Localization with a holding field and a magnetic field

An electron inside a trough of a helium film that wets a periodically struc-tured substrate, in the presence of a holding field and a magnetic fieldB=

0, 0, B

, is characterized by the hamiltonian H^ = p^² of angular momentum. The classical picture of electron motion implied by the hamiltonian of eq. (4.37) is that of revolutions in circular orbits with cyclic frequencyΩ. The hamiltonian of eq. (4.37) is obtained from the more general form of the hamiltonian in eq. (B.1), the potential of eq. (4.25) and the selection of the symmetric gaugeA= −r×B/2for the vector potentialA.

Apart from the shift Λs/(dtop+2δ0) −mω²_Eb²/8, the hamiltonian of eq. (4.37) is identical to the analytically solvable Darwin-Fock hamiltonian⁹. Therefore, the eigenenergies ofH^ in eq. (4.37) will be given by

Eν,l= (2ν+|l|+1)hΩ−l

Instead of the precise form of theDarwin-Fockwavefunctions, which will not be reproduced here, we will make use of themagnetic lengthλB, defined by

λB= h

mΩ 1/2

. (4.40)

The magnetic length characterizes the spatial extent of Darwin-Fock wavefunc-tions, in the sense that quantum objects described by theDarwin-Fock wavefunc-tions and are someλBapart can be regarded as distinct, particle-like entities.

9The Darwin-Fock hamiltonian was introduced and solved byFockin [Z. Phys.47, 446 (1928)]

and independently byDarwinin [Proc. Camb. Phil. Soc.27, 86 (1930)].

4.5. LOCALIZATION WITH A HOLDING FIELD AND A MAGNETIC FIELD 99

For all measurements presented in this thesis, the magnetic length was never larger than5nm.

On the ridges, the potentialV(x, y)is zero, and electron motion there is described by a hamiltonian identical with that of eq. (B.2). As a result, electron motion on the ridges will correspond to circular motion with cyclic frequency ωc(Landau oscillations) and an energy spectrum given by

Eν= (2ν+1)hωc. (4.41) Cyclotron radiusrchas been defined in eq. (2.13) as the mean classical radius at temperatureTof the circular orbit an electron describes, under the influence of a magnetic field along thez-direction. More generally, the cyclotron radius Rccan be similarly defined as the mean classical radius at temperatureTof the circular orbit an electron describes, under the influence of both a holding field and a magnetic field along thez-direction. It is not difficult to show that the equation that relatesRcandΩis

Rc= 1 Ω

2kBT m

1/2

. (4.42)

The ground Darwin-Fock state (ν=0andl=0) exhibits the lowest cy-clotron frequency, equal toΩ, while excited states (ν=1, 2, . . .andl=0) are associated with progressively higher cyclotron frequencies (3Ω,5Ω, . . .). An inversely proportional relation, eq. (4.42), connects cyclotron frequency to cy-clotron radius, meaning that the ground Darwin-Fock state exhibits the largest cyclotron radius, which becomes progressively smaller for excited states.

Localization of electrons inside the troughs of a periodic substrate is expected forRc6b/2, where it is reminded thatbis the diameter of a trough (for the nanotower surfaceb≈800nm). Electron localization will be complete, if the cyclotron radius of the ground Darwin-Fock state is less than or equal tob/2. This condition, along with eq. (4.42) and the definitions ofωcandωEin eqs. (2.13) and (4.25) respectively, leads after some algebra to a condition for the onset of complete electron localization, of the form

8mkBT

e²b² 664mΛsR e²b⁴ +mE_⊥

eR . (4.43)

Physical factors that induce complete electron localization are those that either minimize the left-hand side, or maximize the right-hand side of eq. (4.43).

For example, an increasing temperatureT, a larger trough radiusb, stronger holding fieldsE_⊥and increasing electron densities (enclosed in the radius of curvatureR) all induce complete electron localization. But when dielectric sub-strates are present, whose influence is enclosed inΛs, the term64mΛsR/e²b⁴ is so large that complete electron localization happens for any magnetic field regardless of its magnitude.

By definition, in complete localization electrons cannot be free and the (dynamic) equilibrium equation (4.20) at any instance of time will involve two systems; electrons instantly localized in a trough and electrons instantly localized on a ridge. Taking electron spin into account, the discrete density of states of the system “electron in a trough” isD_t(E_ν,l) =2(2ν+|l|+1)and the continuous, Hartree-Fock density of states is equal toD_t(E) =1/2hΩ. Again taking electron spin into account, the discrete density of states of the system “electron on a ridge”

isD_r(Eν) =2(2ν+1)and the Hartree-Fock density of states isD_r(E) =1/2hωc. Noting that the area of a trough (which contains the majority of electrons and is much larger than the area of a ridge) is approximatelys=πb²/4, the sum of partial fractions from the ground state (ν=0andl=0) upwards leads to the equilibrium equation

X∞ ν=0

D_r(E_ν) e^β(Eν−µ)+1+

X∞ ν,l=0

D_t(E_ν,l)

e^{β(Eν,l−µ)}+1=πb²

4 ns, (4.44) from where the equilibrium chemical potentialµ0(T, B, E_⊥)can be determined.

Except for temperatureT, magnetic fieldBand holding fieldE_⊥, the equi-librium chemical potential depends also on experimental parameters like the dielectric constantΛsof the substrate and the diameterbof a trough, which cannot be changed in the course of a measurement.

For numerically solving eq. (4.44), the infinite summation must be truncated into a summation that involves a finite number of states. The resulting finite series converges to a progressively more accurate value ofµ0when more excited states are included. It was found by numerical experimentation that the inclusion of excited states up toν=40was always sufficient for numerical convergence.

Provided that the equilibrium chemical potentialµ0is known, the partial fractions of electrons occupying the state with energyEν,linside a trough will then be given by

nν,l

ns

= D(Eν,l)

e^{β(Eν,l−µ0)}+1. (4.45) An identical equation holds for the partial fraction of electrons that belong to the state with energyEνon a ridge, as it suffices to substituteEν,lin eq. (4.45) withEνas given in eq. (4.41).

The partial fractions of electrons in the troughs and electrons on the ridges remain constant when the physical quantities that affect electron potential do not change. One would expect then, that the measured conductivity should be a constant nonzero value, reflecting the random fluctuations of electron motion due to temperature and Coulomb interactions.

However, when the magnetic field is not kept constant the partial fractions of electrons in the troughs and on the ridges readjust to a new equilibrium, following the changes ofB. This readjustment means that electrons should be

4.5. LOCALIZATION WITH A HOLDING FIELD AND A MAGNETIC FIELD 101

diffusively drawn in and out of the ridges as the magnetic field changes, whence a contribution to the admittance of the TDES should emerge.

In the ideal case where localized electrons on either the ridges or the troughs do not conduct at all, a nonzero admittance can only be attributed to electrons that diffuse from the ridges to the troughs and vice versa. If only the magnetic field changes, thisflow admittanceY=I/Vof the TDES can be restated as

Y= 1

wherenis the electron charge (in Cb) on the ridges, or in the troughs. The absolute value of the derivatives was used, because flow admittance should be positive irrespective of the sign of the rate of change of the charge.

Physically, flow admittance arises because a voltage differenceVemerges between a trough and a peak and it is measurable because the diffusive motion of electrons from the troughs to the peaks except for thez-component, which cannot be measured with the experimental apparatus used, contains also axy -component, which can be measured because peaks and troughs are obviously at different locations in thexyplane.

For simplicity,Vin eq. (4.46) can be considered to be the voltage differ-ence between a ridge and the bottom of a trough. The potential on a ridge has been defined to be zero, whereas the bottom of a trough has a potential energy Λs/dtop−mωEb²/8, meaning that the voltage difference is

V= Λs

e(dtop+2δ₀)−mωEb²

8e . (4.47)

The electron charge density (in Cb/m²) inside a single trough is the sum of nν,las given in eq. (4.45) over all states multiplied by the electron chargee. A single trough has an area of approximatelyπb²/4and the nanotower substrate, whose area isA =1cm², contains4A/πb²troughs. Therefore, the electron chargenin the troughs of the nanotower substrate will be

n=eAns

X

ν,l

D(Eν,l)

e^{β(Eν,l−µ0)}+1, (4.48) with a similar equation holding for the electron charge on the ridges.

If a measurement of the admittance of a saturated TDES on the nanotower substrate is performed such that the temperatureT and the holding fieldE_⊥ are kept constant while the magnetic fieldBis changed at a constant rate, then the experimental data (values ofYunder these conditions) can be compared with the flow admittance that is calculated from eq. (4.46). Of course, in realistic experimental conditions, it should be expected that the measured admittance be generally larger than the flow admittance. However, the measured admittance should approach the theoretically calculated flow admittance in the limit of high magnetic fields, low temperatures and low electron densities. Indeed, for high

magnetic fields and low temperatures it is reasonable to expect that electron localization in both peaks and troughs would be strong and therefore that dif-fusive processes there would be weak, meaning that there is very little—if at all–conduction from electrons. Low electron densities have the same effect, as they ensure that Coulomb interactions among electrons are minimal and do not disturb their localization.

To our knowledge, neither the physical motivation nor any mathematical definition of the flow admittance appears in the relevant literature. This can be understood if one recalls that flow admittance is zero either for TDES on flat films whenBchanges (because thenn⁰(B) =0) or TDES on deformed films whenBis constant or zero (because thenB⁰(t) =0). Indeed, flow admittance is peculiar to TDES only on deformed films and only when the magnetic field is altered.

CHAPTER 5

Experiments on Quasi Zero-Dimensional

Localization