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Experimental study of coupled electron-ripplon vibrations of the 2D electron crystal at the surface of
liquid helium
D. Marty, J. Poitrenaud
To cite this version:
D. Marty, J. Poitrenaud. Experimental study of coupled electron-ripplon vibrations of the 2D electron crystal at the surface of liquid helium. Journal de Physique, 1984, 45 (7), pp.1243-1255.
�10.1051/jphys:019840045070124300�. �jpa-00209861�
Experimental study of coupled electron-ripplon vibrations
of 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.
2014Nous 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. Nous
observons 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 signaux
nous permet de déterminer le facteur de Debye-Waller W1, et le temps de relaxation des ripplons.
Abstract.
2014We 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 of the 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.
Classification Physics Abstracts
67.90
1. Introduction.
We describe an experimental study of a part of the
phonon spectrum of the two-dimensional crystal
formed 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, that is 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 energy of an electron F
=e 2 (nn)l 2IkB T ( 1 ). The Kosterlitz Thouless 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 several
authors [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 dispersion
relations 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 phonons gives 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] detected
the 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 to investigate the resonant coupling between ripplons
and 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
presence of the electron crystal and measure the
structure factor (this experiment has been proposed independently by Shikin [14], and Williams (private com.)). In fact, this experiment has not been possible,
but, measuring the signals due to the horizontal electron motion, we have been able by varying the
areal density, the temperature and the magnetic field, to extend the study of the coupled excitation
spectrum initiated by Grimes and Adams [9] (at only
one temperature and electron density) and Gallet
et 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 large distances. Thus it is bound in the direction normal to the surface at about a
hundred angstroems from it. Its binding energy is of the 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, parallel
to 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 parallel
to a helium surface. The actual method to realize this system is to bring up a helium surface between the two 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 electrons
are in a disordered state, the kinetic energy of one
electron is large compared to the deformation energy of 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 the surface 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 helium surface excitation. Then it is possible to write the
surface 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 frequency
electron 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 spectrum is not very much perturbed by the surface coupling at high frequency the calculation shows [19] that for the transverse modes which give the main contribution, W i 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 and parallel to the surface motions can be separated.
Perpendicular motion.
-By neglecting the second
order terms the projection of the motion equation
on 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/nG
of 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 boundary
vectors the longitudinal frequency is given by :
where h is the helium height and d the distance between the helium surface and the capacitor upper
plate, and the transverse frequency is
for a triangular lattice.
This spectrum is notably modified by the presence of 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 dimple
cannot follow the deformation. If k
=0 the electron oscillates in the potential well of the static deformation with the frequency
and at k =A 0 dimple and electron oscillate in opposite phase. It is an optical mode where electron has its free mass and
At low frequency electron and dimple vibrate in phase :
where M is the effective electron mass
For intermediate frequencies additional branches lie
below each of the higher ripplons QG’ We call these
modes, « ripplonic » modes. With a magnetic field H
the low frequency modes with high effective mass
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 spectrum calculated as a function of the different parameters : wavevector, electron density and magnetic field.
Fig. 1.
-Calculated dispersion relation of longitudinal and
transverse 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 the
measurements of Grimes and Adams [9]. A one of the
measurements 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 (that
is to say a fixed frequency) and the use of a lock-in
detection. 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 experiment
showed 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 parasitic
field is sufficient to induce large signals. From figure 3
we see that, in the domain of existence of the electron
crystal that we can experimentally reach, we should
be able to excite the non displaced ripplon mode for
an rf electric field perpendicular to the surface. If the rf
field is parallel to the surface, we should be able to see
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 « optical
mode » (we name this mode « optical » by continuity although 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 the disordered state to the solid. Then the electrons are
submitted to an rf electric field and we measure the derivative of the rf absorbed power with respect to the pressing field El versus the voltage between the capa- citor plates, the temperature being fixed.
It has been found necessary to add a magnetic field H ; when H = 0, we observe a very big plasmon signal
in the liquid phase (we used this phenomenon to detect
the liquid to solid transition in our first experi-
ment [13]), but with lock-in detection this signal
saturates 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. First we 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 32 a 8 2 n 3j2 We show the respective liquid and t2 2= N 3/2 O (8 n2 n)3/2 We 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 fountain effect in the helium fill capillary (see Fig. 4).
Temperatures are measured with two Speer carbon
resistors (470 Q at 300 K) placed above and below the experimental copper cell. We calibrate these resistors between 0.5 K and 1.2 K by means of the
helium 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 0.01 K for the precision of our temperature measure-
ment. We regulate the temperature with a heating
wire stuck on the cell.
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 of helium),
5 : Exchange gas can, 6 : Superconductive coil, 7 : Hall
effect probe.
Charge creation and polarizations.
-Figure 5
shows a schematic of our experimental cell. Electrons
are 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 briefly heated, generating a glow discharge through the
helium 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 liquid
helium 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 pressure has to be weak enough to allow a ionizing discharge
to 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 tight feedthroughs, 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 the
helium 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 the
transformer 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 generator reference 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). This
coil is made of a superconducting alloy and has a
superconductive switch; it gives a magnetic field
Fig. 6.
-Sketch of the spectrometer and polarizations.
P : plate, G : grid, L : resonant circuit coil, L’ : coupling
coil.
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.
Filling of 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 increase in capacitance (or decrease in frequency) is due to the slight difference of dielectric constant between liquid
helium 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 charged
surface 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. That is what we did, in a first approach, taking for h the
value 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 electrons while we scan the plate voltage from its maximum value to zero; furthermore we control it by measuring
the 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 beginning
and 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 electron oscillations have for wavevectors the electron lattice
reciprocal vectors,
and ripplon corresponding frequencies are given by :
Hence measuring the frequencies vG
=QG/2 1t leads
to 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 directly n*, but the limit of the density of our 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 thought preferable to retain the value determined in b) by the ripplon, as it is an absolute measurement. (The para- meters a and p are known.) In determination a), we
may have an uncertainty on the measurement of V
(due to trapped charges) and of h (because of capillary
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 density
is no longer correct on the edges of the electron layer.
We have calculated n(r), the real density profile, within experimental conditions; r is the distance to the centre of the electron layer. This profile is determined by the imposed electrostatic potentials and the cell geometry.
We solved the Laplace equation AV
=0 by an itera-
tive method using a computer. We calculated the surface charge density for different charge pool radii
and the same electrostatic configurations. The surface is always supposed to be charged up to saturation, that
is to say its potential is equal to the grid voltage. A good 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 charge pool 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 mapping
method F.I.B. Williams [private com.] found an analy-
tical formula for R and n(r) as a function of y when the surface is equally distant from the electrodes. Figure 9
shows the calculated variation of the charge pool
radius 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 the
order of 300 V. It can be seen that Reff is nearly equal
to the plate radius (15 mm) and varies only slightly
with Fp when Vp « VA. This is the radius we use in the calculation of the wavevectors excited in our cell.
We suppose that the rf field Erl parallel to the surface is 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 edge
of the charge pool have a zero speed and thus the pro-
pagated phonons have a wave vector such that J 1 (kRr .ff)
=0, J1 1 is the first-order Bessel function.
A modulation voltage is applied between the capa- citor plates and the observed signal is the derivative of the rf power absorbed in the resonator with respect
to the plate voltage when VP is reduced to 0. Figure 8
shows that the area of the electron layer does not vary
very much with VP. We cannot modulate the electron
density in this way. On the other hand the scanning
time of VP (10 min) is much larger than the modula-
Fig. 8.
-a) Plot of the calculated surface charge density n(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 of
modulation, the electrons which leave the surface for decreasing VP cannot come back on it when VP
increases again, the electron density is not modulated, only is the pressing field. Thus the observed signal
is 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 the
electron density, in the presence of a magnetic field,
and for two different temperatures.
Fig. 9.
-Plot of the variation of R and Reff versus plate voltage V p for several values of the guard ring voltage VA
calculated by the conformal mapping method. R is the charge pool radius where the surface density is reduced to 0. Reff
is defined as the radius of a uniformly charged electron layer which would have the same total charge. For a typical height 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 plate voltage (or electron density), at two different temperatures and with a magnetic field of 200 gauss. A, B and C are several acoustical lines. L and S delimit the melting of the
electron crystal.
These traces show two distinct series of signals, that
we interpret as optical signals (at lower density) and
acoustical signals (at higher density), and also the
phase transition. It is worth noting that the optical
and acoustical signals are of opposite sign; as a matter
of fact, if one 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 intensity peak to peak values.) Figure 11 depicts the variation of their positions versus temperature; it also shows the 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 of the magnetic field), + : L points
of 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) only solid, 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 the
acoustical signals). Signals (a) and (b) vary slowly with temperature whereas the phase transition critical
density nc varies like T2. So when we change the tem-
perature, we move the transition density nc, and the
Fig. 12.
-Calculated plots of dPjdE 1-’ derivative of the absorbed power, versus density for T = 0.3 K, H
=200 gauss, k
=2.55 cm-1. (a) : taking the system as liquid throughout
the density scan, (b) : taking the system as solid throughout
the density scan, (c) : setting the phase transition at nc 2.07 x 108 cm-2 (which corresponds to rc
=142).
shape of the resulting signal changes considerably.
That is what appears on the experimental traces 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 signal density : indeed we only see a wing of this signal corresponding 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 defined
than 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 - 2
n 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 the
temperature, detected by measuring the longitudinal
electric susceptibility, as in our previous experi-
ment [13].
During our recordings we never observed any delay
at the melting.
Observation of S points (o beginning of melting »),
which give a mean value TS
=160, seems to indicate that the transition has a relative definite width (in the (n, T ) plane), On/n of the order of 27 %. A small part of 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 melting
transition.
-
The linewidth, of the order of 0.5 x 108 cm-1, changes only slightly with temperature or magnetic
field.
-
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 parameter W, 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 W 1 versus
F,IF; F,IF
=(n:12/Tc)/(nl/2/T), + : calculation from our measurements. of optical signals. The full line corresponds
to W = 2.5 x 104 T/nl/2 or W = 0.521 rc/r for Fr
=142.
0 : calculation from measurements of transverse optical
signals by Valdes [18].
versus re/r (which is proportional to T/nI/2), Fe being
r value at the phase transition. Far from melting, we
find W 1 = 2.5 x 104 T /nl/2 for 210 > F > 170, which is in agreement with (3), where we neglect the variation of the logarithm, and where C2 is proportional to n 1’ 2, according to (7).
In the vicinity of melting, the Debye-Waller factor
increases 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 optical signal 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 compare measured and calculated values for linewidth and intensity, we had to introduce
two 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 ripplon
relaxation time TR, we took TR as an adjustable para- meter in order to fit the calculated and experimental
values of the linewidth. Concerning the electrons, the relaxation 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 obtained from 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 experiment
is satisfactory if we take for the ripplon relaxation
time iR
=3 x 10-’ s, which corresponds to a fre-
quency linewidth 1/2 1ttR of 5 MHz. One must notice that in fact the electronic relaxation time ie contributes little to the linewidth, for it is balanced by the factor
m/M, where M > m is the effective mass of the electron in 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 the absorbed power is reached when Wc 1’-1 (cp 2013 pp)12
which corresponds to a field of 36 G, for n
=3 x 108 cm-2 (and we work generally with magnetic
fields 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 peak
width and intensity Anac and Iac with temperature, and for 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 of
these signals, the variation of their positions with temperature makes it impossible to attribute them to
the uniform vertical motion of the electrons, because
Fig. 14.
-Plot of the experimental variation of the position
of 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 (dashed line). 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 na
for high temperatures.
Fig. 15.
-Plot of the experimental variation of the peak
to peak amplitude lac of the acoustical signals with tempe-
rature compared with the calculated variation (full line) for
the 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 the
same values of parameters as figure 15.
the ripplon frequency Q’ G = " k depends on T only
p
through the capillary constant a. For 4He with less
than 20 ppm of 3He, a is independent of T for 0.2
T 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 and
we search the sequence of wavevectors solutions of
J 1 (kRcff)
=0 which give the same determination of
W 1 in function of TI.,In. Then we find kA
=8,88 CM -’,
kB
=6,78 cm-1, kc
=4,67 cm-1 (they are respectively
the 4th, 3d and 2nd zeroes of J 1) and W I = 7,5 x 104 Tl,,In for 320 > F > 180, a value three times the value we deduced from the « optical » signals. Facing
this difficulty if we suppose that the Debye-Waller
factor 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 calculation shows that the found wavevector varies along a curve.
For example kB varies from 13 cm-’ for T
=0.255 K to 20 cm-’ for T = 0.400 K along curve B. This seems
difficult to interpret even by taking into account the
variation of the charge pool radius with density as
shown in figure 9.
,Concerning the width and the intensity of the signals
the results are the same by using either one or the
other value of W1 1 associated with its corresponding
sequence 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 damping
time
gives a good agreement between experiment and cal-
culation. We supposed that tR is inversely proportional
to 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 intensity
with increasing T (Fig. 15), a calculation of Iac by supposing that electron density is modulated gives an increasing Iac with T. This is a verification that only
the 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 of 4He thus changing its capillary constant a [22]. The ripplon frequency QG changes as (X1/2. The calculation shows 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 comparison
with the zero 3 He concentration. By using Edwards
measurements of a [22] we find a good agreement between 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 density
signals are related to the acoustical branch of the
spectrum.
Table I.
-Comparison between experiment and calculation concerning the position of’the optical signal and the
B acoustical signal for two concentrations of*3He in 4He (T
=0.260 K and H
=200 G). For the acoustical signal
the calculation is done for the two possible values of ’ W 1 : (1) W 1
=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 the results deduced from the optical and acoustical parts of the spectrum, we tried to study the influence of
boundary 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 quite
different boundary conditions.
Guard ring voltage influence.
-The experiment
shows that, at a given temperature, the acoustical
position and width decrease with increasing VA while
to 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 and
the voltage around the electron layer as a function of
VA. The variation of R with VA cannot explain the
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 position
of 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 the
same as that of the grid and of the electron layer one
at saturation. With this cell it is possible to observe signals either at fixed density by scanning the frequen-
cy, or at fixed frequency by scanning the density as we
Fig. 17.
-Schematics of the second cell with the applied
dc polarizations, F : tungstene filament, G : grid, A : guard ring, P : plate.
are used to. We shall not go into all the details of this
experiment. The results are very well explained by taking for the Debye-Waller factor the value of W = 2.5 x 104 T/.,,/n for 4 x 108 n 9 x 108 el/cm2
and 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 with the Kosterslitz-Thouless theory : the value of F is of
a right order. On the other hand we observed an
intrinsic transition width. We cannot say whether it is due to a first-order transition or to two second- order transitions separated by a hexatic phase,
-
