Multielectron Bubbles Research Articles

Cascade of Multielectron Bubble Phases in Monolayer Graphene at High Landau Level Filling

Cascade of Multielectron Bubble Phases in Monolayer Graphene at High Landau Level Filling

Superfluid Rayleigh–Plesset extension of FLRW cosmology

Guided by the analogy with the Rayleigh–Plesset dynamics of multielectron bubbles in superfluid He-4, we consider the cosmological FLRW evolution equation with additional cubic and sixth powers of the inverse of the scale factor of the universe. For the barotropic parameter w=2∕3 (coasting universe), along with zero cosmological constant in the absence of viscous terms, by using the Sundman time as evolution parameter, we present parametric solutions for the scale factor of the universe in terms of rational expressions of Weierstrass elliptic functions and their particular cases thereof. For other values of the equation of state parameter w, such as w=−1, but also the same coasting case, we present a more standard discussion in the conformal time variable using solutions obtained by numerical integration.

Trapping Multielectron Bubbles Using a Point Paul Trap

We demonstrate an experimental setup to trap multielectron bubbles in bulk liquid He4 using a point Paul trap. The experimental configuration is based on dc and rf electric fields applied to planar electrodes such as to trap the bubbles at a certain location. We have performed numerical simulations to estimate the experimental parameters at which stable trapping can occur, which matched with the experimental results. Compared to the linear Paul traps reported earlier by our group, point Paul traps allow more efficient loading and better optical access for viewing the trapped bubbles, and will allow simultaneous storage and investigation of multiple electron bubbles in the future.

Stability of multielectron bubbles in high Landau levels

We study multielectron bubble phases in the $N=2$ and $N=3$ Landau levels in a high mobility GaAs/AlGaAs sample. We found that the longitudinal magnetoresistance versus temperature curves in the multielectron bubble region exhibit sharp peaks, irrespective of the Landau level index. We associate these peaks with an enhanced scattering caused by thermally fluctuating domains of a bubble phase and a uniform uncorrelated electron liquid at the onset of the bubble phases. Within the $N=3$ Landau level, onset temperatures of three-electron and two-electron bubbles exhibit linear trends with respect to the filling factor; the onset temperatures of three-electron bubbles are systematically higher than those of two-electron bubbles. Furthermore, onset temperatures of the two-electron bubble phases across $N=2$ and $N=3$ Landau levels are similar, but exhibit an offset. This offset and the dominant nature of the three-electron bubbles in the $N=3$ Landau level reveals the role of the short-range part of the electron-electron interaction in the formation of the bubbles.

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.

Multielectron Bubbles in Liquid Helium

Multielectron bubbles (MEBs) are cavities in liquid helium that contain electrons and possibly vapour. They provide a versatile platform for exploring properties of interacting electrons in two dimensions in a regime of densities that has not been previously available. Since the electrons are pushed against the deformable curved inner surface of the bubble, MEBs also provide an environment to study the roles of curvatures. In this review, we describe the current understanding of MEBs, with a focus on their stability issues, and highlight recent experimental studies of this interesting system.

Collapse of Vapor-Filled Multielectron Bubbles Held Against a Surface

Multielectron bubbles (MEBs) are cavities in liquid helium containing a layer of electrons pinned to the inner surface of the bubbles. Previous experimental work carried out with MEBs in bulk helium-4 above the lambda point showed MEBs can contain vapor, which condenses in a time approximately proportional to the volume of the bubble, and this observation was further confirmed by numerical simulations. In the present work, we describe experiments where the MEBs are held against a solid substrate. We found the rate of vapor condensation and therefore the speed of collapse of the bubble to be orders of magnitude faster compared to MEBs in bulk. We discuss a numerical model and the associated difficulties to explain this difference.

Observation of Cavitation on Electron Bubbles at Small Negative Pressures

Acoustic cavitation is a powerful technique to probe electron bubbles inside the liquid helium. The critical pressure to explode a bubble depends on the number and quantum state of electrons inside the bubble and if the bubble is trapped on a vortex. Here, we report cavitation events that occur at pressure magnitudes approximately 70% lower compared to single electron bubbles. We have considered various possibilities, e.g., single electron bubbles trapped on vortex lines or primary electrons depositing the energy at the acoustic focus and compared the results of our experiments with past measurements reported in the literature. We consider the possibility these new species of bubbles are multielectron bubbles with a small (< 20) number of electrons and discuss future experiments to confirm the same.

Electron bubbles and the structure of the orbital wave function

Stripe-like and bubble-like patterns spontaneously form in numerous physical, chemical, and biological systems when competing long-range and short-range interactions banish uniformity. Stripe-like and the related nematic morphology are also under intense scrutiny in various strongly correlated electron systems. In contrast, the electronic bubble morphology is rare. Some of the most intriguing electron bubbles develop in the two-dimensional electron gas subjected to a perpendicular magnetic field. However, in contrast to bubbles forming in classical systems such as the Turing activator-inhibitor reaction or Langmuir films, bubbles in electron gases owe their existence to elementary quantum mechanics: they are stabilized as wavefunctions of individual electrons overlap. Here we report a rich pattern of multi-electron bubble phases in a high Landau level and we conclude that this richness is due to the nodal structure of the orbital component of the electronic wavefunction.

Coalescence of Multielectron Bubbles in Liquid Helium

In 1977, Volodin et al. observed that the electrohydrodynamic instability of charged helium surfaces can lead to the loss of the electrons from the surface in the form of bubbles. These are Multielectron Bubbles (MEBs) which contain electrons pinned on their inner surfaces. The MEBs form a model system for studying electrons on curved surfaces, which are predicted to have many interesting properties. Recent experiments showed that above the Lambda point, MEBs could be trapped using a Paul-trap and their sizes are mainly determined by the amount of vapour present inside. Here, we report the experimental observation of the coalescence of two MEBs which were moving upward in bulk liquid helium-4. The charge and radii of the MEBs were determined before and after the coalescence. The merging of two similar charged bubbles was possible because of the presence of vapour inside the merging MEBs.

Collapse of Vapor-Filled Bubbles in Liquid Helium

Multielectron bubbles (MEBs) are charged cavities in liquid helium which provide an interesting platform for the study of electrons on curved surfaces. Very recently, we have reported an experiment to trap these objects in a two-dimensional Paul trap, where they could be observed from ten to hundreds of milliseconds. During this time, the vapor inside the bubble condensed which resulted in a steady reduction in their size such that beyond a certain time the MEBs could no longer be detected. In this paper, we present experimental data on the lifetime of the bubbles as a function of their initial radius and compare the results with a theoretical model.

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.

Stable Trapping of Multielectron Helium Bubbles in a Paul Trap

In a recent experiment, we have used a linear Paul trap to store and study multielectron bubbles (MEBs) in liquid helium. MEBs have a charge-to-mass ratio (between 10(-4) and 10(-2) C/kg) which is several orders of magnitude smaller than ions (between 10(6) and 10(8) C/kg) studied in traditional ion traps. In addition, MEBs experience significant drag force while moving through the liquid. As a result, the experimental parameters for stable trapping of MEBs, such as magnitude and frequency of the applied electric fields, are very different from those used in typical ion trap experiments. The purpose of this paper is to model the motion of MEBs inside a linear Paul trap in liquid helium, determine the range of working parameters of the trap, and compare the results with experiments.

Fission of Multielectron Bubbles in Liquid Helium Under Electric Fields

Multielectron bubbles (MEBs) are cavities in liquid helium which contain a layer of electrons trapped within few nanometres from their inner surfaces. These bubbles are promising candidates to probe a system of interacting electrons in curved geometries, but have been subjected to limited experimental investigation. Here, we report on the observation of fission of MEBs under strong electric fields, which arises due to fast rearrangement of electrons inside the bubbles, leading to their deformation and eventually instability. We measured the electrons to be distributed unequally between the daughter bubbles which could be used to control the charge density inside MEBs.

Stabilization of Multi-electron 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 and the difficulty to localize them. We report the observation of long- lived MEBs in a novel cell filled with superfluid helium at static negative pressures. MEBs were extracted from the electron filled vapor sheath of a heated filament loop embedded in the superfluid helium and observed by high-speed photography. MEBs are 2D electron gases on the 3D surface of hollow helium bubbles. Diameters can range from nanometers to millimeters, depending on the number of enclosed electrons. Electrons move in angular momentum states; deformations of the surface are called spherical ripplons. The attractive electron-ripplon interaction leads to an unusual form of superconductivity. If they can be compressed, Wigner crystallization and quantum melting can be observed, as well as a new phase for localization called the ripplo- polaron lattice. MEBs are unstable to tunneling discharge when pressed against a surface. Just as Bose gases are captured in a trap for study, MEBs must also be localized away from walls. We shall discuss methods of capturing them in an electromagnetic trap embedded in the liquid 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.

Surface Instabilities and Generation of Multielectron Bubbles Under Pulsed Electric Fields

We studied the development of surface instabilities leading to the generation of multielectron bubbles (MEBs) in superfluid helium upon the application of a pulsed electric field. We found the statistical distribution of the charge of individual instabilities to be strongly dependent on the duration of the electric field pulse. The rate and probability of generation of these instabilities in relation to the temporal characteristics of the applied field was also investigated.

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.