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Physica E 18 (2003) 184-1 85
Cyclotron resonance for 2D electrons on helium films above rough substrates
aDepartment of Pliysics, Uninersiljj of' Konsrunz. 0-78457 Konstanz, Gexnluny
"SSP, 142432 Ckemogolovk~, M o . r c o ~ ~ Di.~trict, Rus~:sla
Abstract
An invest~gat~on of the microwave absorption for two-dimennonal electron systems (2DES) on helium films and in the presence of a cyclotron resonance (CR) inagietic field are presented. Measured data are expIained by a recently proposed two-fraction model of the 2DES, which makes the general structure of the microwave absorption understandable. The fraction of localized and free electrons can be precisely determrned and rts dependence on the thickness of the helium film above the roughncss of the underlying solld substrate is understood.
O 2003 Elsevier Science B.V. A11 rights reserved.
Keyword.7: 2D electron systems; Cyclotroll resonance; Thin helium films; Surface roughnaqs
A two-dimensional (2D) sl~cct of clectrons 011 thin helium films forms an interesting herd for studying low-dimensional systems. So there is, e.g., the "diinple'Yomation, the high level of stab~lity (with respect to the bulk situation), the d i p o l d i p o l e crystallization, layering effect in the electron mobility and so on, see Ref. [I]. A11 these phenomena are developed under the assumption that the solid substrate is flat. However, in reality solid surfaces are not perfect and the level of roughness is usually not small {the roughness amplitude IS comparable to the thickness of the helium film), So the question arises, how the 2D clectroil system on a thin helium film "feels" the random roughness of the substrate.
A preliminary answer to this question is presented in Rcf.
[ 2 ] . Using quite general assumptions 2D electron layers on t h ~ n hellurn films arc represented as a two-fraction system which leads to various consequences of the understanding of these electron layers [ 2 ] . In this paper the two-fraction scenano is used for the cyclotron resonance (CR) problem.
We explarn, how the free electron motion and localization phenomena can coexist In the prescncc of randomly rough solid subs~ates under CR cond~tions.
* Corresponding author.
E-marl address: juergen.klier@,uni-konstanz.de ( J . Kller).
The experiments are done such that a thin liquid 4 ~ e film is formed inside a microwwe resonator, see inset in Fig. I.
This helium film is covering a high-ohmic %wafer on which a varying potential is applied and so bullding a liold~ng electric ficld for the conservation of the 2DES. A magnetic field R is applied perpendicular to the electron layer. The temperature o f the system is x 1.3 K.
The total CR absorption Q-' call be presented as a corn- binatlon of two fractions, i.e.,
Q-' = Q;' t Q;', (1)
where
QL'
cr n,p[w,, 7 , w , ) ( 2 ) I+
w;r2+
w yp ( w o , ~ , o c ) =
( 1
-
OI:'C2+
0 1 : ~ ~ ) ~ f 4&'C2 ' andQ;' x
nlq(co:z2, w:t2f2 2 2
with x = w,z , z = w,?,
and c(x,z) = 71 -{I
-
sgn((1+
x ) f i-
2 6 ) ) .2
1386-9477/0315-see front matter 0 2003 Elsevier Science B.V. All doi:10.10 t61S1386-9477(02)01087-1
rights reserved
First publ. in: Physica / E [Low-dimensional Systems and Nanostructures], Vol. 18 (2003), 1-3, pp. 184-185
Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/2749/
URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-27498
A. Wiirl et ul. E Physrca E 18 (2003) 184-185 185
0
0 0.2 0.4 0.6 0.8
Magnetic field IT)
Fig. 1. Shown is the absorption Q-I as function of magnetlc field for up ( A ) and down
(v)
sweeps The dashed and dotted llnes represent the free and localized electron fraction, the full line 1sthe sum of both fitted to the data. Here 6
--
6 m m and n, % 60%This data set ( 1 ) corresponds to the same labeIcd data point in Fig. 2. Inset shown is the resonator as used in the experiment In the center 1s the dletectric Si substrate, on which a thin helium film 1s adsorbed. &rr; indicates the microwave electnc field whlch is parallel to the electron layer
The ftlnction c(x,z) is only needed to get the right branch of the arctan frmction.
The average total electron density n, is given by
whcrc n, and nl corresponds to the free and localized eIectron densities.
The total absorption has now two fit parameters: r and n,/n,. To extract these numbers from the experimental data it is convenient to fit the combination
Q - I ~ ~ ; ~ = I
1
Q - l ( w c = 0)
wherc vc = nF/nS, \I, = nl/nF and v,
+
vl = I , together with the definition of mirn"',The fits, bascd on definitions ( 1 )-(6), for the data using the sctup shown in the inset of Fig. 1, are presented in Fig.
1 together with the measured data. One can see that both Qc (dashed h e ) and QI (dotted line) cannot separately fit the data. But their combinat~on (solid linc) with the flexible parameters z. and v, fits the data qulte good. The dependence of wr on h is s h o w in the inset of R g . 2.
Thc results of fimng to CR data like in Fig. I but for a wide range of h are presented in Fig. 2. This shows clearly
Fig. 2. Dependence of free electron fraction n, as funct~on of dlstance of bulk helium level beIow the substrate h. (-) and (o) arc from fitting t o all measured data. The solid line presents the best fit t o all measured fractions The (o), labeled (11, corresponds to the d a t a - p n t s shown in Fig. 1. Inset: the dependence o f w t as function of h 1s shown.
lo
the influence of the substratc roughness {in the case of a regular cormgation such a dependence would not exist).
Within the scenario of Ref. [2] for the gaussian roughness dis~ibntion, with A' as a mean-square roughness amplitude in vertical direction, and the same gaussian correlation in horizontal direction (the corresponding correlation Tength IS
equal to
C,
see Ref. [2]), we can fit the data (the soIid line in Fig. 2) if A=
1 0 nm andi
x: 6 nm. These numbers are typlcal values for common soIid substrates.In conclusion we have shown that with the recently pro- posed two-fraction scenario of electrons on thin helium films, the measured CR-data can be understood. The asym- inetry of the absorption line and its dependence on the thickness of the helium film is modeled. From this, the free and localized electron density is found and one can also ob- tain information a b u t the roughness of the helium-covered substrate.
- 30
-
h 5 [mmlThis activity is supported by the DFG, Forschergruppe
"Quantengase" and the EU-RTN "surface electrons on mesoscopic structures".
OO 2 4 6 8 10 12
bulk level below substrate (h) [mm]
References
[ I ] E.Y. Andrei (Ed.), Two-Drmensional Electron Systems on Helium and other Cryogenic Substrates, Kluwer Academic Publishers, Dordrecht, 1997.
[2]J. Klier, T. Ginzler, A. Wiirl, P. Leiderer, 6. Mistura, E Teske, P. Wyder, V. Shikin, 3. Low Temp Phys. 122 (2001 ) 451.
