Multielectron Bubbles Research Articles

Novel methods to create multielectron bubbles in superfluid helium

An equilibrium multielectron bubble (MEB) in liquid helium is a fascinating object with a spherical two-dimensional electron gas on its surface. We discuss two ways in which they have been created. For MEBs that have been observed in the dome of a cylindrical cell with an unexpectedly short lifetime, we show analytically why these MEBs can discharge by tunneling. Using a novel method, MEBs have been extracted from a vapor sheath around a hot filament in superfluid helium by applying electric fields up to 15 kV∕cm, and photographed with high-speed video. Charges as high as 1.6×10(-9) C (∼10(10) electrons) have been measured. The latter method provides a means of capture in an electromagnetic trap to allow the study of the extensive exciting properties of these elusive objects.

“Magic numbers” in the melting of a cluster of point charges on a sphere

The thermodynamic properties of a cluster of point Coulomb charges on a sphere have been analyzed using the Monte Carlo method for the number of charges 20 ≤ N ≤ 90. The ground state of the system of charges is described in the model of a closed quasi-two-dimensional triangular lattice with topological defects. We have determined the dependence of the Lindeman parameter δL of this system on N and on the dimensionless parameter $$ \tilde T $$ , which is proportional to the temperature T and to the radius R of the cluster: $$ \tilde T = {{k_B T\varepsilon R} \mathord{\left/ {\vphantom {{k_B T\varepsilon R} {e^2 }}} \right. \kern-\nulldelimiterspace} {e^2 }} $$ , where ∈ is the dielectric constant of the medium and e is the charge of a particle. The “magic numbers,” i.e., the N values, for which the melting point of the closed triangular lattice of charges is much higher than those for neighboring N values, have been found. The evolution of the lattice-melting mechanisms with an increase in the number of charges N in a mesoscopic cluster has been analyzed. For N ≤ 32, the melting of the lattice does not involve dislocations (nontopological melting); this behavior of the mesoscopic system of charges on the sphere differs from the behavior of the extended planar two-dimensional system. At N ≳ 50, melting is accompanied by the formation of dislocations. The mechanism of dislocation-free non-topological melting of a closed lattice, which occurs at small N values and is associated with the cooperative rotational motion of “rings” of particles, has been analyzed. The model has various implementations in the mesoscopic region; in particular, it describes the system of electrons over the liquid-helium cluster, the liquid-helium cluster with incorporated charged particles, a multielectron bubble in liquid helium, a charged quantum dot, etc.

Thermionic emission and a novel electron collector in a liquid helium environment

We study two techniques to create electrons in a liquid helium environment. One is thermionic emission of tungsten filaments in a low temperature cell in the vapor phase with a superfluid helium film covering all surfaces; the other is operating a glowing filament immersed in bulk liquid helium. We present both the steady state and rapid sweep I-V curves and the electron current yield. These curves, having a negative dynamic resistance region, differ remarkably from those of a vacuum tube filament. A novel low temperature vapor-phase electron collector for which the insulating helium film on the collector surface can be removed is used to measure emission current. We also discuss our achievement of producing multielectron bubbles in liquid helium by a new method.

Theory of the stability of multielectron bubbles in liquid helium

We have performed numerical calculations to investigate the stability of multi-electron bubbles in liquid helium-4. In our calculations, for each bubble shape considered we find the equilibrium distribution of the electron gas on the inside surface of the bubble. For a stationary bubble, we find that whenever the pressure in the liquid is positive the bubble is unstable against fission into two bubbles. We find that a moving bubble can be stable within a certain region of the pressure-velocity plane. At the high pressure boundary of this region, the bubble undergoes fission, and on the low pressure boundary the bubble explodes.

Stability of multielectron bubbles in liquid helium

The stability of multi-electron bubbles in liquid helium is investigated theoretically. We find that multi-electron bubbles are unstable against fission whenever the pressure is positive. It is shown that for moving bubbles the Bernoulli effect can result in a range of pressures over which the bubbles are stable.

On the Correlation between Properties of Multielectron Dimples and Bubbles

Characteristics of multielectron dimples on the surface of liquid helium and those of multielectron bubbles arising in its bulk when the charged surface develops electrohydrodynamic instability are strongly interrelated. This circumstance allows one to explain a number of observable properties of multielectron bubbles in helium.

Critical Radius and Magnetic Field for Vortex Nucleation on a Superconducting Spherical Surface

Spherical, two-dimensional electron systems are realized in nanoshells and multielectron bubbles in helium. In this contribution, we investigate the superconducting state of such shells using a variational Ginzburg-Landau formulation. A phase diagram as a function of the sphere radius and the magnetic field is derived, and we calculate a critical radius below which vortices cannot be nucleated on the superconducting spherical surface.

Critical temperature and specific heat for Cooper pairing on a spherical surface

Based on an exact solution of the Bardeen-Cooper-Schrieffer type Hamiltonian on a spherical surface, we calculate the specific heat for the electron system with pair correlations on a sphere. We find that it is possible to extract from the specific heat a temperature above which many-body states with broken Cooper pairs get populated. Therefore, we define this temperature as the characteristic temperature signalling the onset of a BCS-type pair-correlated state for electrons on a spherical surface. Such spherical electron systems are realized in multielectron bubbles in liquid helium, for which the above-mentioned characteristic temperature is found to be of the order of 10-100 mK. Both the specific heat and the critical temperature show a pronounced (4-6%) odd-even parity effect that persists even for numbers of electrons as large as 10$^6$.

Cooper pairing and superconductivity on a spherical surface: Applying the Richardson model to a multielectron bubble in liquid helium

Electrons in a multielectron bubble in helium form a spherical, two-dimensional system coupled to the ripplons at the bubble surface. The electron-ripplon coupling, known to lead to polaronic effects, is shown to give rise also to Cooper pairing. A Bardeen-Cooper-Schrieffer (BCS) Hamiltonian arises from the analysis of the electron-ripplon interaction in the bubble, and values of the coupling strength are obtained for different bubble configurations. The BCS Hamiltonian on the sphere is analysed using the Richardson method. We find that although the typical ripplon energies are smaller than the splitting between electronic levels, a redistribution of the electron density over the electronic levels is energetically favourable as pairing correlations can be enhanced. The density of states of the system with pairing correlations is derived. No gap is present, but the density of states reveals a strong step-like increase at the pair-breaking energy. This feature of the density of states should enable the unambiguous detection of the proposed state with pairing correlations in the bubble, through either capacitance spectroscopy or tunneling experiments, and allow to map out the phase diagram of the electronic system in the bubble.

Multielectron Bubbles in Helium and Wigner Crystallization

Multielectron bubbles (MEBs) in superfluid helium support a two-dimensional electron gas (2deg) on the surface of a sphere. In principle, electron densities can range from ∼ 1010 to 1014/cm2. Wigner crystallization has been observed on a flat bulk surface of helium, but the maximum density of ∼ 2 × 109 is insufficient to observe quantum melting. MEBs have a number of fascinating properties. They should not only allow observation of the Wigner lattice and quantum melting, but at very high densities a new type of electron localization into a lattice called the ripplopolaron lattice is predicted. We discuss a number of phenomena and experimental techniques aimed at long-time stabilization of MEBs.

Magnetic Properties of a Multielectron Bubble in Liquid Helium

Multielectron bubbles in liquid helium contain a spherical two-dimensional shell of electrons at the bubble surface. We investigate the properties of this electron gas when the bubble is placed in a homogeneous magnetic field. Results are reported for the Fermi level and the magnetization of the bubble. At high field, we find the formation of Landau bands rather than Landau levels.

Sonoluminescence and collapse dynamics of multielectron bubbles in helium

Multielectron bubbles (MEBs) differ from gas-filled bubbles in that it is the Coulomb repulsion of a nanometer thin layer of electrons that forces the bubble open rather than the pressure of an enclosed gas. We analyze the implosion of MEBs subjected to a pressure step, and find that despite the difference in the underlying processes the collapse dynamics is similar to that of gas-filled bubbles. When the MEB collapses, the electrons inside it undergo strong accelerations, leading to the emission of radiation. This type of sonoluminescence does not involve heating and ionisation of any gas inside the bubble. We investigate the conditions necessary to obtain sonoluminescence from multielectron bubbles and calculate the power spectrum of the emitted radiation.

Properties of the spherical 2D electron system in a multielectron bubble in liquid helium

Multielectron bubbles in liquid helium contain a spherical two-dimensional (2D) electron gas. By varying the number of electrons or the pressure on the bubble, the surface density of electrons can be varied over a much larger range than that available for a 2D electron gas on a flat helium surface. The existence of a ripplopolaron solid phase for pressurized bubbles is discussed, and results are presented on the effective mass and the energy of the ripplopolarons. The predicted Wigner lattice of ripplopolarons will melt by dissociation of the ripplopolarons, resulting in an spherical 2D electron gas. We investigate the direct and exchange contributions of the interaction energy in this state in the Hartree–Fock formalism.

Stability of multielectron bubbles against single‐electron split‐off

AbstractRecently, the stability of multielectron bubbles against fissioning into two fragments with large numbers of electrons each was investigated by J. T. Devreese and two of the present authors (I. S. and J. T). In the present work we study a decay mechanism not included in the earlier model on bubble stability: splitting off of single electrons from the multielectron bubble. We find that bubbles with more than a critical number of electrons are stable against this type of decay. The pressure dependence of this critical number was studied. We find that this particular decay mechanism reveals a similar pressure dependence to other decay mechanisms such as shape instabilities in that increasing the pressure destabilizes the bubble whereas a negative pressure increases bubble stability.

Wigner lattice of ripplopolarons in a multielectron bubble in helium

The properties of ripplonic polarons in a multielectron bubble in liquid helium are investigated on the basis of a path-integral variational method. We find that the two-dimensional electron gas can form deep dimples in the helium surface, or ripplopolarons, to solidify as a Wigner crystal. We derive the experimental conditions of temperature, pressure and number of electrons in the bubble for this phase to be realized. This predicted state is distinct from the usual Wigner lattice of electrons: it melts by dissociation of the ripplopolarons when the electrons shed their localizing dimple as the pressure on the multielectron bubble drops below a critical value.

Coupled ripplon–plasmon modes in a multielectron bubble

In multielectron bubbles, the electrons form an effectively two-dimensional layer at the inner surface of the bubble in helium. The modes of oscillation of the bubble surface (the ripplons) are influenced by the charge redistribution of the electrons along the surface. The dispersion relation for these charge redistribution modes (‘longitudinal plasmons’) is derived and the coupling of these modes to the ripplons is analysed. We find that the ripplon–plasmon coupling in a multielectron bubble differs markedly from that of electrons on a flat helium surface. An equation is presented relating the spherical harmonic components of the charge redistribution to those of the shape deformation of the bubble.

Hexatic undulations in curved geometries.

We discuss the influence of two-dimensional hexatic order on capillary waves and undulation modes in spherical and cylindrical geometries. In planar geometries, extended bond-orientational order has only a minor effect on the fluctuations of liquid surfaces or lipid bilayers. However, in curved geometries, the long-wavelength spectrum of these ripples is altered. We calculate this frequency shift and discuss applications to spherical vesicles, liquid metal droplets, bubbles and cylindrical jets coated with surface-active molecules, and to multielectron bubbles in liquid helium at low temperatures. Hexatic order also leads to a shift in the threshold for the fission instability of charged droplets and bubbles, and for the Plateau-Rayleigh instability of liquid jets.

Fission of multielectron bubbles in liquid helium

The stability of multielectron bubbles (MEB's) in liquid helium is investigated using the liquid-drop model for fissioning nuclei. Whereas a critical positive pressure can make the bubble unstable against fissioning, a small negative pressure suffices to introduce a restoring force preventing any small deformation of the bubble to grow. We also find that there exists an energy barrier making MEB's metastable against fissioning at zero pressure. The results obtained here overcome the difficulties associated with the Rayleigh-Plesset equation previously used to study bubble stability, and shed new light on the limits of achievable bubble geometries in recently proposed experiments devised to stabilize MEB's.

Many-body properties of a spherical two-dimensional electron gas

We investigate the many-body properties of a two-dimensional electron gas constrained to the surface of a sphere, a system which is physically realized in, for example, multielectron bubbles in liquid helium. A second-quantization formalism, suited for the treatment of a spherical two-dimensional electron gas (S2DEG), is introduced. Within this formalism, the dielectric response properties of the S2DEG are derived, and we identify both collective excitations and a spectrum of single-particle excitations. We find that the single-particle excitations are constrained to a well-defined region in the angular-momentum--energy plane. The collective excitations differ in two important aspects from those of a flat 2DEG: on a sphere, the ``spherical plasmons'' have a discrete frequency spectrum and the lowest frequency is nonzero.

Effect of pressure on statics, dynamics, and stability of multielectron bubbles.

The effect of positive and negative pressure on the modes of oscillation of a multielectron bubble in liquid helium is calculated. Already at low pressures of the order of 10-100 mbar, these effects are found to significantly modify the frequencies of oscillation of the bubble. Stabilization of the bubble is shown to occur in the presence of a small negative pressure, which expands the bubble radius. Above a threshold negative pressure, the bubble is unstable.