Final Scholium

The theoretical and experimental analysis presented in this thesis, outlines a coherent framework for working with, and understanding dynamic and kinetic properties of TDES on periodically deformed liquid helium films.

Original motivation of these experiments was that electrons of TDES in the horizontal plane would start moving in circular orbits when a uniform magnetic field in thez-direction were applied, the radius of this circular orbit becoming smaller with an increasing field magnitude. The application of such a magnetic field would not have been yet sufficient to ensure electron localization, because the various scattering mechanisms would cause the centers of the circular orbits to obviate perpetually and execute a diffusive, spatially unconstrained random walk motion. This spatially unconstrained motion of the centers of circular orbits was thought it could be averted when the surface of liquid helium were no longer flat, but instead exhibited periodically positioned lows (troughs) and highs (peaks). Such a periodical structure were to be ensured by using a substrate whose surface had been crafted with periodically positioned troughs and peaks.

Since superfluid liquid helium is a universal wetting agent, it follows that a

5.6. FINAL SCHOLIUM 135

liquid helium film (whose thickness can be easily regulated) will always wet the substrate surface and have the same periodic structure.

Period, depth (height) and diameter of the troughs (peaks) are important, as they influence film structure before and—more critically—after a TDES has been formed (TDES also influence film structure). Capillary effects must be taken into account in the design of a structured substrate, as well as the range of TDES density one wishes to study and the range of the experimentally feasible magnetic field for the onset of localization. These considerations limit the dimensions of surface characteristics in the mesoscopic range (tens or hundreds of nanometer).

An important finding in this thesis is that the use of dielectric substrates greatlyinfluences electron localization, not only because the developing image charges on the surface of the substrate reduce electron mobility: the effect of a structured dielectric substrate is especially pronounced when a holding field is applied. It was shown in the theoretical part of this thesis that electron localization in the troughs of the periodic substrate in this case is almost completeeven in the absence of a magnetic field. This is in sharp contrast to nondielectric structured substrates, where electron interaction with the substrate is minimal or nonexistent and extremely strong holding fields must be applied in order to localize a minute fraction of electrons in the troughs and only for temperatures in the10⁻²K range.

Electron localization due to the combination of a dielectric substrate and a holding field is problematic, because it greatly reduces the measurable admittance signals and meddles with electron localization from the magnetic field. Perhaps contrary to intuition, the theory and the experimental results presented suggest that the magnetic field actuallydelocalizespart of TDES electrons, especially for weak magnetic fields, and so enhance the measured admittance.

Indeed, the model developed in the present thesis predicts that a constant holding field would localize most electrons of a TDES inside the troughs in some Fock-Darwin state. If, however, an increasing magnetic field is applied in addition to the holding field, the fraction of electrons on the peaks of the substrate will increase. Equivalently, localized electrons from the troughs will move to the peaks and occupy a regular Landau state there⁹.

The diffusive electron flow from the troughs to the peaks, induced by the changing magnetic field¹⁰, is a current and it should therefore contribute to the admittance. The developed model gives a quantitative result with regard to that contribution—theflow admittanceYf—, which can be compared with the experimental data. Moreover, it provides an amended Drude model of the form

9Electron potential on a peak can be arbitrarily defined as zero. A consequence of such a selection is that a trough obtains a negative potential, becoming thus a potential well.

10The motion from the troughs to the peaks is the result of a complicated summation of quantum states. Nevertheless, it can be crudely motivated by the fact that larger magnetic fields increase the angular momentumLz, which directs electrons ‘upwards’ (out of the troughs).

1+µ²_pB²np(B)for the normalized admittanceY(0)/(Y(B) −Yf(B)), whereµp

is peak electron mobility andnp(B)the partial fraction of peak electrons as a function of the magnetic field.

For regimes other than the low temperature and low electron density, flow admittance is only a small part of the measured one. This means that most of the measured admittance should rather be attributed to electrons in the troughs, or electrons on the peaks, or both. However, the evolution of the partial fraction of electrons on the peaks with respect to the magnetic field indicates that the fraction of electrons on the peaks is the active part responsible for most of the measured admittance. The main reasons are that first, the partial fraction of peak electrons increases when the admittance increases or retains a large value (at weak magnetic fields) and second that the partial fraction declines when the admittance decreases (at strong magnetic fields).

Even in the exceptional case where the partial fraction of peak electrons increases at a lower rate in strong magnetic fields, Landau localization of peak electrons becomes much better and reduces their contribution to the admittance.

Overall, at the limit of strong magnetic fields the admittance always declines and its evolution tends asymptotically to flow admittanceYf(B). Therefore, for strong magnetic fields electrons are well localized either inside the troughs or on the peaks and the measured admittance should only be due to electrons which flow from the troughs to the peaks or conversely.

The preceding analysis has made clear that quasi zero dimensional electron localization is possible, and that transport properties of localized electrons are heavily influenced by the interplay of temperature, magnetic field, holding field and electron density. Nontrivial physical insight was gained even by using a simple theoretical model that ignores Coulomb interactions, because the com-parison of experimental data on TDES where Coulomb interactions dominate (Wigner crystallized TDES) with the model, revealed the precise form of the influence of Wigner crystallization on the admittance. It was shown that TDES statistical mechanics are critical, in the sense that the TDES can no longer be regarded as a monolithic entity but rather as a system made up of three distinct subsystems (peak electrons, trough electrons, flow electrons) whose dynamic equilibrium determines transport properties.

Future experimental research, benefiting from the presented framework, would undoubtedly extend the measurements of admittance for higher magnetic fields and many more combinations of electron density and temperature. A more elaborate theoretical modeling could perhaps incorporate Coulomb interactions and propose a quantitative result for the admittance of electrons in the troughs (in the context of the Drude model or beyond). This would allow a comparison with the admittance of peak electrons, explaining thus completely the features of the measured admittance.

APPENDIX A