Multielectron Bubbles In Liquid Helium Research Articles

Ultrasound-induced generation of multielectron bubbles in liquid helium

Multielectron bubbles provide a unique platform to study electrons in two dimensions and on curved surfaces, at densities which cannot be accessed using electrons on bulk helium or in semiconductor interfaces. Usually, MEBs are created by applying a large electric field and thereby inducing electrohydrodynamical instability on a charged surface of liquid helium. In the present study, we describe a method to create instability of the charged surface using ultrasound, in the presence of small electric fields. The ultrasound was applied close to the charged liquid-vapor interface, resulting in the formation of a liquid column, which breaks into liquid droplets. The mechanical impact of the droplets falling back into the bulk liquid resulted in the formation of highly charged multielectron bubbles. We estimated the initial charge density of the bubbles above the lambda point to be close to ${10}^{13}\phantom{\rule{0.16em}{0ex}}\mathrm{electrons}/{\mathrm{m}}^{2}$.

Kudryashov and Sinelshchikov’s method for solving the radial oscillation problem of multielectron bubbles in liquid helium

This work investigates the exact general analytical solution of the Rayleigh equation for multielectron bubbles in liquid helium using Kudryashov and Sinelshchikov’s method. We firstly obtain its first integral involving three negative exponential powers, then a specific Sundman transformation is proper established to transform it into an equation for the elliptic functions, and finally the analytical solution expressed by Weierstrass elliptic function is constructed appropriately. As applications, the derived analytical solution is used to test numerical algorithm, and also to construct the analytical expressions of the bubble oscillation period and the derivatives of the bubble radius. Further, the influence of the pressure on the helium is also discussed.

The Creation of Long-Lived Multielectron Bubbles in Superfluid Helium

Multielectron bubbles (MEBs) in liquid helium were first observed in the late 1970s, but their properties have never been explored experimentally due to their short lifetimes. MEBs in liquid helium are predicted to have dynamic instabilities for zero or positive pressures, and stability for negative pressures. We report the production of long-lived MEBs in a novel cell filled with helium at static negative pressures. MEBs were extracted from the vapor sheath of a heated filament loop embedded in the superfluid helium and were observed by high-speed photography as they rose in the helium under buoyant forces. In earlier studies we found that MEBs created in this way had large amplitude oscillations and were unstable to decay. By creating MEBs at temperatures just under the lambda point, these oscillations are rapidly damped and the MEBs relax toward a spherical shape and stability as they rise in the helium.

Studying electrons on curved surfaces by trapping and manipulating multielectron bubbles in liquid helium.

Investigations of two-dimensional electron systems (2DES) have been achieved with two model experimental systems, covering two distinct, non-overlapping regimes of the 2DES phase diagram, namely the quantum liquid phase in semiconducting heterostructures and the classical phases observed in electrons confined above the surface of liquid helium. Multielectron bubbles in liquid helium offer an exciting possibility to bridge this gap in the phase diagram, as well as to study the properties of electrons on curved flexible surfaces. However, this approach has been limited because all experimental studies have so far been transient in nature. Here we demonstrate that it is possible to trap and manipulate multielectron bubbles in a conventional Paul trap for several hundreds of milliseconds, enabling reliable measurements of their physical properties and thereby gaining valuable insight to various aspects of curved 2DES that were previously unexplored.

High Speed Imaging of Generation and Collapse of Multielectron Bubbles in Liquid Helium

We present experimental results on the generation and collapse of multielectron bubbles in liquid helium. By applying voltage pulses to a tungsten tip above the surface of the liquid, millimetre sized deformations were formed. Using high speed photography, we have imaged the disintegration of these deformations into bubbles of sizes ranging from ten to few hundred microns. At temperatures less than 2 K, the bubbles split into smaller bubbles and then disappeared in a time scale of few milliseconds. Smaller bubbles were formed at temperatures around 3 K, but were visible for more than hundreds of milliseconds. Although we have not been able to measure their charge directly, some of these bubbles responded to electric fields, implying these were indeed multielectron bubbles. With the existing theoretical picture, it is not possible to understand the strong dependence of the lifetime of multielectron bubbles on temperature.

Homotopy Analysis Method for the Rayleigh Equation Governing the Radial Dynamics of a Multielectron Bubble

The homotopy analysis method is used to obtain analytical solutions of the Rayleigh equation for the radial oscillations of a multielectron bubble in liquid helium. The small order approximations for amplitude and frequency fit well with those computed numerically. The results confirm that the homotopy analysis method is a powerful and manageable tool for finding analytical solutions of strongly nonlinear dynamical systems.

“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.

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.

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.

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.

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.

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.

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.

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.

Hexatic order and surface ripples in spherical geometries.

In flat geometries, two-dimensional hexatic order has only a minor effect on capillary waves on a liquid substrate and on undulation modes in lipid bilayers. However, extended bond-orientational order alters the long-wavelength spectrum of these ripples in spherical geometries. We calculate this frequency shift and suggest that it might be detectable in lipid bilayer vesicles, at the surface of liquid metals, and in multielectron bubbles in liquid helium at low temperatures. Hexatic order also leads to a shift in the threshold for the fission instability induced in the later two systems by an excess of electric charge.

Unexpected hydrodynamic behavior of multielectron bubbles in liquid helium I

We have investigated multielectron bubbles (with charges Z ~ 106 electrons) in normal liquid 4He. The drift velocity in an electric field and the charge of the bubbles were measured simultaneously. We found the drag coefficient to be over an order of magnitude higher than expected from ordinary hydrodynamics. The velocities did not show a correlation with the charge and could vary up to a factor of 3 for equally charged bubbles.

Properties of multielectron bubbles in liquid helium

We report charge and velocity measurements on multielectron bubbles in liquid helium. Oscillations and fissioning of bubbles are observed.