Theory of Quasi Zero-Dimensional Localization
5.1. General considerations
Quasi zero dimensional localization was investigated experimentally in TDES on liquid helium films that wet the nanotower periodic structure, depicted in fig. 4.1.
Two dimensional electron systems of various electron densities held at var-ious temperatures underwent a magnetic field sweep at a constant rate while the admittanceY=I/V(with obvious notation) was measured by means of a Sommer-Tanner electrode geometry (see fig. 3.7). Temperature and electron density were kept constant throughout the magnetic field sweep.
The conductivity of TDES on liquid helium in the presence of a magnetic field depends on temperature, electron density and the magnitude of the magnetic field. Temperature dependence of conductivity reflects the strength of the various scattering mechanisms (ripplons, helium gas atoms, substrate), while electron density affects conductivity by means of the Wigner phase transition, as discussed in the relevant sections of chapters 1 and 2. The application of a magnetic fieldB. 1.5T leads to a resistanceR(B) =R(0) 1+µ2B2
, whereR(0)is the resistance in zero magnetic field andµis electron mobility (see section 2.5.1). A plot of R(B)as a function ofB2should then be a straight line whose slope isµ2R(0), as shown in fig. 5.1 which depicts a measurement of the resistance of a TDES on bulk liquid helium above the nanotower substrate. It should be noted that for stronger magnetic fields or higher electron densities deviations from this simple Drude behavior occur, which are reviewed both theoretically and experimentally byMonarkha et al.[82].
The use of suitable periodically structured dielectric substrates wet by helium films, in order to achieve electron localization of TDES, was explored extensively byKovdryaand coworkers in the one dimensional case [62, 63, 85] andValkering et al.[110] in the zero dimensional case.
Valkering et al.[110] measured the admittance of a TDES on a thick helium film with estimated thickness0.4µm that wet a two dimensional grating similar to the nanotower substrate, as a function of a magnetic field perpendicular to
103
Figure 5.1. ResistanceRof a TDES on bulk liquid helium as a function of the square of the magnetic field,B2, measured at a temperatureT=500mK.
A linear Drude behavior is ob-served for weakB, with zero field resistanceR(0) =9.45× 107Ωand electron mobility µ=1020m2/Vs.
0 10 20 30 40
R[GΩ]
0 0.2 0.4 0.6 0.8 1
B2[T2]
Magnetoresistance atT=500mK
the surface of liquid helium. A linear Drude behavior with respect toB2could be established, but the resulting electron mobility was one order of magnitude lower than for a comparable TDES on bulk liquid helium at the same tempera-ture. To our knowledge, the ideas and results ofValkering et al.[110] were not expanded further and we could not find other instances in the literature where zero dimensional electron localization with periodically structured substrates is attempted.
The application of a magnetic field perpendicular to the surface of liquid helium forces electrons to revolve along circular orbits whose classical radiusrc, defined in eq. (4.42), becomes smaller for increasing magnetic field magnitudes.
If the nanotower substrate is wet with a sufficiently thin helium film, the surface of liquid helium will follow the curvature of the substrate, developing thus troughs and ridges. As the magnitude of the magnetic field increases, the radii of electron orbits will become equal or smaller than the radii of the troughs and electrons will be trapped in there, revolving around the ‘walls’ of the troughs.
Conductivity in the presence of a magnetic field is largely determined from the diffusive motion of the centers of the circular electron orbits, which is much less pronounced for trapped electrons. As a result, in the ideal case of very low temperatures, it should be expected that the conductivity (or equivalently, admittance1) would abruptly decrease when the magnetic field makesrcequal to the trough radius and this abrupt decrease would signal the onset of zero dimensional localization.
For higher temperatures, the thermal energy might be enough to permit elec-trons (especially those belonging to the ground Landau state) to be drawn in and out of the troughs, but this effect should progressively diminish asrcdecreases further and electrons are drawn towards the bottom of the trough. This means
1A direct equivalence exists between conductivity and admittance for TDES with the same geometry and in the following, depending on the context, they will be referred to as equivalent.
5.1. GENERAL CONSIDERATIONS 105
that the abrupt decrease of conductivity at lower temperatures would be softened at higher temperatures and would resemble an activation-type Arrhenius law.
This simple qualitative picture of electron localization must be amended in view of the experimental results to be presented and the model that was developed in chapter 4.
According to the theoretical results of chapter 4, the interaction of electrons with the dielectric substrate is so strong that most electrons are localized in the troughs under the influence of the holding field even before a magnetic field is applied. The conductivity of a TDES under these conditions should be close to zero, with any nonzero conductivity arising due to thermally induced fluctuations in the motion of localized electrons plus the conductivity due to the small fraction of free electrons.
If, in addition, a magnetic field is applied, the density of states is altered and the partial fractions of free (delocalized) versus trapped (localized) electrons are altered as well. This means that as the magnetic field changes, electrons localized in the troughs move diffusively to the ridges, in order to account for the alteration of the partial fractions, so that in the end a diffusive current associated with a nonzero admittance develops.
From the definitionY=I/Vof admittance,Vcan be identified as the voltage difference between the bottom of the trough (the trough is a well not only in geometric but also in potential terms) and the ridge, whileIis the current of electrons moving diffusively from a trough to a ridge. Elementary manipulations then lead to eq. (4.46), where one sees thatYdepends on the rate of change of the magnetic field (which is known and constant) and the rate of change of the partial fraction of electrons on the ridges (or the troughs) with respect to the magnetic field.
The partial fraction of electrons on the ridges can be calculated numerically for any given set(ns, T, B), wherensdenotes the saturated electron density of the TDES (determined from the holding field voltage),Tis the temperature and Bthe magnitude of the magnetic field. In a magnetic field sweep,ns andT are constant throughout, whereasBis measured in discrete steps. Therefore, it suffices to obtain a discrete set of pairs(B, n), wherenis the partial fraction of electrons on the ridges as calculated numerically from eq. (4.45)2, which can then be interpolated by a continuous functionn(B). Numerically differentiating n(B)provides the rate of change of the partial fraction of electrons on the ridges with respect to the magnetic field and allows the admittanceYto be determined theoretically. The theoretical values can thereafter be compared with actual measurements. The procedure is easier to implement in symbolic programming
2This equation actually refers to the partial fraction of electrons in the troughs, but an almost identical formula holds for electrons on the ridges, as explained in the text accompanying the equation.
languages likeMathematicaand execution time for 200 points and the first 40 quantum states should not be more than a minute for vintage PCs.
Electron interactions were ignored in the derivation of the theoretical model and electrons residing in the ridges and the troughs were assumed not to conduct at all. As a result, the admittance of a TDES when the magnetic field remains constant should be zero according to the model, a prediction which is obviously false.
Indeed, electrons in both the ridges and the troughs conduct and electron interactions are important for high electron densities, making the admittance nonzero even when the magnetic field does not change. However, the main result of the model is that a component of the measured admittance should be attributed to a diffusive electron current from and to the troughs as the magnetic field changes. This component will constitute a major part of the measured admittance for low electron densities and low temperatures, but it will be rendered a minority for high electron densities and high temperatures.
Moreover, assuming that the TDES covers the whole area (1mm2) of the substrate and by estimating the number of troughs from the approximation 2r=800nm for the diameter of a trough, the model predicts an admittance which can be directly compared with the measuredY, without the need to employ the transmission line or the equivalent circuit model in order to calculate the conductivity of the TDES.