Two estimates for the normalized admittance

128 128.5 129 129.5 130

σ[nS]

−4 −3 −2 −1

Vg[V]

Vg↑ Vg↓

Effect of Guard Voltage atT=1.2K

Figure 5.4. Conductivity_σ of a TDES on the nanotower substrate against guard voltage V_g. Saturated electron density is_7.29×10¹²m⁻²(_≈6.4eper trough). Holding field volt-age is₆V, excitation voltage is ₁₀mV and excitation fre-quency is₅₅kHz. Initial value was_Vg=1.4V.

weak, and film thickness cannot be varied in a wide range without encountering problems in charging the liquid helium surface with electrons. For these reasons, guard voltage and bulk liquid helium level—which determines film thickness—

remained unaltered throughout most of localization experiments.

5.3. Two estimates for the normalized admittance

The physical quantity that was measured in our experiments was the admit-tanceYof TDES on deformed liquid helium films. Up to now, apart from the flow admittanceYf, given by eq. (4.46), which is predicted to be a component of the total admittance, the implications of the model with respect to the total admittance were not explicitly discussed but implied at a qualitative level. In this section, an effort is made to derive a quantitative estimate for thenormalized admittance (a pure number) based on a modification of the standard Drude model, which was described in section 2.5.1, for TDES on deformed liquid helium films. The Drude model estimates the normalized quantityY0/Y, whereY0is the admittance for zero magnetic field. In addition, by means of a different physical reasoning, a proportionality is derived for the normalized quantityY_∞/Y, where Y_∞is the limiting admittance for extremely strong magnetic fields.

5.3.1. Drude normalized admittance. The basic prediction of the Drude model is the relationσ0/σ=1+µ²B², connecting the normalized conductivity σ0/σwith electron mobilityµand the magnetic fieldB(the same relation holds for the normalized admittance as well). In the original formulation of the Drude model, the TDES resides on a flat surface and its electrons undergo Landau oscillations. However, the TDES for the particular deformed films considered in chapter 4 partition themselves in two fractions, peak electrons and trough electrons.

If the partial fraction of peak electrons is denoted withnp, the partial fraction of trough electrons will necessarily be1−np. Peak electrons perform Landau oscillations and the original Drude model applies without modification, giving a contribution 1+µ²_pB²

npto the total normalized admittance, where µpis electron mobility on the peaks.

Trough electrons on the other hand perform Darwin-Fock oscillations and the derivation of the Drude model outlined in section 2.5.1 no longer applies.

Nevertheless, it can be expected that a contributionF(µt, B) (1−np)to the total normalized admittance should arise, whereF(µt, B)is an as yet unknown function of the magnetic fieldBthat contains electron mobilityµtin the troughs as a parameter.

Then, the total normalized admittance according to the amended Drude model we propose for TDES on deformed films should be of the form

Y0

Y−Yf

=

1+µ²_pB²

np+F(µ_t, B) (1−np). (5.2) In the denominator of the normalized admittance the flow admittanceYfis subtracted from the measured total admittanceY, because its physical origin is different and cannot be described by a Drude model. In the discussion of the experimental data in section 5.5 and especially fig. 5.6 one can easily conclude that the agreement of the model with the experimental data is improved when Y0/(Y−Yf)instead ofY0/Yis considered.

The unknown functionF(µt, B)could in principle be calculated by starting off from the Boltzmann equation

ma= −mv

τ −e∇(V−v·A), (5.3)

which is just eq. (2.14) with the Lorentz force written in terms of the potential Vand the vector potentialA, and following through the steps of the derivation in section 2.5.1 in order to determine the conductivity tensor. However, this approach is no longer straightforward because the potential, except for the AC componentV0e^iωtof the lockin amplifier, also contains the DC component given by the general eq. (4.22) due to the deformed topology of the film. For this reason, an approximation ofF(µt, B)shall be employed.

More specifically, one can see from eq. (5.2) by settingB=0thatF(µt, 0) =1, since it isnp=0andYf=0, while by definitionY=Y0in that case. Let us now generalize and approximateF(µ_t, B) =1even forB > 0. Then, one can immediately obtain the estimate

Y0

Y−Yf≈1+µ²_pB²np when F(µt, B) =1, (5.4) for the normalized admittance, which is very similar to the original Drude expression for TDES on flat films.

5.3. TWO ESTIMATES FOR THE NORMALIZED ADMITTANCE 111

The effect of the approximation ofFwas not to ignore any contribution of trough electrons to the total normalized admittance. This would have been the case if it wereF(µt, B) =0, leading to the equationY0/(Y−Yf) = 1+µ²_pB²

np

for the normalized admittance. Essentially, the approximation was to regard the contribution of trough electrons as constant for all magnetic fields. This is correct only if the said contribution is minimal, or equivalently, when trough electrons are very well localized regardless of the magnetic field. It was explained in section 4.4 that for deformed films wetting a dielectric substrate most electrons are already very well localized inside the troughs (and their mobility is corre-spondingly very small) when a holding field is present even before a magnetic field is applied. The approximationF(µt, B) =1forB>0should therefore not be unjustified.

It has been shown in chapter 4 thatnpis not constant but a function ofB among other things. As a consequence, the right-hand side of eq. (5.4) already anticipates that the behavior of the normalized admittance in the left-hand side, which is calculated from the experimental data, will deviate from the straight line one would expect from the normal Drude model, when normalized admittance is plotted againstB². The open question is whether the anticipated deviation will be predicted in its exact form or not.

5.3.2. Cyclotron radius normalized admittance. One of the initial mo-tivations of localization experiments came about from the intuition that the cyclotron radius of electrons under the influence of a magnetic fieldB= (0, 0, B) decreases and promotes their localization. This means that the total measured admittance depends on the cyclotron radius and the form of this dependence should be explored further.

In the context of the model proposed in chapter 4, let peak electrons be characterized by a cyclotron radiusr^(p)c and trough electrons byr^(t)c . Then, a proportionality

Y∝r^(p)c np+r^(t)c (1−np) (5.5) should hold for the total admittanceY.

The generic cyclotron radius isr= 2kBT/mω²1/2

, whereωis to be substituted withωc=eB/mfor peak electrons andΩ= ω²_E+ω²_c1/2

for trough electrons (ωEis defined in eq. 4.26). Even for magnetic fields of several Tesla it isωcωEand one can writeΩ≈ωEwith good accuracy. Then, by replacingrcin eq. (5.5) with the proper expression with respect toω, it is found that For obtaining the normalized admittance one could proceed to divideY0

byYas given in eq. (5.6). However, a more elegant proportionality turns out if

instead the limiting admittanceY_∞for very strong magnetic fields is used. Of course, at the limitB−→∞it isY_∞=0and therefore a more realistic limit, like the (very strong)Bwhereωc≈ωEshould be used, which then leads to the limiting value

Y_∞−→ 1 ωE

2kBT m

1/2

for ωc≈ωE. (5.7) The proportionality that connects the normalized admittance withωc= eB/mand the partial fraction of peak electronsnpcan then be written down very elegantly as

Y_∞

Y−Yf∝ 1 1+

ω_E ωc−1

np

. (5.8)

Comparing eqs. (5.4) and (5.8) the following comments can be made:

• Normalized admittance of eq. (5.4) begins with a value of unity and is always a numbergreaterthan unity for strongerB.

• Normalized admittance of eq. (5.8) is always a numberlessthan unity and approaches unity for strongB.

• The opposite behavior of normalized admittance exhibited by eqs. (5.8) and (5.4) can be traced back to the fact thatY0> Y_∞.

It should be kept in mind that eq. (5.4) is an approximation, while eq. (5.8) aproportionality. By plottingY0/(Y−Yf)there should be an appropriate fit with 1+µ²_pB², from which electron mobilityµpcan be determined. In contrast, a plot ofY_∞/(Y−Yf)must be multiplied by a proportionality factor in order to become comparable with the right-hand side of eq. (5.8). However, after the multiplication by this factor, which cannot be predicted by the model, the form of the two curves should be the same.