1 Introduction

In this article, we shall be concerned with two simplest species of singly charged ions in He II. The positive ion is formed when an electron is removed from helium, e.g., by field ionization at a sharp tungsten tip. It can be viewed as a “snowball"—a molecular He\(^+_2\) ion surrounded by solid “sphere" consisted by about 20 helium atoms, pulled together by inhomogeneous electric field. Due to electrostriction, the local pressure exceeds the solidification pressure within a sphere of radius about \(0.6\,\)nm [1]. The effective mass of the positive ion is about \(30~m_4\) at very low temperature and saturated vapor pressure, and increases with temperature [2,3,4], where \(m_4\) is the mass of \(^4\)He atom. The negative ion is produced when an electron is added to He II, e.g., by a sharp field-emission tip. Due to the Pauli principle, the electron is trapped in an empty bubble of radius \(\approx 1.7\,\)nm, which is determined by a balance between the zero-point energy of the electron and the surface energy of the bubble. The effective mass of the negative ion is almost purely of hydrodynamical origin, the so-called hydrodynamically added mass, at very low temperature and saturated vapor pressure about \(230\,m_4\). Although a variety of charged (even multi-charged [5,6,7]) and neutral ion species [8] exist in liquid \(^4\)He as well as liquid phases of \(^3\)He [9], possessing rich internal structure and interesting dynamical properties, in this article we shall take a simplified view that both positive and negative ions are singly charged solid spheres with the appropriate radii and effective masses.

Since these ions are charged, they can be trapped in the vicinity of interfaces in liquid helium possessing different electric permittivities (such as the phase-separation boundary in \(^3\)He–\(^4\)He mixture) [10], the most natural being the free surface of He II. One of the extraordinary physical properties of superfluid \(^4\)He (He II) is extremely high thermal conductivity, thanks to which it evaporates only from the surface, which is therefore atomically smooth. At sufficiently low temperature, there is effectively a vacuum above the surface, with dielectric constant \(\varepsilon =1\), while He II possesses a slightly higher dielectric constant, \(\varepsilon \approx 1.057\) at very low temperature [11].

It is easy to show that both the positive and negative charges are attracted to the surface if placed above it and repelled from it if placed below. This enables to trap them in the vicinity of the helium surface (both \(^4\)He and \(^3\)He [9]). For charges above the surface, the trapping occurs naturally, as electrons (or, in principle, e.g., positrons) thanks to the Pauli principle cannot easily penetrate through the surface. This is widely used to create sheets of electrons, whose extraordinary physical properties are the subject of intense ongoing experimental as well as theoretical research by a number of investigators [10, 12,13,14,15,16]. In order to trap charges below the surface of superfluid helium, a vertically oriented external electric field \(E_0\) must be applied so that the charge e (i.e., the ion) experiences a potential

$$\begin{aligned} V(z)= eE_0z+ \frac{(\varepsilon -1)e^2}{16\pi \varepsilon _0 \varepsilon (\varepsilon +1)z}\,. \end{aligned}$$
(1)

The last term reflects the charge density induced by an ion on the surface, which can be represented by an image charge. The ion becomes trapped at the minimum of this potential, at the depth

$$\begin{aligned} z_0= \left[ \frac{(\varepsilon -1)e}{16\pi \varepsilon _0 \varepsilon (\varepsilon +1) E_0} \right] ^{1/2}\,. \end{aligned}$$
(2)

For a typical experimentally applied electric field of \(5\,\)kVm\(^{-1}\) the depth is \(43\,\)nm. The potential well, Eq. 1, can be filled by ions up to areal number density \(2 \varepsilon _0E_0/e\) (typically corresponding to spacing about \(1 ~\mu\)m) when the field produced by ions themselves cancels the holding field \(E_0\). Additionally, the temperature must be low enough, typically below \(\approx 1\,\)K, that the ions are not kicked from the well thermally. Energy levels populated by ions in the (slightly anharmonical) well are spaced typically by \(\approx 2\,\)mK, while experiments are performed at much higher temperatures with significant excitation out of the ground state. The amplitude of vertical motion is, however, much less than typical interion spacing, and for many purposes, the system can be considered as two-dimensional (2D).

Figure 1 shows the typical arrangement of electrodes used for most of the experiments by the group of Joe Vinen in Birmingham. The circularly symmetric experimental cell is half-filled by pure He II. The electrodes are shaped in the form of a pill box and serve not only for applying the static electric field, but also for application of the ac drive and detection of the induced ac signal. Charging the surface by either negative or positive ions created by a sharp tungsten tip through the fine grid covering the central opening of the bottom electrode is possible for positive ions below 1K and for negative ions only within the temperature window 0.9–1.1 K, when they have sufficient time to thermalize and, on the other hand, the amplitude of the thermally excited vertical oscillation does not bring the electron in the bubble too close to the surface, enabling electron to tunnel out of the liquid through the free surface [18].

Fig. 1
figure 1

Left: Schematic diagram of the experimental cell, half-filled with pure He II, showing the electrodes A, B, C and D, shaped in the pill box form. The hole in the middle of the bottom electrode is covered by a fine grid, which allows charging the surface by either negative or positive ions created by a sharp tungsten tip placed below it (not shown). Right: The charge density of the ion pool of circular symmetry is nearly constant and drops rapidly, within the distance d of order the half-height of the cell, to zero near its edge. Reproduced with permission from [17]

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In the experiment, the pool of ions must be confined in the horizontal direction by a suitable fringing field. The equilibrium distribution of the trapped ions is determined by the condition that the horizontal component of the electric field acting on any ion, composed of that due to potentials applied to the electrodes and that due to the charge distribution of the ions themselves, vanishes. An analytic solution of this problem, for the special case of the pool exactly in the middle between the top and bottom electrodes spaced by \(d \ll R_c\), where \(R_c\) is cell radius, was given by Glattli et al. [19]. In most cases, this analytical solution does not appreciably differ from numerical solutions provided by Lambert and Richards [20] or by Barenghi et al. [3].

We note in passing that both positive and negative ions can be trapped based on the same principle below the surface of normal as well as superfluid phases A and B of \(^3\)He, as it was shown by Kono and coworkers. They charged both positive and negative ion pools at temperature about 100 mK and investigated their mobility down to 250 \(\mu\)K [9].

2 Oscillatory Modes of the Ion Pool

The ion pools support several collective modes of oscillation, which serve as an excellent tool to study their physical properties. In the early experiments, Poitrenaud and Williams [2] studied, oscillation of both positive and negative ions in the trapping potential, Eq. 1, and used the observed frequency of oscillation to deduce the effective ionic masses. The modes of vertical oscillation in an anharmonic potential well are in fact quite complex, as they are inherently coupled to surface waves–ripplons as well as to other vibrational modes of the ion pool.

Plasma resonances can be excited as well as detected electrically, using electrodes shown in Fig. 1. These resonances–plasma waves–involve motion of ions in the horizontal plane. 2D plasma waves were first observed by Ott-Rowland et al. [21] for a pool of positive ions and by Barenghi et al. [4] for pools of negative ions. In the absence of dissipation, the dispersion relation between the angular frequency \(\omega _p\) and wavenumber k reads

$$\begin{aligned} \omega _p^2=\frac{n_0 e^2 k}{2\varepsilon _0 M} \,\,\frac{2\sinh (kd) \sinh [k(h-d)]}{\sinh (kh)+(\varepsilon -1) \cosh (kd) \sinh [k(h-d)]}\,, \end{aligned}$$
(3)

where \(n_0\) is the areal density of ions, M stands for the effective ionic mass, h is the distance between the top and bottom electrodes, and d is the distance between the layer of ions from the bottom electrode [22]. Assuming that the dielectric constant of liquid helium differs from unity only slightly and that the ion pool is localized midway between the top and bottom electrodes, we arrive at simplified relation

$$\begin{aligned} \omega _p^2=\frac{n_0 e^2 k}{2\varepsilon _0 M} \tanh \left( \frac{kd}{2} \right) \,, \end{aligned}$$
(4)

In the presence of vertical magnetic field B the dispersion relation becomes modified:

$$\begin{aligned} \omega ^2=\omega _p^2+ \omega _c^2\,, \end{aligned}$$
(5)

where \(\omega _c= eB/M\) is the cyclotron frequency.

For a circular pool of fixed radius R, under approximation of constant areal density of ions, \(n_0\), and assuming that the edge of the pool does not radially move, the perturbed charge density in any plasma mode is of the form [23]

$$\begin{aligned} J_m(k_{m,n}r)\exp [i(m\theta -\omega t)]\,, \end{aligned}$$
(6)

where m is a positive or negative integer and the discrete sets of wavenumbers \(k_{m,n}\) are determined by the boundary conditions requirering that the ionic displacement vanishes at \(r=R\). For axisymmetric modes, \((m=0)\) this boundary condition yields k independent of B; for \(B=0\) they are degenerate with respect to the sign of m. The frequency of axisymmetric modes increases with B. The behavior of non-axisymmetric modes is more complicated–here the degeneracy is lifted by the magnetic field and wavenumbers \(k_{m,n}\) are \(B-\) dependent. Behavior of modes with no nodes in radial direction is peculiar—while the frequency of the mode with negative m increases with B, the frequency of the mode with positive m falls with increasing B and for sufficiently large B drops below \(\omega _c\), where k becomes imaginary. In the limit of large B the frequency of the edge mode propagating around the edge of the pool is given by Eq. 4, providing that k is taken to be the component of the wavenumber parallel to the pool edge – for \(m=1\), k equals to 1/R [23]. Such modes are known as edge magnetoplasma waves, in analogy with edge modes in 2D heterostructures.

In order to resolve the plasma resonances experimentally, they have to be sufficiently sharp. The plasma waves can be damped by two mechanisms: that associated with a finite ionic mobility within the helium and any internal dissipation, due to, for example, 2D viscosity, within the ionic plasma itself. The damping gives rise to a finite linewidth in the plasma resonant modes. These two contributions to the damping can be distinguished by their different dependences on plasma frequency—in most cases, the finite ionic mobility is dominant. Measurements of the linewidth can therefore be used to deduce values of the mobility. While above about \(100\,\)mK the mobility is nearly independent of trapping depth and is due to the scattering of the ions off phonons and rotons, at lower temperatures, the mobility strongly depends on trapping depth \(z_0\); at very low temperature, the ripplon-limited mobility decreases approximately as \(\mu _{ripp} \propto z_0^{-7/2}\) [24, 25] which suggests that it becomes limited by the scattering by thermally excited capillary waves, or ripplons, on the He II surface. For a deep liquid, their dispersion relation reads

$$\begin{aligned} \omega ^2=\frac{\sigma }{\rho } k^3 +gk\,, \end{aligned}$$
(7)

where \(\sigma\) is the surface tension, \(\rho\) is the density of He II and g stands for the acceleration due to gravity. For the case of ion pools, the effect of gravity can be neglected. As we shall see later, the interaction of ion pool with ripplons is essential in various ways.

For low drive, the in-phase response of any plasma mode is linear and approximately of Lorentzian shape if the drive frequency is swept across the resonance; however, it becomes nonlinear if the drive is increased, the resonant peak broadens, becomes asymmetric and the maximum response is appreciably shifted to lower frequency. The effect arises from nonlinear terms in the equations of motion of the ion fluids, and in the case of positive ions, probably also from a velocity dependence of the effective mass.

For the phenomena described in this paper, the nonlinear effects are important in that they lead to a coupling between modes. The excitation of one mode at a relatively large amplitude gives rise to a change in response of another mode excited simultaneously, in both the frequency of its maximum response and linewidth. The mode coupling enables very sensitive detection or various modes of response [26]: scanning the drive frequency over the frequency interval of interest results in easily detectable changes in the in-phase response of the simultaneously driven axisymmetric mode, typically the (0,1) fundamental mode, see Fig. 2. The exciting drive needs to be some thousand times larger than is necessary for the excitation of an axisymmetric plasma mode, and it presumably relies on small departures from symmetry in the real cell arising from machining errors, lack of exact leveling, and perhaps variations of contact potential from place to place on the electrodes. This so-called double-drive technique is very useful, as it enables detection of non-axisymmetric plasma modes in a cell of approximately circular symmetry (such as shown in Fig. 1); electrostatic coupling of non-axisymmetric modes to the electrodes is weak, and their direct detection experimentally very difficult.

Fig. 2
figure 2

Left:Typical in-phase response of the fundamental axisymmetric plasma mode, driven at small amplitude, when a second drive at relatively large amplitude is swept through the frequency range up to \(50\,\)kHz. Detected modes of response are indicated by arrows. Upper spectrum: \(T=109.5\,\)mK, lower spectrum: \(T=145.0\,\)mK. \(B=1.41\,\)T, \(R=12.14\,\)mm, \(n_0(0)=2.89 \times 10^{11}\,\hbox {m}^{-2}\), \(z_0=53.77\,\)nm, \(T_m=122.5\,\)mK. Reproduced with permission from [27]. Right: A typical observed spectrum showing several peaks due to the new type of edge magnetoplasma mode for the pool of positive ions of areal density \(n_0 =8.8\times 10^{10}\,\)m\(^{-2}\), radius \(R = 12.34\,\)mm; trapped below surface of helium at the depth \(60\,\)nm; \(T = 60\,\)mK; \(B = 1.2\,\)T. Reproduced with permission from [28]

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Spectra of the type as shown in Fig. 2 contain four different families of modes, which can be distinguished by the range of frequency in which they occur, by their dependence on magnetic field and on the level of the second drive. Consider first the spectral features that appear in the frequency range above about 10 kHz. At relatively low second drive levels (in the range 1–10 mV rms applied to the wall electrode), the only features present are those indicated by heavy vertical arrows and they can be identified as non-axisymmetric conventional bulk and edge magnetoplasma modes discussed in the approximation of constant charge density above.

At higher drive levels, other “satellite” features appear, as indicated by the small closed arrows. As can be clearly seen, one of these satellites is split by about \(100\,\)Hz.

At frequencies below about 10 kHz, the situation is more complicated and depends crucially on the temperature. The reason is that the 2D ion pool undergoes the phase transition, and below the melting temperature \(T_m\) it exists in a form of Coulomb crystal, which supports additional family of modes, the shear modes (see later). Above the melting temperature and at a low drive, the only visible spectral feature is very broad, with its peak slightly below 1 kHz—this feature will be discussed later. With increasing drive, sharp spectral features (marked by open arrows in Fig. 2) appear on top of this broad feature: the multipole edge modes reported in Ref. [28, 29] (see the right panel of see Fig. 2), theoretically predicted to exist by Nazin and Shikin [30]. Their resonant frequencies are shown in Fig. 3. We note in passing that this family of modes has been independently observed by Kirichek et al. in 2D sheets of electrons on the top of helium surface [31].

Fig. 3
figure 3

Edge modes measured on a sheet of positive ions trapped below the surface of liquid helium (\(T=37\,\)mK, \(R=13.93\,\)mm, \(n_0= 6.563 \times 10^{11}\,\)m\(^2\), \(z=36.53\,\)nm, \(T_m=189\,\)mK), plotted against externally applied vertical magnetic field. Numerical calculation results are plotted by solid lines. Conventional (full symbols) and soft dipole (open symbols) edge magnetoplasmon frequencies (different curves correspond to different azimuthal indexes of the mode). Reproduced with permission from [29]

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Figure 4 shows how the frequencies of the conventional magnetoplasma modes for one particular sheet of positive ions (sheets of negative ions behave similarly) vary with externally applied vertical magnetic field. The simplest theoretical model described above yields the broken lines in the left panel. There is a qualitative agreement with the experimental result. The essential physics of the conventional magnetoplasma modes is correctly described, but the predicted frequencies are too high. An improvement was achieved by recognizing that the rigid boundary condition is unrealistic [28, 32]. For even better quantitative agreement between the theory and experiment, proper account of the screening by the actual electrode system must be taken into account; for further details of these complex approaches, we direct the reader to original publications [19, 27].

Fig. 4
figure 4

Measured mode frequencies above \(10\,\)kHz, plotted against magnetic field. \(T=37\,\)mK, \(R=13.93\,\)mm, \(n_0= 6.563 \times 10^{11}\,\)m\(^2\), \(z=36.53\,\)nm, \(T_m=189\,\)mK. The same experimental data in both panels are shown together with theoretical calculations using various models. Left panel: Broken lines–simple theory with rigid boundary condition, solid lines–with movable boundary. Right panel: The solid lines–theory of Ref. [28], broken lines– theory of Ref. [27]. The solid straight line in both panels is the value of \(\omega _c\). Reproduced with permission from [27]

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3 Crystallization of the Ion Pools

Our everyday experience tells us that a rich variety of 3D crystals exists. Their constituents occupy positions in a crystal lattice, satisfying long-range order. One way to examine the order in a system is to look at the Bragg diffraction peak, which for a 3D infinite perfect crystal takes the form of a delta-function. Any disturbances to the order should manifest themselves in the Debye-Waller factor, which controls the height of the Bragg peak. As for the 2D case, Mermin [33] showed that there cannot be any long-range order in a 2D system of infinite size. It does not mean, however, that crystallization in 2D cannot take place–a finite 2D system can exhibit a quasi-long-range order, where Bragg peaks are power law singularities rather than delta-functions. As we discuss below, crystallization in 2D is generally more subtle and differs from the 3D case in a number of important ways.

The possibility that a 3D system of particles interacting via repulsive Coulomb forces might under some conditions form a crystal was first discussed by Wigner [34] in the context of the three-dimensional Fermi-degenerate electron fluid, hence the term Wigner crystal. The crystallization occurs upon reaching the critical value of the ratio of the Coulomb potential energy of the system to its kinetic energy \(E_{kin}\). While for a degenerate quantum system of fermions, \(E_{kin}\) is the Fermi energy \(E_F\), Crandall and Williams [35] considered classical 2D Coulomb fluids, such as the sheets of ions below the helium surface or electrons above it of sufficiently low areal density \(n_0\). Their kinetic energy is just the thermal energy, \(k_B T\) and the abovementioned energy ratio is conventionally expressed in the form of the so-called plasma parameter

$$\begin{aligned} \Gamma =\frac{n_0^{1/2} e^2}{4 \pi ^{1/2} \varepsilon _0 E_{kin}}=\frac{n_0^{1/2} e^2}{4 \pi ^{1/2} \varepsilon _0 k_B T} \,. \end{aligned}$$
(8)

The critical value of \(\Gamma\) at which both the positive and negative 2D ion pools below the helium surface or low density electron sheets above it are expected to crystallize, \(\Gamma _c \approx 130\), follows the prediction of numerical simulations for truly 2D systems of Morf [36]. As for terminology, for the charged 2D ion or electron pools, we prefer the term Coulomb crystal rather than Wigner crystal, which assumes compensating of point charges by the existence of a continuous “soup" of charge of opposite sign and is therefore electrically neutral.

The observation of 2D crystallization is fairly easy in the case of electron sheets above the surface of helium, where it relies on the strong electron-ripplon coupling that significantly effects the plasma mode dispersion relation. The interaction between an electron and the helium surface causes a dimple to form under each electron and couples the motion of ripplons propagating on the helium surface with the motion of electrons in the sheet. In the 2D electron fluid phase, the electron motion is rapid and irregular, so that dimples on the helium surface cannot keep up with the electrons. As a result, no well-defined dimples are formed in the 2D fluid phase and the plasma mode dispersion relation, Eq. 3, is not strongly affected. In the crystal phase, however, electrons become located at its lattice sites and a dimple is formed below each of them. Grimes and Adams [37] observed the effect of crystallization on the dispersion relation of the longitudinal plasma modes discussed above and found \(\Gamma _c=137 \pm 15\). Further evidence of crystallization was provided by Mehrotra et al. [38], who observed an anomaly in ripplon limited mobility at the melting transition.

For completeness, we note here also later additional experimental observations of the electron crystallization by Rees et al. [39], from the Bragg-Cherenkov scattering. At rest, the Coulomb solid is “dressed” by a cloud of quantized capillary waves–ripplons, however, under a driving force, the repeated ripplon decoupling leads to stick–slip current oscillations, the frequency of which can be tuned by adjusting the temperature, pressing electric field, or electron density. The system is analogous to polaron states in solids, in which electrons are dressed by a cloud of virtual phonons, or lattice deformation [40, 41]. Joe Vinen contributed in the significant way to understanding of this interesting problem [42].

Although is was clearly anticipated that sheets of both positive and negative ions would display 2D crystallization, its observation proved to be more difficult. In contrast to the electron sheet, no observable change was seen in effective mass of ions while crossing the expected melting transition, as the ion-ripplon coupling has much weaker effect, due to much larger ionic mass. Still, following the long-lasting effort of the group of Joe Vinen, experimental evidence of crystallization of the positive ion pool was provided in 1990 by Mellor and Vinen [43] (for details, see also [44]), who excited horizontal oscillatory motion of ions which provided generation of ripplons. Left panel of Fig. 5 shows that this causes an enhanced energy absorption from the electric field driving the ionic motion, driving the so-called Shikin mode [45,46,47].

Fig. 5
figure 5

Left panel: Absorption versus frequency for a pool of positive ions with areal density \(n_0=4.8 \times 10^{10}\,\)m\(^2\), measured at \(T= 10\,\)mK. Right panel: Melting curve of ionic crystal. Solid line is \(\Gamma _c = 130\). Reproduced with permission from [43]

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The Shikin mode can be introduced as follows. Suppose that the ion crystal is driven by an electric field oscillating at angular frequency \(\omega _E\). Oscillatory motion of the ions will cause each ion to emit capillary waves at the same frequency, obeying the dispersion relation, Eq. 7. If the wave number \(k_r\) is equal to the magnitude of a reciprocal lattice vector \(\textbf{G}\) of the 2D Coulomb crystal, then the capillary waves from the different ions will interfere constructively. The result is the formation of a standing capillary wave above the ion sheet, with the antinodes of the capillary wave coinciding with the ionic positions in the Coulomb crystal lattice. This coupled ion-capillary-wave mode is the Shikin mode. In practice, there are several reciprocal lattice vectors satisfying the condition \(k_r= G\), so that the standing wave is the superposition of several standing waves traveling in different directions. The right panel of Fig. 5 confirms that crystallization indeed occurs at plasma parameter value \(\Gamma _c \approx 130\) previously evaluated in the case of electrons above the helium surface. We note, however, that the increase in absorption is very small and its observation therefore requires high drives and long averaging times. Moreover, the peak at a Shikin mode frequency is observed to be very broad, which may be associated with a crystal that has been partly damaged by the high Shikin drive applied in the plane of the ion pool (for damage and annealing of the ion crystal, see later).

Fortunately, it is possible to observe the excitation of a Shikin mode much more easily, by application of the double drive technique described above, i.e., via its effect on a chosen (typically the fundamental axisymmetric (0,1) mode) simultaneously driven plasma mode. The experimental protocol of what can be called 2D ripplon crystallography, an analogue of conventional 3D x-ray crystallography, is as follows [17]. The lowest axisymmetric plasma mode (0,1) (in some cases (0,2) mode) is driven at angular frequency \(\tilde{\omega }\) at or very near its resonant angular frequency \(\omega _p\) by applying a rf voltage to the electrode D, and the response is measured by the current induced on the grounded electrode B. At the same time, a rf voltage at frequency \(\omega _E\), at or near a Shikin frequency \(\omega _S\), is applied to electrode A. Figure 6 is a 3D plot of the observed plasma mode response as a function of the frequencies \(\tilde{\omega }/2\pi\) and \(\omega _E/2\pi\) in the vicinity of \(\omega _S(G1)/2\pi\), where \(\textbf{G1}\) is the smallest reciprocal lattice vector of the ion crystal.

The physical content of Fig. 6 can be summarized as follows. When the frequency \(\omega _E\) is well removed from any Shikin frequency \(\omega _S\), the plasma mode response is a simple Lorentzian centered on the frequency \(\omega _p\). When \(\omega _E\) is close to \(\omega _S\), there is a relatively small effect on the plasma mode response, which consists primarily of an increase in the plasma mode resonant frequency, with some increase in attenuation. However, other features occur when \(\omega _E\) is close to a frequency that is, to a good approximation, either \((\omega _S -\omega _p)\) or \((\omega _S +\omega _p)\). In the former case, the effect is primarily to increase the attenuation of the plasma mode; in the latter case, the effect is primarily to decrease the plasma mode attenuation. With a Shikin drive of only modest amplitude, these changes in attenuation are very large; the increase in attenuation at \((\omega _S -\omega _p)\) can be sufficient to quench the response almost completely, and the decrease at \((\omega _S +\omega _p)\) can be sufficient to increase the peak plasma mode response by a factor as large as 10. These experimental observations convincingly prove the existence of the Coulomb crystal. They, however, rely on the use of the sensitive double-drive technique and still await detailed quantitative theoretical considerations.

Fig. 6
figure 6

Three-dimensional plot showing the in-phase response of the (0,1) plasma mode plotted against the frequency, \(\omega /2\pi\), of the plasma mode drive and the frequency, \(\omega _E/2\pi\), of the Shikin drive. The ion crystal characteristics: \(n_0=7.14 \times 10^{11}\,\)m\(^2\), \(T=36\,\)mK, \(T_m=193\,\)mK, trapping depth \(z_0=35.9\,\)nm, and magnetic field \(B=1.213\,\)T. Reproduced with permission from [17]

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We see from the left panel of Fig. 7 that similar features are seen when \(\omega _E\) is close to \(\omega _S(G_n)\), where \(\textbf{G}_n\) are larger reciprocal lattice vectors. Observation of the frequencies of the first four reciprocal lattice vectors allows to find the ratios of the magnitudes of the first four reciprocal lattice vectors: the observed ratios are 1:2.26:2.80:4.28. This unequivocally confirms that the crystal has a triangular lattice, as these ratios with errors of not more than 1%, agree with the theoretical values 1:2.28:2.83:4.31 for a triangular lattice.

Furthermore, as the plasma mode is driven more strongly, emission peaks and absorption dips in the plasma mode response become visible not only at the frequencies \(\omega =\omega _S \pm \omega _p\), but also at the frequencies \(\omega =\omega _S \pm n\omega _p\), where n is an integer; an example is shown in the right panel of Fig. 7. These features appear as a consequence of inelastic scattering of ripplons, when n plasmons can be absorbed of emitted simultaneously. For more detailed theoretical description of 2D ripplon crystallography, we direct the reader to the original publication of the Vinen’ group, Ref. [17], where the inelastic scattering and role of the Debye-Waller factor are taken into account.

Fig. 7
figure 7

Left panel: The observed in-phase response of the (0, 1) plasma peak plotted against Shikin drive frequency, \(\omega _E/2\pi\), for a fixed amplitude of Shikin drive. The arrows show the frequencies of the four lowest Shikin modes, corresponding to the four smallest reciprocal lattice vectors. Right panel: The observed in-phase responses of the (0,1) and (0,2) plasma modes plotted against the Shikin drive frequency for a large plasma mode drive. \(n = 3.853\times 10^{11}\,\)m\(^{-2}\), \(T=37\,\)mK, \(T_m= 141.1\,\)mK, \(z=37.5\,\)nm, and \(B=1.34\,\)T. Reproduced with permission from [17]

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The 2D ion crystal supports transverse shear modes involving ionic motion in the plane of the ion pool. Shear modes in a 2D classical electron crystal above the surface of liquid helium were observed by Deville et al. [50]. They had high frequencies which were shifted significantly by interaction of the electrons with dimples in the helium surface. Even so, good measurements of the shear modulus of the crystal and its temperature dependence were obtained.

In the ease of the ion crystal, the corresponding surface (anti)dimples are very small and have a negligible effect. It is therefore possible to study shear modes of low frequency without interference from them. The transverse shear modes have the dispersion relation

$$\begin{aligned} \omega ^2=\frac{\mu k^2}{n_0 m} \end{aligned}$$
(9)

for small wavenumbers k. At absolute zero temperature, the value of the shear modulus \(\mu\) for the purely 2D system has been predicted by Bonsall and Maradudin [51]

$$\begin{aligned} \mu =\frac{0.24 e^2 n_0^{3/2}}{4 \pi \varepsilon _0} \,, \end{aligned}$$
(10)

and its temperature dependence calculated by Chang and Maki [52]. In order to drive shear modes electrostatically in the cell of circular symmetry shown in Fig. 1, a vertical magnetic field must be applied as in this case the shear modes cease to be purely transverse and couple to the axisymmetric drive. The detection of the shear modes occurs through a non-linear coupling that exists between different modes of oscillation, in this particular case between the shear modes and the plasma modes, typically the fundamental (0,1) mode. In other words, to detect the shear modes experimentally, the sensitive double drive technique was employed once again.

In the approximation of constant areal density of the ion crystal, two boundary conditions must be satisfied. The first (only approximately valid, see above) is that the radial displacement of the ions vanishes at the edge of the pool with constant charge density; the second is that the shear stress at the pool edge will also vanish. Since the shear (transverse) modes must be observed in the presence of a magnetic field, they must have associated with them a small longitudinal component. It is easy to show [22] that no mode with a single wavenumber can satisfy both the boundary conditions. The resonant modes of the pool are therefore formed from mixtures of two wavenumbers: for shear modes in the presence of a large magnetic field, one of these wavenumbers is real and the other is purely imaginary (i.e., an edge mode). The predicted frequencies shown in the top panel of Fig. 8 are based on these considerations—the observed temperature dependence of the mode frequencies is consistent with that predicted from the temperature dependence of \(\mu\) [52].

The ions have a finite mobility which is determined by collisions with ripplons. Within experimental error, the observed damping of plasma waves in the ion pools is independent of frequency and due entirely to this finite mobility. Contrary to that, the observed damping of the shear modes at low temperatures (see the bottom left panel of Fig. 8) increases with increasing frequency and is significantly larger than can be accounted for by the finite mobility. The extra damping is presumably due to internal friction in the crystal [48].

Fig. 8
figure 8

Left: Observed spectrum of sheer modes (top) and their linewidths in an ion pool with \(n_0 =7.51 \times 10^{11}\,\)m\(^{-2}\), \(T=30\,\)mK. Reproduced with permission from [48]. Right: The time recovery of the shear spectrum after damaging the 2D ion crystal by applying a high shear drive at 17 mK in an applied vertical magnetic field of 1.35 T for an ion pool of charge density \(2.9 \times 10^{11}\,\)m\(^{-2}\), radius 12.1 mm and melting temperature \(T_m=125\,\)mK, trapped \(53\,\)nm below the helium surface. Reproduced with permission from [49]

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There is experimental evidence [49, 53] that a carefully produced crystal can be deliberately damaged, see right panel of Fig. 8. The upper spectrum is obtained for a carefully prepared crystal at a low drive level (20mV). At time t=0 the crystal is subjected to a high shear drive (2V), the frequency of this drive being swept from 0.1 to \(3\,\)kHz within one minute. The subsequent spectra are taken at the low drive level at the times indicated. Initially, the shear mode has practically disappeared, but recovery occurs gradually over a period of many minutes. The 2D crystal of positive ions is therefore found, similarly as many conventional 3D crystals, to anneal after damage. The authors defined a recovery time \(\tau\) as the time required for the amplitude of the detected signal at the shear mode resonant frequency to recover to 1/e of its original amplitude. Observed values of \(\tau\) have some tendency to decrease with increasing temperature, but the temperature dependence is surprisingly small, suggesting that the relevant activation energies are also small.

So far, we considered the properties of ion crystals at temperatures much lower than the melting temperature \(T_m\). The 2D melting process, discussed in detail in a review by Strandburg [54], is believed to take place through a Kosterlitz–Thouless transition [56, 57] and ought to take place in two steps, via the so-called hexatic phase. While the crystalline phase of the 2D ion crystal at low temperature contains defects of the lattice that can be viewed as bound dislocation pairs, the hexatic phase is characterized by the dissociation of these thermally excited pairs of dislocations, see the left side of Fig. 9. Further, the dislocation can be viewed as a pair of disclinations and dissociation of these then subsequently leads to fluid or plasma phase at high temperature. If this is indeed the case, then the molten phase of an ion crystal should upon increasing temperature become a hexatic, crudely described as a crystal containing a finite density of free dislocations that above \(T_m\) rapidly increases [58, 59].

Fig. 9
figure 9

Left: Positive (a) and negative (b) disclinations in a triangular lattice. Note the rotation of the triangular cells by 60 degrees a clockwise, and b counterclockwise, as a clockwise path around the disclination is traveled. Note that these disclinations may also be described as particles having a five, and b seven neighbors, respectively, rather than six. Dislocation (bottom left) may be viewed as a bound disclination pair. A path around the dislocation fails to close, as shown. The two disclinations in a square lattice, one having five nearest neighbors and one having three, are also shown. Reproduced with permission from [54]. Right: Theory fitted to observed shear mode response. Reproduced with permission from [55]

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As we just discussed, the modes of transverse response at low temperatures, in the crystal phase, are elastic shear waves, characterized by a temperature-dependent shear modulus \(\mu (T)\). The modes of transverse response in the hexatic phase are expected to be more complicated. They have been studied theoretically by Sonin and Vinen [60], who argued that they can be shown to include viscous waves characterized by a kinematic viscosity, \(\nu\):

$$\begin{aligned} \nu =\frac{e n_0^{-1/4}}{16\Gamma (\pi \varepsilon _0 M^*)} \frac{n_0}{n_f} \exp \left( \frac{W}{k_B T} \right) \,, \end{aligned}$$
(11)

where \(n_f\) is the concentration of free dislocations, and W is the height of the barrier that impedes motion of a dislocation. W is not known, but it is likely to be rather small, at least for dislocation motion in a slip plane. For small values of \(t=(T-T_m)/T_m)\), the ratio \({n_f}/{n_0}\) is expected to be equal to \(\exp (-2bt^{0.37})\) [58], where b is a constant of order unity, that depends on the core energy of a dislocation, but which is not known with any precision near \(T_m\). The dominant temperature dependence in Eq. 11 is likely to be in the factor \(n_0/n_f\).

The effective viscosity is modified by viscoelastic effects and by effects due to the relaxation of residual dislocation pairs [59]. Experimentally [55], at temperatures above the melting temperature, the sharp shear modes discussed above are observed to be replaced by a broad response, as shown in the right panel of Fig. 9, together with the theoretical fit based on estimated values of quantities entering Eq. 11.

The fit is rather good; however, the broad viscous response is observed to be present not only above \(T_m\) but also just below the melting temperature. This most likely means (in view of the damage and annealing processes just discussed) that the drive applied to the system was too large and led to shear strains that broke a significant number of the bound dislocation pairs. Detailed answers to complex questions associated with the existence of the hexatic phase and two-step melting of the crystalline phase of 2D sheets of ions will have to await further dedicated investigations.

4 Conclusion

We have reviewed the main results on investigation of pools of positive and negative ions trapped at milliKelvin temperatures underneath the surface of isotopically pure superfluid helium, obtained by the Birmingham experimental low temperature physics group led over many years by Joe Vinen. For more than a decade, besides the author of this review, a number of graduate students supervised by Joe and younger members of the group including N.J. Appleyard, C.F. Barenghi, P.L. Elliott, C.J. Mellor, J. Meredith, C.M. Muirhead, C.I. Pakes, P.K.H. Sommerfeld as well as visitors A.A. Levchenko, V.B. Shikin, E.B. Sonin and others participated in these investigations. The reviewed results on ionic masses, mobility, oscillatory modes of response of the ion pools, and 2D crystallization have been achieved under true leadership of Joe, who at the same time was contributing in important ways to the research fields of superconductivity and quantum turbulence. The author highly appreciates the possibility to participate in the research described, which subsequently led to fruitful long-lasting scientific collaboration, and thanks Joe and Susan Vinen for sharing their wisdom, kind help, and three decades of friendship.