Cyclotron resonance for two-dimensional electrons on thin helium films

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Cyclotron resonance for two-dimensional electrons on thin helium films

J. Klier, A. Wu¨rl, and P. Leiderer

Fakulta¨t fu¨r Physik, Universita¨t Konstanz, D-78457 Konstanz, Germany G. Mistura

Universita di Padova, 35131 Padova, Italy V. Shikin

ISSP, 142432 Chernogolovka, Moscow District, Russia 共Received 19 January 2002; published 10 April 2002兲

We present a systematic investigation of the microwave absorption for two-dimensional electron layers on thin helium films and in the presence of a cyclotron resonance共CR兲magnetic field. To explain the measured data, a recently proposed two-fraction structure of the electron system is used and here described in detail.

Hereby the problem of substrate roughness, usually always present for electrons on thin helium films, is taken into account and it turns out to be an important parameter. Within this model the general structure of the microwave absorption becomes understandable and the fraction of localized and free electrons can be precisely determined. The details of the observed asymmetry and shift of the CR line shape are discussed.

DOI: 10.1103/PhysRevB.65.165428 PACS number共s兲: 67.70.⫹n, 72.10.⫺d, 72.60.⫹g, 73.50.⫺h I. INTRODUCTION

A two-dimensional共2D兲sheet of electrons on thin helium films forms traditionally an interesting field for studying low-dimensional systems. So there is, e.g., the ‘‘dimple’’

formation,^{1– 4}the high level of stability共with respect to the bulk situation兲,^{5– 8} the dipole-dipole crystallization,^9,10 the layering effect in the electron mobility,^11,12 and so on. All these phenomena are developed under the assumption that the solid substrate is flat. However, in reality solid surfaces are not perfect and the typical level of roughness is usually not small 共the roughness amplitude is comparable to the helium film thickness兲. Under these conditions the question arises of how the 2D electron system on a thin helium film

‘‘feels’’ the existing random roughness of the substrate. A preliminary answer to this question is presented in Ref. 13.

Using quite general assumptions 2D electron layers on thin helium films are represented as a two-fraction system which leads to various consequences of the understanding of these electron layers.¹³ In this paper the two-fraction scenario is systematically developed for the cyclotron resonance 共CR兲 problem. We explain, how the free electron motion and localization phenomena can coexist in the presence of randomly rough solid substrates under CR conditions.

II. CORRUGATED HELIUM FILM WITHOUT 2D ELECTRONS

To describe the behavior of a helium film adsorbed on a corrugated substrate we first consider the neutral situation, i.e., without electrons on the helium surface.

共1兲Let us assume the periodic substrate profile␦^{(x) as}

␦共x兲⫽h⫹

再

^␦^⫺⁰^␦^,⁰^, ⁰â^⭐^⬍^x^x^⭐^⬍â,^共â^⫹^b^兲^. ^共¹^兲

Here (a⫹b) is the corrugated structure period and h (⬎2␦0) is the distance between the average structure of the substrate and the bulk liquid helium surface.

For simplification we only discuss the limiting case

b²⬍␴lv

␳^g ^, ^共²^兲

i.e., the width of the structures is assumed to be small com- pared to the capillary length of the liquid helium. ␴lv is the surface tension,␳ the bulk helium density, and g the accel- eration due to gravity. The properties of a neutral helium film d(x) on a rough substrate can be extracted from

␴lv

d

⬙

共x兲

关1⫹共d

⬘

兲²兴^3/2⫺␳^gd共x兲⫹ C₃

d³共x兲⫽␳^gh, 共3兲 where C₃ is the van der Waals constant of the helium-solid substrate boundary.

In the case of a uniform helium film, when␦0→0, Eq.共3兲 is reduced to the conventional definition of the helium film thickness d 共for simplicity the retardation effect for very thick films,¹⁴where d depends with the fourth power on h, is not considered here兲

d³⫽ C₃

␳^gh^. ^共⁴^兲

In the presence of a regular corrugation␦^(x)⫽0 the question arises about the properties of the nonuniform thickness d(x). One can show that then

d共x兲⫽

再

^d^R^top^⫹^,^d^min^⫺

^冑

^R²^⫺^共^x^⫺^c^兲²^, ⁰â^⭐^⬍^x^x^⭐^⬍共â,â^⫹^b^兲^, ^共⁵^兲

where

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d_top³ ⫽ C₃

␳^g共h⫹␦0兲, 2␴lv

R ⯝␳^gh, ^c⫽a⫹b 2, d_min⫽2␦0⫺␨0, 共6兲 and

␨0⫽R⫺

冑

^R²⫺b²/4. 共7兲 R is the radius of curvature of the capillary condensed thick helium film, see Fig. 1. The structure of helium films of this kind has been visually investigated in Ref. 15.

Using definition共5兲we can formulate the conditions for a weak and strong helium film corrugation; the corrugation is weak, if

␦0

2Ⰶd² or ␨0Ⰶd. 共8兲

In the opposite limiting case we have the strong corrugation situation.

The first case of condition共8兲is evident. It corresponds to the perturbative coexistence between the helium film and the corrugation of the solid substrate with zero approximation for d from Eq.共4兲. The details and behavior of d(x) versus

␦(x) in this limit are investigated, e.g., in Ref. 16.

More interesting for the roughness problem below is the second case, i.e., when the helium film profile does not follow the corrugation ␦(x) of the solid substrate. In the limit RⰇb, when ␨0→0, then d_min→2␦0. Therefore in this limit the definition of d共4兲correlates only with d_top 共6兲. The be- havior of d(x) between the tops is controlled mainly by the Laplace pressure with the essential screening of the profile

␦(x) below the helium surface. Such a behavior has interest- ing consequences for a rough substrate, because a small amount of the strong random deviations in the distribution

␦(x) together with the effect of the Laplace screening be- tween these tops can control all the behavior of the helium film d(x) along the rough substrate.

共2兲Roughness is a property of most solid substrates. Usu- ally it is assumed that a one-dimensional random roughness behavior␦(x) can be described by a Gaussian distribution of the amplitudes

G共␦兲⫽ 1

共2␲⌬²兲^1/2exp

冉

^⫺²^␦^⌬²²

冊

^, ^共⁹^兲

where⌬²⫽具␦²典 is the mean-square roughness amplitude in vertical direction.

In terms of Eq.共9兲the influence of the roughness is weak, if

⌬²Ⰶd². 共10兲

But, after the discussion of Eqs. 共1兲–共10兲 it becomes clear that the limitation共10兲is not enough to avoid the problem of roughness in the behavior of d(x). Even a small amount of high roughness tops with ␦²Ⰷ⌬² can be important for the details of the d(x) dependence.

To introduce the Laplace screening effect in the calcula- tion of d(x) we need some additional basic definitions. One of them is the general expression for the average density of the high enough tops n_␦ above the fixed level␦⬎0 共Ref. 17兲

n_␦⫽ 1

2␲

冑

⫺Z

⬙

共0兲exp

冉

^⫺²^␦^⌬²²

冊

^, ^共¹¹^兲

where

Z

⬙

共0兲⫽⫺

冉

²^⌬^␲

冊

²

^冕

⁰^⬁^wS^共^w^兲^dw.

S(w) is the spectral density function for the roughness dis- tribution, Eq.共9兲, with the correlation function

具␦共x兲␦共x⫺x

⬘

兲典⫽⌬²exp

冉

^⫺²^x^␩

^⬘

²²

冊

^. ^共¹²^兲

Here具␩²典 is the correlation length in horizontal direction.

After calculations, see Ref. 17, one gets from Eqs. 共11兲 and共12兲

n_␦⫽ 1

s␩^exp

冉

^⫺²^␦^⌬²²

冊

^, ^s²^⫽²^␲^, ^␩^⫽

^冑

^具^␩²^典^. ^共¹³^兲

Here ⌬² is from Eq.共9兲and具␩²典 is from Eq.共12兲.

To couple the arbitrary level ␦⬎0 关Eq. 共13兲兴 with the Laplace radius R, we assume that for the Laplace length b we have

b⫽n_␦^⫺¹. 共14兲 Figure 2 explains the definition of the geometric size b and helps to understand the correlation between b, R, and␨0, i.e.,

␨0⫽R⫺

冑

^R²⫺b²/4, see Eq.共7兲and also Fig. 1. In addition, we require

db d␨0

⫽dn_␦^⫺¹

d␦ ^. ^共¹⁵^兲

This condition couples the characteristics of R with the

‘‘speed’’ of the change of n_␦ versus ␦^.

The four definitions 共7兲 and 共13兲–共15兲, are sufficient to express␦^{, b,}␨0, and n_␦ versus R,␩^{, and}⌬. All these defi- nitions are labeled by the index ‘‘a’’ to indicate the connec- tion of the density of active tops n_a and the position of the active level ␦a to the problem of a helium film on a rough substrate with

FIG. 1. Shown is a schematical sketch of a corrugated surface where, due to capillary condensation, a suspended thick liquid helium film exists. The symbols are explained in the text.

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R⫽b_a

2

冑

¹^⫹2^b⌬^a²²lnb_a

s␩ ^共¹⁶^兲

and

␦a 2

2⌬²⫽lnb_a

s␩^, ⁿ^a^⫺¹^⫽^b^a^, ^s␩ⁿaⰆ1.

The screening effect共i.e., decreasing of n_aversus R) is most pronounced if RⰇ⌬. In this case, and in the reasonable in- terval of ‘‘h’’, we have b_a²Ⰷ⌬². Under these conditions we get

R⯝ b_a

2

冑

²

冉

^b^⌬^a

冊

^ln^1/2^s^b␩^a^. ^共¹⁷^兲 So for the situation of the one-dimensional roughness one gets b_a⬇R^1/2.

In the two-dimensional case we have 共if fluctuations are independent兲

n_a^⫺¹⬇b_a². 共18兲 In addition to n_a it is reasonable to use n_a^T as density of active tops. The reason is that all active tops above the level

␦aare sensitive to the pressing electric field E_⬜. Under these conditions the highest tops are unstable, and all local electrons will be localized in a narrow interval of ␦⬍␦a with density n_a^T such that

n_a^T⯝⫺ T eE_⬜

dn_a d␦ ^⫽ⁿ^a␦a

⌬² T

eE_⬜. 共19兲 The definition of n_a^T 共19兲has practically the same R depen- dence as n_a 共because ␦a is a ‘‘weak’’ function of R). By summarizing we can see that 共a兲 the helium film above a rough substrate causes a finite density of active tops n_a, see Eqs.共18兲or共19兲, even under condition共10兲;共b兲n_a of these

tops is a continuous function of h (n_a→0 if h→0), and it can be comparable with the total number of the 2D electrons n_s 共see Sec. III兲;共c兲the helium film thickness on the tops is practically not sensitive to the distribution of ␦ ^with ␦

⬎␦a. Indeed, if⌬²Ⰶh²共such a requirement is fulfilled with high accuracy兲, the local helium film thickness, d_a, above the active roughness tops can be estimated from

C₃

d_a³⯝␳^g共h⫹d_a兲⫹␴lv 关d_a

⬙

^da

⬙

兴^1/2. 共20兲 If⌬²Ⰶh² and the local derivatives d_a

⬙

above the active tops are less or comparable to␦

⬙

^{, then}

d_a

⬙

^da

⬙

⭐具^d

⬙

^d

⬙典

⭐具␦

⬙

␦

⬙典

^, 具␦

⬙

␦

⬙典

⫽3⌬²

␩⁴ ^. ^共²¹^兲 In this case definition 共20兲is reduced to

C₃

d_a³⯝␳^gh⫹␴lv具␦

⬙

␦

⬙典

^1/2^, 共22兲 which is not sensitive to␦a. The definition for具␦

⬙

␦

⬙典

^{in Eq.}

共21兲follows from the combination of Eq.共9兲and the definition of the correlation function, Eq.共12兲, see Ref. 17.

III. 2D ELECTRONS ON A ROUGH HELIUM FILM, TWO-FRACTION STATISTICS

In the presence of a substrate with active tops the 2D- electron system 共2DES兲 on a thin helium film above this substrate is separated in two fractions. One fraction of the electron density n_ecorresponds to free electron motion along the helium surface. In Fig. 2 these electrons form the electron puddle between the active tops. The second fraction n_l rep- resents the density of electrons localized to potential wells of the roughness of the solid substrate. These electrons are localized in the vicinity of the tops above the helium film. It is evident that

n_e⫹n_l⫽n_s, 共23兲 where n_s is the total 2D electron density, which is typically fixed. But the relationship between these fractions is flexible.

This follows from the behavior of the equilibrium chemical potential␮0. To define␮0 we follow the procedure used for semiconductors,¹⁸ i.e.

n_l⫽ n_a

exp关共V_a⫺␮0兲/T兴⫹1, V_a⬍0,

n_e⫽ n₀^eexp共T_e/T兲

exp共⫺␮0/T兲⫹1, n₀^e⫽ mT

共2␲ប²兲, 共24兲

T⭓ប␻c, T_e⫽ eE_⬜⌬ 2

冑

^{2 ln}^1/2共

冑

^R⌬/␩兲, and

FIG. 2. Shown is a schematical cross section through a typical randomly rough substrate. Above the active tops only a thin van der Waals helium film exists with the localized electrons (䊉) above the tops. In between the active tops a suspended thick helium film is formed with the quasifree electrons (䊊) above. The symbols are explained in the text.

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V_a⭓⫺⌳

d_a with ⌳⫽e²共⑀d⫺1兲

4共⑀d⫹1兲. 共25兲

⑀d is the dielectric constant of the solid substrate and d_a is defined in Eq. 共22兲. n₀^e is the free electron density of states, and the factor exp(T_e/T) in the definition of n_e reflects the average energy difference between local and free states in the presence of E_⬜⫽0. So putting definitions 共24兲 and 共25兲 in Eq. 共23兲, we can derive␮0 versus V_a, T, n_a, and n_s.

In some limiting cases we get for␮0, from Eq.共24兲,

␮0→

再

^⫺^Vâ^{T ln}^⫺^{T ln}^关共ⁿ^关共⁰ê^⫺ⁿâⁿ^⫺^s^兲ⁿ^/n^s^兲^s^/n^兴^,^s^兴^,ⁿâ^→ⁿâ^0,^⬎ⁿ^s^. ^共²⁶^兲

One can see that a␪-like behavior of the chemical poten- tial, with the amplitude of the jump dependent on V_a, has developed, see Ref. 19, around

n_a⫽n_s. 共27兲

A more detailed form of␮0 is 2⑀ⁿsx⫽关⑀共n₀^e⫺n_s兲⫹共n_a⫺n_s兲兴

⫹

冑

关⑀共n₀^e⫺n_s兲⫹共n_a⫺n_s兲兴²⫹4⑀ⁿs共n₀^e⫹n_a⫺n_s兲, 共28兲 where

x⫽exp

冉

^⫺^␮^T⁰

冊

^, ^⑀^⫽^exp

冉

^V^T^a

冊

^, ^V^a^⬍^0.

The second term in Eq. 共28兲 is always positive. The first term, however, changes sign 共in the limit ⑀→0) when n_a crosses the value n_s. Before this point there is a strong compensation between the two terms. Such a compensation corresponds to the first asymptote in the definition of ␮0 共26兲. Just after condition 共27兲 the compensation stops, and we have the second asymptote in the definition of ␮0 共26兲.

From Eqs.共24兲 and共26兲 we can write the useful asymp- tote of n_l in the limit n_lⰆn_a

n_l⯝ n_a

共n₀^e/n_s兲exp关共V_a⫹T_e兲/T兴⫹1 共29兲 with

n₀^eⰇn_s and V_a⬍0.

The situation n_lⰆn_a is possible, if n₀^e

n_sexp

冋

^V^a^⫹^T^T^e

册

^⬎^1.

IV. TWO-FRACTION ELECTRON SYSTEM IN THE AC REGIME

The two-fraction structure of a 2DES is important both in the dc and the ac regime. In the first case the fraction n_l is practically immobile and the conductivity ␴xx can be described 共in the Drude approximation兲as¹⁹

␴xx

dc⫽n_ee²␶ m . The thickness dependence of␴xx

dc follows both from the parameters␶^{(d) and n}e(d).

In the ac regime, however, the local fraction n_l is more

‘‘active’’ because the localization does not stop the dynamics of the n_l electrons. To introduce the ‘‘top’’ eigenfrequencies

␻a, we use the coupling electron energy in the form

V_a⯝⫺ ⌳

d_a共x兲, 共30兲 where

d_a共x兲⯝d_a共0兲

冉

¹^⫹^2d^␦

^⬙

^a^x^共⁰²^兲

冊

and an expansion of this expression near the minima V_a共x兲⯝⫺ ⌳

d_a共0兲⫹k_ax²

2 , 共31兲

where

k_a⫽ ⌳␦

⬙

d_a共0兲² and ␻a 2⫽k_a

m.

In the definitions 共30兲 and共31兲 we assume that the helium surface is practically flat (R²Ⰷ⌬²) and so the local distance d(x) between electron and top profile is only sensitive to␦

⬙

^. To describe the properties of ␦

⬙

we have to use the same definition as in Eqs.共20兲and共22兲

具␦

⬙

␦

⬙典

^⫽␤²⫽3⌬²

␩⁴ ^. ^共³²^兲 The corresponding Gaussian distribution␦

⬙

of the curva- tures is

D共␥兲⫽ 1

共2␲␤²兲^1/2exp

冉

^⫺²^␥^␤²²

冊

^, ^␥^⫽^␦

^⬙

^. ^共³³^兲

All ␻a modes 共31兲 are split in a magnetic field ␻a

→␻a⫾. The existence of this splitting is included in the cal- culation of dissipation in the low frequency limit ␻a⫾⭐␻^, where ␻ is the external frequency. For the eigenmodes ␻a

⬎␻the possible soft contribution of these modes in the general dissipation 共50兲 is neglected due to numerical reasons 共such a statement has been formulated in Ref. 20兲 and the possibility to simplify the algorithm of the fit共see below兲.

We now make some remarks regarding the well known dimple effect in the ac-free electron behavior: On one hand, there is an experimental indication²¹with respect to the CR- dimple shift on the bulk helium surface. There is also the

‘‘dimple’’ interpretation of a quasi-dc-electron mobility on a thin helium film.²² On the other hand, theoretical estimations⁴show a small probability for the existence of the single electron dimple under the typical helium conditions.

Precise dimple effect measurements²³within the crystalliza-

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tion problem show a complete correlation between the existence of dimples and the Coulomb crystallization in the 2DES. Our search for the dimple effect in the CR data for electrons on thin helium films on Hostaphan¹³lead to a negative answer. The same negative result can be extracted from the CR measurements in Ref. 24 for electrons on a helium film with solid hydrogen as a substrate. And finally there is no indication of a CR-dimple shift in the most recent CR measurements²⁵with electrons on bulk helium.

So the conclusion with respect to the CR-dimple shift is mainly negative. To understand the positive conclusions^21,22 we believe that the data of Ref. 22 can be explained by an alternative scenario, see Ref. 4, where a sharp decrease of mobility versus helium film thickness without a self-trapping effect is proposed. The same motivation is applicable to the interpretation of Ref. 26 for the data of Ref. 27. The data of Ref. 21 partly reflect the influence of the Coulomb crystallization. Besides, also the contribution of the resonator effect 共see below兲is possible.

V. CR-MEASUREMENTS IN RESONATORS CR for electrons on helium is typically investigated by using a resonator. In this case the conventional way to fix the resonance is reduced to measurements and calculations of the resonator reflection 共transmission兲 versus electron density, magnetic field, helium film thickness, etc. The experiments usually follow these ways. For the calculations, however, the existing ways are quite limited. One uses either the transmission line model²¹or the simple possibility following from the classical共quantum兲motion equations without any indication how the resonator background becomes important.^25,28,29

The dissipation of the free electrons Q_e^⫺¹ is defined as 共see, e.g., Ref. 28兲

Q_e^⫺¹⫽ReE_储j_x*_共␻^,␻c兲, 共34兲 j_x共␻^,␻c兲⫽共␴xx

⬘

⫹i␴xx

⬙

兲E_储共35兲

␴xx

⬘

⫽n_se²␶ m

共1⫹␻²␶²⫹␻c 2␶²兲共1⫺␻²␶²⫹␻c

2␶²兲²⫹4␻²␶² ^共³⁶^兲

␴xx

⬙

⫽⫺共␻␶兲n_se²␶ m

共1⫹␻²␶²⫺␻c 2␶²兲共1⫺␻²␶²⫹␻c

2␶²兲²⫹4␻²␶²^. ^共³⁷^兲 Here E_储 is the effective electric field along the helium film, n_s is the electron density,␶ is the elastic time relaxation,␻ the external frequency, and␻c the cyclotron frequency.

In reality the electron motion in a resonator is not free.

There is a coupling between the electron motion and the resonator mode. The level of this coupling is the essential characteristic of the system 2D electrons in a resonator.

To estimate such a coupling we have to solve the corresponding eigenproblem. In the case of the resonator in Fig. 3 the lowest eigenmode is共see details in Ref. 30兲

cot共kh兲兵^cos关d共q⫺k兲兴⫺i␴^tan共kd兲其

⫺sin关d共q⫺k兲兴⫹i␴⫽0, 共38兲

k²⫽␻²

c², q²⫽⑀d

␻²

c², ␴⫽4␲

c ␴xxcos共kd兲sin共qd兲. The substrate has the thickness 2d, the size of the empty resonator is 2h, and ␴xx is from Eq.共34兲.

From Eq.共38兲the role of the coupling constant is shown by the combination

␴0⫽4␲␴xx 0

c , ␴xx 0 ⫽n_se²␶

m . 共39兲

The critical scale for this parameter is␴0⬇1. This is suitable for electrons on bulk helium under the condition: n_s

⭓10⁸ cm^⫺² and ␶⭐10^⫺⁷ s 共these conditions are presented in Refs. 21 and 29兲. In this case ␴0⭐1. However, for thin helium films with the electron density from above and typical values ␶⭐10^⫺¹⁰s, ␴0Ⰶ1 and so the resonator effect is not important.

Under the conditions␴0Ⰶ1 we can therefore use the conventional perturbation theory. Indeed in this case

k⫽k₀⫹␦^k, ␦^k⫽␦^k

⬘

⫹i␦^k

⬙

^, ␦^kⰆk₀, 共40兲 where k₀ corresponds to the cavity eigenmode without electrons

cot共k₀h兲⫺tan关d共q₀⫺k₀兲兴⫽0. 共41兲 If in addition dⰆh then Eqs. 共38兲, 共40兲, and 共41兲 are reduced to

h␦^k

⬙

⫽⫺4␲

c cos共k₀d兲sin共q₀d兲␴xx

⬘

共␻0␻c兲, 共42兲

h␦^k

⬘

⫽⫺4␲

c cos共k₀d兲sin共q₀d兲␴xx

⬙

共␻0␻c兲. 共43兲 The structure共42兲of the damping of the eigenmode is the same as the absorption Q_e^⫺¹ 共34兲,共36兲. Therefore definition 共34兲is reasonable for a 2DES in a resonator, if␴0Ⰶ1.

FIG. 3. a兲 Experimental setup, 共b兲 schematical sketch of the cavity with the dielectric substrate of thickness 2d in the center. On the right a typical profile of the electric field E is shown.

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In addition to Eqs. 共38兲–共43兲 predict the perturbation of the eigenmode 共43兲. This perturbation is shown by our experimental data共see Fig. 4兲and also corresponds to the situation␴0Ⰶ1.

Now we go back to the two-fraction problem. After discussion of Eqs. 共34兲–共43兲 we can present the free electron absorption in the two-fraction model as

Q_e^⫺¹⬀n_ep共␻0,␶^,␻c兲, 共44兲

p共␻0,␶^,␻c兲⫽ 1⫹␻0 2␶²⫹␻c

2␶² 共1⫺␻0

2␶²⫹␻c

2␶²兲²⫹4␻0

2␶²^, ^共⁴⁵^兲 where␻0 is the frequency from Eq.共41兲, and n_e is the free electron fraction from Eq.共24兲.

The corresponding absorption Q_l^⫺¹(␻0,␻c) due to local states is

Q_l^⫺¹共␻0,␻c兲⬀n_l

冕

0

⬁

D共␥兲f共␻0 2␶²^,␻c

2␶²^,␻a 2␶²兲d␻a

共46兲 and

f共z,x,t兲⫽ 共z⫺t兲²⫺z⫺zx 关共z⫺t兲²⫺z⫺zx兴²⫹4z共z⫺t兲² with

z⫽␻0

2␶²^, ^x⫽␻c

2␶²^, ^t⫽␻a 2␶²

where n_l is from Eq. 共24兲, the function ␻a(␥) from Eqs.

共31兲,共33兲, and D(␥^{) from Eq.}共33兲. The total absorption Q^⫺¹⫽Q_e^⫺¹⫹Q_l^⫺¹ 共47兲 contains six external parameters d, n_s, ␶^,␶a,⌬, and␩^{. It is} reasonable to assume that d and n_sare well defined indepen- dently. The scale of␶can be estimated using known mobility calculations for 2D electrons on helium. As a result the

above calculations have two fit parameters ⌬ and ␩ ^which have to be extracted from the experimental data.

In reality this program is too complicated共especially with the introduction of ␶a). So some simplifications are necessary. First we cut the integration in Eq.共46兲by the value␻0

共the reasons for such a simplification have been discussed earlier兲and assume that within the interval between 0 and␻0

all ␻a are equally probable. Due to the probability distribution of all␻a that is true if

具␻a 2典⭓␻0

2. 共48兲

The additional assumption

␶a⬇␶ 共49兲

cannot be directly proven. But when taking Eq.共49兲, we can calculate the integral in Eq.共46兲in explicit form. In addition the fit shown below demonstrates that for the absorption data condition共49兲is valid.

After the simplifications given above the integral in Eq.

共46兲is transformed into the form

Q_l^⫺¹⬀ n_lq共␻0,␶^,␻c兲, 共50兲 q共␻0,␶^,␻c兲

⫽

arctan

冑

^z

1⫹x⫹

冑

^xz^⫹^arctan

冑

^z

共1⫹x兲

冑

^z⫺z

冑

^x^⫹^c^共^z,x^兲

2

冑

^z

共51兲 and

c共z,x兲⫽␲

2 兵¹^⫺^sgn^关共¹^⫹^x^兲

冑

^z⫺z

冑

^x兴其^.

The function c(z,x) is only used to switch to another branch of the arctan() function. The total absorption has now two fit parameters␶ ^{and n}e/n_s. To extract these numbers from the experimental data it is convenient to fit the combination

Q^⫺¹共␻c (max)兲

Q^⫺¹共␻c⫽0兲⫽␯e p共␻0,␶^,␻c

(max)兲⫹␯lq共␻0,␶^,␻c (max)兲

␯ep共␻0,␶^,0兲⫹␯lq共␻0,␶^,0兲 , 共52兲 where

␯e⫽n_e

n_s, ␯l⫽n_l

n_s, and ␯e⫹␯l⫽1, 共53兲 together with the definition of␻c

(max)

.

␦^Q^⫺¹

␦␻c

兩max⫽0. 共54兲 Under these conditions there is no guarantee for a good enough reproduction of the absorption line shape. Neverthe- less the fit shows a reasonable solution of this problem, see Fig. 5.

FIG. 4. The transmitted signal 共left y axis兲 and the resonance frequency共right y axis兲as function of the magnetic field are shown.

The minimum in the transmission, around 0.4 T, determines the cyclotron resonance ␻c. The change in frequency, around ␻c, gives the evidence of the perturbation of the eigenmode共for further explanation, see text兲.

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VI. FITTING THE EXPERIMENTAL DATA To investigate the influence of the roughness of the under- lying substrate on the CR absorption we have set the helium film thickness by adjusting the bulk helium level below the Si substrate. So varying the distance h from small to large values changes the thickness of the helium film d from thick to thin values. The typical Q^⫺¹ lines for the different film thicknesses are shown in Fig. 5. It is evident that the CR quality decreases and the asymmetry of the absorption line increases as the helium film gets thinner.

The fit procedure can be split into two parts: first we fit the shape of the Q^⫺¹ data using Eqs. 共44兲,共47兲,共50兲,共52兲, 共54兲. The details of this fitting are presented in Fig. 5. One can see a quite good reproduction of the features of Q^⫺¹in a wide range of the helium film thicknesses. In addition, one can extract information about the electron fractions and the corresponding time relaxation as function of the helium film thickness, see Figs. 6 and 7. The data are, at least qualita- tively, understandable: n_l monotonically grows and␻␶^goes down.

Secondly we try to explain the data shown in Fig. 6 and 7.

For the progress in the interpretation of ␻␶, our initial description is too crude 关see, e.g., the simplification 共49兲兴. Therefore we cannot explain the details of the behavior of

␻␶, Fig. 7. However, within the framework of the fractional structure of the 2DES this problem can be, at least qualitatively, described.

We start from the simplification共48兲, which looks reasonable considering the excellent fit in Fig. 5, using Eqs. 共28兲, 共44兲, 共50兲. In explicit form the inequality 共48兲 is reduced to the estimation of 具␦

⬙

␦

⬙典

. Using definition ␻a

2 共31兲 with d_a⭐10^⫺⁶ cm, ⑀d⬇10, and the experimental value ␻ FIG. 5. Shown is the absorption Q^⫺1 as function of magnetic

field for up (䉭) and down (䉮) sweeps. The dashed and dotted lines represent the free and localized electron fraction, the full line is the sum of both fitted to the data. From共1兲–共3兲the helium film thickness decreases. In 共1兲 n_e⬇67%, in 共2兲 n_e⬇60%, and in 共3兲 n_e⬇49%. These three data sets correspond to the same labeled data points in Figs. 6 and 7.

FIG. 6. Dependence of free electron fraction n_eas function of distance of bulk helium level below the substrate h. (䊉) and (䊊) are from fitting to all measured data. Both in 共a兲and 共b兲the solid lines present the best fit with the same parameters of a and T₀. The dashed and dotted lines in共a兲show fits with different T₀but keep- ing a fixed, and in共b兲with fixed T0but varying a. This shows the good agreement with one set of parameters to describe the measured data. The (䊊), labeled共1兲to共3兲, correspond to the same data points as shown in Figs. 5 and 7.

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⬵2␲¹⁰¹⁰ ^s^⫺¹ we estimate from具␻a典^⭓␻ ^{the value} 具␦

⬙

␦

⬙典

^⫽³^⌬

2

␩⁴^⭓¹⁰

13cm^⫺². 共55兲

This estimation shows that a smooth approximation 共25兲 for V_a is not realistic enough. It has to be reduced, using, e.g., the potential energy between the pointlike charge and the spherical dielectric instead of the semi-infinite dielectric.

Unfortunately the spherical image problem has no analytic solution, see Ref. 31. To obtain an analogy to the problem of the cylindrical image potential, we use for V_a the expression

V_a→⫺ 2⌬ 共d_a⫹a兲

a²

关共d_a⫹a兲²⫺a²兴, 共56兲 where

a²⫽具␦

⬙

␦

⬙典

^⫺¹^.

In the limit aⰇd_aexpression共56兲is reduced to V_a from Eq.

共25兲.

If so, we go to the definition of n_e共24兲with n_afrom Eq.

共18兲, E_⬜⫽2␲^ens and V_a from Eq. 共56兲. Using T_efrom Eq.

共24兲 and a from Eq.共56兲 as a fit parameter we can explain the data in Fig. 6 with

T_e⬵3.19 K and a⬵2.25⫻10^⫺⁷ cm.

The fitting lines, Fig. 6, are very sensitive to variations both in T_e 关Fig. 6共a兲兴 and a 关Fig. 6共b兲兴. Therefore the level of accuracy for the above numbers is not realistic, but the scale of these numbers is reliable.

It is necessary to note that the scenario presented in Fig. 6 has some artificial correction. The ‘‘beginning’’ of all these lines corresponds to the free electron behavior of the 2DES when regime n_lⰆn_a takes place. But in this case the ‘‘pla- teau’’共i.e., the first three data points from the left side of Fig.

7兲should be around 100% instead of the measured 70%. To explain such a shift, which is not sensitive to the thickness of the helium film, we have to remember the geometry of our setup. The silicon substrate has a finite area with a sharp perimeter. The equilibrium helium film around this perim-

eter, just after crossing the point h⫽0, is inevitable thinner than on the main silicon area. And this should cause localization of the edge electrons while the electrons above the substrate are still free. This, we expect, causes a sharp drop in the free electron fraction which may well be around 30%.

Exact measurements in this range, however, still have to be done.

Finally from the above numbers for T_e and a, using defi- nitions共56兲,共32兲, and共24兲, and n_s⬵10⁹ cm^⫺² we get

⌬⬵ 2T_e

␲^e²ⁿs

⬵8 nm,

␩⁴⫽3⌬²a²⬵共6 nm兲⁴.

These values, being in the nm regime, are reasonable and typical for a real surface.

VII. CONCLUSIONS

The consideration above is related to a quite general problem of the behavior of a 2DES on a liquid helium film in the presence of a substrate with the usual surface roughness. The use of a thin helium film is desirable to increase the values of the critical electron density. However, simultaneously the influence of the surface roughness grows and becomes an important factor. The interaction of the 2DES with the substrate roughness is not direct. Rather it is modified by the nonuniform profile of the helium film. Such a ‘‘screening’’ effect is especially strong if the roughness amplitude ␦(x) exceeds the equilibrium helium film thickness d: ␦^(x)Ⰷd. Under these conditions the conventional perturbation theory does not work, and the question arises, how to describe such a quite typical situation.

We here propose the two-fraction model to be suitable to describe the behavior of a 2DES when ␦^(x)Ⰷd. We have defined the density of the active tops n_a, which is dependent on the characteristics of the random rough substrate. The definition of the localized n_l and free electron fraction n_eas a function of n_a, n_s, T, and d, as well as the description of the dynamics of the localized electrons is given. All this information has been used to interpret the CR data for 2D electrons on thin helium films. The most prominent feature of these data is the unusual asymmetry of the absorption line shape and its growing as the helium film thickness decreases.

The two-fraction picture explains this asymmetry quite well, see Fig. 5. The developed fit program helps to extract impor- tant information about n_land n_efrom the experimental data.

In addition, we can explain, at least qualitatively, the behav- ior of n_e(h), see Fig. 6.

In the scenario presented here several approximations and assumptions are made: we take the substrate roughness to be Gaussian-like and one dimensional, we use the simple ac- Ohm’s law instead of the self-consistent resonator response, we neglect a possible distribution in the time relaxation ␶a, propose a quite naive modification共56兲of the potential V_a, and introduce the influence of possible localized states along the perimeter of the substrate. Within these approximations FIG. 7. Shown is the dependence of␻␶on the level of the bulk

helium below the substrate surface, h. The (䊊), labeled共1兲to共3兲, correspond to the same data points as shown in Figs. 5 and 6.

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the obtained quantitative information might be not very reliable. However, the qualitative conclusions, i.e., 共a兲the two- fraction model has a field of application,共b兲 the CR asymmetry has its origin in the two-fraction scenario, and共c兲there is an overlap between the dc- and ac- two-fraction predic- tions, is very reasonable.

ACKNOWLEDGMENTS

This activity is supported partly by the Deutsche Fors- chungsgemeinschaft, Forschergruppe ‘‘Quantengase,’’ the INTAS Network 97-1643, and the RFBI Grant No. 02 02 17082.

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