TWO DIMENSIONAL ADSORBED CHARGES

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TWO DIMENSIONAL ADSORBED CHARGES

F. Williams

To cite this version:

F. Williams. TWO DIMENSIONAL ADSORBED CHARGES. Journal de Physique Colloques, 1980,

41 (C3), pp.C3-249-C3-262. �10.1051/jphyscol:1980338�. �jpa-00219857�

J O U R N A L DE PHYSIQUE Colloque C3, supplPmenr au n " 4, Tome 4 1, avril 1980, page C3-249

TWO DIMENSIONAL ADSORBED CHARGES

F.I.B. Williams

Conanissariat d Z'Energie Atomique, Division de l a Physique, DPh-G/PSRM, Boite PostaZe no 2 , 91190 GIF SUR YVETTE, France.

I. Introduction.- While trying not to en-

-

¹

k~ cm "thermal" wave-vector satis- comber with too much detail but not to fying

- H~

k2 = T

2m T leave out what is essential, I shall try to

sketch some of what seems to me to be L cm linear dimension of sample size

interesting in the behaviour of two dimen-

sional systems of charged particles. In the

-

⁹

-

n

(z)

^cm areal charge density = context of a colloquium on adsorbed phases,

I shall use the similarities and diffe-

2 In

^{(g) eis'~d2g}

( 2 ~ )

_.-

_,

-

rences with neutral adsorbed systems

to illustrate my propos, but with the war- _{n cm}

-

² 1

- n(r) e-ig-rd2r ning that I know even less about such phases 9 L

-

(of which the archetype for me are rare

N gases adsorbed on graphite).

The first part is devoted to setting ₉ up the problem and a framework in which to discuss it before describing some physically li realisable systems. It is only in the last

R . section that one will find a discussion of -I the experimental methods used -or proposed-

s (9) and a r6sumi of some results already ob-

tained.

-

number of particles in system

cm

-

¹ wave vector-reserved for electromagnetic field cm position vector of ith

charge

cm vector of real space latti- ce poi9t j

<nL>

1 = N -3- = ~ - 1 < l l eigsril >

n i

structure factor

It will become rapidly obvious that s 1 exponent appearing in struc- this brief review has no pretension of being ture factor as q+G

exhaustive, even as regards the bibliography, erg temperature

- - -

where I shall rely heavily on the listings erg = 4 n y / ~ ~ temperature below which X; +-

in the various review articles /17,23,24/.

T~ erg melting temperature

11. General description.- u

-i cm displacement of ith charge from its ideal lattice po- (a) Lexicon

sition lXi.gi=13i+~i.

a cm (n n) -1/2

b cm Burgers vector

-

^{U(T) cm} idem expressed as a field

-

_variable

E (r) statvolts cm-' electric field

- X I Y cm position coordinates in

D dimensionality plane

z cm position coordinate perpen- g ( R ) ^<n tr) n(r+R) > density correlation dicular to plane

<n2 (5) > function

gi

cm-' reciprocal lattice Greek symbols vector

a erg ~ r n - ~ surface tension q cm-' wave vector in reci-

procal space-reserved

r

s ec- damping rate for ripplons for charges

-

¹

k cm wave vector in reci- cm- Debye-Hiickel screening

-

procal space- (general) parameter

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980338

C3-250 ^JOURNAL DE PHYSIQUE

p gm cm-' density

gi em spatial position vector of ith charqe (3-D)

D erg chemical potential

v dyne cm-' shear modulus

O O erg zero-point enfrgy of loca- lisation =

XT

^{= Fermi}

energy for non-interacting ma 2-dim collection of fermions O D erg Debye temperature for trans-

verse sound

:c,(I) a n deviation of surface from equilibrium position z (r) =

constant Subscripts + superscripts

classical cyclotron core

charge distribution longitudinal

plasmon ripplon

electro-magnetic transverse

indicates unit vector parallel to vector so hatted

(b) Non-interacting particles at an inter- face

-

Consider two polarisable media with an interface. The polarising action of a charge on a material medium reduces its energy so that it is attracted to the more polarisable region. (The so-called "image force" from the calculational trick of using an image charge to produce a field with the correct boundary conditions). For the planar interface between two semi-in- finite media of dielectric constants € 1 and

c 2 , this polarisation potential is

if e is the charge and z the distance to the interface. In addition there is usually a short range potential representing the energy of dissolution of the particle in the medium. The combination of the two can lead to a potential well into which the charge can condense leading to adsprption on the interface.

For a host medium homogeneous in the x-y plane, the potential for a charge is a

function only of the z coordinate, which allows us to separate its z motion from its motion in the x-y plane. A single particle is described In terms of a series of quan- tum states and energies

Y (g) = @,(z) exp ik.r

PI2

2

E(n,&) = En

+

~ i ; i k

In the case of a potential well, cer- tain of the states Qn(z) of lowest En are localised in the well, representing adsorp- tion.

Figure 1 illustrates the situation for a sharp interface between two dielec- tric media.

Fig. 1 : Potential of a charge near an interface at z=0 between dielectric media of dielectric constants

€ 1 and E 2 . p 1 and p7 are the energies of dissolution

of the charqe in each medium so that

v I+

Thequantumlevels localised near the interface are also shown ("surface states").

(c) Inter-particle interaction

On writing pi = r . + zit, the Coulomb

-1

interaction between particles becomes

- -J

^{C i €,}

- - -

surface s t a t e s C -

To the ektent that

1

zi-Z,

1

^{' C} <zi> <<

1 ~ ~ - rj

n-' we can neglect the second and higher order terms and the problem becomes two-dimensional.

Pa

However the non-interacting particle description

{ ~

is no longer very useful

~ ~ l

-

when < V 2 > h

-

kL and we must look for ano-

% 2m

ther description which better reflects the correlations introduced by V,. The quantum number n retains its significance to the extent that the higher order terms in (3) do not mlx the +n (buckling of the charges) , but

k,

while remaining a good quantum number for the collection of particles, can no lon- ger be associated with a single particle.

At low densities, one expects a liquid-like isotropic correlation where the charges have reduced probability of approaching one another closer than R,,+ ~/nne'=y-'; the Debye-Hiickel screening approximation in 2- dimensions gives

On increasing the density to n > > ~ i ' ~

,

the particles avoid another more and more stron- gly so that g(R) becomes strongly peaked at R = a , the steric radius, much as in a more usual liquid with short range interactions /1,2/. On increasing even further the den- sity, there will come a point where the particles aquire an orientational correla- tion to better maximise the distance between all pairs. That is the classical solidifi- cation, but if one tries to localise the particles too much by further compression, they will pick up too much (quantum)kinetic

h2 1

energyoflocalisation, E Q a

2

~ , compared with the gain in potential energy, EP a

a ' to remain solid giving rise to the solid- liquid transition predicted by Wigner /3/

movement has no elastic resistance, only viscous damping.

In the solid phase, the description of the longitudinal modes does not change, at least for k a < < 1 on a rigid interface, but the shear deformations meet an elastic resistance. Furthermore, although its ori- gin is still the coulomb interaction, it is short ranged : shear corresponds to slipping periodic lines of charge with respect to one another, and the periodic part of the potential of such a line falls off exponen- tially with distance from the line. Such modes therefore show the usual linear dis- persion relation

where B % 1 (B = 0.372 for a triangular lattice /6/).

The solid also has "topological"

excitations constituted by "edge" disloca- tion points of energy

where

uco

is the core energy, the Burgers vector. Pairs of such dislocations of equal and opposite Burgers vector separated by s avoid the logarithmic divergence of

(4) by cancelling one another's fields at distances greater than s to give

for 3-dimensions, but the dimensionality plays

We remark that most of the dislocation ener- only a quantitative role in the argument.

gy is in shear, since the effective compres- ( c ) Collective description-excitations/4,5,6/ sional modulus is high for long distances.

A first approxima~ion is ,-hat of a char- We notice once more that the characteristic ged fluid. If the equilibrium configuration energy is e2/a since bza.

is described by a uniform charge density, All these excitations are illustrated the longitudinal hydrodynamic excitations in On figure *

infinite geometry are density waves : The charge affords the possibility of acting strongly on the system with a magne- n(r)

-

= exp i(5.1-wk t'

1

tic field. The Lorentz force on a single

This unsual uk % k1/2 relation reflects the long range nature of the Irl Coulomb repul- 4

sion. In this fluid phase, the transverse

electron confined to a plane gives rise to the cyclotron motion of period

JOURNAL DE PHYSIQUE

Fig. 2 : Vibrational excitation spectrum for the 2-dimensional Coulomb system in fluid and solid phases. The hatched x-axis represents the viscous damped transverse motion in liquid phase. h=magnetic field applied perpendicular to plane.

H . being the component of magnetic field I

perpendicular to the plane. If the electron is also subject to (electrical) longitudinal and transverse forces from collective exci- tations, these two polarisations arecoupled;

their modified dispersion relations are given /5,7/ by the roots of

where it is understood that each w=u(k), the excitations still being characterised by a wave vector as the translational invariance has not been broken. To the extent that

ut << w (k a < < ^l), to a good approximation P

This modification is illustrated by the dashed curves on figure 2.

(d) Phase diagram

Th: fact that the correlation energy, e

Ec;4r

-

,varies more slowly than the kine- tic energy of localization,

hZ 1

Qo %

m

T,adds the novel feature of quan- tum melting at T a = 0 as n is increased. This is the Wigner transition, and must be added to the purely classical melting with in- creasing temperature at low density which is characterized by the classical thermal energy T overwhelming the extra correla- tion energy. We can describe the critical densities involved by dimensionless cons- tants

(1 2 )

- e2/a (= r = a/ao) quantum T=O

'Q -

00

^{s melting}

classical T > > O melting 0 There have been several attempts /8, 9,10,2/ to calculate the values of T Q and Tc for the solid-liquid transition. The most recent results of numerical calcula- tions (Monte-Carlo) give

Because these two criteria must define a single phase transition line on an n

-

^T

plot, we see that there must be some tempe- rature T" above which solidification cannot occur. A schematic phase diagram is sket- ched on figure 3.

Fig. 3 : Phase diagram of the 2-dimensional Coulomb system. CL=classical liquid, CS=classical solid, QS=quantum solid, QL=quantum liquid, T* is maximum temperature at which system may be solidi- fied, nW is Wigner density above which the solid melts at T=O due to quantum effects. n =1013 C B I - ~

for m* = m W

el'

The value of = 137 +- 15 observed in the recent experiment of Grimes and Adams /11/ for electrons on liquid helium for 3 < n < 9x10' cm-'and 0 . 4 < T < 0 . 7 K supports this estimate rather well.

(f) Models of classical melting

<u2 >

- - -

^{j l} + - e n k ^T

In the classical limit, starting from (15)

a OD

the solid phase with no topological defects,

where kT is the wave vector of the excita- Platzmannana Fukuyama /8/ calculated the

tion such that Pok = T.

point at which the transverse sound disap-

Most experiments, however, measure the rela- pears because of the renormalization of the

tive positions of particles and a more shear modulus by thermal fluctuations,

sensible question to ask onself is themselves stronger as the shear resistance

becomes smaller. They found a first order _< (gi-gj) 2 >

transition at Tc

-

^4.5. More recently,

a KosterlitzandThouless /12/ proposed a me-

chanism of melting by dissociation of dislocation pairs. As the temperature of the solid increases, the mean separation

between thermally activated dislocationpairs The consequence of this in reciprocal increases so increasing the probability of _space is that the factor

pairs of smaller separation finding them-

selves between components of another pair S (9) = < n l > = <

I

exp ig (zi-cj) >

and screening their interaction, furthering i j

their mean separation. ~t some temperature has the form,/l3/ near a reciprocal lattice TM the pairs dissociate and render the solid - G=9-K -

plastic or liquid (zero elastic shear resis- S ( 9 ) % S ( G ) ( K L ) - ~ ( ~ ) (17) tance). They give a very simple argument for

this critical temperature based on the remark that the entropy and the energy of a free dislocation point depend on dimension size in the same logarithmic way :

i.e. the &-function behaviour of a 3dimen- sional harmonic solid is replaced by a power law divergence. For the harmonic model with transverse sound only

-

the longitudinal branch contributes little for a charged so- lid because of the w

-

^{k 1 I 2} dispersion

-

we can write

When the free energy F = U

-

TS changes v (T) being the shear modulus. To estimate sign, the appearance of free dislocations the behaviour as T ^-+ TM, we remark that becomes rapidly favorable and the solidmelts Nelson's /14/ calculations for a triangular

(l-_ = 80). lattice, based on the Kosterlitz-Thouless

L

For quantum melting, I don't think that give any such specific model has been proposed.

(g) Fluctuations and structure factor in the solid phase

As the dimensionality D is reduced,the number modes.of small wave vector becomes relatively more important

IW--

dN "(D-l)].

In 2 dimensions, for a harmonic mogel whose excitations have alinear dispersion relation, this leads to a logarithmic divergence in the mean. square displacement about an equili- brium site. Explicitly

(h) Interface roughness

When the interaction with the subs- trate is no longer uniform over the plane of confinement, we can write its potential V(x)=

I

VG exp igs.r+

-

G ^-S

-9 (20)

G are the reciprocal lattice vectors of the

-S

substrate in the plane, so that the VGs re-

c3-254 JOURNAL DE PHYSIQUE

present the periodic part of the interaction while

I v ( ~ )

1 is the aperiodic part represen- ting the rugosity. V(h) for a solid subs- trate has a static part due to imperfections and a dynamic part due to vibrations whe- reas for a liquid substrate the dynamic part is due to surface waves (capillary- gravity waves) but the static part is nor- mally zero unless some non-uniform static force acts on the surface. This aperiodic component scatters and/or traps the super- ficial charges. A free electron at suffi- ciently high temperature so as not to be trapped by a fluctuation undergoes dissipa- tive transport by quasi elastic scattering with a high frequency ( W T > ~ ) mobility

and _{9 = 2} k sin 2 0

,

k being the electron wave vector.

In the case of a liquid substrate where

V(k) - ⁵

^{e E, 5 (gi)}

+

arises from

1

thermal capillary wave fluctuations of ef- fective mass >>mel! /15,16/ one has once again quasi-elastic scattering and

a being the surface tension, TR and TE the rippion and electron temperatures. References to experimental work in this domaln may be found in the review article by Grimes /17/.

The periodic potential has little effect if its period is rapid compared to the thermal de Broglie wavelength, except in modifying the effective mass. As far as I know, no calculation have been done for dense systems where one should take into account correlation between charges.

It would be interesting toinvestigate the behaviour of the solid-liquid transition

*TO simplify the discussion, we have neglected the non-local effect of polarization of the substrate ripples/lS/ which tends+O as kT+O. In practice, the form cited dominates for n

;

^{3 x 10' cm-' for}

T 2 ^{0.5 K.}

in the presence of random potentials such that the probability of trapping at the transition temperature becones non-negli-

rjible.

The periodic part VGs is that which gives rise to registry ;

-

it is probable, however, that this part is less important for electronic systems where the basic in- tercharge distance is always considerably larger than the substrate periodicity (even in the most dense systems of MOS structures

-

^{see below}

-

there remains a factor of 20-30 in linear dimensions).

111. Some physical examples

(a) Electrons on liquid helium surface One of the simplest examples is un- doubtedly that of electrons condensed onthe liquid-vapour interface of helium /17/.The substrate is a liquid dielectric, fhere is no intrinsic substrate periodicity and the surface roughness is due to fluctuations resulting from thermal excitations of the capillary waves.In finite geometry (L > >

capillary length), the surface density is limited to n

;

3 x109 electron charges

~ m : -beyond this limit the coulomb repul- ~ sion softens the surface to the point where the surface tension and gravity can no longer stabilise it /18/. A typical experi- mental configuration for the production and confinement is illustrated on figure 4.

A liquid substrate presents the advantages of absence of intrinsic periodi- city and of low rugosity

,

but the dis- advantage of being deformable. Nonetheless, we shall see how this "disadvantage" per- mitted the first observation of the liquid

solid transition. It will be remarked that He is the only liquid dielectric which is still liquid at the temperature of surface charge solidification at the maximum charge density it can support.

(b) Ions under liquid helium surface Charges in liquid helium behave llke heavy ions with effective m a s s e s ~ l 0 0 h e l i u m masses /19/. These ions can also be confi- ned to the liquid-vapour interface from which they are repelled by means of a sui- table electric field. Apart from the mass

% lo6 met which eliminates all quantum ef- fects and reduces the frequencies of vibra- tional modes by l o 3 and cyclotron frequen- cies by 10'

,

the system is similar to that of electrons above the surface ; in parti- cular the classical melting temperature is identical, as is the limiting denslty attainable.

M O S

F i g . 4 : Two r e a l l s a b l e p h y s i c a l bldimenslonal coulomb s y s t e m s . EL HE = e l e c t r o n s on l i q u i d vapour i n t e r f a c e of helzum ; f r e p r e s e n t s a thermionic e m i t t e r e l e c t r o n s o u r c e ; V > vc > v a .

.

^The

z - d i m e n s ~ o n s a r e t p l c a l l y

3

^mm , a t t a l n a b l e den- s l t l e s n<3x109 cm-! MOS = metal o x i d e s u r f a c e i s shown ; VG > V, = Vd ; t h e i n s u l a t o r t h i c k n e s s i s

t y p i c a l l y 1500 , l a t e r a l dimensions t y p i c a l l y 1 / 2 nun ; a t t a l n a b l e densities n < 1 0 ' ~ cm-2.

(c) Ion puddles at the helium 3-4 demixion interface /20/ and electron puddles at liquid helium-vapour interface /21/

A variant of the previous system.

Under certain conditions near the surface instability, the ions concentrate in pud- dles of some N % lo6 ions and these puddles arrange themselves into a triangular lattice solid. The melting temperature of this arrangement is presumably higher by N ~ / ~ than that of a uniform distribution of sin- gle charges on the same area.

(d) Electrons on liquid helium film on solid substrate

In principle, one can condense elec- trons on any solid surface ; the difficulty

perfect not to trap them on fixed sites. Is should be possible to get around this pro- blem by interposing a film of liquid between the electrons and the solid surface. An example might be a liquid helium film some 100-500 A thick which would attenuate the effect of the solid substrate rugositywhile maintaining enhanced film surface rigidity due to the Van der Waals attraction between the liquid and the substrate. In this way one can vary the effect of the substrate rugosity by varying the film thickness, and indeed monitor the V(k) seen by the elec- trons by measuring their mobility (formula

(21)). One should note, however, that inas- much as the film thickness is small compa- red with interparticle spacing, the inter- action between particles is reduced from l/r to predominatly l/r2 (monopole-dipole).

(e) Electrons or holes at a metal-oxide- semiconductor interface

An example of a solid substrate is provided by the condensation of electrons on the semiconductor side of a metal-oxide- semiconductor sandwich commonly used in MOSFET (field effect transistor) devices.

Figure 4b schematically illustrates the situation, with approximate potentials and z-dimension energy levels for an inversion layer in n-channel MOSFET /22,23/. The sur- face densities attainable are n4,l0l3 ~ m - ~ and so attain the Wigner quantum limit for formation of a solld. The difficulty to date has been excessive random potentials, but if this problem can be solved oneshould have access to the Wigner transition at T = 0.

IV. Experiments.- We distinguish somewhat arbitrarily three categories of experiment:

thermodynamic, transport and diffraction (structure), measurements.

The omnipresent handicap in dealina with two dimensional systems is the small number of particles. This is overcome either by multiplying the surface to volume ratio with e.g. a foliated substrate, or by employing very sensitive measuring techni- ques (e.9. Auger electron, LEED for neutral absorbates) /24/.

is to find or prepare a surface sufficiently

C3-256 JOURNAL DE PHYSIQUE

A bulk liquid substrate, with its type of measurement exceedingly sensitive, single planar interface,demands sensitive

techniques ; a solid substrate, or a liquid film absorbed an a solid, can often be foliated to permit the somewhat less sensi- tive thermodynamic measurements like speci- fic heat vs temperature or mass of adsorbate vs volume vapour pressure (a measure of the chemical potential).

(a) Thermodynamic measurements

A single layer charged adsorbate has typically some l o 9 e cm-2 for a liquid helium substrate of 10 cm2 to % 1013 e cm-2 for an MOS structure of 0.1 cm2. But the charge, while limiting the density also affords an easy and sensitive way of mea- suring that density either by the field induced by the surface charge (when the surface charge field annuls the externally applied field, it is not possible to con- dense further charge) or by dischargingthe surface and measuring the total charge so extracted. Instead of measuring the vapour pressure as for neutrals, it should be possible to measure directly the ratio of adsorbed to vapourized electrons by using dlrectly the 2-dimensionality of their motion : electrons in 3-dimensions have a cyclotron frequency w,= e H whereas those confined to move on the surface have a frequency w2 = e H cos 0 (uniform mode k=O) w h e r ~ 0 is the angle between H and the nor- mal to the plane of confinement. These two cyclctron resonance lines can be made quite distinct and the ratio of their integrated intensities gives the information sought.

The sensitivity can be made sufficient to see typically %lo2 electrons. The two resonance lines have been seen by Edel'man /25/ but not exploited for their equilibrium thermodynamic potential. In the same manner as for coverage vs vapour pressure curves of e.g. Duval and Thomy/26/ for neutral adsorbates, this type of measurement could explore the phase diagram of the adsorbed electrons.

(b) Transport measurements

The Pavolvian reflex on seeing a charge is to measure its mobility ; the strong electric interaction renders this

particularly for high mobility, low mass carriers. Measuring the resonances in this electrical susceptibility x(k,w) yields

- -

very direct information on the excitations of the system.

The z-excitations, which give infor- mation on the separation of the quantum

levels in the electron substrate potential V ( z ) , show up in xZZ(k = 0,w). This has been measured for ions under the liquid helium surface /27/ (w % l o 2 MHz), elec- trons on the liquid helium surface : /28/

(w % lo5 MHz) and electrons at the MOS in- terface / 2 9 / (w 2.10' MHz)

.

ii) Logg~fudina~-mgtign

xXx(k 8,w) gives the compressional wave spectrum. It has been measured for electrons on helium to demonstrate the existence of two-dimensional plasmons. (k

= 5 cm-', w : 100 MHz) /30/ and for elec- trons in the MOS structure / 3 1 / ( k a 2x10' cm-', w = 1 0 ' ~ HZ).

In magnetlc field, this susceptibiliy shows also the cyclotron- or plasmo-cyclo- tron-modes, as demonstrated by Brown and Grimes /32/ for electrons on helium and by Allen et a1 /33/ and Abstreiter et a1 /33/

for electrons in MOS structures.

It isalso this susceptibility whose low-k behaviour signals the onset of an electron crystal on helium by virtue of the coupling with the deformable liquid surface /11,34,35/. A single electron localised in the x-y plane and held on the surface by a vertical electric field

E

indents the sur- face, but the energy gain of such a dimpled state is typically of the order of a few mK and so is not stable against thermal activation at higher temperatures. If howe- ver, N electrons in a correlated lattice structure indent the surface, it is the total trapping energy of waffling (forma- tion of dimple lattice commensurate with electron lattice) of N x a few m K which must nowbecompared with the energy of the centre of mass motion kgT as the criterion for self-trapping. This latter condition

is well satisfied for typically realisable electron crystals on helium (10' < n < 10' electrons cm-'), so one must then associate such a crystal with a commensurate waffle- like surface deformation, whereas for the liquid state, where there remalns onlyshort range scalar correlation, the therval agi- tation prevents the formation of ln-register dimples.

Figure (5.SE) schematizes in one di- mension the static lattice situation. The

uniform electrlc fleld E' acting on the electron distribution n(g) exerces a pres- sure. n = e 'E n(r)i = e E Z na ei9'f: on the surface which therefore deforms to a shape

e Ezn

z = ~ ( g ) = Z 5 eig.r where t, =

-

4 ⁹

^-

a g

g are the reciprocal lattice vectors of the structure.

Frg. 5 : Schematic illustration of behaviour o f electrons o n llquid hellurn ln llquid L phase and solid S phase. In L phase, there is no deformatlon of liquld locked In registry to electron positions;

ln the solid phase, the electrons move a s a corre- lated block, the liquid deforms under the pressing pressure and an in-reglstry w a f f l e is ~ n d u c e d (SE=

solid equilibrium conf~guratlon). SO (solid optical mode) and SA (solid acoustical mode) illustrate the relative motlons of the deformatlon and the elec- trons in the two principal vibrational modes.

The consequence on the dynamics is to replace the unperturbed modes (electronic

phonons and surface waves) by an opticaland an acoustical mode (electrons and transla- tion of surface deformation in antiphase

(fig. (5.SO) ) and in phase (fig. (5.SA)) res- pectlvely. If we represent the translation of a surface undulation 5 by 5 parallel

g 9

to q

,

we must associate with it an effec- tive mass

m

' = o/g (g ig)2

9 (24)

reflecting the necessary movement of fluid to achleve the translation. The lattice and the surface deformation are coupled by the potential of the electron riding on the un- dulatlon

V = e E'

1

^L, (ri)=

where gi is the displacement of the ith electron from its lattice point. This poten- tlal gives directly the restoring force for the k=O optical mode : for k#O we must add the electrostatic (plasma) restoring force, whence

W 2 0 (k) - a w;

+

W 2 P ( k ) -

2

where w i : (e --G' - (26)

and w 2 ( k ) is the unperturbed plasmon fre-

P

-

quency

.

The k = O acoustical mode, being a uni- form displacement of the whole system, has no restoring force unless k#O, when the com- pression of the electron lattice imposes the electrostatic (plasmon) restoring force. The result is a longitudinal acoustical mode of the same k-dependence as the unperturbed plasmon mode but with a modified effective mass :

for w < < w ( G I , the ripplon frequency at R

-

wave-vector G

.

There are other modes where w f 0 as k + 0 having ripplon-like frequencies which are also coupled to a horizontal-electron motion due to the multiplicity of surface modes coupled by harmonics in the charge on

*

Once again, for simplicity, the non-local term is figure 6. It could be remarked that the

neglected.

C3-258 JOURNAL DE PHYSIQUE

coupling is a sort of unklapp process where- by the periodic potential couples ripplon modes separated by reciprocal lattice vectors ; these modes are folded into the reduced zone scheme of the figure.

'% long modor

F i g . 6 : S c h e m a t i c c o m p a r i n g t h e longitudinal v i b r a - t i o n a l s p e c t r a i n l i q u i d a n d s o l l d p h a s e s f o r e l e c - t r o n s o n hellurn. P l i q = u n p e r t u r b e d p l a s m o n mode I n l i q u i d p h a s e (w 2. k a s k -+ 0 b e c a u s e o f f i n l t e geo- m e t r y e f f e c t s ; a p l a n a r c o n d u c t o r i s s u p p o s e d t o l i e u n d e r t h e e l e c t r o n s a s l n f i g u r e ( 4 ) w *I. k1I2 f o r k h > > 1 ) . RGn a r e t h e r i p p l o n modes a t t h e r e c i p r o - c a l l a t t i c e v e c t o r s Gn o f t h e e l e c t r o n s o l i d . T h e s e a r e c o u p l e d t o l n p l a n e m o t i o n o f t h e e l e c t r o n s b y t h e p o t e n t l a 1 o f t h e e l e c t r o n s r r d l n g on t h e w a f f l e .

These modes are excited by a longitu- dinal electric field wave (in the x-y plane) The coupling is strong because of the low effective mass

-

approximately the free electron mass for the optical mode and ty- pically some 5x free electron mass for the acoustical mode. To appreciate the sensiti- vity of such resonances, the surface conduc- tance presented on resonance

0 mho

for a free electron mass and the ripplon

-

²

limited mobility at n'L 5 x 10' cm for electrons an helium below 0.7 K.

Grimes and Adams /11/ made the first observation of the solidification of elec- trons on helium by discovering these modes excited by ameander line inserted in their parallel plate condenser where the longitu- dinal field was provided presumably by the uniform (solenoid-type) field of the zig- zag, if not by the fringing fields. The k- vectors of the resonances corresponded to resonances in the pill-box witha fundamental k = ² cm-I and the resonances seen were the principal acoustical mode and some ripplon like modes ; they did not report seeing the

optical resonance.

(c) Transverse motion

Transverse modes should make themsel- ves felt in

x

(kG,w) and as such have not

YY

yet been observed. Once again, the low k spectrum is perturbed for charges on a de- formable liquid substrate. Similarly to the longitudinal modes, the optical and acousti- cal transverse modes as k-0 should be given by

Experimentally it is difficult to create an electric field transverse to the propagation vector for w/k < < c , particular- ly in the region where w (k) > w

t - ' L 0 -

It might be possible to use an electro- mechanical wave as in a piezo-electric. It happens that the phase velocity of a trans- verse wave in an electron solid at n = l o 7

~ m is vt - ~ 2 4 . 3 ~ 1 0 ~ cn set-' and varies like nl/' : this is close to the sound ve- locities in solids. In a sufficiently low symmetry piezo-electric it is possible to excite a sound wave with transverse electric field using an electromagnetic structure : if the electron solid and piezo-electric wave velocities can be tunned to one ano- ther, energy is fed from one to the other and can be detected by an additional atte- nuation of the piezo-electric wave. For n

;

l o 7 cm-2 using a Rayleigh wave on LiNbO, an additional attenuation of %10 db cm-l should be induced by coupling to the transverse sound of an electron solid at 100 MHz.

An experimentally more convenient way of detecting transverse sound, however, is to introduce it into xXx(k x , w ) and to de- tect this with a purely electro magnetic structure operating in the principal mode to produce a longitudinal electric field.

This is a much simpler proposition, with the convenience of being able to fix

k

^geo-

metrically, independently of w (typical structures are ladder lines, meander lines or interdigital lines).

The Lorentz force in a normal magnetic field mixes the transverse and longitudinal modes. This can easily be seen by observing that a longitudinal electric field

E~ = ex exp i I q x

-

^wtl

imposes a longitudinal velocity on the electrons

The resulting Lorentz force

excites the transverse mode with a coupling constant

of the direct coupling that would exist if E had been transverse ; the frequency of the

-

mode is only shifted by a factor For 5x10' electrons cm-2 coupled

wide X = 1 0 0 ~ structure with similardistance to the ground plane, the additional attenua-

wc 1

tion for

-

_U ^% is a % 3 db cm-'

.

P

This method of detectingthetransverse mode is applicable in principle to both the electrons on helium and the inversion layer MOS systems. In the latter case, it becomes an extension of the plasmon experiment of Allen et a1./31/, but where either a sample with longer relaxation times must be found or a shorter wavelength grating must be deposited if one to meet the wtr > 1 condi- tion for a clear resonance.

One can extract the shear modulus from this experiment (v = for w > > wlocal)I.

v"n

so affording a quantitative test as to the validity for this system of the dislocation pair model of melting such predicts a relationship between v(TM) and TM and the behaviour of v (T) as T + T M (equation (19) ),

in much the same way as pS (T) for the superfluid film experiments of Bishop and R ~ P P Y / ~ ~ / .

(d) Diffraction measurements

It is reassuring, even instructive, to see the structure of a phase. The charged

systems we have cited have interparticle distances ranging from lo-' cm for the pud-

0

dle crystal through 10' A for the mono-

0

charge particles on liquid helium to l o 2 A

for carriers at MOS interfaces.

Light of appropriate wavelength seems the simplest probe. For the puddle crystal, the interaction of light with the dimped surface in a geometrical optics regime suf- fices to image it clearly. The monocharge system on helium poses greater difficulty.

The seemingly obvious technique of Bragg scattering suffers from lack of intensity ;

the Thomson cross-section is too small

( a cm2) and the surface is not suf-

ficiently waffled at these wavelengths

( 0

;

cm2 electron). For ions in the

liquid, the bubbles or snowballs are not big enough (a % cm2 Ion)

.

Neutron scattering cross-sections are of the same order.

Electrons would seem more promising with their large Rutherford cross-section at low energies, but these same low energies may be difficult to handle.

Another approach is to use substrate or interface excitations as "projectiles".

An example is ripplon lattice scattering for the electrons on -or ions under- the liquid helium/38/. Since the charges couple the electro magnetic field to the ripplons, one has a double handle on the system. The surface excitation -charge system- electric field interaction is, once again,

On averaging over the ground z-state, writing the surface deformation z(f,t) = z0

+

⁵ (l,t) and the x-y plane density opera- tor ? ^{6 (r-ri (t)} ) = n ( I , t)

,

we can rewite

1

(33) V = Vo + dV where

having taken time and spatial Fourier com-

C3-260 ^{JOURNAL DE} PHYSIQUE

ponents. As usual, there are non-zero con- tribution to the averaged interaction when

If the electron structure is static,

wed=

0 for reciprocal vector g and the electric field is produced -or detected- by uniform parallel capacitor plates q=O at frequency w e-m- - wO, surface deformations of wave vector

k=-CJ

at frequency o R- - -wO, in- teract with Eo via the density modulation n This is of course just a formal way of

9 '

representing the motion of corks bobbing on water : when the corks are spatially com- mensurate with the water waves, they all bob in phase.

The eigenmodes of surface waves have frequencies

for k 2 >:

a

If then, one produces eigenwaves at frequency wR, one could detect vertical motion of the spatial q=O average of the

charge distribution proportional to its Fourier components n However in the case

s=lr'

of a lattice structure where n is strongly g

peaked around CJ = 5 , this technTque would require collinearity as well as equal ma- gnitudes of

s

^and &. If instead one excites the ripplons via the lattice using a uniform electric field at frequency wo, one excites the surface at wave vectors CJ and frequency wO, proportionally to n There is resonant interaction when the eigenfrequency wR(k=lgl) 9'

-

= wO. This avoids problems of orientation, but sacrifices information. The power so fed into the ripplons -and which can be detected as a power loss from the electro-magnetic system- is

/ ^s

^{(ik cos (}

⁺

^{yk sin ( ) (d (37)}

where k o is defined by wo = wR(k0). We have lost angular information on the structure factor. It is evident that this method also works for a liquid, and as the peaks in the liquid structure factor occur at the same g as the peaks in the angular average of the triangular lattice structure factor, the difference between liquid and solid will show up in the width and intensity rather than the positions of the peaks in w. Des- pite the large effective mass (mass of a cube of helium of lattice cell dimension per electron), such an experiment is feasible*

V. Summary.- Charges condensed on a surface show several particularities in comparison with neutral adsorbed systems : the long- range Coulomb interaction limits the densi- ties attainable, modifies considerably the compressional excitations and renders interactions via the electromagnetic field very strong ; in the case of electrons the low mass introduces the zero-temperature quantum melting phenomenon of Wigner above a density of the order of 1 0 ' ~ cm-'.

Although the experimental knowledge of these systems is so far less rich, perhaps, than that for neutrals, a certain number of experiments have already been done such as the observation of individual excitations perpendicular to interfaces and of those along the interface in a magnetic field, of collective longitudinal excitation and of classical solidification. Some of the expe- riments to be done are : to observe trans- verse excitations, to measure structure factor, to measure thermodynamic properties

(cAr vapour pressure). An extension of these studies to controllably non-homogeneous substrates would seem to be another realm of interest.

*

For capacitor plates of 10 cm2 separated by 3 mm forming the capacitance of a resonant circuit of Q=Qc and containing n=109 electrons cm-2 one can theoretically detect the power absorbed from a 1 b W input using a room temperature detector of band width 10 Hz with a signal to noise of approximately Qs-RQc/30 where Qs-R is the relative width of the integrand for P(w) in ( 3 7 )

.

Acknowledgements.- It is a pleasure to /18/ Gor'kov, L.P. and Chernikova, D.M., acknowledge interesting conversation with Pis'ma Zh. Eksp. Teor. F i 2 . G (1973)

119 , JETP Lett. 18 (1973) 68. _- many of my colleagues at Saclay -in parti-

Chernikova, D.M., Zh. Eksp. Teor.Fiz.

cular Jacqueline Poitrenaud, G6rard Deville, - 68 (1975) 249, Sov.Phys. JETP

41

⁽¹⁹⁷⁵⁾

Francois Gallet. Didier Martv and Michel 121.

Devoret. I have also enjoyed and benefited Gor'kov, L.P. and Chernikova, D.M., Dokl, Akad. Nauk SSR

228

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