Energy relaxation between an electron crystal and liquid helium

Physics LettersA 159 (1991) 277—278 North-Holland

PHYSICS LETTERS A

Energy relaxation between an electron crystal and liquid helium Yu.P. Monarkha Physical-Technical Institute of Low Temperatures, 47 Lenin Avenue, 310164 Kharkov, USSR

V.B. Shikin Institute of Solid State Physics, 142432 Chernogolovka, MoscowDistrict, USSR

D.C. G~attliand F.I.B. Williams Service de Physique du So/ide ci de Resonance Magnetique, Orme de Merisiers, 91191 Gif-sur- Yvette Cedex, France Received 15 July 1991; accepted for publication 6 August 1991 Communicated by V.M. Agranovich

A study is presented of the energy relaxation in a two-dimensional electron crystal on the free surface of liquid helium. The relaxation rate is determined in terms of the dynamic form-factor of surface electrons which can be written in the framework of the self-consistent theory of coupled phonon—ripplon modes.

The information about the energy loss rate which determines the establishment of the temperature equilibrium between an electron crystal and a liquid substrate is one of interesting results of the measurements of the electron capacitance of the Wigner crystal [1]. The aim of the present note is to calculate this rate and to compare the theory with the experimental data [1]. Since the Wigner crystal formed by surface electrons can exist only at sufficiently low temperatures, T
total momentum s= I q+ q’ <
[3,41. The role of the correlation effects in the processes of energy relaxation of surface electrons has been studied in ref. [41. For this, the energy relaxation rate also for the Wigner crystal. In this case neglecting multi-phonon processes, we have for a(s, w), Ns2F~ 4 m X

~ w~([nP.kTC+

llô(w—wP,k)

—

Elsevier Science Publishers B.V.

+

~

ôk_~,

5(W+WP,k)

~ ö~±~,g). (2)

5+flP,k(T~)

Here ~J~p,kis the spectrum of the longitudinal (p=Q) and transverse (p=t) phonons of the 2D Wigner crystal, g are the vectors of the reciprocal lattice, s= )g+g’ I 277

Volume 159. number 4,5

PHYSICS LETTERS A

4 October 1991

2(u2>),

F~(Te)=exp(—~s

(3) (it2> is the mean square displacement of the dcctrons. According to ref. [5]

-

-~

x[(u~>+u~ln(T/hw~/B)1,

B=

1’

(01 =Wq.

~

=

l’,~ at q=g

1

(tm(oT

.

(4)

Here u~= (2~,n,i’~/T)— and (us> is the mean

square amplitude of the zero point oscillations, c,

~

theistransverse sound velocity, g1 = gI, t>, is from (1), a the surface tension coefficient. Now using the general expression for (F> from ref. [41 in terms of a(s, w), the explicit expression for ~from (2)—(4) and the smallness of the parameter (i~— T) / 7’<< I which makes it possible to ne-

~‘

/1,)

Fig. I Temperature dependence of energy 2. loss(I) ratethe fortheoretithe 2D cal curve for L~=0.97eV and f(0) 56.6 cm~2: (2) the same Wigner crystal, ;(T) for n~=I.02x 108 cm curve for 10=1 eV and((0) 58.8 cm~1/2 The crosses correspond to experimental data [1].

also shown. It is easily seen that the given theoretical

glect the contribution of longitudinal phonons, one

description of the energy relaxation in the 2D elec-

can find the following expression for (F>.

tron crystal at the liquid helium surface gives not only a qualitatively correct temperature dependence which is in a good fit with the experimentally obtained de-

(Er> =NvJT, TJ(T~.— T),

pendence i~(T)but gives also numerical values for the relaxation time without any fitting parameters. which are very close to the measured values. The latter is an additional argument in favour of the very simple self-consistent description of coupled phonon— ripplon modes [51 which we applied here in order to determine w~and (ui>. Summing up we would like to underline again that

3~4’27

~

i’~~•T~

~ g2F~(TJ . 0

i-f

1~2//i

0 =X~ l~f~ (0) , = Bw~exp( — ~

,

(2in J~~) (ui>), =

where J( is the barrier height for the electrons at the liquid helium surface, [(0) is the wave function at this boundary. In the range of applicability of formulae (5)with in the of g we may restrict ourselves to terms Ig~sum =g~. In order to obtain a correct temperature dependence for t~ it is necessary to take into account the softening of the transverse phonon modes, according to ref. [61. c~(i>)=c~(0)(l_30.8 C,2(Thfl)l/2)

(6)

Taking into account all this the graph ç ( T) determined by (5) when T~is close to T is given in fig. 1 for two models of the potential energy of the surface electrons above the flat helium surface: (1) V 0~0.97eV,f~(0)~56.6 cm /2 and (2) V0~1 eV, .f~(0) 58.8 cm — /2 The experimental data [11 are 278

the results obtained here, especially expression (5) for r 0, are only valid if the condition w~>>wholds and so it ceases to be2)valid forthe veryratio low electron since w~/w~densities (n1 < I cm— and attains large values (of the order of 10) at

o~

2.

n 1=r4X l0~cm

References [1] D.C. Glaitli, E.Y. Andrei and FIB. Williams. Phys. Rev. leti. 43 (1988)420; Surf. Sci. 196 (1988)17. [2]V.ShikinandYu.Monarkha,J.LowTemp.Phys. 16(1974) 193. [3] Yu. Monarkha. Soy. J. Low Temp. Phys. 4 (1978) 515

[4] Yu. Vil’k and Yu. Monarkha. Soy. J. Low Temp. Phys. IS (1989). [5]Yu. Monarkha and V. Shikin. Soy. J. Low Temp. Phys. 9 (1983) 471. [6] R. Morf. Phys. Rev. Leti. 43 (1979) 931