Physical intuition for the behavior of admittance

From the theoretical developments so far, a physical intuition emerges for the behavior of the admittanceY, which will be summarized here for the sake of clarity before we proceed with the discussion of the measurements.

5.4.1. Finite admittance is a result of diffusion. First and foremost, a fi-nite nonzero admittance is the result of electron diffusion and at the limit where diffusion ceases to exist admittance can only be zero or infinite [82]. This means that the weakening of diffusive processes leads ultimately to a limiting value of the admittance: if the limiting value is infinite, then admittance necessarily increases, otherwise, admittance must decrease.

For TDES admittance arises from scattering processes between electrons and helium gas atoms (forT > 0.7K) or ripplons (forT < 0.7K) and its limiting value in the absence of diffusion is zero (this is a general result for two-dimensional

5.4. PHYSICAL INTUITION FOR THE BEHAVIOR OF ADMITTANCE 113

systems [1]). Scattering of electrons by helium gas atoms becomes exponentially weaker with decreasing temperature, while scattering by ripplons only polynomi-ally so³. As a result, the admittance of TDES can only decrease when temperature decreases, because diffusive processes, or equivalently scattering mechanisms, weaken.

This behavior is the complete opposite of what is observed for the system of the three-dimensional electron gas in the conduction band of metals, where the admittance of courseincreasesfor a decreasing temperature. The main (diffusive) scattering mechanism of electrons is collisions with the ion grid, or equivalently the coupling of electrons to phonons, which decreases with decreasing temperature exactly like the coupling of TDES to ripplons. But due to the fact that the limiting value of the admittance in the absence of diffusion is infinite for this system, weakening of diffusive mechanisms actually increases the admittance.

5.4.2. The natural triplet and its effects on the admittance. The physical properties of the admittance in our system are a function of the triplet(T, B, ns), namely temperature, magnetic field and saturated electron density. Certainly, ad-mittance depends also on parameters like the dielectric constant of the substrate and the thickness of the liquid helium film, or excitation voltage and frequency, but these cannot or will not change in the course of a measurement, in the par-ticular way it was performed at least. Let us further analyze the expectedgeneric effect each member of the triplet has on the admittance when the other two are kept constant. Generic means in this context that the complications of electron kinetics, especially diffusion of electrons to and from the peaks of the substrate, which alters the partial fractions of electrons in troughs and peaks, are ignored for the time being. Then, we have the following:

Increasing temperatureshould lead to a monotonically higher admittance because it weakens electron localization in both troughs and peaks. This can be understood, if one recalls from previous chapters the definition of the cyclotron (classical) radiusrc= 2kBT/mω²1/2

, which is a measure of electron local-ization in the sense that largerrcimply less localized electrons, more prone to diffusive processes and therefore more conductive (see eq. 4.42 and the relevant discussion). The proportionalityrc∝T^1/2means thatYshould increase for an increasing temperature and from eq. (5.6) it is seen thatY∝T^1/2as well.

Increasing magnetic fieldon the other hand should lead to a monotonically lower admittance because it enhances electron localization in both troughs and peaks. Again, this can be understood from the definition ofrc if one recalls

3An equivalent formulation occasionally used in the literature is in terms of thecouplingof TDES to helium gas atoms and ripplons. Then one could say, for example, that the coupling of electrons with ripplons is stronger than the coupling with helium gas atoms for lower temperatures, or that the coupling of electrons with ripplons exhibits a polynomial dependence on temperature.

thatω=eB/mfor electrons on peaks andΩ > ωfor electrons in troughs⁴. The dependence of cyclotron radius withBnow has the formrc∝B⁻¹and from eq. (5.6) one getsY∝B⁻¹for the admittance.

Increasing saturated electron densityenhances Coulomb interactions, which were left out of the theoretical treatment so far. It is therefore not easy to predict their influence when they become important. A quantitative measure of the strength of Coulomb interactions is theΓparameter, discussed in sections 1.4 and 2.3, which can be calculated by means of the eq. (2.10). A Wigner crystal is theoretically expected forΓ > 125but experimentally it is found that it forms forΓ&137. Gamma parameter is always provided in the caption of measure-ment plots. For the measuremeasure-ments with a combination of(ns, T)that allows the formation of a Wigner crystal, admittance does exhibit deviations from what is expected from the theoretical model, the deviations’ features and physical inter-pretation being extensively discussed in sections 5.5.2 and 5.5.3. The conclusion of the relevant discussion is that Coulomb interactions are important when a Wigner crystal is formed but much less so otherwise.

5.4.3. Admittance is mainly due to peak electrons. Another intuition is that electrons in peaks should contribute more to the conductivity—and there-fore also the admittance—than electrons in troughs. This follows immediately from the fact thatrc(the classical radius of electron motion) is much greater for peak electrons than troughs electrons becauseωc< Ω. In turn, this is a consequence of the fact that peak electrons performLandauoscillation while trough electrons performDarwin-Fockoscillations. Therefore, the following are completely equivalent formulations:

• Trough electrons are better localized than peak electrons.

• Peak electrons diffuse more than trough electrons.

• Peak electrons contribute more to conductivity (admittance) than trough electrons.

It is important to realize that the kinetics of electrons proposed here are in sharp contrast to that of the two-fraction model [56], where the emerging physical picture considers electrons on rough peaks as essentially “pinned” due to the very strong interaction of those electrons with the underlying dielectric substrate. This, effectively, means that only the “unpinned” part of TDES should be active (i.e trough electrons in our case), or that electrons on rough peaks should not contribute to properties such as electron mobility and conductivity.

However, the size of the peaks, the sufficient thickness of the film on the peaks (approximately25nm) and the periodical nature of the substrate’s surface leads us to adopt a completely different physical picture, namely, that electrons

4Electrons in troughs performDarwin-Fockoscillations due to the presence of holding field andΩis the respective quantum of oscillation.

5.4. PHYSICAL INTUITION FOR THE BEHAVIOR OF ADMITTANCE 115

should be essentially unpinned and mobile in both peaks and troughs, or, to for-mulate more drastically, the effects of substrate roughness should be disregarded.

Therefore, according to our physical picture, which is corroborated byValkering et al.[110], electrons are in principle free to move from a peak to a trough and vice versa.

5.4.4. Expected phenomenology. The physical intuition developed so far suggests a rather simple phenomenology for what could be expected from mea-surements of the formY=Y(B): admittance should monotonically decrease for increasingBwithout exhibiting any maxima or minima. Moreover, forT1<

T2< . . . < Tmthere should be a regular orderingY1(B)> Y2(B)> . . . > Ym(B), while forn⁽¹⁾s < n⁽²⁾s < . . . < n^(m)s there should also be a general ordering Y1(B)< Y2(B)< . . . < Ym(B), which could be irregular.

However, a simple inspection of the measurements in figs. 5.5 through 5.11 is enough in order to realize that TDES on a deformed surface seem to exhibit a much richer phenomenology, unless one would adopt thetrivialapproach of dismissing the maxima ofYin several measurements as artifacts, or else adopt thedeus ex machinaapproach and attribute them to uncontrolled experimental factors like substrate roughness, pinned charges and the like.

A physical factor that was left out until now is the effect of TDES statistical mechanics and in particular its effect on the partial fractions of electrons in troughs and peaks together with the resulting flow admittance. Indeed, it was shown for TDES on deformed liquid helium films that when the magnetic field changes the distribution of electrons on the peaks and in the troughs changes as well. A TDES should then no longer be regarded as a monolithic entity with a uniform density but rather as made up of three component systems.

5.4.5. The three component systems. A TDES on a deformed liquid he-lium film and in the presence of holding and magnetic field partitions in three component systems with different contributions to the admittance and different densities. These component systems are thetrough electrons, thepeak electrons and theflowing electrons.

Flow Electrons: This system comprises of those electrons that moveto orfromthe troughs of the deformed film at any instant of time. They belong neither to the troughs nor to the peaks and their contribution to the admittance is the flow admittance defined in eq. (4.46). This system ceases to exist when dB/dt=0and flow admittance becomes zero in that case.

Peak Electrons: This system comprises of all electrons localized on the peaks of the deformed film. They performLandauoscillations and their contribution to the admittance is expected to be of the formY∝B⁻¹.

Trough Electrons: This system comprises of all electrons localized inside the troughs of the deformed film. They performDarwin-Fock oscil-lations and as a resultrcis much smaller than that of peak electrons.

Consequently, they are better localized and contribute less to the admit-tance than peak electrons and flow electrons, although it is expected thatY∝B⁻¹as in the case of peak electrons.

A much richer phenomenology can be accounted for, when the effect of TDES statistical mechanics is included, because the origins of the measured admittance become diverse and can counteract with each other.

In particular—and this is a perfectly possible scenario—, the general de-crease of the admittance when the magnetic field inde-creases could be counteracted not only by the flow admittance but also by an increasing partial fraction of peak electrons, which as it was repeatedly stated are more conductive than trough electrons. From eq. (5.6) it is seen that this scenario comes about whennp(B)/B is an increasing function for an interval ofB, for example whennpis of the form np(B) =aB²+bB+c.

Of course, in the end, the induced localization from the magnetic field takes over in both peaks and troughs of the deformed charged film and for sufficiently highB(say, whenYfrom peak and trough electrons approaches zero) only flow admittance should be measured. Thus, in that way, a means of calibration and a reality check is obtained for the proposed model because the numeric value of flow admittance should not be far from the measuredYin the limit of strong magnetic fields.

The amended Drude model of eq. (5.4) and the cyclotron radius approxima-tion of eq. (5.8) are complementary with each other, in a sense which we shall try to make more precise. More specifically:

• The amended Drude model, eq. (5.4), could explain minima (values Y(B)< Y0) in the evolution of the admittance, where by definition Y0/Y > 1. But itcannotexplain or even predict admittance maxima (valuesY(B)> Y0) because the right-hand side of eq. (5.4) is always greater than unity.

• The cyclotron radius model, eq. (5.8), could explain admittance max-ima (valuesY(B)> Y_∞), where by definitionY_∞/Y < 1. But itcannot explain or even predict minima in the evolution of the admittance (val-uesY(B)< Y_∞) because the right-hand side of eq. (5.8) is necessarily less than unity.

These important observations should be kept in mind in the analysis of the experimental data, where one encounters cases whereY(B)> Y0andY(B)< Y_∞ (see for example fig. 5.9A and the relevant discussion).