In the following, we present results from these calculations for the electrostatic potential in the sample cell, the charge-induced surface deformation of the liquid and the inhomogeneous charge density underneath the tip. Two geometries are studied: One is similar to the experiments reported in Ref. , where the height of the pin above the liquid surface and the pin diameter were of the order of 1 mm (geometry A); in the second part (geometry B), these typical values are reduced by one order of magnitude in order to see the influence of the length scales.
The diameter of the pin d and its height above the helium level h are chosen as d = 1.0 mm and h = 1.3 mm. For the sake of simplicity and in order to keep the number of variables small, we fix the potential of the bottom plate to zero in what follows.
We start with the potential distribution. Figure 3 shows an example where the potential at the pin is 700 V. In addition to the color-coded potential values, some field lines (white) are plotted, which indicate the local direction of the electric field. Also shown is the profile of the liquid surface. The total number of elementary charges in this simulation was N = 2.28 × 108.
An important quantity which can directly be derived from Fig. 3 is the size of the ion pool. It follows from the shape of the electric field lines at the He surface: In the region of surface charge, the field lines have to be perpendicular to the surface, because any field component parallel to the surface would lead to a redistribution of the charges, until the field parallel to the surface vanishes. Outside of the charge pool, however, this condition does no longer hold, and the field lines in general are inclined to the surface. In addition, they exhibit a slight kink at the surface, arising from the discontinuity in the dielectric constant.
The surface deformation of the liquid below the pin, generated by the electrostatic pressure of the ion pool, is hardly discernible on the scale of Fig. 3. We therefore show representative examples for the surface profile on a vertically expanded scale in Fig. 4 for two cases: Fig. 4a is for a relatively small total number of elementary charges N = 1.23 × 107, Fig. 4b for N = 2.28 × 108. First of all, the hillock is much higher for the larger charge, as expected. As the graphs further indicate, the full width at half maximum (FWHM) of the hillock profile in Fig. 4b is larger than in 4a, implying that the hillocks also grow in width as more charges are added to the pool.
It is revealing to compare the surface deformation in Fig. 4 with the charge distribution in the ion pool, as it is plotted in Fig. 5 for the same parameters. A first glance already shows that the FWHM of the charge profiles is similar to the FWHM of the surface profiles, but a closer inspection discloses a distinct difference between the two profile types: While the surface deformation varies smoothly as a function of radial position, the charge distribution displays a kink at the edge of the charge pool, rather prominent in Fig. 5a, with a nearly vertical tangent. For the larger pool in Fig. 5b, the difference between charge and surface profile is less pronounced, but the finite slope of the charge profile at the edge of the charge pool persists. This implies a well-defined, sharp boundary of the charge pool.
For a given geometry, the height of the Taylor cone depends on the applied potential difference ΔU = Ubottom − Upin between tip and plate, and on the number of elementary charges in the ion pool. As ΔU is raised, the deformation increases nonlinearly, shown in Fig. 6a for several constant pool charges between 107 and 5 × 108 e. It is to be noted that there are no hillock heights above about 300 µm; the curves end because the self-consistent calculations do no longer converge. This is an indication that the Taylor cone becomes unstable, in principle similar to the already mentioned EHD instability of a plane liquid surface. The numerical value of this critical hillock height should not be taken too literally, because, as pointed out earlier, the approximations entering the simulations do not hold for large deformations with a relatively sharp apex. Qualitatively, however, such an instability is expected and is also observed experimentally, as described in Ref. : Above some critical deformation, the surface becomes unstable, a liquid jet develops, and charges are lost from the surface. This EHD phenomenon is called a Taylor jet in the literature .
In Fig. 6b and c, we have plotted the hillock width and the inverse radius of curvature of the hillock apex for the same set of parameters as in Fig. 6a. As the voltage is increased at constant hillock charge, the FWHM decreases, and the cone profile is getting sharper. Again, the terminations of the curves at high voltage mark the points where the calculations do not converge any more. There is no big change in the values of the width, just a reduction by about a factor of 2. The inverse radii of curvature in Fig. 6c, however, exhibit pronounced divergences close to the end points of the curves, a clear signal for approaching the instability.
It has already been pointed out that one of the aims of this study is to determine the maximum charge density that can be achieved in the apex, as compared to the maximum charge density on a flat surface. Results for this quantity are plotted in Fig. 7, as a function of total hillock charge and for a set of voltages ΔU. A similar behavior as for the hillock height in the previous graphs is observed: As the voltage ΔU or the charge Q is increased, also the charge density n rises, until at some critical values the curves terminate, because the simulations do no longer converge. The bending upward in the curves for U > 500 V indicates that the system starts to develop into a positive feedback runaway situation: The charged surface is attracted by the tip, which brings the electrons closer to the tip, which attracts more electrons, thereby increasing the charge density in the center, which moves the surface closer to the tip, and so on. As long as the fields are small, this feedback is significantly less than unity and gravity alone is enough to stabilize the system. The bend upwards is the regime where gravity is no longer sufficient to prevent the jet instability; instead, the surface tension contributes the main part for stabilizing the small area under the tip. If the field or charge is increased further, even the surface tension is not enough, and the instability develops.
The maximum charge densities found in the calculations are on the order of 2.5 × 1013 e m−2. This is close to the critical value obtained for a plane, homogeneously charged He surface , which means that using the combined action of surface tension and gravity in an inhomogeneous electric field on bulk helium does not allow a noticeable increase in n beyond what is accessible in a homogeneous field, at least not for the geometry investigated so far.
For charge pool dimensions smaller than the capillary length a, the forces due to surface tension will dominate over gravity. One might expect that under such a condition higher electron densities will be stable at the liquid surface. We have therefore repeated the simulations for a reduced distance h = 130 µm between pin and (unperturbed) helium surface, i.e., a factor of ten smaller than in the previous chapter, and also the pin diameter was reduced by about an order of magnitude (to 160 µm). This should allow one to confine the charge pool to a radius well below a.
Results for the calculations of the height and charge profiles on this restricted length scale are shown in Figs. 8 and 9. In contrast to Fig. 5, the charge profile is now indeed distinctly narrower than the surface profile, its radius being a factor of 5 below the capillary length. Data for the hillock height, width and curvature as a function of the applied voltage are plotted in Fig. 10, again for a set of pool charges. Qualitatively, the behavior is similar to the results in the previous chapter, in particular the nonlinear growth of the height and the inverse radius of curvature with increasing voltage, terminating at end points given by the non-convergence of the simulation. Due to the smaller length scales, the absolute values of the applied voltage and the profile height are clearly reduced compared to Fig. 6.
The influence of the smaller geometry on the maximum charge density in the apex, however, is quite small, as is illustrated by a comparison of Fig. 11 with Fig. 7. A slight increase from about 2.5 × 1013 m−2 to 3 × 1013 m−2 is discernible, but this is not a big step toward a more complete investigation of the phase diagram of surface state electrons. In order to achieve bigger effects, one obviously would have to go to still smaller distances between tip and helium surface, which, however, would imply serious experimental challenges for the configuration “pin—bulk He surface” which is considered here. Even slight vibrations will lead to liquid surface waves with amplitudes larger than a tip-surface distance in the few micrometer range, and if both come into contact the charges will be lost. A way out could be fixing the helium surface as a thick “suspended film” by means of capillary forces in a narrow mesoscopic channel [4, 16]. In fact, first experiments in such a channel geometry have demonstrated that electron densities up to 1014 m−2 can be reached . This should allow one to study signatures of quantum corrections in the SSE system. Efforts to increase the density to even higher values, which would be desirable for investigating the full phase diagram of SSE, have so far led to sudden irreversible losses of charge, probably at some irregularities of the conductive channel walls. Therefore, better nanoscale control of the He films will be required in order to achieve higher electron densities up to the degenerate Fermi regime.