Layering transitions in superfluid helium adsorbed on a carbon nanotube mechanical resonator

TL;DR

This study investigates layering transitions in superfluid helium films on a carbon nanotube, revealing first-order phase transitions and high-quality multilayer formation, with implications for quantum fluid dynamics on cylindrical surfaces.

Contribution

It provides the first direct observation of layering transitions in helium films on nanotubes using mechanical resonance measurements.

Findings

Demonstrated helium layer formation up to five atoms thick

Observed layer-by-layer growth with discontinuities indicating first-order transitions

Achieved unprecedented quality of helium multilayers on nanotubes

Abstract

Helium is recognized as a model system for the study of phase transitions. Of particular interest is the superfluid phase in two dimensions. We report measurements on superfluid helium films adsorbed on the surface of a suspended carbon nanotube. We measure the mechanical vibrations of the nanotube to probe the adsorbed helium film. We demonstrate the formation of helium layers up to five atoms thickness. Upon increasing the vapour pressure, we observe layer-by-layer growth with discontinuities in both the number of adsorbed atoms and the speed of sound in the adsorbed film. These hitherto unobserved discontinuities point to a series of first-order layering transitions. Our results show that helium multilayers adsorbed on a nanotube are of unprecedented quality compared to previous works. They pave the way to new studies of quantized superfluid vortex dynamics on cylindrical surfaces, of the Berezinskii-Kosterlitz-Thouless phase transition in this new geometry, perhaps also to supersolidity in crystalline single layers as predicted in quantum Monte Carlo calculations.

Figures

Equations

$$\delta f_0(T)=-0.074\frac{1}{m_{\mathrm{NT}}{f}_0}\frac{r_{\mathrm{He}}}{L}\frac{(k_{\mathrm{B}}T{)}^3}{(\hslash c{)}^2}$$

Full text

When exposing graphite to a helium vapor at low temperature, a helium film forms on the graphite surface. Several experiments have shown that the thickness of this film grows layer by layer as a function of the vapor pressure. These layers are one atom thick Greywall1991 ; Greywall1993 ; Chan1992 ; Reppy1996 ; Saunders2017 , and there are successive “layering transitions” between layers n and n+1. Clements et al. Clements1996 ; Clements1993 had predicted that the layering transitions are first order transitions, meaning that the coverage should show a series of sharp discontinuities as a function of the helium pressure. However, such discontinuities could not be observed, probably because previous techniques required large surface areas, which were obtained e.g. by chemically exfoliating graphite and by subsequently pressing the material into sheets. As a result, the substrate surface was made of small crystalline platelets of $\sim(10$nm$)^{2}$ area Reppy1996 and probably contained a sizeable amount of defects including wedges where liquid helium could accumulate. Despite the modest quality of these surfaces, evidences for layer-by-layer growth were reported but without any visible discontinuities Godfrin1995 ; Chan1992 ; Reppy1996 ; Saunders2017 ; Menachekanian2019 .

By studying the mechanical resonance of a single wall carbon nanotube (NT) we have observed discontinuities in the adsorbed mass as a function of the injection of helium atoms and demonstrated for the first time that the layering transitons are indeed first order. A carbon nanotube is an excellent substrate, since the crystalline quality is high, and such nanotubes can be made free of adsorbed contamination Tavernarakis2014 . Our results prove the very high quality of carbon nanotubes as potential substrates for new original studies like quantized superfluid vortex dynamics on cylindrical surfaces, or the Berezinskii-Kosterlitz-Thouless phase transition in a new geometry, perhaps also of supersolidity in crystalline single helium layers of variable density as predicted by Gordillo et al.Gordillo2011 .

Recent advances in nanomechanics have already allowed studies of liquids at small scales Favero2015 ; Kashkanova2017 ; Carmon2016 ; Harris2016 ; Bowen2016 but the study of the adsorption of superfluid helium on a single carbon nanotube is original (Fig. 1a). We fabricate mechanical resonators by suspending a single NT of 3 nm diameter and 1.1 $\mu$m length. This NT is contacted to two platinum electrodes and is capacitively coupled to a gate electrode (Fig. 1b). In order to suppress any surface contamination, we grow the nanotube in the last step of the fabrication process and we anneal it by passing a large current (6 $\mu$A) through it in the dilution refrigerator at 20 mK prior to the adsorption of helium. This procedure already allowed us to grow xenon monolayer crystals that were commensurate with the nanotube lattice Tavernarakis2014 . The nanotube vibrates as a doubly-clamped string. The mechanical vibrations of the fundamental mode of the nanotube are driven capacitively and measured electrically (Fig. 1c) McEuen2004 ; Moser2014 ; Bonis2018 . The resonance frequency can be tuned by a large amount around 30 MHz when tuning the gate voltage (Figs. 1c and 1d). The quality factor deduced from the thermal motion linewidth is $\approx 2\cdot 10^{5}$. We inject 4He from room temperature into the sample cell through a capillary.

A mechanical resonator based on a suspended nanotube is a remarkable sensor of atoms adsorbed on its surface Chaste2012 ; Bockrath2008 ; Cobden2010 ; Cobden2012 . The mass detection can reach single atom resolution Chaste2012 . Driving the resonator to detect adsorbed atoms does not affect the dynamics of the adsorbates, since the amplitude of the driven vibrations can be kept smaller than the amplitude of thermal vibrations. The resonance frequency $f_{0}=\frac{1}{2\pi}\sqrt{\frac{K}{M}}$ depends on the ratio of the string elasticity also called spring coefficient $K$ to the effective mass $M$. The coefficient $K=k_{\mathrm{NT}}+k_{\mathrm{He}}$ is the sum of the respective elasticities of the naked NT- also called pristine - and of the adsorbed helium. Similarly the total mass $M=m_{\mathrm{NT}}+m_{\mathrm{He}}$ is the sum of the respective masses of the NT and of the adsorbed helium. With helium adsorbed, both $K$ and $M$ increase but their contributions to the resonance frequency have opposite signs, so that we could distinguish between them by varying the gate voltage. This is a great advantage of our experiment. Previous studies of two-dimensional helium superfluids often used torsional oscillators Reppy1978 ; Reppy1996 ; Saunders2017 . Contrary to our case, these torsional oscillators were macroscopic with not much possibility to change their resonance frequency so that mass effects were difficult to distinguish from elasticity effects Reppy2014 . As we shall see below, we have observed the two effects in two different temperature regimes.

By studying the transport properties of electrons inside the NT, we have demonstrated that our nanotubes are free of disorder. As a function of the gate voltage, the conductance is periodically modulated due to quantum electron interference Liang2001 . This periodic modulation would be deteriorated by a tiny amount of disorder (see section I of Supplementary Information).

We measured the nanotube length $L\simeq 1.1~{}\mu$m from the characteristic source-drain voltage $V_{\mathrm{C}}=1.5$ mV of the electron interference pattern. This length is $L=hv_{\mathrm{F}}/2eV_{\mathrm{C}}$ Liang2001 where $v_{\mathrm{F}}=8\times 10^{5}$ m/s is the Fermi velocity of nanotubes. It is consistent with the width of the trench and we obtained a similar but less reliable length ($L\approx 1~{}\mu m$) from the Coulomb blockade measurements of the conductance.

We observe a series of steps when measuring the resonance frequency of the nanotube as a function of the number $N_{\mathrm{He}}$ of helium atoms injected into the sample chamber (Fig. 1e). We proceed by successive injections of $\approx$ 76 cm3 of helium gas at a pressure set between 0.08 and 3 mbar. When we increase $N_{\mathrm{He}}$, the vapour pressure $P$ in the surrounding of the nanotube gets larger, but $P$ cannot be quantitatively estimated due to the unknown adsorption of helium atoms onto the cell walls. We attribute the reduction of the resonance frequency $\Delta f_{0}$ measured in Fig. 1e to the helium mass $m_{He}$ adsorbed on the NT. Each plateau in $f_{0}$ is assigned to a different layer adsorbed on the nanotube. At the completion of each layer, a new layer starts and the resonance frequency jumps down. It means a divergence of the coverage at constant helium pressure, that is, a divergence of the compressibility and a vanishing of the sound speed in the helium layer. This is in agreement with the first-order transition predicted for the layering transition Clements1996 . The helium density at each discontinuity agrees within 6 % with the established density of helium adsorbed on graphite (Section III of Supplementary Information). During the completion of each layer, helium atoms must be homogeneously distributed over the surface and the chemical potential of the helium vapor needs to increase to equilibrate the increase of the repulsion betwen atoms. This explains the observed finite slope $\partial f_{0}/\partial N_{\mathrm{He}}$ of the plateaus (Fig. 1e).

Figure 2a shows the layer-by-layer desorption of helium from the surface of the nanotube, but now as a function of temperature up to 10 K and for 5 successive amounts of helium in the cell. The frequency of the steps are similar to those measured while varying $N_{\mathrm{He}}$ at constant temperature (Fig. 1e). One temperature sweep takes a few days to ensure the reproducibility of measurements. The layering transitions are rounded, probably because the layering transitions have a critical point of order 1 K (see Ref. Ramesh1984 ). By varying the gate voltage, consequently the spring constant, we have confirmed that the variation of the resonance frequency $f_{0}$ in Fig. 2a is due to the change in the resonator mass. Indeed the large frequency change $\Delta f_{0}$ is an increasing function of $f_{0}$ (Fig. 2b and Supplementary Section IV).

The first and second layers show some peculiar structures which would need further study to be compared with predictions by Boronat et al. (see Ref. Boronat2012 and Section VI of Supplementary Information). We thus focus on the third and fourth layers. By looking more precisely at the frequency variation below 0.7 K, we discovered the existence of a small minimum near 0.5 K (see Fig. 2c). The rise of $f_{0}$ above $\sim$0.5 K in Fig. 2c signals the desorption of helium atoms from the nanotube. Figure 2d shows the dependence of the depth $\delta f_{0}$ of this small minimum on the gate voltage. Contrary to the case of the large frequency shift $\Delta f_{0}$ of Fig. 2b, the slope is negative (see Fig. 2d and Supplementary Section IV). It means that this small minimum is due the helium contribution to the total spring constant of the resonator.

We attribute it to the temperature variation of the helium surface tension, that is, to the growing entropy of the helium film Atkins1965 . It is due to the third sound states in the case of thin superfluid films Atkins1959 . These waves are predicted to be longitudinal with no sizeable displacement of helium atoms in the direction perpendicular to the surface, except for densities near the layering transition Clements1994 . The third sound has a linear energy dispersion given by the speed $c$. The resulting dependence of $\delta f_{0}$ on temperature $T$ is

[TABLE]

where $m_{\mathrm{NT}}$ is the mass of the pristine nanotube, $L$ the suspended nanotube length, $k_{\mathrm{B}}$ the Boltzmann constant, and $\hbar$ the reduced Planck constant. See Section V of the Supplementary Information for a detailed derivation.

Figures 3b and 3e show a very good agreement with the $T^{3}$ law of Eq. (1) from which we could extract the sound velocity $c$ (Figs. 3c and f). The layer fraction are computed by comparing the resonance frequency at 20 mK to the frequency of the jump indicated by the dashed line on Fig. 1e. We notice a large jump in sound velocity from the end of the 3rd layer (30 m/s) to the beginning of the 4th layer (210 m/s), in agreement with the predictions of a first order transition by Clements et al. Clements1993 ; Clements1996 . Although we found no evidence of ripplon excitations in our system, studying thicker helium layers may lead to their observation in future work.

Our work shows that helium layers adsorbed on carbon nanotubes are of unprecedented quality. This system is of great interest for the study of different quantum phenomena. The cylindrical boundary condition of the superfluid imposes quantized translational velocities of a vortex around the circumference Guenther2017 . This boundary condition also modifies the vortex-vortex interaction energy Guenther2017 , so that it might alter the Berezinskii-Kosterlitz-Thouless topological phase transition. According to Gordillo et al. Gordillo2011 , helium monolayers adsorbed on nanotubes may become supersolid at low temperature. In summary, carbon nanotubes could be used for original studies of helium-based quantum phenomena.

Acknowledgments We thank N. Guenther, P. Massignan, A. Fetter, and G. A. Williams for fruitful discussions. This work is supported by the ERC advanced grant 692876, the Foundation Cellex, the CERCA Programme, AGAUR, Severo Ochoa (SEV-2015-0522), the grant FIS2015-69831-P of MINECO, and the Fondo Europeo de Desarrollo Regional (FEDER). M.C.G. acknowledges partial financial support from the MINECO (Spanish Ministry of Economy) Grant No. FIS2017-84114-C2-2-P. M.C.G. also acknowledges the use of the C3UPO computer facilities at the Universidad Pablo de Olavide. J.B. acknowledges financial support from MINECO Grants FIS2014-56257-C2-1-P and FIS2017-84114-C2-1-P.