Microwave absorption of electrons bound to films of helium

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Microwave absorption of electrons bound to films of helium

Beate Lehndorff, Klauss Dransfeld

To cite this version:

Beate Lehndorff, Klauss Dransfeld. Microwave absorption of electrons bound to films of helium. Jour-

nal de Physique, 1989, 50 (18), pp.2579-2586. �10.1051/jphys:0198900500180257900�. �jpa-00211083�

Microwave absorption ^of electrons bound to films of helium

Beate Lehndorff and Klauss Dransfeld

Fakultät für Physik, ^Postfach 5560, ^Universät Konstanz, ^D-7750 Konstanz, ^F.R.G.

(Reçu ^{le 14}

1989, révisé le 29 mai 1989, accepté ^{le 2} juin 1989)

Résumé.

Nous

mesuré l’absorption micro-onde, pour des températures inférieures à 1,75 ^{K et à la} fréquence ^{fixe de 9} GHz, d’un gaz d’électrons

film d’hélium superfluide. ^Les

électrons sont attirés

la surface par les forces images ^et par

champ électrique ^continu

additionnel d’intensité variable et dirigé selon la normale du film. A

de la formation de fossettes dans la surface de l’hélium, les électrons sont aussi localisés dans la direction latérale.

Pour

champ micro-onde orienté parallèlement

^{film et} pour des températures en-dessous de 1,4 K,

observé

maximum large ^de l’absorption micro-onde quand ^le champ ^de

rétention varie. Nous pensons avoir observé la vibration latérale de résonance des électrons localisés dans leur fossette, qui peut être accordée électriquement par le champ ^continu pressant.

Abstract.

We have measured the microwave absorption ^of

^two dimensional electron gas

film of superfluid ^{helium at} temperatures below 1.75 K and at

fixed frequency of 9 GHz. The electrons

attracted to the surface by image forces and in addition by

variable dc-electric field directed normally ^to the film. Due to the formation of dimples ^{in the} helium surface the electrons

are

also localized in the lateral direction. For the microwave field oriented parallel ^to the film and for temperatures below 1.4 K

have observed

broad maximum of the microwave absorption

when

varied the clamping field. We believe that

have observed the lateral

vibration of electrons localized in their dimples, ^which

^be electrically ^tuned by ^the dc-holding

field.

Classification

Physics ^Abstracts

71.55

73.20

67.70

Since the invention of the transistor in 1948 by Schockley and Pearson [1] ^{and the} development of mosfet-structures two dimensional electron systems have been of growing

interest. As demonstrated by ^von Klitzing [2] ^{the two} dimensional electron gas -(2 DEG) ^can

show a completely ^{new and} unexpected behavior known as the quantum Hall effect (QHE).

In the last few years ^not only the two-dimensional electron gas but also one-dimensional

«

quantum ^wires

[3] ^{and even zero} dimensional

quantum ^{dots »} [4] ^{have been} investigated.

Since the early seventies it has become clear that electrons on the surface of liquid ^helium

are attracted to the liquid ^surface by ^an image potential, ^while they remain free to move

parallel ^to the surface. Electrons bound on liquid helium have served as an ideal modelling

system ^{for the} study ^{of the} general properties ^of the 2 DEG. One very interesting phenomenon discovered in this system ^{was the so-called}

Wigner-cristallization

^{of the}

electrons forming ^a two-dimensional electron solid [5]. ^The Wigner-crystal ^{was observed to}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180257900

2580

form on the surface of liquid ^{helium at} temperatures of the order of 0.5 K and for electronic densities of up ^to 109 _per cm2. At these densities the potential energy of the electrons exceeds their kinetic energy (about k7) considerably ^{and the} Wigner crystallization ^can be induced by cooling ^{down to a} few tenths of a degree.

At electronic densities of 109 _per cm 2 which can be obtained on the surface of liquid helium,

the Fermi energy of the electrons ^amounts only ^{to 30} mK, which is much less than the thermal energy of 0.5 K. In this ^sense electrons on liquid ^{helium at} temperatures of the order of 1 K

can be considered as a classical Coulomb system. ^The quantum ^limit

^{where the} Fermi

energy exceeds the thermal energy

requires temperatures well below 30 mK or

at the

more convenient temperatures of about 1 K - electronic densities of about 1011 _per cm2.

In their pioneering experiments Etz et al. [6] showed that such densities ^can indeed be attained and remain stable on helium films of about 100 Â thickness on substrates like Mylar

foils or silicon. In the case of electrons on thin helium films the strong image potential ^{of the} substrate, however, ^{leeds to a} decrease of their mobility ^also parallel ^{to the surface. After the}

theoretical prediction of this effect by several authors [7-9] this so-called polaronic ^transition

was first experimentally discovered by ^Andrei [10]. ^{In her} experiments the electron mobility dropped by four orders of magnitude when the thickness of the helium film was decreased from about 1 000 Â to 700 Â. At about 900 Â ^an interesting ^{resonance like structure was} superimposed. This behavior was theoretically analyzed by several authors [11-14]. ^Peeters

and Jackson [14] ^{achieved a rather} good qualitative agreement ^{with the} experimental

observations. They calculated the effective _mass, low frequency mobility ^{and excited states of}

the electron-ripplon-complex. ^{The basic} picture ^{is the} following : while electrons undergo ^this polaronic transition a dimple ^{is formed on} the helium surface underneath each electron. Since the velocity ^of capillary ^waves (ripplons) ^on liquid helium is very low (about ⁵ m/s) compared

to the thermal velocities of the electrons the dimple ^can be assumed to be quasi ^{static even at}

the rather low frequencies of 27 kHz used by ^Andrei [10]. The formation of dimples ^{thus leads}

to a strong localization of the electrons also parallel ^to the surface. If this picture ^is adequate

we thought it should be possible by ^a lateral electric field of suitable frequency ^{to induce}

electronic transitions between the energy ^states in the localization potential.

It was therefore the principle aim of this investigation ^to search for these resonance

transitions which we expected ^{to occur in the} GHz-frequency range [14]. Only ^{few other}

microwave experiments with electrons on helium films have so far been published [15, 16].

For our experiments ^{we used a} cylindrical ^microwave cavity having ^an

empty

^resonance

frequency ^{of 12 GHz} (Fig. la). ^After loading it with the dielectric substrate the resonance

frequency shifted down (mainly ^{due to the} high dielectric constant of Si) ^{to 9} GHz and the final Q-factor ^{was about 5 000 at} 4.2 K. The microwave resonator was used in the

TEmo’mode with the ac-electric field of axial direction parallel ^to the surface of the silicon wafer carrying the helium film. This undoped silicon wafer also served as an electrode to

apply ^the dc-holding field necessary needed ^to keep up ^to 1011 electrons per cm2 on the helium film. The microwave conductivity of silicon at helium temperatures ^is neglegible while its dc-

photoconductivity under illumination is high enough ^to serve as a dc-electrode. Nevertheless in our first preliminary experiments pure silicon ^or oxidized silicon (carrying ^a nearly 1 000 À

silicon dioxide film) ^{turned out to be} unsuitable as a substrate for the following unexpected

reason : as soon as charges ^were deposited ^{on a} He-film carried by the Si-wafer we observed

an extremely strong ^microwave absorption spoiling the microwave resonance of the cavity

almost completely. ^This strong ^microwave absorption also occurred in the absence of ^a He-

film when the charges ^were deposited directly ^on the oxidized silicon. We believe that by ^the

charge deposition ^a conducting ^inversion layer ^{is formed} underneath the silicon surface facing

each deposited charge resting immobile and at close distance ^on the Si02-filM surface. We have not pursued ^this interesting phenomenon, but rather tried to avoid the formation of an

inversion layer ^{in Si} by increasing the distance between the Si-surface and the deposited charges. For this purpose ^we used a polymer ^foil (of Mylar ^or Teflon) ^{as an addional dielectric}

spacer medium between the He-film and the Si-electrode as indicaded in figure ^lb.

The thickness of the absorbed He-film (about ³⁰⁰ Â) ^was controlled by ^the position ^{of the}

substrate (1 ^{cm above the} liquid ^helium level). ^The helium film was electrically charged using

a corona discharge ^{from a} tungsten tip extending ^a few tenths of a millimeter into the cavity (see Fig. la). ^The charge density ne deposited ^on the He-film is defined by ^the clamping voltage ^U applied during ^the deposition process :

Fig. ^la.

^{Microwave resonator} with substrate. The cavity dimensions

20

in length ^{and 20}

diameter. The substrate (enlarged cross-section in Fig. lb)

made up by

silicon wafer

(30

¹⁰

^0.7 mm) carrying

polymer foil and the helium film

top of it. The silicon wafer is

electrically ^connected by

Al-electrode to the clamping voltage outside the cavity.

Fig. ^lb.

Enlarged cross-section of the substrate showing ^the top part of the silicon wafer carrying ^the

polymer foil and the helium film.

2582

Here _Eo and

are the permittivity of free space and the dielectric ^constant of the polymer ^foil respectively and where d stands for the thickness of the foil. The sign ^{of the} applied voltage ^is always such that the electrons are pressed against ^the He-film. After completing ^the deposition ^the voltage ^{could be} slowly reduced and increased again

within certain limits

without strongly changing the electronic density. Even when the voltage ^{of the} clamping ^field

was reduced to zero not all electrons are lost. Some remain on the film due to the image ^forces

of the substrate and perhaps ^{also due to the} pinning by charged surface defects.

After electrically charging ^{the film as} described above the microwave absorption ^was

measured as a function of the dc electric clamping ^{field at} temperatures between 1.12 and 1.75 K. The microwave bridge ^{we used had a} resolution of 10- 4 of the incident power and is described elsewhere in detail [17].

Figure ^{2 shows}

^{as a} typical example

the observed microwave absorption ^caused by

electrons of a density ^{of 2 x} 101° CM- 2 which where bound to a helium film condensed on a

6 _1£m thick Hostaphan (Mylar) foil stretched over the silicon wafer. When the clamping ^field

was varied the microwave absorption clearly ^{showed a} broad maximum at about 15 volts. For

voltages above 15 volts the absorption drops rapidly. ^When starting ^with larger initial electron densities the position of the maximum shifts to higher clamping fields. The measured

absorption without any previous ^electron loading is also included in figure ^{2 for} comparison.

Fig. ^2.

The microwave absorption ^at

frequency ^{of 9 GHz and}

temperature ^{of 1.12 K}

^function of the applied clamping voltage. ^A polymer foil, ⁶ mli thick, of Hostphan (Mylar) ^served

^substrate

for the helium film. The electron density

^{about 2}

^{101° cm-’} (top curve). ^A

without electrons

(bottom curve) is shown for comparison.

The absorption ^{is also} strongly temperature dependent. Figure ^{3 shows} measurements at

three different temperatures below 1.75 K with an electron density ^{of about} 1011 cm- 2. At

1.75 K and above no maximum could be seen. Using ¹³ >m Teflon instead of Mylar ^{a similar}

Fig. ^3.

^Similar

figure 2 but for three different temperatures and for the higher ^electron densities of about 1011 cm- 2.

behavior could be seen as is shown in figure ^{4. At} higher electronic densities and clamping

fields below 50 volts we observed ^an additional absorption peak ^{which is not} yet understood.

It is not clear whether this additional peak ^{is due to a} Wigner transition or caused by ^a detrapping ^{of some} of the electrons from their individual dimple. ^{It seems} reasonable, however, ^{that in} figure 4 the maximum occurs at higher clamping voltages because for the Teflon - having ^a larger ^{thickness and a} lower dielectric constant than Hostaphan ^{- one has}

to apply ^a higher voltage ^{U to reach the same} external field and force acting ^on the electron

according ^{to :}

It should, however, also be mentioned that the surface qualities ^{of the} amorphous Hostaphan - foil and the more crystalline ^{Teflon are} probably different, which may also be of importance ^when comparing the results with both materials.

The observed density dependent maximum of the microwave absorption may be discussed in terms of a resonance transition of the electrons between the energy levels resulting ^{from the}

harmonic potential of the deformation. The lateral binding energy of the electron in its dimple

can be expressed ^as [17] :

where Eim is the electric field created by ^the image charge of the substrate (dependent ^{on its}

dielectric constant : 3.7 for Hostaphan ^{and 2.1 for} Teflon) ^and EeXt the external clamping

field. The surface tension of liquid ^{helium is} given by cr

^{3.78 x} 10-4 N/m2. With these numerical values we find for the binding energy of the electron ^to its dimple ^on Hostaphan

W

1.9 x 10-22 j (W/k

¹⁵ K) ^{for zero} clamping field and at the maximum shown in

figure ^{3 we} get ^W

^{7.7 x} 10-22 J (W/k

⁵⁵ K).

2584

Fig. ^4.

^The microwave absorption ^at

frequency of 9 GHz and at

temperature ^{of 1.21 K}

function of the applied clamping voltage. ^Here

polymer ^{foil of} Teflon, ¹³ m thick, served

substrate for the helium film. The electron density

^{about 7}

1010 cm- 2 (top curve). ^{The bottom}

curve was

measured without electrons pesent.

This binding energy is high compared ^{to kT at the} working temperature of T = 1 K and much

higher ^{than h v} ( v

^microwave frequency), the assumed spacing of the energy levels.

Therefore the harmonic approach ^{shown in} figure ^{5 seems to be} justified ^{at low} temperatures.

For the lateral _range

of the potential (see Fig. 5) ^we follow Ikezi and Platzman [18] ^and

assume that

is comparable ^{to the} capillary length of the helium film which for ^a film thickness of 100 Â is about 600 À.

The harmonic part ^{of the} potential ^{V shown in} figure ^{5 can} be written as :

The eigenfrequency v of the electron (effective ^mass meff) ^{in the} potential ^{than is :}

For a lateral binding energy of W

7.7 x 10-22 j (at the maximum of absorption ⁱⁿ Fig. 3)

and for the free electron mass we find the eigenfrequency of the electron to be

v ⁼

75 GHz which is nearly ^{one order of} magnitude higher than the microwave frequency ^of

9 GHz used in our experiment.

In our view this discrepancy ^{is not too} surprising : ^the assumption ^{of a static} dimple potential ^{and the use} of the free electron mass as effective mass in (4) may be correct at higher frequencies ^{but is} hardly justified ^{at a} frequency of 9 GHz where

according ^{to the} dispersion ^relation [18]

^a helium film can still support capillary ^waves (having ^a wavelength

of about 60 Â at 9 GHz). ^{Thus even at a} frequency of 9 GHz the dimple ^is probably ^not completely static. A certain mobility ^{of the} dimple ^{has to} be considered even at microwave

frequencies probably leading ^{to an} increase of the electronic effective mass in (4) ^{to a value of}

about 50 free electron masses as we observed. This is in quite satisfactory agreement ^with

results of Kovdrya ^{et al.} [15] and Mende et al. [16] ^who derived from their experiments ^at

Fig. ^5.

^Schematic representation of the lateral localization potential ^for

^{electron in its} dimple

helium film.

9 GHz an effective mass which was about one order of magnitude higher than the bare electron mass.

The fact that the maximum of the microwave absorption ^{shifts to} higher clamping voltages

if higher electronic densities are used appears also plausible ^{to us :} if the mutual distance between neighboring ^electrons approaches ^the length scale of each dimple, ^one expects ^the dimple potential ^{shown in} figure ^{5 to} become shallower at higher electronic densities, ^and consequently higher clamping ^{fields are needed to} achieve the same transition frequency.

Perhaps the increased screening of the Coulomb interaction at the higher densities, ^{which has}

been described by ^Peeters [19], may also be of importance ^here.

The resonance maximum can best be seen at the lowest temperatures (see Figs. ^{2 and} 3)

and disappears ^at temperatures above 1.7 K presumably because of the increased rate of

scattering between electrons and gas ^{atoms at} elevated temperatures ^as also observed

experimantally by Iye [20]. ^{But we} may have ^to consider also other possible contributions to the experimental line width : the thermal energy of the electrons is ^{at our} present working temperatures still larger than the energy splitting corresponding ^to the microwave transitions.

- Finally, ^the geometry ^{of our} silicon electrode carrying ^the polymer foil and the He-film is not perfect : if the distance between the Si-substrate and the He-film is not identical accross

the sample, ^the clamping field will be inhomogeneous : showing higher values where the effective distance is smaller and vice versa. Variations of the distance by ^{several f.Lm} (i.e.

about 10-20 %, ^see Fig. 1b) ^cannot be excluded with our present set-up and may - in part

be responsible for the observed linewidth at the lowest temperatures. ^{We are therefore} planning ^to carry ^our experiments ^{to lower} temperatures ^{and to use more} uniform electrodes for the application ^{of the} clamping fields. We have also applied magnetic ^fields perpendicular

to the He-film ; these results will be published separately ^lateron.

In conclusion, ^our experiments have shown that electrons on a thin helium film, ^{which are} localized in quasi ^zero dimensional dimples, ^{can be excited to resonance} vibrations by ^lateral

electric fields at microwave frequencies. ^{Their resonance} frequencies ^{can be tuned} by ^a

vertical dc-electric field. For increased electron densities the resonance shifts to lower

frequencies. The electronic resonance absorption ^{can best be observed at} temperatures ^below 1.2 K where collisions with atoms in the gas phase ^{can be} neglected. ^The origin ^{of the line}

width is not yet fully understood.

2586

Acknowledgements.

We gratefully acknowledge fruitful discussions with Prof. P. Leiderer, ^his encouragement ^and advice. We are also grateful ^to J. Lôhle for experimental assistance and discussions. Valuable discussions with Dr. M. Lehndorff and Dr. F. Peeters contributed to our theoretical

understanding. Ute Petzold has helped ^{with the} preparation ^{of the} figures. ^We benefitted also from the helpful remarks and questions ^{of the} referee. This work was supported by ^the

Deutsche Forschungsgemeinschaft.

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