13]. At lower coverages, a mixture of 1/3solid and fluid phases has been observed [18]. On a CNT, the true hexagonal symmetry ofthe 1/3 phase is broken as the equilibrium distance between helium atoms becomes directiondependent. This influences the structures in equilibrium, and it may lead to modified attributesfor the 1/3 phase on a CNT, for example, reduced stability against other possible configurations.At higher coverages, other commensurate phase structures are possible including ones com-posed of dimers, as illustrated in Figs. S3B and S3C, or even chains, as in Figs. S3D and S3E2336]. However, in the planar geometry of graphite, these phases are most likely not stablebecause the hard core diameter of helium atom is larger than the distance between centers ofneighboring hexagons as discussed in the Main text. Nevertheless, even in the planar geometryof graphite, Greywall seemed to observe an onset of specific features at the corresponding cov-erages and suggested that there could exist 2/5 or 3/7 phases [15] illustrated in Figs. S3C andS3D.In contrast to the planar geometry of graphite, CNT can facilitate additional commensuratephases: the distance between helium atoms is now larger and exceeds the hard core diameterwhen the orientation of dimers is inclined with respect to the axis of the tube. The dimer crys-talline phases illustrated in Figs. S3B and S3C become then possible. Thus, we may concludethat CNT, due to its curvature, supports stable solid commensurate dimer phases in which theorientation of dimers is perpendicular or inclined with respect to the tube’s axis. For example,when the structures shown in Figs. S3B and S3C are wound into a CNT around their horizontalaxes, as depicted in Fig. 4B in the Main text.
VI: Tension induced by adsorbed helium (Figs. S4, S5)
Similar to what has been observed with other substrates and adsorbents [22, 37], He atomsadsorbed on CNT contribute both to the mass and to the spring constant of the tube, therebychanging the resonant frequency. The actual frequency vs. mass relation depends on the locationof the mass on the oscillation tube. For example, if the mass is located at the ends of the tube,the influence of the mass is small compared with the same mass located near the center. Hence,a transition between islanded film (liquid clusters near the ends of the tube) and a uniformlydistributed fluid becomes quite prominent.In constructing our coverage scale in Sect. I, we assumed that the change in the elastic stateof the tube can be neglected in fluid phase at high temperatures, and that the frequency shift is24overned by the mass of helium atoms M He added to the mass of the carbon lattice M C , seeEq. (3). At high T the distance between helium atoms and the substrate is the largest, whichminimizes the influence of helium atoms on the elastic properties of the CNT. Upon localizationof the atoms in solid phases, the free surface energy becomes lowered, which leads to a changein the spring constant of the tube as observed in the experiments as a frequency shift.At larger coverages ρ > . nm − , the transition from uniform fluid to solid is not asso-ciated with a change in the effective mass on the tube but only with a change in the effectivespring constant of the mechanical resonator. In fact, our coverage scale is corroborated by theexperimental findings at 1/3 filling at ρ = 6 . nm − coverage, at which the ” √ × √ ” com-mensurate phase induces the strongest observed frequency jump between the liquid and solidphases.The contribution of helium to the nanotube spring constant can be viewed as a change in thetension F , which relates directly to the mechanical resonant frequency, F = (1 / L ) (cid:112) F /µ ∼ MHz; here we have used the length L = 700 nm measured with SEM after the experiment, µ (cid:39) . · − kg / m for the mass per unit length of the tube with diameter d = 1 . nm,and F (cid:39) . nN. In the solid phase, helium atoms are preferably localized above the centersof carbon hexagons, see Fig. S3. Upon localization of the atoms, there occurs a change inthe surface energy which enhances the tension F of CNT. The magnitude of this additionaltension can be estimated approximatively using the total energy density of adsorption, δ F ∼ πd E ρ (cid:39) pN where we have taken coverage / ( ρ = 6 . nm − ), and the adsorption energy E = − E , = 137 K per atom [14].One can obtain a more accurate estimate for the helium-induced tension from the well-known adsorption potential for helium atom on graphite U ( z ) [14]. Owing to the zero pointmotion, and due to the asymmetry of the adsorption potential, the equilibrium location of theHe atom is not at the potential minimum, but at an average height about 0.2 ˚A further from the25arbon lattice [14, 38], as it is seen in the Fig. S4A. Consequently, there is an attracting forcebetween He and C atoms, which is balanced by the force coming from the zero point motion.This force results in a change of the unit cell size and inducing an extra tension in the lattice.The magnitude of the force can be estimated by differentiating the potential U ( z ) : f || ( z ) =(1 / dU/dz ( z ) · tan φ = (1 / dU/dz ( z ) · ( a/z ) where f || is the longitudinal component of the(attracting) force acting from helium atom at the average height z on a single neighboring carbonatom, see Fig. S4. With the interatomic distance in carbon lattice a = 1 . ˚A and averagedistance from nanotube to helium atom z = 2 . ˚A we find f || = 1 . pN. Now, assumingagain the 1/3 phase, we calculate the additional net tension over the cross-section of the tube δ F = F || = ( πd ) / (3 √ a/ f || = 28 pN (see Fig. S4), in a close agreement with the crudeevaluation made in the previous paragraph. The estimated additional tension gives about 2 %for the relative change of the resonant frequency. This change is close to what we observe inthe frequency across the quantum phase transition (QPT) from / phase to gliding solid stateas seen in Fig. 3 of the Main text.We note that both estimations set an upper limit for the helium-induced tension: the energydensity approach disregards the mutual disposition of helium and carbon atoms, while the gra-dient method takes into account only nearest neighbors. With these reservations, we assert thatthe estimations are well in accordance with our experimental results displayed in Fig. 3 of theMain text. The additional tension is strongest in the most symmetric 1/3 phase, while it is muchweaker in the fluid and in the ”gliding solid” states in which helium atoms are delocalized.We emphasize that the additional tension due to adsorbed helium atoms is inherently aquantum effect. A classical particle would remain at the minimum of the adsorption potentialand would not exert a perpendicular force on the adsorbing atoms. A quantum particle, however,cannot be localized, and its average position in an asymmetric potential will be shifted awayfrom the minimum due to zero-point oscillation.26nlike normal solid phase, uniform fluid and gliding solid states do not produce additionalnet force on individual carbon atoms, as helium atoms are mobile in these phases and, on aver-age, distributed uniformly. However, all phases possess internal pressure which also increasesthe tension F of the nanotube. To estimate this contribution, we assume that the characteristicpressure in the 2D solid helium is of the order of bulk He solidification pressure, 34 bar, mul-tiplied by the thickness of helium layer, c ∼ ˚A, P D = P sol c ∼ . N / m. For a CNT withdiameter of d = 1 . nm, this contributes δ F = πdP D = 5 pN to the tension of the tube. Bycomparing the additional tension δ F with the tension F (cid:39) . nN of the bare tube, we find thatthe pressure of adsorbed helium layer can give upto 1 MHz increase of the resonant frequency.The additional tension due to helium pressure depends both on temperature (due to thermalexpansion) and on helium density of the adsorbed solid helium (due to compressibility). Indeed,in solid phases at ρ > . nm − at T < . K we have observed intriguing dependence of theresonant frequency on temperature, which includes both linear and quadratic behavior (Fig. 4Aof the Main text and Fig. S5A).Our nanotube is operating mechanically in the regime where the gate voltage dominates thetension in the tube and the frequency increases with V g . In this regime, the resonant frequency F = (1 / L ) (cid:112) F /µ of the tube will then be changed according to δF F = 12 δ FF . (5)The two most important contributions from the adsorbed helium to F ( F ) of the oscillator are:the additional force F || in the commensurate normal solid described above, and the heliumpressure P which contributes directly to the tension, δ F ( T ) = F || + 2 πP ( T ) . (6)The first term is temperature-independent at low T , and it provides the large shift of the centralfrequency F in the commensurate monomer 1/3 phase, which disappears at the transition to27he liquid (with the increase of temperature) or to the gliding dimer solid (with the increase ofdensity). The second term is present in all phases, and it can be estimated as proportional to T in normal solid due to one-dimensional phonons, as we have indeed observed in the experiment(see Fig. 4A of the Main text). Transverse phonons with characteristic energies of (cid:126) c/d ∼ Kare not excited at our temperatures; c (cid:39) m / s denotes the estimated speed of sound [13].At coverages ρ > . nm − the quadratic behavior of the resonant frequency F as a func-tion of temperature abruptly changes to a linear one (Fig. 4A of the Main text and Fig. S5A)pointing to a quantum phase transition to another solid phase where the excitation spectrum isquite different. The pressure P which is proportional to δF is now linear in temperature, cor-responding to a phase with constant specific heat. One such phase could be the supersolid state,while another possible solid state with P ∝ T stems from the proposal by Andreev and Lif-shitz on a delocalized system of weakly interacting vacancies which have an exitation spectrum ε = ε + (cid:126) k / m ∗ ( ε is the activation energy of vacancy) [5]. In solid He on CNT the vacan-cies are topologically stable due to the mismatch of solid helium superlattice, and as such, theyare attributes of the ground state and have ε = 0 . In this case, they would have exactly the sameproperties as an ideal gas with constant number of particles, having temperature-independentheat capacity. 28IG. S1: Electrical characteristics of the nanotube sample.
Most of the data was measuredon a parallel combination of two chips with preselected tubes: ( A ) Decimal logarithm of thetransconductance dI/dU gate in Siemens as a function of the bias and gate voltages. The twinstructure of the transconductance peaks reflects the fact that there are two parallel tubes withslightly different Coulomb energies. One example of working points [ U gate , U bias ] is marked bya red circle. ( B ) DC I-V curves at two different values of the gate voltage.29IG. S2: Mechanical resonance spectra.
The two mechanical resonances on which most ofthe data were obtained. Thin curves are fits using the differentiated Lorentzian form of Eq. 2,while the fits illustrated by the thick lines include also dephasing fluctuations. For determinationof the resonance frequencies in this work, the former method without dephasing was employed.The data was measured on He coverage ρ = 1 . nm − at the base temperature 20 mK of thedilution refrigerator. 30IG. S3: B asic commensurate solid phases. Patterns of helium atoms adsorbed on a planehexagonal carbon lattice. ( A ) Monomer 1/3 phase observed on graphite [13]. ( B ) The intro-duced here most symmetric 3/8 dimer phase. ( C ) 2/5 dimer phase proposed by Greywall [15];both 3/8 and 2/5 phases, together with topologically-induced defects, may form the backbone ofthe gliding solid. ( D ) 3/7 phase suggested by Greywall [15]. ( E ) Striped 1/2 phase. In the planargeometry this is the maximum density of helium on the first layer, above which the second layerstarts to form. 31IG. S4: Tension of the nanotube due to adsorbed helium atoms. ( A ) Adsorption potential U ( z ) (thick curve) [14], the ground state, and the wave function profile along z -axis. Projectionof attractive force f z acting from single carbon atom of CNT to an adsorbed helium atom iscalculated as (1/6) of the derivative − dU/dz of the adsorption potential. ( B ) Longitudinalcomponent of the force f || = f sin φ = − (1 / dU/dz tan φ . ( C ) Setting of the calculation ofthe additional tension due to the adsorbed helium in the 1/3 phase. The net force acting on thecross-section of the tube is ( πd/l ) · f || = 4 πdf || / (3 √ a ) ( d is the diameter of the tube, l is theperiodicity of the 1/3 superlattice marked in the picture).32IG. S5: Power law fits across quantum phase transition. ( A ) Exponent β of the power lawfit F ( T ) = F (0) + a β T β of the data shown in Fig. 4A of the Main text. ( B ) Prefactor a β of thefit F ( T ) = F (0) + a β T β with exponents distributed around β = 2 . ± . for normal solid at ρ < . nm − , and with β = 1 . ± . for gliding solid state at ρ > . nm − . References [1] P. Kapitza, Viscosity of liquid helium below the λ -point. Nature , 74 (1938).[2] J. F. Allen, A. D. Misener, Flow of liquid helium II.
Nature , 75 (1938).[3] D. D. Osheroff, R. C. Richardson, D. M. Lee, New magnetic phenomena in liquid He3below 3 mK.
Phys. Rev. Lett. , 885-–888 (1972).[4] A. J. Leggett, Interpretation of recent results on He3 below 3 mK: a new liquid phase? Phys. Rev. Lett. , 1227–123 (1972).[5] A. F. Andreev, I. M. Lifshitz, Quantum theory of defects in crystals. Sov. Phys. JETP ,1107–1113 (1969). 336] A. J. Leggett, Can a solid be ”superfluid”? Phys. Rev. Lett. , 1543–1546 (1970).[7] G. V. Chester, Speculations on Bose-Einstein condensation and quantum crystals. Phys.Rev. A , 256–258 (1970).[8] E. Kim, M. H. W. Chan, Probable observation of a supersolid helium phase. Nature ,225–227 (2004).[9] E. Kim, M. H. W. Chan, Observation of Superflow in Solid Helium,
Science , 1941–1944 (2004).[10] J. Day and J. Beamish, Pressure-driven flow of solid helium.
Phys. Rev. Lett. , 105304(2006).[11] M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof’ev, B. V. Svistunov, M. Troyer, Thefate of vacancy-induced supersolidity in He.
Phys. Rev. Lett. , 080401 (2006).[12] S. Sasaki, R. Ishiguro, F. Caupin, H. J. Maris, S. Balibar, Superfluidity of Grain Bound-aries and Supersolid Behavior, Science , 1098–1100 (2006).[13] M. Bretz, J. G. Dash, D. C. Hickernell, E. O. McLean, O. E. Vilches, Phases of He and He monolayer films adsorbed on basal-plane oriented graphite.
Phys. Rev. A , 1589–1615 (1973).[14] H. Godfrin, H.-J. Lauter, ”Experimental properties of He adsorbed on graphite” in
Progress in Low Temperature Physics, Volume XIV , W. P. Halperin Ed. (Elsevier, 1995),pp. 213–320.[15] D. S. Greywall, Heat capacity and the commensurate-incommensurate transition of Headsorbed on graphite.
Phys. Rev. B , 309–318 (1993).3416] S. Mishra, D. Beyer, K. Eimre, S. Kezilebieke, R. Berger, O. Gr¨oning1, C. A. Pignedoli1,K. M¨ullen, P. Liljeroth, P. Ruffieux, X. Feng, R. Fasel, Topological frustration inducesunconventional magnetism in a nanographene. Nature Nanotechnology , 22–28 (2020).[17] J. L´eonard, A. Morales, P. Zupancic, T. Esslinger, T. Donner, Supersolid formation in aquantum gas breaking a continuous translational symmetry, Nature , 87-–90 (2017).[18] K.-D. Morhard, J. Bossy, H. Godfrin, NMR measurements on He adsorbed on Grafoil atsubmonolayer coverages and millikelvin temperatures.
Phys. Rev. B , 446–454 (1995).[19] D. Sato, K. Naruse, T. Matsui, H. Fukuyama, Observation of self-binding in monolayer He.
Phys. Rev. Lett. , 235306 (2012).[20] J. Chaste, A. Eichler, J. Moser, G. Ceballos, R. Rurali, A. Bachtold, A nanomechanicalmass sensor with yoctogram resolution.
Nature Nanotechnology , 301–304 (2012).[21] Z. Wang, J. Wei, P. Morse, J. G. Dash, O. E. Vilches, D. H. Cobden, Phase transitions ofadsorbed atoms on the surface of a carbon nanotube. Science , 552–555 (2010).[22] G. Y. Chen, T. Thundat, E. A. Wachter, R. J. Warmack, Adsorption-induced surface stressand its effects on resonance frequency of microcantilevers.
J. Appl. Phys. , 3618-3622(1995).[23] A. F. Andreev, ”Quantum crystals” in Progress in Low Temperature Physics, Volume VIII ,D. F. Brewer Ed. (North Holland, 1982), pp. 67–132.[24] N. Wei, P. Laiho, A. T. Khan, A. Hussain, A. Lyuleeva, S. Ahmed, Q. Zhang, Y. Liao, Y.Tian, Er-X. Ding, Y. Ohno, E. I. Kauppinen, Fast and ultraclean approach for measuringthe transport properties of carbon nanotubes.
Adv. Funct. Mater. , 1907150 (2019).3525] J.-P. Kaikkonen, A. T. Sebastian, P. Laiho, N. Wei, M. Will, Y. Liao, E. I. Kauppinen, P. J.Hakonen, Suspended superconducting weak links from aerosol-synthesized single-walledcarbon nanotubes. Nano Research , 1–6 (2020).[26] V. Gouttenoire, T. Barois, S. Perisanu, J.-L. Leclercq, S. T. Purcell, P. Vincent, A. Ayari,Digital and FM demodulation of a doubly clamped single-walled carbon-nanotube oscil-lator: Towards a nanotube cell phone. Small , 1060–1065 (2010).[27] A. N. Cleland, Foundations of Nanomechanics (Springer, 2003), pp. 233–236 .[28] G. A. Steele, A. K. H¨uttel, B. Witkamp, M. Poot, H. B. Meerwaldt, L. P. Kouwenhoven, H.S. J. van der Zant, Strong coupling between single-electron tunneling and nanomechanicalmotion.
Science , 1103–1107 (2009).[29] B. Lassagne, Y. Tarakanov, J. Kinaret, D. Garcia-Sanchez, A. Bachtold, Coupling mechan-ics to charge transport in carbon nanotube mechanical resonators.
Science , 1107–1110(2009).[30] B. Dzyubenko, H.-C. Lee, O. E. Vilches, D. H. Cobden, Surface electron perturbationsand the collective behaviour of atoms adsorbed on a cylinder.
Nature Physics , 398–402(2015).[31] J. Beyer, M. Schmidt, J. Engert, S. AliValiollahi, H. J. Barthelmess, Reference measure-ments of SQUID-based magnetic-field fluctuation thermometers. Supercond. Sci. Tech. ,065010 (2013).[32] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics, Volume III (Quantum Me-chanics) (Pergamon Press, 1969). 3633] W. E. Carlos, M. W. Cole, Band structure and thermodynamic properties of He atoms neara graphite surface.
Phys. Rev. B , 3713–3720 (1980).[34] B. Brami, F. Joly, C. Lhuillier, Is there a liquid phase in a (sub)monolayer of He adsorbedon graphite?
J. Low Temp. Phys. , 63–76 (1994).[35] P. A. Whitlock, G. V. Chester, B. Krishnamachari, Monte Carlo simulation of a heliumfilm on graphite. Phys. Rev. B , 8704–8715 (1998).[36] A. D. Lueking, M. W. Cole, Commensurate phases of gases adsorbed on carbon nanotubes. Phys. Rev. B , 195425 (2007).[37] G. Y. Gor, N. Bernstein, Revisiting Bangham’s law of adsorption-induced deformation:changes of surface energy and surface stress. Phys. Chem. Chem. Phys. , 9788-9798(2016).[38] F. Joly, C. Lhuillier, B. Brami, The helium-graphite interaction. Surf. Sci.264