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Experimental study of coupled electron-ripplon vibrationsof the 2D electron crystal at the surface of liquid helium

D. Marty and J. Poitrenaud

Service de Physique du Solide et de Résonance Magnétique, CEN-Saclay, 91191 Gif-sur-Yvette Cedex, France

(Reçu le 28 décembre 1983, révisé le 5 mars 1984, accepté le 16 mars 1984)

Résumé. 2014 Nous étudions les spectres d’absorption du système d’électrons à deux dimensions formé par les élec-

trons à la surface de l’hélium liquide, en travaillant à fréquence fixe et en balayant la densité électronique. Nousobservons la transition liquide-solide du système d’électrons à deux dimensions. Dans la phase solide, nous obser-vons deux séries de signaux du spectre des vibrations couplées électron-ripplon. L’interprétation de ces signauxnous permet de déterminer le facteur de Debye-Waller W1, et le temps de relaxation des ripplons.

Abstract. 2014 We study radio frequency absorption spectra of the 2D electron system at the surface of liquid helium,

by working at fixed frequency and scanning the electron areal density. We observe the liquid to solid transition

ofthe 2D electron system. In the solid phase, we observe two series of signals of the coupled electron-ripplon vibra-

tion spectrum. The interpretation of these signals leads to a determination of the Debye-Waller factor W1, and of

the ripplon relaxation time.

J. Physique 45 (1984) 1243-1255 JUILLET 1984,

Classification

Physics Abstracts67.90

1. Introduction.

We describe an experimental study of a part of the

phonon spectrum of the two-dimensional crystalformed by the electrons at the surface of liquid helium. A theory of two-dimensional melting has been

worked out by Kosterlitz and Thouless [1, 2], then

developed by Halperin and Nelson [3, 4] and

Young [5] : according to this theory, the melting takes

place by the unbinding of pairs of dislocations and

the emergence of free dislocations, above a certain

temperature, and this transition is continuous, thatis to say the thermodynamic functions have no dis-

continuity at the transition and there is no latent heat

nor surfusion.The thermodynamic state of a classical Coulomb

system is determined by the value of the parameter r,the ratio of the potential energy to the kinetic energyof an electron F = e 2 (nn)l 2IkB T ( 1 ). The KosterlitzThouless theory [1] leads to a calculated value of

F > 80 at the transition.

Halperin, Nelson and Young also predict the exis-

tence of an intermediary phase between liquid and

solid, called hexatic phase », which displays an

orientational ordering and no translational ordering.

Computer simulations performed by severalauthors [6, 7] have indeed shown the existence of a

phase transition, but rather suggest a first order one.

Theoreticians [8] have predicted that, if the elec-tronic crystal exists, it should have a triangular pri-mitive cell, and they have calculated the dispersionrelations of the longitudinal and transverse phonons. Actually, for the system we study, the situation is

more complicated because of the coupling between

the electron system and surface waves (or ripplons).The free surface of liquid helium is subject to exci-

tations, the ripplons, whose dispersion relation (at

high wave vector), is Q 2 = (fY../ p) k3, where a is the

capillary constant and p the liquid density. Coupling, between these ripplons and the longitudinal and

transverse electronic

phononsgives rise to a coupled

excitation spectrum which will be explained in part 2.

Analysing several modes of these coupled excita-

tions, Grimes and Adams [9] revealed the existence

of 2D electron ordered phase. Gallet et al. [10] detectedthe transverse optical mode at low k and derived the

temperature variation of the shear modulus, parti-cularly near melting. Mehrotra et al. [11], and Eselson

et al. [12] measured the 2D electron mobility, which is

related to the coupling with ripplons.Our first experiment [13] allowed us to detect the

liquid-solid transition, by measuring the electron

longitudinal susceptibility. The experiment described

here (which is the continuation) was initially set toinvestigate the resonant coupling between ripplonsand the vertical electron motion, and to detect the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045070124300

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presence of the electron crystal and measure thestructure factor (this experiment has been proposedindependently by Shikin [14], and Williams (privatecom.)). In fact, this experiment has not been possible,but, measuring the signals due to the horizontalelectron motion, we have been able by varying the

areal density, the temperature and the magneticfield, to extend the study of the coupled excitation

spectrum initiated by Grimes and Adams [9] (at onlyone temperature and electron density) and Galletet al. [10] (at higher frequencies); from this study we

have derived the value of the ripplon relaxation time.

2. Short theory of the coupled electron-ripplon vibra-

tional spectrum.

An electron above a helium surface is submitted to a

strong repulsive force at small distances and to a pola-rization attraction at

largedistances. Thus it is bound

in the direction normal to the surface at about a

hundred angstroems from it. Its binding energy is ofthe order of 8 K. This electron hollows a local defor-

mation, « a dimple », in the helium surface whose

depth is related to the electron localization, parallelto the surface. The energy gained in this way is very

small, a few mK [15]. We look now at an assembly of

such electrons constrained to move in a plane parallelto a helium surface. The actual method to realize this

system is to bring up a helium surface between thetwo plates of a capacitor between which an electric

field Ei is applied which draws the electrons emitted

above the surface to this surface. When the electronsare in a disordered state, the kinetic energy of one

electron is large compared to the deformation energyof the localized surface state and the electrons do not

see its effect ; on the other hand if the electrons are

strongly correlated, the deformation energies of each

electron add up and their sum can be compared to the

global translation kinetic energy. The electrons are

trapped in their dimple lattice. Let us assume such an

electron crystal is realized. The Hamiltonian for thesurface electron interaction can be written :

where z(ri) is the distance between an electron i and

the surface and ri its position in the horizontal plane.Hp.,(ri) depicts the polarization interaction between

liquid helium and charges.The basic hypothesis is [16-18] that the characteristic

frequency of the electron oscillations around their

average position is much larger than that of the heliumsurface excitation. Then it is possible to write thesurface deformation and the electron density as a

function of the reciprocal electron lattice vectors G

[for a triangular lattice G 2 = N G 8 1t2 n/,,/3- with

NG=

1, 3, 4, 7,...] and to average the high frequencyelectron modes on a time scale small compared with

the period of the surface motion but longer than the

electron characteristic time. Thus the « Debye Waller

factor » W is introduced :

u’ >is the contribution to the mean

square displace-ments from the fast modes.

By taking into account that the phonon spectrumis not very much perturbed by the surface coupling at

high frequency the calculation shows [19] that for thetransverse modes which give the main contribution,

Wi is given by :

where c, is the transverse sound velocity and WM and

Wm two cut-off frequencies. The static surface defor-mation can be written

with zv = nJ/L-tG’ where

The second term in fi comes from Hp.,. fl-’ is a

characteristic distance of the electron from the sur-

face. Following these ideas the electron oscillation

equation can be solved and the perpendicular andparallel to the surface motions can be separated.

Perpendicular motion. - By neglecting the second

order terms the projection of the motion equationon the z axis can be written

If an electric oscillating field El is applied perpendicu-lar to the surface, it is found that z includes two terms,one being the static deformation already given(Eq. (4)), while the other is :

corresponding to the resonant excitation of the capil-lary waves. The effective mass for this motion is p/nGof the order of 1014 times the free electron mass.

Horizontal motion.- Without the ripplon inter-

action, the longitudinal and transverse phonon dis-

persion law is well known [8]. For small wave vectors

compared with the first Brillouin zone boundaryvectors the longitudinal frequency is given by :



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where h is the helium height and d the distancebetween the helium surface and the capacitor upper

plate, and the transverse frequency is

for a triangular lattice.

This spectrum is notably modified by the presenceof the substrate. By introducing the coupling factor

Fisher et al. [16] found the secular equation of longi-tudinal and transverse motions :

The dispersion relations when a magnetic field is added

perpendicular to the surface are :

We is the cyclotron frequency.The spectrum splits up into several branches :

without magnetic field at high frequency the dimplecannot follow the deformation. If k = 0 the electronoscillates in the potential well ofthe static deformation

with the frequency

and at k =A 0 dimple and electron oscillate in oppositephase. It is an optical mode where electron has its

free mass and

At low frequency electron and dimple vibrate in phase :

whereM is the effective electron mass

For intermediate frequencies additional branches liebelow each of the higher ripplons QG’ We call these

modes, « ripplonic » modes. With a magnetic field H

the low frequency modes with high effectivemass

are little affected, but the longitudinal and transverse

high frequency modes are coupled by H and two new

frequencies appear w, and w- which move respecti-vely towards high and low frequency when H increases.The w - branch meets the different acoustical and

ripplonic modes. Figures 1, 2, 3 show the spectrumcalculated as a function of the different parameters :wavevector, electron density and magnetic field.

Fig. 1.- Calculated dispersion relation of longitudinal andtransverse coupled modes. T = 0.3 K, n = 2.9 x 108 cm - 2,H = 0. Debye-Waller factor = 0.44. The dashed lines show

the uncoupled mode spectrum. W, X, Y, Z represent themeasurements of Grimes and Adams [9]. A one of themeasurements of the Saclay group [10, 17, 18].

3. Principle of our measurements.

The cell has been conceived for the study of resonant

capillary waves excited by the vertical electron motion.

The study of the signal to noise ratio S/N for such a

signal shows the necessity of a resonant cavity (thatis to say a fixed frequency) and the use of a lock-indetection. As a matter of fact S/N - 1,5 if the resonant

cell quality factor Q is equal to 400, the rf power 1 nW

and the detection band width 10 Hz. The experimentshowed that actually the electrons were also submitted

to a small electric field parallel to the surface because

of inhomogeneity of the electric field Elf . This parasiticfield is sufficient to induce large signals. From figure 3we see that, in the domain of existence of the electron

crystal that we can experimentally reach, we should

beable to excite the

non

displaced ripplon mode foran rf electric field perpendicular to the surface. If the rf

field is parallel to the surface, we should be able to see



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Fig. 2. - Calculated variations of the frequencies of the

coupled electron-ripplon modes versus magnetic field.

k = 2.55 cm -’, n = 2.9 x 10’ cm-2, T = 0.3 K, Wi= 0.44.

m is the working frequency. (The lowest frequency mode is

not represented.)

either the acoustical longitudinal mode or the « opticalmode » (we name this mode « optical » by continuityalthough it no longer has this characteristic). An

experiment is conducted as follows : first an electron

monolayer is created above the helium surface, and is

cooled to temperatures low enough to go from thedisordered state to the solid. Then the electrons are

submitted to an rf electric field and we measure thederivative of the rf absorbed power with respect to the

pressing field El versus the voltage between the capa-citor plates, the temperature being fixed.

Ithas been found necessary

to add a

magneticfield

H ; when H = 0, we observe a very big plasmon signalin the liquid phase (we used this phenomenon to detectthe liquid to solid transition in our first experi-ment [13]), but with lock-in detection this signalsaturates the amplifier and becomes troublesome.

Besides, magnetic field is a useful parameter to test the

theory of coupled modes.

4. Experimental apparatus.

Cryogenics. - We work at temperatures between

0.275 K and 1.2 K. Firstwe

use a helium 4 cryostat,which is pumped to primary vacuum and provides a

temperature of 1.2 K. Then, starting from this tempe-

Fig. 3. - Calculated variations of the coupled electron

ripplon modes versus electron density for two wavevectors

k = 2.55 (full line), k = 8.88 (dashed line). T = 0.3 K,H = 200 G, W = 2.5 x 10-4 Tll. m is the working fre-

quency. The dotted lines represents ripplon frequency

S c 2 = 1V c 32a 8 2 n 3j2 We show the respective liquid andt2 2= N 3/2 O

(8 n2 n)3/2We show the respective liquid and

solid domains for T = 0.3 K. Circles are resonances corres-

ponding to the horizontal motion of the electrons, the square

is the resonance due to the vertical motion at 17 MHz.

rature, we cool down to 0.25 K with a one shot helium 3

refrigerator pumped to secondary vacuum, and in

direct thermal contact with the experimental cell. A

second helium 3 refrigerator, working continuously,provides a thermal anchor at 0.7-0.8 K, in order to

limit thermal leaks, particularly due to the fountaineffect in the helium fill capillary (see Fig. 4).

Temperatures are measured with two Speer carbonresistors (470 Q at 300 K) placed above and below

the experimental copper cell. We calibrate theseresistors between 0.5 K and 1.2 K by means of thehelium 3 vapour pressure temperature scale, then

extrapolate the values down to 0.25 K. Using the

empirical formula :

A, B, C are deduced from the calibration between

0.5 K and 1.2 K. We expect an absolute value of

0.01K

for the precision of our temperature measure-

ment. We regulate the temperature with a heatingwire stuck on the cell.



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Fig. 4.- General view of the experiment. 1 : Main helium 3

refrigerator 0.25 K T 1.2 K, 2 : Secondary helium 3

refrigerator T - 0.7 K, 3 : Helium 4 fill capillary of the cell,4 : Experimental cell (with electrons at the surface ofhelium),5 : Exchange gas can, 6 : Superconductive coil, 7 : Hall

effect probe.

Charge creation and polarizations. - Figure 5shows a schematic of our experimental cell. Electronsare generated by a tungsten filament : the filament

-

which is biased to a high negative voltage (300 to

400 V) (compared with the grid potential)- is brieflyheated, generating a glow discharge through thehelium 4 vapour. The plate is positively biased with

respect to the grid, creating an electric field E 1. which

draws the electrons from the grid towards the plate :they are stopped by the potential barrier due to liquidhelium and spread over the surface. A guard ring is

negatively biased with respect to the grid. Charging of

the surface is only possible at a temperature neigh-bouring 1.2 K, because of the effect of the helium 4

vapour pressure filling the top of the cell; the pressurehas to be weak enough to allow a ionizing dischargeto be started, but not too weak, otherwise the electrons

Fig. 5. - Cross sectional view of the cell, 1 ; Tungsten fila-

ment, 2 : Grid, 3 : Plate, 4 : Guard ring, 5 : Coaxial tightfeedthroughs, 6 : Resonant circuit coil, 7 : Helium fill capil-lary.

are accelerated by the electric field in the absence of

collision with the gas molecules and they cross thehelium surface.

The grid and the plate are separated by 2.7 mm,

and the plate area is 7 cm2.Radiofrequency and magnetic field. - A schematic

diagram of the spectrometer is shown in figure 6. It

works in a reflexion mode, with homodyne detection,at a fixed frequency of 17 MHz. The resonant circuit

(which has a quality factor of 400 at T 1.2 K) is

coupled to the high frequency circuit by means of thetransformer LL‘ ; this allows the isolation of the spec-trometer from d.c. voltages, and also it lowers the reso-

nant circuit impedance to the transmission line impe-dance (50 Q).We refine this adjustment with an impedance

transformer, at room temperature. The signal reflected

by the resonant circuit is mixed with the generatorreference signal (whose phase can be varied) in order

to detect the absorption signal. The voltage between

the plate and the grid is modulated at 450 Hz, and we

measure the derivative of the absorbed power with

respect to plate voltage Yp (the grid is grounded)by means of a lock-in detection.The guard ring is used to confine electrons above

the plate, in a region where the rf electric field is

homogeneous and has no extraneous horizontal

component. A vertical magnetic field is generated by a coil

placed around the exchange gas can (see Fig. 4). Thiscoil is made of a superconducting alloy and has a

superconductive switch; it gives a magnetic field



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Fig. 6. - Sketch of the spectrometer and polarizations.P : plate, G : grid, L : resonant circuit coil, L’ : couplingcoil.

which can be varied up to 1 500 G and which is

measured, either following the dc supply current, or

by means of a Hall effect probe, located beneath the

exchange gas can.

Fillingof the cell. - We fill the cell with helium 4

at 1.2 K, by injecting known quantities of helium 4,filtered through active charcoal cooled at 77 K,and we measure the corresponding grid-plate capaci-tance, or frequency of the resonant circuit. The increasein capacitance (or decrease in frequency) is due to the

slight difference of dielectric constant between liquidhelium 4 (s = 1.057) and vapour (8=1). Thus one

obtains a filling curve of the cell (see figure 7) which

allows us, by measuring the frequency or capacitance,to deduce the liquid helium height above the plate.

Determining the electron density.- One of the

experimental difficulties in the study of 2D electrons

is the accurate measurement of the electron density.a) A first method is to deduce the density from the

helium height, when the surface is saturated with

electrons, i.e. when the electric field above the chargedsurface is zero. Then the electric field below the sur-

face is 8 V pi h = 4 nne (14). In order to apply this

method, one has to know precisely the height h and

one has to make sure that the surface is saturated. Thatis what we did, in a first approach, taking for h thevalue deduced from the cell filling curve (see Fig. 7).We make sure of the saturation of the surface, because

we progressively empty the surface of its electronswhile we scan the

plate voltagefrom its maximum

value to zero; furthermore we control it by measuringthe grid current while discharging the surface. (We

Fig. 7.- Filling curve of the cell : frequency of the resonant

circuit versus the liquid helium 4 volume injected in the cell.

Points A and B correspond respectively to the beginningand the end of filling of the volume between plate and grid.

have also a rough measurement of the total surface

charge by integrating the grid current.)

b) An alternative method is to measure the electron

density of the electron crystal by determining the size

of its elementary lattice cell, using the coupling with

ripplons : the only ripplons which couple with electronoscillations have for wavevectors the electron lattice

reciprocal vectors,

and ripplon corresponding frequencies are given by :

Hence measuring the frequencies vG = QG/2 1t leadsto a determination of the areal density n.

In our experiment, we work at fixed frequency and

scan the electron density n ; the equality (a/p) G 3(n) =W2 defines a value n* for n ; we do not measure directlyn*, but the limit of the density ofour acoustical signals(see Fig. 14) is precisely n*.

Experimentally, we noted a shift of about 20 %between the values of the densities determined in a)and b). We do not explain this shift, but we thoughtpreferable to retain the value determined in b) by the

ripplon, as it is an absolute measurement. (The para-meters a and

pare

known.)In determination

a),we

may have an uncertainty on the measurement of V

(due to trapped charges) and of h (because of capillary



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effects point A of figure 17 may not be exactly the

beginning of the filling of the plate to grid volume).So, practically, we have multiplied by 1.18 the densities

deduced from the expression gvplh = 4 nne. We

estimate the precision to be better than 20 % for the

absolute measurement of the electron density (whereas

the relative error is one or two percent).The maximum electron density reached in our

experiment is 109 cm - 2 (the calculated value for the

maximum areal density is 2.25 x 109 cm-2).

5. Density profile. Wavevector. Modulation effect.

Equation (14) which gives the electron surface densityis no longer correct on the edges of the electron layer.We have calculated n(r), the real density profile, withinexperimental conditions; r is the distance to the centre

of the electron layer. This profile is determined by the

imposedelectrostatic

potentialsand the cell

geometry.We solved the Laplace equation AV = 0 by an itera-

tive method using a computer. We calculated thesurface charge density for different charge pool radii

and the same electrostatic configurations. The surfaceis always supposed to be charged up to saturation, thatis to say its potential is equal to the grid voltage. Agood choice for R will give n(R ) = 0. The main para-

meter is y = VP VA VA . For different values of y we

calculated R, n(r) and the voltage around the chargepool in its plane. Figure 8 shows examples of the den-

sity profile and voltage around the electron layer for

several values of y. By using the conformal mappingmethod F.I.B. Williams [private com.] found an analy-tical formula for R and n(r) as a function of y when thesurface is equally distant from the electrodes. Figure 9shows the calculated variation of the charge poolradius R and its effective radius as a function of the

plate voltage for different guard ring voltages. All our

measurements have been performed with VA = - 5 V

and typically the maximum plate voltage was of theorder of 300 V. It can be seen that Reff is nearly equalto the plate radius (15 mm) and varies only slightlywith Fp when Vp « VA. This is the radius we use inthe calculation of the wavevectors excited in our cell.

We suppose that the rf field Erl parallel to the surfaceis due to a deformation of the field lines of Elf at the

place of the discontinuity between the plate and the

guard ring. We assume that electrons on the edgeof the charge pool have a zero speed and thus the pro-

pagated phonons have a wave vector such that

J 1 (kRr .ff) = 0, J11 is the first-order Bessel function. A modulation voltage is applied between the capa-citor plates and the observed signal is the derivativeof the rf power absorbed in the resonator with respectto the plate voltage when VP is reduced to 0. Figure 8shows that the area of the electron layer does not vary

very much with VP. We cannot modulate the electrondensity in this way. On the other hand the scanningtime of VP (10 min) is much larger than the modula-

Fig. 8. - a) Plot of the calculated surface charge densityn(r) (normalized to the surface charge density at the cell axis)versus the distance r to the centre of the cell y = (V p - V A)/V p.For example y = 1.025 when VP = 200 V and VA = - 5 V

and y = 2 when Y,=5Vand V A = - 8 V.b) Plot of the confining potential V in the plane of the

charge pool versus the distance r’ to the charge edge. The

potential of the electron layer is 0.

tion period (1/450 s). Since, during one period ofmodulation, the electrons which leave the surfacefor decreasing VP cannot come back on it when VPincreases again, the electron density is not modulated,only is the pressing field. Thus the observed signalis the derivative of the rf power with respect to the

pressing field.

6. Experimental results and interpretation.

Figure 10 shows two experimental traces of the deri-

vative of the absorbed power, dPjdE.1’ versus theelectron density, in the presence of a magnetic field,and for two different temperatures.



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Fig. 9. - Plot of the variation of R and Reff versus platevoltage V p for several values of the guard ring voltage VAcalculated by the conformal mapping method. R is the chargepool radius where the surface density is reduced to 0. Reffis defined as the radius of a uniformly charged electron

layer which would have the same total charge. For a typicalheight h = 1.5 mm, n = 4.59 x 108 cm-2 when VP = 100 V.

Fig. 10. - Experimental traces of dPjdE 1- (derivative of the

absorbed power with respect to pressing field) versus platevoltage (or electron density), at two different temperatures

and with a magnetic field of 200 gauss. A, B and C areseveral acoustical lines. L and S delimit the melting of the

electron crystal.

These traces show two distinct series of signals, thatwe interpret as optical signals (at lower density) and

acoustical signals (at higher density), and also the

phase transition. It is worth noting that the opticaland acoustical signals are of opposite sign; as a matter

of fact, ifone increases the pressing field, and therefore

the coupling between the surface and the electrons,the frequency of the optical mode increases, whereas

the frequency of the acoustical modes decreases.We have studied the variation of these signals

(position, width, intensity) versus temperature and

magnetic field. (We take for the width and intensitypeak to peak values.) Figure 11 depicts the variationof their positions versus temperature; it also showsthe points in the (n, T ) plane of the phase transition.

Fig. 11. - Experimental results reported in the (n, T)(density, temperature) plane. 0 : acoustical signals, 0 : opti-cal signals (with indication ofthe magnetic field), + : L pointsof the « end of melting ». Full lines are the values calculated

for the parabolas defined by r = n"’ n1/2 e2/kT for r =142

and r = 160.

6.1 PHASE TRANSITION. - Figure 12 shows the cal-

culated variation’ of dP/dE 1. versus n, for only one

wavevector, and in the presence of a magnetic field :

in (a) we assume the system is only liquid, in (b) onlysolid, then in (c) we plot the resulting signal when we

place the phase transition at nc = 2.07 x 101 CM-2

(corresponding to T = 142) (we did not plot here theacoustical signals). Signals (a) and (b) vary slowly with

temperature whereas the phase transition criticaldensity nc varies like T2. So when we change the tem-

perature, we move the transition density nc, and the



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Fig. 12. - Calculated plots of dPjdE 1-’ derivative of theabsorbed power, versus density for T = 0.3 K, H = 200 gauss,k = 2.55 cm-1. (a) : taking the system as liquid throughoutthe density scan, (b) : taking the system as solid throughoutthe density scan, (c) : setting the phase transition at nc2.07 x 108 cm-2 (which corresponds to rc = 142).

shape of the resulting signal changes considerably.That is what

appearson the

experimentaltraces of

figure 10 : at T = 0.256 K the phase transition occurs

at a density lower than the optical signal density,which indeed is nearly completely visible; at T =

0.368 K, on the contrary, the phase transition occurs

at an electron density higher than the optical signaldensity : indeed we only see a wing of this signalcorresponding to n > nc.We also note that the phase transition has a certain

width, and we observe two transition densities nL

and ns (ns > nL). Generally, the L point (« end of

melting ») is more clearly experimentally definedthan is the S point.

We have checked that the coordinates (nL, TL)(density, temperature) obey a law n’ L /2 / T L = constant,which is characteristic of the phase transition of a 2D

Coulomb system (see (1»). With 1.4 x 108 cm - 2n 5 x 108 CM-2, we have measured a mean value

of the F parameter TL = 142 ± 5 (with the a) deter-mination of the density we should get r=131±5).We verified that this transition line fitted well

with the transition points observed by the solidification

of the crystal at constant density by lowering thetemperature, detected by measuring the longitudinalelectric susceptibility, as in our previous experi-ment [13].

During our recordings we never observed any delayat the melting.

Observation of S points (o beginning of melting »),which give a mean value TS = 160, seems to indicatethat the transition has a relative definite width (in the

(n, T ) plane), On/n of the order of 27 %. A small partof this width An/n = 3 % has been attributed to den-

sity inhomogeneity (this has been estimated from

measurements at low helium

level).6.2 OPTICAL SIGNALS. - The main features of this

signal are the following :- The position (density) varies rapidly with the

magnetic field; slowly with temperature, but this

effect is amplified in the proximity of the meltingtransition.- The linewidth, of the order of 0.5 x 108 cm-1,

changes only slightly with temperature or magneticfield.- The intensity decreases when the magnetic field

increases (for H > 100 G).

To calculate the position of the optical mode wop,we solve the secular equation (8), taking into account

the coupling with the three lower modes of ripplons(Qt, 03, 04) and taking the Debye-Waller parameterW, as adjustable parameter. For the wave vector k,we take kReff as the first zero of Bessel function Jl,Reff being the effective radius (Reff = 1.5 cm).The results of the calculation are indicated on

figure 13, where we have plotted the variation of W,i

Fig. 13. - Variation of the Debye-Waller factor W1 versus

F,IF; F,IF = (n:12/Tc)/(nl/2/T), + : calculation from our

measurements. of optical signals. The full line correspondstoW

=

2.5 x 104 T/nl/2 or W=

0.521 rc/r for Fr = 142.0 : calculation from measurements of transverse opticalsignals by Valdes [18].



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versus re/r (which is proportional to T/nI/2), Fe beingr value at the phase transition. Far from melting, we

findW1= 2.5 x 104 T /nl/2 for 210 > F > 170, which

is in agreement with (3), where we neglect the variationof the logarithm, and where C2 is proportional to n1’ 2,according to (7).

In the vicinity of melting, the Debye-Waller factorincreases more rapidly, which reflects the rapid enhan-

cement of electron fluctuations near the phase transi-

tion (as W1 = G, U2 )/4).These results are in good agreement with those

resulting from measurements of the transverse opticalsignal performed by Valdes [18], and which are

reported in figure 13.

The calculation accounts well for the shift of the

optical signal with magnetic field, observed experi-mentally (and we have verified that the magnetic field

has no influence on the value of Debye-Waller factor).Then, in order to

comparemeasured and calculated

values for linewidth and intensity, we had to introducetwo relaxation times : TR, the relaxation time for

ripplons alone, and ie the relaxation time of non

ordered electrons. As, to our knowledge, there is at

present no calculation nor measurement of the ripplonrelaxation time TR, we took TR as an adjustable para-meter in order to fit the calculated and experimentalvalues of the linewidth. Concerning the electrons, therelaxation time or mobility have been studied both

experimentally and theoretically by many authors :at the pressing fields and temperatures (T 0.6 K)of our experiment, the prevailing process is electron-

ripplon scattering; we took for T, the value obtainedfrom the empirical formula indicated by Grimes and

Adams [20] for the mobility :

with C = 9.3 x 1011 Vs-’ and Eo = 230 Vcm-.’.We calculate the absorbed power : the longitudinal

displacement of the electron is :

with

(the sum is over the three ripplon modes 01, 03, Q4)and the absorbed power is :

Then we compute dP/dE 1- versus density n.

The agreement between calculation and experimentis satisfactory if we take for the ripplon relaxationtime iR = 3 x 10-’ s, which corresponds to a fre-

quency linewidth 1/2 1ttR of 5 MHz. One must noticethat in fact the electronic relaxation time ie contributeslittle to the linewidth, for it is balanced by the factor

m/M, where M > m is the effective mass of the electronin the coupled electron-ripplon motion.

(For example for n = 3 x 108 cm-2, T, = 2.1 x 10-9 s,

1/2 nte = 75 MHz, but, with m/M 10-2, (m/M )/2 RT,0.75 MHz, whereas 1/2 nLR = 5 MHz.)The decrease of the signal intensity with magnetic

field is explained by the fact that the maximum of theabsorbed power is reached when Wc 1’-1 (cp 2013 pp)12which corresponds to a field of 36 G, for n = 3 x

108 cm-2 (and we work generally with magneticfields higher than 100 G).We also observed a weak signal (see Fig. 10) at a

density slightly smaller than that of the main signal;we think it is an optical signal with a higher wave-

vector.

6.3 ACOUSTICAL SIGNALS. - As opposed to « opti-cal » signals, the signals which appear at higher den-

sity vary very little with magnetic field at least if H

is lower than about 1 000 G, but they vary conside-

rably with temperature. Figures 14, 15 and 16 show

the variation of their position nac’ their peak to peakwidth and intensity Anac and Iac with temperature, andfor different concentrations of helium 3 in helium 4,for a magnetic field H = 200 G. Three series of signals

A, B, C are visible. Notwithstanding the number ofthese signals, the variation of their positions with

temperature makes it impossible to attribute them tothe uniform vertical motion of the electrons, because

Fig. 14.- Plot of the experimental variation of the positionof the acoustical signals with temperature compared with

the calculated variation : 1) for W= 7.5 x 104 T /jn and

kA = 8.88 cm-1, kB = 6.78 cm-1 and kc = 4.67 (full line),

2) for W= 2.5 x 104 T/,/n and k, = 15.17 cm-1, k2 =

13.08 cm-1, k3 = 10.98 cm-1 and k4 = 8.88 cm-1 (dashedline). 40 represents the results for electrons on 4He with less

than 1 ppm of 3He, 0 the results for 4 He + 200 ppm of 3He.n* = 6.05 x 108 electrons/cm2 is the asymptotic limit of nafor high temperatures.



1253

Fig. 15. - Plot of the experimental variation of the peakto peak amplitude lac of the acoustical signals with tempe-rature compared with the calculated variation (full line) forthe same values of parameters as figure 14 and

and

Fig. 16. - Plot of the experimental variation of the peak to

peak width Ana, of the acoustical signals with temperature

compared with the calculated variation (full line) for thesame values of parameters as figure 15.

the ripplon frequency Q’ = " k depends on T onlyGp

through the capillary constant a. For 4He with less

than 20 ppm of 3He, a is independent of T for 0.2T 0.8 K. We interpret them as the excitation of

acoustical longitudinal phonons of wavevectors kA,kB, kc.

Variation with temperature. - We use the same

method as described previously. But it can be seen

from figure 3, that acoustical signals appear at increas-

ing densities for decreasing wavevector. So we do not

know a priori which wavevector k to attribute to curves

A, B or C. Therefore we solve equation (9) with the

Debye-Waller factor W, as the unknown quantity andwe search the sequence of wavevectors solutions of

J 1 (kRcff) = 0 which give the same determination of

W1 in function of TI.,In. Then we find kA = 8,88 CM -’,

kB = 6,78 cm-1, kc = 4,67 cm-1 (they are respectivelythe 4th, 3d and 2nd zeroes of J 1) and WI = 7,5 x

104 Tl,,In for 320 > F > 180, a value three times thevalue we deduced from the « optical » signals. Facingthis difficulty if we suppose that the Debye-Wallerfactor is the same for the whole spectrum and has its

optical value, we can solve the secular equation for a

point n, T of a curve A, B or C by taking the wave-

vector as the unknown parameter. This calculationshows that the found wavevector varies along a curve.

For example kB varies from 13 cm-’ for T = 0.255 Kto 20 cm-’ for T = 0.400 K along curve B. This seems

difficult to

interpreteven

by takinginto account the

variation of the charge pool radius with density as

shown in figure 9.,

Concerning the width and the intensity of the signalsthe results are the same by using either one or the

other value of W11 associated with its correspondingsequence of wavevectors. We find by taking the elec-

tron relaxation time measured by Grimes (15) and

with the help of equation (16) that a ripplon dampingtime

gives a good agreement between experiment and cal-

culation. We supposed that tR is inversely proportionalto the square of the ripplon wavevector [21]. This time

is five times larger than that which accounts for the

optical signal width (for example, here TR = 1.3 x

10 -’ s for n = 8 x 108 CM - 2). Concerning the spec-tacular experimental decrease of the signal intensitywith increasing T (Fig. 15), a calculation of Iac bysupposing that electron density is modulated gives an

increasing Iac with T. This is a verification that onlythe pressing field is modulated (cf. § 5).

Addition of small amounts of 3He. - At low tem-

peratures T 0.5 K He condenses at the surface of4He thus changing its capillary constant a [22]. The

ripplon frequency QG changes as (X1/2. The calculationshows that the change in QG will affect the acoustic

signals much more than the optical ones. We add

from 30 to 400 ppm 3He in liquid helium 4 and we

observe (see Fig. 14) that the lower is T, the higher is

the density at which the signal occurs by comparisonwith the zero 3 He concentration. By using Edwards

measurements of a [22] we find a good agreementbetween experiment and calculation. We give an

example concerning the position of the signals at

T 0.260 K and H = 200 G (see Table I).

We think that this confirms that the high densitysignals are related to the acoustical branch of the

spectrum.



1254

Table I. - Comparison between experiment and calculation concerning the position of’the optical signal and theB acoustical signal for two concentrations of*3He in 4He (T = 0.260 K and H = 200 G). For the acoustical signalthe calculation is done for the two possible values of ’ W 1 : (1) W1

= 7.5 X 104 T/.jn and kB = 6.78 cm-1, (2)WI = 2.5 x 104 T/fi and kB = 14 cm-l.

In order to explain the discrepancy between theresults deduced from the optical and acoustical partsof the

spectrum,we

triedto

study the influence ofboundary conditions. We made two additional expe-riments the first one by varying the guard ring voltage,and the second one by using another cell with quitedifferent boundary conditions.

Guard ring voltage influence. - The experimentshows that, at a given temperature, the acoustical

position and width decrease with increasing VA whileto a first approximation, the optical signals and tran-

sition are insensitive to VA. Figures 8 and 9 show

the calculated variation of the charge pool radius andthe voltage around the electron layer as a function of

VA.The variation of R with

VAcannot

explainthe

observed effect and we are not able to take into account

theoretically the variation of the potential which

confines the electron. But qualitatively it seems that

increasing VA reduces the variation of the positionof the acoustical signals with temperature which is

coherent with a lower Debye-Waller factor.

A cell with different boundary conditions. - Figu-re 17 shows the diagram of this cell [8] ; the electron

layer goes up to the guard ring whose potential is thesame as that of the grid and of the electron layer one

at saturation. With this cell it is possible to observe

signals eitherat fixed

density by scanningthe

frequen-cy, or at fixed frequency by scanning the density as we

Fig. 17.-

Schematics of the second cell with the applieddc polarizations, F : tungstene filament, G : grid, A : guardring, P : plate.

are used to. We shall not go into all the details of this

experiment. The results are very well explained by

takingfor the

Debye-Waller factorthe value

ofW = 2.5 x 104 T/.,,/n for 4 x 108 n 9 x 108 el/cm2and 0.075 K T 0.3 K (here we used a dilution

refrigerator).

7. Conclusion.

With the experimental study of absorbed power

spectra at 17 MHz versus electron density, we observe :

- a liquid to crystal phase transition of the 2D

electron system. The results are partially coherent withthe Kosterslitz-Thouless theory : the value of F is ofa right order. On the other hand we observed an

intrinsic transition width. We cannot say whether itis due to a first-order transition or to two second-

order transitions separated by a hexatic phase,- two series of signals of the spectrum of the elec-

tron-ripplon coupled excitations : optical signals andacoustical signals. These signals can be accounted forwithin the frame of the theory of coupled electron-

ripplon vibrations, first developed by Fisher, Halperinand Platzman [16].From the optical signals, we determine the value

of the Debye-Waller factor Wl,= 2.5 x 104 T/nl/2(far from melting), which agrees with other measure-

ments.

But we cannot interpret acoustical signals in a simpleway by taking this value for W 1 (we tried to studyexperimentally the influence of boundary conditions;but the theoretical problem remains unsolved).We determine the ripplon relaxation time at 17 MHz,

which may be approached by the interval :1 MHz

1/2 7rTR 5 MHz.

Acknowledgments.

We wish to thank F. I. B. Williams for the initial ideaof this experiment and his help throughout the investi-

gation ;we

also acknowledge for fruitful discussionsour colleagues G. Deville, F. Gallet, C. Glattli, A. Val-

des and C. Heyer-Ricoul for technical aid.



1255

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[17] GALLET, F., Thèse de 3e cycle, Université Paris 6,Novembre 1980.

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d. marty and j. poitrenaud- experimental study of coupled electron-ripplon vibrations of the 2d...

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