Revista Brasileira de Física, Vol. 16, n? 2, 1986
Surface Electrons on Helium Films*
NELSON STUDART'
Departamento de Fisica, Universidade Federal de São Carlos, Caixa Postal 676, São Carlos, 13560,
SP, Brasil
and
Departamento de Fisica e Ciências dos Materiais, Instituto de Fkica e Química de São Carlos, USP,
Caixa Postal 369, São Carlos 13560, SP, Bmsil
Recebido em 29 de julho de 1985; 22 versão em 22 de novembro de 1985
Abstract
T h e o r e t i c a l c a l c u l a t i o n s o f some p r o p e r t i e s o f two-dimensional
e l e c t r o n s on a l i q u i d helium f i l r n adsorbed on a s o l i d s u b s t r a t e a r e r e viewed here. We describe the spectrum o f e l e c t r o n bound s t a t e s on b u l k
helium as w e l l on h e l i u m f i l r n s . The c o r r e l a t i o n a l p r o p e r t i e s ,
s u c h as
the s t r u c t u r e f a c t o r and c o r r e l a t i o n energy, a r e determined a s f u n c t i o n s
o f the f i l m thickness f o r d i f f e r e n t types o f substrates i n t h e f r a m e work o f a General i z e d Random-Phase Approximation. The c o l l e c t i v e e x c i t a t i o n s o f t h i s system a r e a l s o described. The r e s u l t s f o r e l e c t r o n s on
the s u r f a c e o f t h i n f i lms and b u l k h e l ;um a r e e a s i l y obtained. We exami n e the e l e c t r o n i n t e r a c t i o n w i t h the e x c i t a t i o n s o f the l i q u i d helium
s u r f a c e r e s u l t i n g i n a new p o l a r o n s t a t e , which was observed v e r y r e c e n t l y . The ground s t a t e energy and the e f f e c t i v e mass o f t h i s polaron
a r e determined by u s i n g the p a t h - i n t e g r a l formal ism and uni t a r y - t r a n s f o r m a t i o n method. Recent s p e c u l a t i o n s about the phase diagram o f elect r o n s on the helium f i l m a r e a l ç o discussed.
1. INTRODUCTION
The e x t e n s i v e study o f the p r o p e r t i e s o f the system formed by
e l e c t r o n s on t h e s u r f a c e o f l i q u i d h e l i u m began i n the e a r l y
w i t h the t h e o r e t i c a l p r e d i c t i o n made by Cole and cohen1,
and
seventies
indepen-
d e n t l y by s h i k i n 2 , about t h e p o s s i b i l i t y o f e l e c t r o n s being trapped on
a d i e l e t r i c s u b s t r a t e where they could be conf ined by a p o t e n t i a l well i n t h e
z - d i r e c t i o n and c o u l d move more o r l e s s f r e e l y i n the
x-y plane o f the
surface. A f t e r t h i s p r e d i c t i o n , a s e r i e s o f experiments were performed
t o i n v e s t i g a t e t h i s system u n t i l Grimes and 6rown3 c o n c l u s i v e l y
served t h e e x i s t e n c e o f these e l e c t r o n i c s u r f a c e s t a t e s i n a
ob-
beautiful
spectroscopic experiment, This experiment c o n s i s t s i n the m e a s u r e m e n t
of the microwave a b s o r p t i o n as a f u n c t i o n of an a 1 t e r n a t i n g e l e c t r i c
*This is an invited review article
' C N P ~Research Fellow
Revista Brasileira de Fisica, Vol. 16, n? 2, 1986
f i e l d a p p l i e d i n t h e d i r e c t i o n perpendicular t o the l a y e r o f
on t h e s u r f a c e o f l i q u i d helium. They measured t h e
electrons
s p l i t t i n g s due t o
t r a n s i t i o n s between d i f f e r e n t e l e c t r o n i c s t a t e s i n the
well
potential
by observing the wave a b s o r p t i o n as the s p l i t t i n g s were tuned i n t o resonance w i t h t h e frequency o f the i n c i d e n t r a d i a t i o n . The m o t i v a t i o n f o r
s t u d y i n g t h i s system f r o m a fundamental p o i n t o f view i s t h a t an immense
v a r i e t y o f w e l l - d e f i n e d and c l e a n experiments can be done, which are i m p o r t a n t f o r t h e o r e t i c a l progress i n areas such as p h a s e - t r a n s i t i o n s
two dimensions, many-body theory, the p o l a r o n s t a t e , t r a n s p o r t
ena, the Wigner c r y s t a l l i z a t i o n , e t c . Furthermore,
in
phenom-
r e a l i z a b l e techniques
o f measuring t h e m o b i l i t y o f s u r f a c e e l e c t r o n s were employed f o r studyi n g the thermal s u r f a c e e x c i t a t i o n s o f quantum l i q u i d s
and
s o l i d s . We
would l i k e t o emphasize the pedagogical appeal o f t h i s e l e c t r o n i c s y s tem. For example, the image- potential- induced s u r f a c e s t a t e s
are
ob-
t a i n e d f rom a very s imple one-dimens i o n a l ~ c h r g d i n ~ eequation.
r
Moreover,
a l a r g e amount o f t h e o r e t i c a l and experimental work could be done u s i n g
techniques s u c e s s f u l l y a p p l i e d t o o t h e r systems and t h e r e a r e e x c e l l e n t
reviews devoted t o t h i s f i e l d
4-9
,
I n t h i s paper, we would l i k e t o surnmarize some r e c e n t t h e o r e t i c a l work, emphasizing t h e case o f e l e c t r o n s on the s u r f a c e oF l i q u i d
h e l ium adsorved on a s u b s t r a t e .
Theoutlineof
I n s e c t i o r i 2,
t h i s paper i s a s f o l l o w s .
b r i e f l y discuss the general p r o p e r t i e s o f s u r f a c e
h e l ium, such as the spectrum o f bound s t a t e s
,
electrons
evaluated
we
on
bulk
througli
model
The d i e l e c t r i c
c a l c u l a t i o n s and compared t o the experimental r e s u l t s .
f o r m u l a t i o n o f t h i s two-dimensional ( 2 ~ many)
electron system
i s des-
c r i b e d and we emphasize the breakdown o f t h e Random-Phase Approximation
(RPA) by t r e a t i n g the c o r r e l a t i o n s o f the 2D c l a s s i c a l plasma.
Hn
Sec-
t i o n 3, we analyse t h e new experimental s i t u a t i o n wi t h the e l e c t r o n s on
the s u r f a c e o f a l i q u i d helium f i l m w e t t i n g a s o l i d s u b s t r a t e ,
sent the change i n the spectrum o f t h e e l e c t r o n i c s t a t e s due
We preto
and s u b s t r a t e , By v a r y i n g the f i l m thickness and t a k i n g d i f f e r e n t
f i 1m
sub-
s t r a t e s one m o d i f i e s s i g n i f i c a n t l y the n a t u r e o f the
electron interac-
t i o n , from a s t r i c t l y d i p o l a r p o t e n t i a l t o t h e usual
Coulomb
t i o n f o r e l e c t r o n s c o n f i n e d i n a plane. I n s e c t i o n 4 , we
interac-
analyse
in
Revista Brasileira de Ffsica, Vol. 16, n? 2, 1986
d e t a i l the i n f luence o f the f i l m thickness and severa1 k i nds o f
sub-
s t r a t e s on the many-body p r o p e r t i e s o f the system, by u s i n g a k i n d o f
General i zed RPA,
the se1 f -cons i s t e n t f i e ?d m e t h ~ d ' ~'de
, o b t a i n the c o r -
r e l a t i o n a l p r o p e r t i e s f o r e l e c t r o n s on the s u r f a c e o:
b u l k helium.
thin
f i lms
I n s e c t i o n 5, we i n v e s t i g a t e the s e l f - l o c a l i z a t i o n
ãnd
o f an
e l e c t r o n i n a p o l a r o n i c s t a t e by c o n s i d e r i n g i t s i n t e r a c t i o n w i t h a
cloud o f r i p p l o n s , the quantum e x c i t a t i o n s o f the helium s u r f a c e .
Our
c a l c u l a t i o n s a r e c a r r i e d o ~ i tn the framework o f Feynman's path i n t e g r a l
formalism"
Low and
the
as w e l l as i n the m o d i f i e d v a r i a t i o n a l scheme o f
P i nes theory
'" '
trans i t ion
, We discuss the i n t r i g u i n g
Lee,
in
which the e l e c t r o n i c s t a t e transforms from a n e a r l y f r e e t o a l o c a l i z e d
s t a t e , w i t h a dramatic increase o f the e f f e c t i v e mass.
This
polaron
t r a n s i t i o n has been observed q u i t e r e c e n t l y i n the e l e c t r o n system l e v i tatir;g above a helium f i l m l ' .
And, f i n a l l y , we
d e s c r i b e soine r e c e n t
speculations about the shape o f the phase diagram f o r
l i u m f i l m s , wi:h
e l e c t r o n s on he-
the p o s s i b i l i t y o f c r e a t i n g a new 20 f l u i d w i t k an i n -
t e r e s t i n g quantum Wigner t r a n s i t i o n n o t y e t w e l l s t u d i e d .
2. SOME PROPERTIES OF THE SURFACE ELECTRONS
The e l e c t r o n i c s u r f a c e s t a t e s a r e induced by
p o t e n t i a l between t h e e l e c t r o n and the helium s u r f a c e ,
from two d i f f e r e n t c o n t r i b u t i o n s : ( i ) a long-range
the
interaction
which a r ises
interaction
coming
from the p o l a r i z a t i o n o f the l i q u i d s u r f a c e and g i v e n by an a t t r a c t i v e
image p o t e n t i a l and,
( i i ) a p o t e n t i a l b a r r i e r due t o P a u l i ' s
exclusion
principie which does n o t a l l o w an e x t r a e l e c t r o n i n the helium atom.The
model p o t e n t i a l can be c o n s t r u c t e d as
where t h e d i e l e c t r i c occupies the h a l f - s p a c e z < 0 . For helium,
E
we have
= 1 . 0 5 7 and V . = 1 eV. I f one assumes t h a t the b a r r i e r c a n b e approxi-
mately taken as i n f i n i t e and the p e r p e n d i c u l a r and para1 l e l motions can
Revista Brasileira de Ffsica, Vol. 16, no 2, 1986
be separa ted, the ~ c h r g dnger
i
equa t i o n
where
I s solved e x a c t l y w i t h the boundary c o n d i t i o n s O n ( 0 ) = Qn(m) = O ,
s o l u t i o n s o f the unidimens i o n a l ( z - d i r e c t i o n )
The
~ c h r g d i n ~ e equation
r
are
g i v e n by
O (2) =
n
where R
nvo
Z
R
(2)
n,o
(2.3)
( z ) a r e t h e wavefunct ions o f the r a d i a l equa t i o n o f the hydro-
gen atom w i t h zero angular momentum. The energy spectrum
is
then
the
wel 1-known s p e c t r i m o f the hydrogen atom,
For the lowest subband, one has
where a o is t h e e f f e c t i v e Bohr r a d i u s and
= 7x10-'
eV f o r
elestr~ns
on helium.
Spectroscopic measurements by Grimes
e t ~ 2 . 'showed
~
t r a n s i t i o n s between the ground s t a t e and the f i r s t e x c i t e d
about
5%
l a r g e r than those p r e d i c t e d by the hydrogenic model
eq. ( 2 . 4 1 , as s h w n i n the t a b l e below
Transi t i o n
11-3
2
Theory
Experiment
119.7 GHz
125.9 2 0.2 GHz
141.8 GHz
148.6
0 . 3 GHz
that
the
states
are
given
by
Revista Brasileira de Física,Vol. 16,n'?
2, 1986
This small discrepancy is attributed to interface effects such
as the f ini teness of the potential barrier and the true behavior of the
polarization interaction at small distances. In order to take into account these effects severa1 model calculations have been performed to
fit the experimental results, In particular, HipÕlito et a2.16 have obtai ned an exact sol ut ion, in terms of conf luent hypergeometric functions,
for the following model potential
where B is an adjustable parameter. This model potential corresponds to
treating the potential outside the surface as an image potential with
the origin at a small distance B inside the dielectric. In fact, B is
interpreted as being the posi tion of the center of mass of the induced
charge, That is the effective position of the interface.
Thus, this
phenomenological potential e1 iminates the non-physical d i vergence of
the classical image potential at the real interface. When the eigenvalue equation is solved as a function of B for the levels n = 1,2 and
3 and the energy separations between these levels are set equal to the
takes the value 1.01 8.
experimental values, the parameter
In the experimental situation there is an additional potential
coming from an external electric field F pressinç the electronsagainst
the surface and distorting the fosm of the potential to
Since there is no exact analytical solution for this problem,
we have performed numerical calculations of the bound-state energies
for the levels 1, 2 and 3, in the presence of the applied electricfield
and with the value of B previously calculated. In fJg. 1, we show the
trans i tions , from the ground-s tate to exci ted levels as observed in the
experiments of Grimes et aZ. 1 5 . The spectrum looks I ike the Lyman series
of the hydrogen atom. We plot in fig. 2 the transition frequencies as
a function of the external electric field applied between
two planar
Revista Brasileira de Física, Vol. 16, n? 2, 1986
f = 220.0 G H Z
*
n
POTENTIAL DIFFERENCE ACROSS CELL IVOLTS)
-
Fig.1
Experimental r e c o r d i n g o f mm-wave
absorption d e r i v a t i v e
v s the p o t e n t i a l
d i f f e r e n c e across t h e e x p e r i m e n t a l c e l l
taken a t a frequency o f 220 GHz and temp e r a t u r e o f 1.2 K. The l i n e a r S t a r k e f f e c t i s employed t o tune t h e s p l i t t i n g s
between bound e l e c t r o n i c s u r f a c e s t a t e s
on l i q u i d h e l i u m t o resonance w i t h t h e
a p p l i e d r a d i a t i o n . The 1 -t 2, 1 -t 3 ,
t r a n s i t i o n s a r e analogous t o the Lyrnan
a, 8 ,
transitions o f t h e hydrogen
atom ( A f t e r C.C. Grimes et aZ, ref.15).
....
...
e l e c t r o d e s one l o c a t e d above t h e surface and another i n s i d e
the l i q u i d ,
The energy s e p a r a t i o m between the lowest subband and the lowest few exc i t e d subbands are cornpared w i t h those obtained n u m e r i c a l l y . As onesees,
the experimental r e s u l t s a r e w e l l f i t t e d by the calcu.lation.
The rnost s t r i k i n g f e a t u r e o f t h i s systern i s
a c c e s s i b l e range o f electron densities n
2D Fermi energy i s g i v e n by
-
10'
-
Lhe
lo9
experimental
crn-*.
As
the
Revista Brasileira de Física, Vol. 16, no 2, 1986
120~-
50
I
I
100
150
ELECTRIC FIELD (VOLTS!CM )
-
Fig.2
Frequencies f o r the t r a n s i t i o n s 1 -t 2 and 1 + 3
as a f u n c t i o n o f externa1 f i e l d . The crosses a r e
the
experimental r e s u l t s o f Grimes e t d.15 whi l e the dashed
curves a r e t h e i r model c a l c u l a t i o n . The s o l i d curvesare
the r e s u l t s o f H i p ó l i t o e t a1.l6 based on the e x a c t l y
s o l u b l e model p o t e n t i a l g i v e n by eq. ( 2 . 6 ) .
F << kgT
then E
f o r temperatures above a few m i l l i k e l v i n s .
Indeed,
the
e l e c t r o n system behaves I i k e a non-degenerate plasma described by a two
-dimensional
Rolczmann d i s t r i b u t i o n as a l i m i t o f the Fermi d i s t r i b u t i o n
a t experimental d e n s i t i e s and temperatures.
be c h a r a c t e r i z e d by a plasma parameter
r
I n t h i s way, the system can
d e f i n e d as t h e
ratio
between
t h e averages o f p o t e n t i a l and k i n e t i c energies,
(T
i n energy u n i t s )
.
For smal 1 values o f
,'i
the Coulomb i n t e r a c t i o n i s
less important and we have a d i l u t e system a t h i g h temperature.
termediate d e n s i t i e s ,
1 <
A t in-
< 100, the system becomes h i g h l y correlated,
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
o r l i q u i d - l i k e . A t h i g h d e n s i t i e s and low temperature, a
s i t -i o n t o an ordered s t a t e , the c l a s s i c a l Wigner c r y s t a l
phase- t r a n -
,
I n f a c t , one o f the most important development i n
t o occur.
i s expected
thiij
has been the o b s e r v a t i o n o f the c r y s t a l i z a t i o n o f e l e c t r o n s
and ~ d a m s ' ~a t
rm=
field
by Grimes
137. This r e s u l t i s c o n s i s t e n t w i t h c a l c u l a t i o n s
simu~
c a r r i e d o u t us i ng molecular dynamics18 and Monte ~ a r l o ' computer
l a t i o n s , We s h a l l come back t o t h i s p o i n t i n s e c t i o n 6 i n the c o n t e x t o f
the h e l i u m f i l m s .
The s i m p l e s t approximation t o t r e a t a 2D c l a s s i c a l plasma w i t h
p a r t i c l e s i n t e r a c t i n g v i a a p a i r p o t e n t i a l +(r) = e * 2 / ~ , i s the Random
.
-Phase Approximation (RPA) The e f f e c t s o f the h e l ium s u b s t r a t e a r e i n cluded i n t o the renormal i z e d e l e c t r o n charge e* = (2/1+€)'12
sense,
e , I n some
t h i s approximat i o n i s e q u i v a l e n t t o the ~ebye-HÜckel method
I"
i t i s known t o be v a l i d i n the low d e n s i t y regime.
f o r m u l a t i o n o f the many-body problem,
the
dielectric
the d i e l e c t r i c f u n c t i o n
+
r e l a t e d t o t h e dens i ty-dens i t y response f u n c t i o n x(q,w)
and
.+
E(,~,W)
is
by
where $ ( q ) i s t h e F o u r i e i transform of the bare p o t e n t i a l .
The ~ J i e l e c -
t r i c response f u n c t i o n i n RPA i s u s u a l l y c a l c u l a t e d on the b a s i s o f Ohe
Vlasov e q u a t i o n and w i l l be d e r i v e d i n a more general c o n t e x t i n section
4. I n RPA, t h e d i e l e c t r i c f u n c t i o n t u r n s o u t t o be
where
x,
($,w)
i s the densi ty- densi t y response f u n c t i o n o f t h e i tleal gas,
g i v e n by
x,
where z = (w/q)(m/~)1'2.
w r i t t e n as
+
(q,d =
-
(~/T)w(z)
Here, ~ ( z i) s t h e plasma d i s p e r s i o n
(2.12)
furiction
Revista Brasileira de Fisica, Vol. 16, n? 2, 1986
The s t a t i c s t r u c t u r e f a c t o r
~ ( q )i s
related t o
the
imaginary
p a r t o f t h e d e n s i t y - d e n s i t y response f u n c t i o n o f the system through the
well-known f l u c t u a t i o n - d i s s i p a t i o n theorem
I n the c l a s s i c a l l i m i t hw << T, t h i s f u n c t i o n can be w r i t t e n
by
means
o f the Kramers-Kronig r e l a t i o n as
where
kD =
2 ? r n e 2 / ~i s the Debye wavevector.
,
The p a i r c o r r e l a t i o n f u n c t i o n g(r) wh i c h represents the propa b i l i t y o f f i n d i n g one p a r t i c l e a t a d i s t a n c e r from another
one,
is
obtained f rom the inverse F o u r i e r t r a n s f o r m as
I n RPA, g ( r ) i s g i v e n by20
where
H, (x) and
Y O(x) a r e r e s p e c t i v e l y the zero- order Struve and Bessel
f u n c t i o n s o f second k i n d . So, the p a i r c o r r e l a t i o n f u n c t i o n diverges as
- e 2 / ~ near
r
t h e o r i g i n , c o n t r a r y t o the exponential 3D behavior.
This
divergence i s s t r o n g l y manifested i n the c o r r e l a t i o n energy Ec g i v e n by
wfiich diverges l o g a r i t m i c a l l y . T h i s i s a m a n i f e s t a t i o n o f
o f t h e RPA i n d e s c r i b i n g the f e a t u r e s o f the 2D e l e c t r o n
the
failure
plasma
even
Revista Brasileira d e Ffsica, Vol. 16. n?
f o r small values o f
r.
2. 1986
So, we cannot use t h i s approximation i n t h e s h o r t
-range domain r 5 e 2 / ~ ,where the e l e c t r o n s a r e s t r o n g l y
correlated,
suggesting the importance o f the short- range c o r r e l a t i o n s i n a 2D
sys-
tem. Furtherrnore, t h e . p a i r c o r r e l a t i o n f u n c t i o n which must be p o s i t i v e
def i n i t e f o r a11 separations takes on negative values i n t h i s r e g i o n .
I n 3D, a s i m i l a r f a i l u r e has l e s s s i g n i f i c a n t e f f e c t s o n
the
corre-
l a t i o n energy, because the short- range domain does n o t c o n t r i b u t e t o i t s
l e a d i n g terms. Then, t h e inadequacy o f RPA i s much more s i g n i f i c a n t
in
two than three- dimensional system,
From the zeros o f the d i e l e c t r i c f u n c t i o n one can d e t e r m i n e t h e
c o l l e c t i v e e x c i t a t i o n s o f the system.
I n the long-wavelength
l i m i t and
f o r small Landau damping, the plasma d i s p e r s i o n r e l a t i o n can be w r i t t e n
and
Thi s two-dimens i o n a l p!asmon
i s very i n t e r e s t i n g
because
d i s p e r s i o n and damping r e f l e c t the incomplete screening i n 20.
its
Physi-
c a l l y , plasmons a r e plane d e n s i t y waves propagating l i k e l i nes 01' charge
i n 2D and sheets o f charge i n 3D. While the e l e c t r i c f i e l d requ i' red
to
r e s t o r e charge u n i f o r m i t y i s independent o f the wavelength X i n 3 D ,
it
f a l l s o f f as h-'
i n 2D. T h i s gives a plasmon frequency
pendent o f q i n 3D and behaves l i k e eq. (2.18)
whi ch
i:; inde-
i n 2D.
3. ELECTRONS ON THE SURFACE OF HELIUM FILMS
The spectrum o f bound s t a t e s changes s i g n i f i c a n t l y i f e l e c t r o n s
a r e p u t on a h e l i u m f i l m c o v e r i n g a s o l i d s u b s t r a t e
constant
E
S'
as shown i n f i g . 3 .
with a dielectric
I n t h i s case, the s u b s t r a t e i i i d u c e s
a l t e r a t i o n s i n t h e image f o r c e s and one m o d i f i e s the e l e c t r o n i c s t a t e s
by vary i n g o n l y the t h i ckness
2 of
the f i l m . The image p o t e n t i a l can be
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
r ELECTRON LAYER
-
Fig.3
Schernatic i l l u s t r a t i o n o f the
geometric arrangement f o r t h e s u r f a c e
e l e c t r o n s on f i l m o f l i q u i d h e l i u m adsorbeb on a s s b s t r a t e .
w r i t t e n as
where Q,
= s 6 / ( ~ + 1 ) and
~
a = 4 ~ 6 , w i t h 6 = ( c s - E ) / ( E ~ + E ) ,Since the s e r i e s
i s r a p i d l y convergent, one can r e t a i n o n l y the f i r s t term.
A d e t a i l e d a n a l y s i s o f the spectrum o f the e l e c t r o n i c s t a t e s o n
a helium f i l m adsorved on a metal (6=1) was g i v e n by Gabovich
For t h i c k f i l m s ,
t h e c o n d i t i o n <z>
n
et
~ 2 . ~ 3
<< d i s s a t i s f i e d and one canexpand
as
i n this l i m i t ,
f i e l d Fd
-
the s u b s t r a t e a c t s as an externa1 p r e s s i n g
electric
l/d2. The presence o f a l a r g e constant negative c o n t r i b u t i o n
&,e2/d t o the p o t e n t i a l energy o f the e l e c t r o n g r e a t l y
b i n d i n g energy o f e l e c t r o n s (11.6 meV f o r
d
= 300
8)
increases
as
compared
the b i n d i n g e n e r g y o f e l e c t r o n s o n b u l k helium (0.7meV).
e f f e c t i v e e l e c t r i c f i e l d Q1.s/d2
and t h e van der Waals
Due
forces
the
with
t o the
coming
from t h e s u b s t r a t e which s t a b i 1 i z e the e l e c t r o n - f i l m system (see sectíon
6) the maximum d e n s i t y t o which the s u r f a c e can be
charged i n c r e a s e s
drastical lyZ4.
The c o r r e c t i o n t o the spectrum o f e l e c t r o n s on b u l k h e l ium (eq.
2.4)
i s g i v e n by
A& (d) =
n
-
Revista Brasileira de Ffsica, Vol. 16, n? 2, 1986
which corresponds t o a f i r s t - o r d e r S t a r k e f f e c t o f the
clampinq
field
Fd'
For t h i n f i lms, one can n e g l e c t the image f o r c e s i n the h e l ium
f i l m and the spectrum i s determined by the image c o n t r i b u t i o n
substrate.
of
the
I n t h i s s i t u a t i o n the e l e c t r o n p o t e n t i a l t u r n s o u t t o be the
same as g i v e n by eq. (2.6)
w i t h Q and f3 replaced b y
Q,
d respec-
and
t i v e l y . So, the energy l e v e l s d i f f e r markedly from the hydrogenüc spectrum and a r e g i v e n a p p r o x i m a t e d ! ~by
f o r ã >> a, and h i g h quantum numbers.
I t i s obvious t h a t the s u b s t r a t e wi11 a l s o a f f e c t
appreciably
the magnitude and n a t u r e o f the i n t e r a c t i o i ; between the e l e c t r o r i s i n the
plane. The bare p o t e n t i a l between the eleccrons can now be
foun~d f rom
the i d e n t i t y
w i t h a p p r o p r i a t e boundary c o n d i t i o n s f o r the
f i e l d Z 5 . The r e s u l t, i n q-space,
potential
a n d e1 e c t r i c
is
where
w i t h 6 defined through eq. (3.1).
(d*)
For e l e c t r o n s on t h e
bulk
Iielium
we recover t h e usual p o t e r i t i a l f o r e l e c t r o n s conf ined i n e plane
I n the 1 i m i t o f t h i n h e l ium f i lms (qd <<
11,
~ ( q d )assumes the fclrm
205
Revista Brasileira de Física, Vol. 16, n? 2, 1986
For a m e t a l s u b s t r a t e , one o b t a i n s a p o t e n t i a l w n i c h does n o t depend on
q, t h a t i s
Then, i n t h i s case t h e i n t e r a c t i o n between e l e c t r o n s i s s t r i c t 1 y
di-
p o l a r , s i nce i n r e a l space one can w r i t e
We rnust n o t e t h e d r a s t i c s c r e e n i n g o f t h e Coulcinb
interaction
comi ng
from the substrate.
For a s u b s t r a t ' e w i t h l a r g e d i e l e c t r i c c o n s t a n t
cS >>
for
E,
)
the form
example a semimetal, ~ ( q d assumes
As one can see, t h i s system c o n s t i t u t e s an e x c e l l e n t l a b o r a t o r y
t o t e s t many-body t h e o r i e s , s i n c e , by v a r y i n g t h e f i lm t h i c k n e s s and t h e
p a r t i c l e d e n s i t y w h i c h a r e t h e e x p e r i m e n t a l l y a c c e s s i b l e pararneters, one
can change t h e n a t u r e o f t h e b a r e i n t e r a c t i o n between t h e p a r t i c l e s and
t h e plasma p a r a m e t e r .
4. CORRELATIONS I N THE 2D CLASSICAL ELECTRON PLASMA
The s t a t i c and dynamic p r o p e r t i e s o f e l e c t r o n s on t h e
surface
o f 1i q u i d h e l ium were s t u d i e d i n r e f e r e n c e s 26-28, As d i s c u s s e d i n sect i o n 2, t h e s h o r t - r a n g e c o r r e l a t i o n s a r e q u i t e i m p o r t a n t i n t h i s system,
so t h a t an improvement o f RPA i s necessary. The c a l c u l a t i o n s have been
c a r r i e d o u t on t h e b a s i s o f a k i n d o f g e n e r a l i z e d random-phase a p p r o x i mation,
col
t h e s e l f - c o n s i s t e n t - f i e l d method (SCFA)developed by Singwi
and
labor ator^'^,
We s t a r t w i t h t h e L i o u v i l l e e q u a t i o n f o r t h e N - p a r t i c l e
dis-
Revista Brasileira de Fisica, Vol. 16, n? 2, 1986
-+
3
p,.
t r i b u t i o n f u n c t i o n fN(rl,
4
-+
..
-+
-+
rN,
pNlt),
i s def i ned as
where the o p e r a t o r &(ri ,pi)
(G.,t) i s an externa1 p o t e n t i a l . I n t e g r a t i ng t h e L i o u v i 1 l e
and V
ext z
equation w i t h respect t o the coordinates and rnomenta o f N-1 p a r t i c l e s ,
one o b t a i n s the equation of motion f o r the o n e - p a r t i c l e
3
distribution
-+
f u n c t i o n fl (r,p,t)
3 + +
where f2(r,p,r1
,pi I t) i s
3
the t w o - p a r t i c l e d s t r i b u t i o n f u n c t i o n .
The
equation o f motion f o r f, c o n t a i n s , i n t u r n , the t h r e e - p a r t i c l e d i s t r i b u t i o n f u n c t i o n and so on.
I n the SCFA,this
n f i n i t e h i e r a r c h y cjf equa-
t i o n s i s t r u n c a t e d by decoupling the t w o - p a r t i c l e d i s t r i b u t i o n f u n c t
through the ansatz
-+ -+
where g ( r - r ' )
i s the s t a t i c p a i r c o r r e l a t i o n f u n c t i o n , With t h i s ansatz,
t h e short- range c o r r e l a t i o n s a r e included, i n an approximate way,thraigh
-+ -+
g ( r - r ' ) which g i v e s a measure o f t h e probabi l i t y o f f i n d i ng a s e c o n d
+
3
r . Assuming g ( r ) = 1
-+
f o r a11 r i n eq. (4.4) i s e q u i v a l e n t t o the decoupling o f f, i n RIPA and
-+
p a r t i c l e a t a p o s i t i o n r: when the f i r s t one i s a t
leads t o the Vlasov equation.
We assume now t h a t
Revista Brasileira de Física, Vol. 16, n? 2, 1986
where f , ( p ) i s t h e equi 1 i b r i u m d i s t r i b u t i o r ! f u n c t i o n and
the d e v i a t i o n o f
f,
6f r e p r e s e n t s
induced by the weak externa1 p o t e n t i a l . Using
eqs.
( 4 . 4 ) and ( 4 , 5 ) one l i n e a r i z e s the e q u a t i o n o f motion. Proceeding i n the
standard way, one o b t a i n s t h e induced p a r t i c l e d e n s i t y i n F o u r i e r space
-+
where x(q,w)
iç
t h e dens i ty- densi t y response f u n c t i o n g i v e n by
The e f f e c t i v e p o t e n t i a l
$ ( q ) i s r e l a t e d t o the s t r u c t u r e f a c t o r
the F o u r i e r t r a n s f o r m o f g(r) , through the expression
The second term o f eq.
( 4 . 8 ) i s r e s p o n s i b l e f o r the l o c a l f i e l d e f f e c t s
due t o short- range c o r r e l a t i o n s neglected i n RPA,
by s e t t i n g
a(;)
= $();
i n eq.
which
( 4 , 7 ) . I n t h i s sense RPA
is
is
recovered
a
trivial
s p e c i a l case o f the SCFA.
As before, the f l u c t u a t i o n - d i s s i p a t i o n theorem (eq. (2.13)) and
Krarners-Kr&-iig
r e l a t i o n p u t together a l l o w us t o w r i t e
the
density-
- d e n s i t y response f u n c t i o n a t zero frequency as
so t h a t the s t r u c t u r e f a c t o r t u r n s o u t t o be
This completes the SCFA scheme. The equations ( 4 . 2 ) - and
have t o be so l v e d numer i c a l l y i n a s e l f - c o n s i s t e n t way. From
208
(4.4)
~ ( q ) t,h e
Revista Brasileira de Flsica, Vol. 16, nQ 2, 1986
p a i r c o r r e l a t i o n can be evaluated s i m p l y by a F o u r i e r t r a n s f o r n i a t i o n o p e r a t ion.
For the e l e c t r o n
gas
on
bulk
h e l ium,
we
t h e s e l f - c o n s i s t e n t equations i n a s i m i l a r way t o the
cain
wri te
3D c l a s s i c a l e l e c -
t r o n .plasma, i .e.
where G ( ~ )i s the l o c a l f í e l d f u n c t i o n d e f ined by t h e r e l a t i o n
such t h a t
where K ( x ) and E(x) a r e the complete e l l i p t i c i n t e g r a l s o f the F i r s t and
second k i n d r e s p e c t i v e l y .
The s e l f - c o n s i s t e n t s o l u t i o n i s o b t a i n e d by the s tandard method
of i t e r a t i o n . W i t h a reasonable i n p u t ~ ( q ) , t h e e f f e c t i v e p o t e n t i a l $ @
i s calculateci and from i t a new ~ ( q i)s obtained. The procedure
i s re-
peated u n t i l s e l f - c o n s i s t e n c y i n ~ ( 4 i)s achieved.
For the b u l k helium case the p a i r c o r r e l a t i o n f u n c t i o n i s p l o t t e d i n f i g . 4 f o r severa1 values o f t h e plasma parameter
r.
The r e s u l t s
agree s a t i s f a c t o r i l y w i t h those o b t a i n e d from Monte C a r l o ~ i m u l a t i o n s ~ ~ .
Note t h a t , c o n t r a r y t o RPA r e s u l t s , our g(r) i s p o s i t i v e f o r
a r a t ions
a111 s e p -
.
The i n f l u e n c e o f t h e f i l m t h i c k n e s s on the
self- consistent
s t r u c t u r e f a c t o r f o r two d i f f e r e n t s u b s t r a t e s (metal and glass) i s s h w n
i n f i g . 5 and 6.
I n these f i g u r e s , the f i l m thickness i s i n u n i t s o f the
-112
core radius a = (m)
. For d > 100,
S ( q ) i s independent of the n a t u r e
o f the s u b s t r a t e and s i m i l a r t o t h a t o f the s t r i c t l y 2D
electron
gas.
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
-
Fig. 4
P a i r c o r r e l a t i o n f u n c t i n g ( r ) as a
f u n c t i o n o f r i n u n i t s o f (m)-f o r s e v e r a l
values o f t h e p l a s m a parameter,
Thepoints
a r e the Monte C a r l o r e s u l t s .
!h
For 1 < d < 100, the d i f f e r e n c e s i n ~ ( 4 f) o r the two
substrates
are
found o n l y f o r smal 1 4.
From S(q) obtained s e l f - c o n s i s t e n t l y , the c o r r e l a t i o n
energy
can be evaluated as
I n f i g . 7, we present the r e s u l t s f o r the
correlat ion
energy
d e n s i t y E /nT as a f u n c t i o n o f the f i l m thickness f o r a meta 1 substrate.
C
Revista Brasileira de Física, Vol. 16, n? 2, 1986
-
Fig. 5
S t r u c t u r e f a c t o r s ( ~ i )n b o t h
Self- Consistent- Field- Approxima t i o n
(SCFA) and Random-Phase- Approxima t i o n
(RPA) f o r severa1 values o f t h e f i l m
thickness (di = 0.01, d2 = 0.1,d3=1.0,
and d,, = 100) and plasma parameteri' =3,
f o r a metal s u b s t r a t e .
de =
The v a l u e
100 i s o u r e s t i m a t e o f t h e c r i t i c a 1 t h i c k n e s s ,
above
which the r e s u l t s t u r n o u t t o be the same as i n the b u l k h e l i u m case.
From the poles o f t h e densi ty-dens i t y response f u n c t i on,
can g e t the d i s p e r s i o n r e l a t i o n f o r the plasmons.
I n the
long
one
wave-
l e n g t h l i m i t and f o r weak damping,
I n f i g . 8, we show the r e s u l t s o f the plasma d i s p e r s i o n r e lation for
r
=
3 and a metal s u b s t r a t e . I n the l i m i t o f t h i n f i l m s ,
we
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
-
Fig.6
S t r u c t u r e f a c t o r s($)
f o r a glass s u b s t r a t e . T h e p a r arneters a r e t h e same as those
i n f i g . 5.
observe an unusual a c o u s t i c a l rnode w
P
where c, = (a121 'I2
9
Here a = s-S(O&S(O) i s the
a t i o n o f the constant S(q) and
velocity.
= cq w i t h
v = (2T/m)'/'
T
f ractional
i s the thermal
devi-
e1 e c t r o n
I n R P A ~ ' , aRpA = =&/E.
I n c r e a s i n g the f i l m thickness, the long range interactioncomes
t o be q u i t e important u n t i l t h e b u l k l i m i t i s reached, w i t h
where
y
=
(1/4im)< cik
-range e f f e c t s .
[ s ( k ) - ~ J i s the c o r r e c t i o n t o RPA due t o
short-
Revista Brasileira de Flsica, Vol. 16, no 2, 1986
THICKNESS (d/a)
-
Fig. 7
C o r r e i a t i o n energy d e n s i t y g c h T as a
f u n c t i o n o f thickness f o r a metal s u b s t r a t e and
= 3 . The l i n e i s a guide t o t h e eye. Note t h a t
the assymptotic l i m i t corresponds t o t h e b u l k
l imit.
r
-
Fig.8
Long-wavelength d i s p e r s i o n r e l a t i o n curves i n u n i t s
a
o f w, = (2.rrne2kD/m)''2 f o r
metal s u b s t r a t e .
The dashed
l i n e s a r e the r e s u l t s from RPA.
The parameters are the (,ame as
those o f f i g . 5.
Observe t h e
t r a n s i t i o n from an a c o u s t i c a l
mode t o an usual 2D plasmon.
Revista Brasileira de F isica, Vol. 16, no 2, 1986
5. POLARONIC EFFECTS
s ec t i o n s
The quantum e l e c t r o n i c s t a t e s described i n previous
a r e formed by assuming a p l a n a r l i q u i d boundary. H o w e v e r , a p e r t u r b a t i o n
o f the s u r f a c e p r o f i l e , due t o the presence o f the normal modes o f
os-
c i l l a t i o n o f the l i q u i d surface, m o d i f i e s the form o f the wave e q u a t i o n
[eq.
(2.20
governing the motion o f e l e c t r o n s . We w i l l d e s c r i b e now the
e l e c t r o n i n t e r a c t i o n w i t h these e x c i t a t i o n s , c a l l e d r i p p l o n s , which a r e
the quantized c a p i l l a r y - g r a v i t y waves o f t h e helium s u r f a c e . F i r s t ,
we
consider a s i n g l e e l e c t r o n p i c t u r e f o r the e l e c t r o n - r i p p l o n c o u p l i n g ,
d i s r e g a r d i n g the e l e c t r o n - e l e c t r o n i n t e r a c t i o n , L a t e r , the
Coulornb i n -
t e r a c t i o n between l o c a l i z e d e l e c t r o n s over the deformed s u r f a c e w i l l be
introduced.
The r i p p l o n s a r e bosons w i t h a d i s p e r s i o n g i v e n by
u2
r = [g'q
+
(5.1)
( õ / ~ ) 4 ~tanh
]
qd
where q i s the r i p p l o n wave v e c t o r , a the s u r f a c e tension, p the d e n s i t y
o f l i q u i d helium. The van der Waals a c c e l e r a t i o n g' = g+f, where
g
the a c c e l e r a t i o n o f g r a v i t y and f = 3a/pd4, w i t h a the
Waals
van
constant responsible f o r the a t t r a c t i v e forces exerted on
der
the
h e 1 i um
atom from the s i d e o f the s u b s t r a t e .
I n terms o f the r i p p l o n c r e a t i o n and a n n i h i l a t i o n o p e r a t o r
and a
4'
is
+
a
+
4
the displacement o f the helium s u r f a c e u(r) can be w r i t t e n as
where
,4I2
= (hq tanh qd)/2wr
The treatment of e l e c t r o n - r i p p l o n s c a t t e r i n g based on the Born
-0ppenheimer a d i a b a t i c p r i n c i p l e , which takes i n t o account the
adjust-
ment o f the e l e c t r o n i c s t a t e s to' the long wavelength o s c i l l a t i o n o f the
l i q u i d , has been discussed i n d e t a i 1 by S h i k i n and ~ o n a r k h a ~ ' .
The e l e c t r o n - r i p p l o n i r i t e r a c t i o n p o t e n t i a l can be w r i t t e n
in
Revista Brasileira de Flsica, Vol. 16, no 2, 1986
the following form
V (z) =
4
A E-&e2/z2 + &e24~,(qz)/z + &,e2/(z+d)
4
+ eFI
-
and Kl(z) is the modified Bessel function of second kind. At
(5.4)
low tem-
perature and high externa1 electric fields, the electron-ripplon interaction can induce a self-localized electronic surface state. l'he electron deforms the hel ium surface forming a smal 1 dimple i n wti i ch i t
localizes itself3lS3'. This electron-ripplon bound complex is cal led a
polaron.
In the long wavelength 1 imi t where experiments have been performed, the motions parallel and perpendicular to the surface can be
separated, and the polaron is described by a Frohlich type Hamilton ian
The path-integral formalism, as introduced by Feynman for the
polaron problem, was first worked out by ar ias^^ and later by HipÕlito
et aZ34 in order to evaluate the ground-state energy and effective mass
of the polaron,
The appl ication of the Feynman theory to this specific problem
was motivated by the fact that Feynman's treatment isthemost successful
overall theory of the optical polaron, giving the lowest upper bound to
the ground state energy, In this method, the ripplon coord inates are
exactly eliminated in favor of a non-linear retarded interaction of the
electron with itself, The physical meaning of this interaction with the
past is that the perturbation caused by the moving electron spend time
to propagate in the medium.
Revista Brasileira de Física, Vol. 16, n? 2, 1986
The path i n t e g r a l a r i s e s i n t h e c a l c u l a t i o n o f the
propagator
( o r the p a r t i t i o n f u n c t i o n f o r f i n i t e temperature)
which i s the p r o b a b i l i t y amplitude f o r the dynamical e v o l u t i o n o f
the
system, I f we know IK, the e i g e n v a l u e s a n d e i g e n f u n c t i o n s o f
the
Hamiltonian would be, i n p r i n c i p l e , determined. I f we l e t t h e " time"
6-
where E i s the ground s t a t e energy.
g
The a c t i o n S, corresponding t o the p o l a r o n
Hamiltonian, is
g i v e n by
Obviously the f u n c t i o n a l i n t e g r a l , b e i n g i n t r a c t a b l e ,
replaced by a simple e x a c t l y s o l u b l e f u n c t i o n a l S,,
The use o f
must be
Jensen's
inequality
F
<F>
< e > ? e
a1 lows us t o o b t a i n a v a r i a t i o n a l upper bound on the ground s t a t e ene
so
where E , = l o g J &(.r)e
.
and <. ,>
means average wi t h w e i g t h e+so,
Feynman was a b l e t o choose a model t o c a l c u l a t e S, which
con t a i ns
the
p h y s i c a l s i g n i f i c a n c e o f the t r u e a c t i o n S . I n t h i s model, the s o - c a l l e d
two-parameter model
,
i n s t e a d o f the e l e c t r o n being coupled t o t h e normal
modes o f the medium, i t i n t e r a c t s w i t h a s i n g l e f i c t i t i o u s
particle via
a s p r i n g , and the p a i r o f p a r t i c l e s are f r e e t o wander, The mass o f
the
Revista Brasileira de Flsica, Vol. 16, no 2, 1986
e x t r a p a r t i c l e and the s p r i n g constant a r e the v a r i a t i o n a l
parameters.
A f t e r e l i m i n a t i n g the coordinates o f the f i c t i t i o u s p a r t i c l e s , the t r i a l
a c t i o n assumes the form
For o u r s p e c i f i c e l e c t r o n - r i p p l o n complex,
where
F(u)
=
v2-Q2
7
(
and 0 i s d e f i n e d i n terms o f the v a r i a t i o n a l paraméter C and Q as
The v a r i a t i o n a l parameters
and C a r e determinedby m i n i m i z i n g
the express i o n f o r t h e energy. The numeri c a l c a l c u l a t i o n s f o r tbie ground
s t a t e energy Óf the p o l a r o n f o r f i l m s on a metal a r e s k w n i n f i g . 9 ,
as
a f u n c t i o n o f t h e f i lm thickness and f o r two values o f the externa1 e l e c
t r i c f i e l d , The energies f o r f i l m s deposited on o t h e r s u b s t r a t i s s w e r e
a l s o c a l c u l a t e d and the r e s u l t i s t h a t the i n f l u e n c e o f t h e s u b s t r a t e
i s significant only f o r d
o r d e r o f 50
8
< 500 8.
When the f i l m thickness
the c o n t r i b u t i o n o f the s u b s t r a t e t o t h e
ts
of
electric:
t o t a l l y dominates the e x t e r n a l e l e c t r i c f i e l d , The main r e s u l t i s
i n t h e l i m i t o f zero e x t e r n a l e l e c t r i c f i e l d @=O), the p o l a r o n
the
field
that
energy
i n a metal s u b s t r a t e i s g r e a t e r than i n any o t h e r s u b s t r a t e .
S i m i l a r c a l c u l a t i o n s have been made by Jackson and Platzman
f o r one p a r t i c u l a r thickness o f the f i l m (d
= 100 8 ) . I n o r d e r
to
get
a n a l i t y c a l r e s u l t s i n the s t r o n g c o u p l i n g and weak c o u p l i n g l i m i t s they
introduced a c u t o f f
kC
= ( p g ' / ~ ) 1 / 2 (a being t h e c a p i l l a r y constant), i n
the e l e c t r o n - r i p p l o n i n t e r a c t i o n and approximated the r i p p l o n d i s p e r s i o n
[eq,(5.1)]
by a sound l i k e d i s p e r s i o n f o r q c
kc, They found a phase-
- t r a n s i t i o n l i k e behavior from an e l e c t r o n i c d e l o c a l i z e d
state
to
a
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
-
Fig.9 Local i z a t í o n energy o f
an e l e c t r o n as f u n c t i o n o f t h e
thickness o f the helium f i l m
on m e t a l l i c s u b s t r a t e . S o l i d
and dashed curves correspond
and
t o F = 20.0 x ] o 3 V/cm
5 . 0 x l o 3 V/cm.
s e l f - t r a p p e d one a t a c r i t i c a l v a l u e o f the e l e c t r o n - r i p p l o n c o u p l i n g
constant. T h i s " l o c a l i z a t i o n " t r a n s i t i o n r e f e r e d here i s n o t t o be taken
l i t e r a l l y , b u t r a t h e r i n terms o f the r a p i d i t y and
m a g n i t u d e of
change of the e f f e c t i v e mass o f the polaron over a narrow range
the
o f the
coupl i n g constant. Our numerical c a l c u l a t i o n s u s i n g the p a t h - i n t e g r a 1
formal ism were performed i n terms o f
experimental
access i b l e
v a r i a b l e s , the externa1 e l e c t r i c f i e l d and f i l m thickness,
and do n o t
the
i n c l u d e any k i n d of approximation i n the e l e c t r o n - r i p p l o n i n t e r a c t i o n
o r ripplon dispersion.
The p r o p e r t i e s of t h i s p o l a r o n i n h i g h magnetic f i e l d havebeen
Revista Brasileira de Flsica, Vol. 16. no 2. 1986
i n v e s t i g a t e d by Sai toh wi t h the use o f p a t h - i n t e g r a l ~ n e t h o d s ~ ~ .
Very r e c e n t l y Degani and H i p ó l i t o 3 7 have s t u d i e d i n d e t a i 1 t h e
ground s t a t e p r o p e r t i e s w i t h emphasis on t h i s nove1 and i n t e r e s t i n g pol a r o n i c phase t r a n s i t i o n . They used a s imple u n i t a r y - t r a n s f o r m a t i o n f o r rnalism, based on a g e n e r a l i z a t i o n o f the v a r i a t i o n a l a p p r o x i m c i t i o n o f
Lee, Low and Pines, which was p r e v i o u s l y a p p l i e d t o t h e s t u d y o f
the
s v r f a c e p o l a r o n by de Bodas and t i i p ó l i t o 3 ' , The advantage o f t h i s method
i s the g r e a t f l e x i b i l i t y f o r a p p l i c a t i o n s t o Hamiltonians
(5.5) w i t h severa1 k i n d s o f i n t e r a c t i c n p o t e n t i a l and
similar
to
appropriate dis-
p e r s i o n r e l a t i o n , I t was demonstrated t h a t t h i s approximation gives t h e
same r e s u l t s as Feynman's formalism i n s t r o n g - c o u p l i n g and weak-coupling
l i m i t s f o r o t h e r types o f polarons. Furthermore, t h i s u n i t a r y - t r a n s f o r mation method has been used i n the study o f t h e p o l a r o n phase t r a n s i t i o n
For t h e e l e c t r o n - r i p p l o n ground s t a t e , t h e v a r i a t i o n a l
function
I$>
wave-
i s p o s t u l a t e d as a product o f an e l e c t r o n wavefunction and
a coherent r i p p l o n s t a t e , T h i s s u r f a c e s t a t e i s n o t an e i g e n s t a t e o f
+
the t o t a l para1 l e l momentum o p e r a t o r p
t'
+
where p i s the e l e c t r o n momentum.
The m i n i m i z a t i o n o f t h e energy should be performed
t r a i n i n g the o p e r a t o r
where
-+
t
by
cons-
so t h a t
i s t h e Lagrange m u l t i p i i e r (which ends up as the p o l a r o n v e l -
o c i t y ) introduced t o keep the e x p e c t a t i o n v a l u e o f the t o t a l
mornentum
equal t o a g i v e n c o n s t a n t ,
The method c o n s i s t s f i r s t i n s u b j e c t i n g
H =H
+ -+
- p , pt t o a canonical t r a n s f o r m a t i o n S I ,
SI = e x p ( - i n
1 +q,r+ a+9 a9 ) ,
+
the
Hami I t o n i a n
(5.18)
c
where
i s a v a r i a t i o n a l parameter whi ch recovers the weak-coupl i ng ap-
Revista Brasileira de Flsica, Vol. 16, no 2, 1986
p r o x i m a t i o n as n=1 and g i v e s the s t r o n g - c o u p l i n g theory i n
q
+
the
l i m i t
o.
Next the e x p e c t a t i o n v a l u e o f the r e s u l t i n g H a m i l t o n i a n i s c a l -
c u l a t e d by choosing a v a r i a t i o n a l wave f u n c t i o n w i t h the f o l l o w i n g form
where 10> i s the r i p p i o n ground- state wave f u n c t i o n ,
1
a ( O >= O
o b t a i ned
+
f rom
and <O
O > = ] , h and p , a r e v a r i a t i o n a l parameters and S2
4
i s t h e well-knownI second Lee-Low-Pines canonical t r a n s f o r m a t i o n given by
S,
= exp [
1 (La i - L aq)I
4
9
4
and F s i s a v a r i a t i o n a l f u n c t i o n t o be determined,
4
M i n i m i z i n g the energy w i t h respect t o the
+
ametersh and p , ,
up t o second o r d e r i n the v e l o c i t y
variat ional
+
p, we f i n a l l y
parob-
t a i n the expressions f o r the ground- stete energy and e f f e c t i v e mass given
T h i s c a l c u l a t i o n f o r m a l l y reproduces the p a t h - i n t e g r a l r e s u l t s
provided one introduces t h e upper c u t - o f f kc =
g r a t i o n over q i n eq.
JpgSríi
int8
the
(5,131,
By analogy w i t h t h e p o l a r o n problem we d e f i n e a c o u p l i n g
s t a n t a as
intecon-
Revista Brasileira de Física, Vol. 16, n? 2, 1986
where
FT
=F
+
& , e / d 2 . I n the weak-coupl i n g approximation
and r~= 1. I n t h i s l i m i t i t i s c l e a r from eq.
E
9
(5.17)
i!;
a
smal 1
t h a t a minimum
of
occurs a t X = O . Then, f o r the energy and e f f e c t i v e mass we o b t a i n
=-2ya
,
y=%m/hkc
mx/m
(5.23a)
1
(5.23b)
I n t h e s t r o n g - c o u p l i n g l i m i t , corresponding t o l a r g e values of
a, r l + O
,
t h e energy behaves l i k e
and the e f f e c t i v e mass goes l i k e
I n general, t h e energy, d e f i n e d i n eq.
w i t h respect t o t h e v a r i a t i o n a l parameters X and
(5.17),
n.
is
miriimized
Hence, w i t h the best
effec-
- f i t values of those parameters t h e ground- state energy and t h e
t i v e mass a r e obtained.
I n f i g s . 10 and 1 1 the r e s u l t s
are
plotted
a g a i n s t the externa1 e l e c t r i c f i e l d f o r v a r i o u s values o f t h e ttiickness
o f the helium f i l m adsorbed i n s o l i d neon (and k
C
= 1.45 109/d2
As we see from f i g . 10, f o r each thickness o f t h e helium l a y e r
cm-'1.
the en-
ergy has two d i s t i n c t branches corresponding t o the s t a b l e s o l u t i o n s o f
eq. (5.21). The f i r s t branch, r e p r e s e n t i n g the w e a k - c o u p l i ng
( 0 = 1)
regime, crosses the second one which corresponds t o the strong-coupl ing,
a t a c e r t a i n c r i t i c a 1 v a l u e o f t h e clamping e l e c t r i c f i e l d
Fc, A t t h i s
the en-
c r o s s i n g p o i n t t h e r e i s a discontinuous change i n t h e slope o f
ergy, c h a r a c t e r i z i n g a f i r s t - o r d e r p h a s e - t r a n s i t i o n s behavior i n
which
the p o l a r o n s t a t e transforms from a n e a r l y - f r e e t o a l o c a l i z e d s t a t e
type. The extremely r a p i d v a r i a t i o n a t the c r i t i c a 1 p o i n t i s more
m a t i c a l l y seen i n f i g . 11, where t h e e f f e c t i v e mass, i n
units
of
drathe
e l e c t r o n mass, changes by severa1 orders o f magnitude.
~
t o corrobRecent experimental measurements by ~ n d r e i 'appear
o r a t e t h i s t h e o r e t i c a l p r e d i c t i o n by observing the l o c a l i z a t i o n o f
the
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
1o-'
L
ELECTRIC FIELD (VOLT/ cm)
-
F i g , 10
Ground-state energy as f u n c t i o n o f the
clamping externa1 f i e l d f o r t h r e e values o f the
f i l m thickness i n t h e case where the s u b s t r a t e i s
s o l i d neon ( E = 1.24). P o i n t s a r e numer i c a 1 r e s u l t s and 1 ings a r e o n l y guides t o the eye.
e l e c t r o n as a p o l a r o n i c s t a t e f o r d
í:
1000
8
and 0.4 < T < 1 K.
I n her
experiments Andrei makes measurements o f the m o b i l i t y and e f f e c t i v e m a s s
o f e l e c t r o n s on a h e l ;um f i l m , She found t h a t a t c e r t a i n c r i t i c a l p o i n t
1
a sharp b1 o r d e r o f magnitude drop i n t h e m o b i l i t y occurs, The e l e c t r o n i c e f f e c t i v e mass on the o t h e r hand increases by 8 orders
mag-
of
n i tude, c h a r a c t e r i z i n g t h e o b s e r v a t i o n f o r the f i r s t time, of t h e t r a n s i t i o n t o a s u r f a c e polaron s t a t e as p r e d i c t e d Dy a
singl e electron
t h e o r y , Although t h e experimental measurements support the basic p o l a r o n
model ideas, many e l e c t r o n s a r e present i n the experimental
situation,
suggesting t h a t Coulomb i n t e r a c t i o n between e l e c t r o n s p l a y s a n important
r o l e , i n t r o d u c i n g then a new s c a l e o f energy i n t o t h e
problem
p o l a r o n s t a t e , w i t h b i n d i n g energy o f the o r d e r o f m i l i k e l v i n s ,
(the
as
we
Revista Brasileira de F lsica, Vol. 16, n? 2, 1986
106
I
I
-
I
I
I
helium films
104-
I
d=100A
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
!I
I
I
I
I
I
I
k&--
-*;*-
t
I
I
1.0
IO*
I
I
I
I
I
I-
I
I
I
I
to2-
I
I
I
I
-
I
1
A
&
-
,
103
io4
105
ELECTRIC FIEL0 (VOLT /cm)
ioEi
-
Fig.11
E f f e c t i v e mass i n u n i t s o f the f r e e e l e c t r o n
mass as f u n c t i o n o f t h e e x t e r n a l e l e c t r i c f i e l d f o r
t h r e e values o f the f i l m thickness i n the case where
t h e s u b s t r a t e i s s o l i d neon. P o i n t s a r e numerical r e s u l t s and l ines a r e guides t o the eye.
have c a l c u l a t e d before,
i s f a r t o o low t o be observable a t
mental temperatures 0.4 K
<
the e x p e r i -
T C 1 K).
With many e l e c t r o n s present, an i n s t a b i l i t y o f t h e chasged he-
l ium s u r f a c e can be generated by forming separate many-electron dimples
w i t h i n an energy s c a l e which might a l l o w e x p e r i m e n t a l
observation.
W i t h i n the c o n t e x t o f A n d r e i ' s experiment, we now describe t h e
proper-
t i e s o f an i n d i v i d u a l dimple f i l l e d w i t h severa1 e l e c t r o n s ( N
40
-
90)
trapped a t the s u r f a c e o f a h e l ium f i l m adsorbed on s o l i d sapphi re3'.
Since t h e e f f e c t i v e charge o f the dimple (Ne) and the niass ( ~ m )
a r e l a r g e , the p o l a r o n c o u p l i n g constant a becames p r o p o r t i o n a l t o
f i f t h power o f the number o f e l e c t r o n s N.
I n t h i s case, the polaron
the
is
s t r o n g l y bound t o the s u r f a c e even a t zero e x t e r n a l p r e s s i n g e l e c t r i c
f i e l d . Thus, from the s t r o n g - c o u p l i n g regime o f our model, we
t a i n the t o t a l energy E
9
can
ob-
associated w i t h a many- electron dimple, j u s t b y
adding t o t h e o n e - p a r t i c l e energy g i v e n by eq.
(5.21) w i t h
= O
the
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
energy o f the Coulomb i n t e r a c t i o n Uc between e l e c t r o n s i n s i d e thedimplc
The t o t a l energy i s then s t r a i g h t f o r w a r d e v a l u a t e d a n d , i n u n i t s o f
h2k:/2~,
has the f o l lowi ng form
B
9
where
EZ(X)
=
a exp( /A)E<(-l/A)
i s the Exponentia
+
%(A)
I n t e g r a l f u n c t i o n and
Also, M = Nm i s the dimple mass, and $ ( r ) i s g i v e n by eq. (5.19).
M i n i m i z i n g t h e ground s t a t e energy, g i v e n by eq.
(5.25),
with
respect t o h we f i n a l l y f i n d the t o t a l energy o f the system
and the Coulomb i n t e r a c t i o n energy
UC
= (N~F~
2 /) 4 ~ a
which i s o f the o r d e r o f magnitude o f K e l v i n s .
From eq.
(5.27) we conclude t h a t f o r a simp l e d i m p l e
e n e r g e t i c a l l y advantageous ( w i t h E- <
o f the h e l ium f i l m i f
F'T > 15.14
Y
O)
i t
is
t o be loca 1 i z e d a t the s u r f a c e
akc. For b u l k h e l ium (d*)
mum externa1 f i e l d necessary t o l o c a l i z e a dimple i s Fc =
the m i n i -
3120
V/cm,
which i s c o n s i s t e n t w i t h the experimental o b s e r v a t i o n o f L e i d e r e r
e t
a~.'+'.
I n f i g . 12 we show t h e r e s u l t s f o r the t o t a l energy, g i v e n
eq.
(5.27), as f u n c t i o n o f t h e h e l i u m f i l m thickness, f o r F =
80
and f o r severa1 values of the number o f e l e c t r o n s i n s i d e
the
N o t i c e t h a t , f o r each number o f e l e c t r o n s i n the dimple,
there
by
V/cm
d impl e.
is
a
c r i t i c a 1 thickness o f the h e l i u m f i l m above which the e n e r g y becomes
the dimple can n o t e x i s t i n a s t a b l e form. The
p o s i t i v e , and therefore,
c r i t e r i o n f o r t h e s t a b i l i t y o f the charged s u r f a c e i s , on t h e otherhand,
r e l a t e d t o the average d e n s i t y o f charges above t h e s u r f a c e , There i s a
c r i t i c a 1 concentration n
C
o f charges,
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
(0) N-40
II ) N a 5 0
( 2 ) N.60
13) N a 7 0
14) N - 8 0
( 5 ) N.90
F = 8OV/crn
-
F i g . 12
Bi nding energy o f a
s i n g l e dimple f i l l e d wi t h N elect r o n s as f u n c t i o n o f the h e l i u m
f i l m thickness f o r a n e l e c t r i c
f i e l d F = 80 V/cm,
The d a s h e d
p a r t o f each curve corresponds t o
an u n s t a b l e s o l u t i o n where the
is
average e l e c t r o n d e n s i t y
l a r g e r than the c r i t i c a 1 d e n s i t y
n
Th e s o l i d curves rspresent
th% f i l m thickness regions where
the dimple i s s t a b l e . Tht: shaded
reg i o n corresponds t o Andrei ' s
experimental r e g i o n (n = '10' crn")
where a dimple w i t h more than 40
e l e c t r o n s can be forrned.
.
FILM THICKNESS (10 %I
above which the dimple i s n o t s t a b l e . For n = n
.
C
the surface
becomes
unstable á t a wave v e c t o r kc = (3g1/o) l l 2 / d 2 I n f i g . 13,
the d e n s i t y
n i s p l o t t e d as a f u n c t i o n o f f i l m thickness, c r i t i c a 1 d e n s i t y nc and
the experimental c o n c e n t r a t i o n n = 108
r e c e n t l y used by ~ n d r e14.
i
I t i s c l e a r from our r e s u l t s t h a t the appearance o f a s t a b l e maiiy-elect r o n dimple i s l i m i t e d by f i l m thickness as w e l l as
by
densi'ty above the surface. From f i g . 13, we note t h a t , a t
the e l e c t r o n
experimental
d e n s i t i e s , o n l y dimples w i t h more than N = 40 e l e c t r o n s can be
formed
on the s u r f a c e o f the helium f i l m . We can a l ç o see the r a p i d i n c r e a s e o f
the d e n s i t y w i t h decreasing c r i t i c a l t h i c k n e s s . C o n s e q u e n t l y , a t
smaller
thickness the e l e c t r o n s become more concentrated a t the c e n t e r
of
the
Revista Brasileira de Física, Vol. 16, n9 2, 1986
i06
1
,
O
40
80
120
160
FILM THICKNESS (10A I
-
Fig.13
Average e l e c t r o n d e n s i t y o f a
s i n g l e dimple as f u n c t i o n o f t h e
film
thickness f o r an externa1 e l e c t r i c f i e 1 d
F = 80 V/m, The curves a r e described i n
the same way as i n f i g , 12.
dimple and the whole s t r u c t u r e o f the dimple becornes sharper, as can be
seen d i r e c t l y from the average r a d i u s R o f t h e m u l t i - e l e c t r o n dimple
F i n a l l y , we can v e r i f y from eqs. (5.27) and (5.30)
e x i s t e n c e of a s t a b l e m u l t i - e l e c t r o n dimple i s p o s s i b l e under
that
the
the con-
d i t i o n kcR 5 1/2. This means t h a t the s i z e o f the r e g i o n c o n t a i n i n g the
Revista Brasileira de Física, Vol. 16, n'? 2, 1 9 8 6
-
0.0
-
Fig. 14
The phase diagram f o r
b u l k h e l ium and f o r a c? = 100 8
f i l m l y i n g on t w o s u b s t r a t e s
(sapphire 6=0.9 and metal A = ] ) .
n, = 2.4 x 1012 c K 2 and T,=33 K.
( A f t e r Peeters and Platzman r e f .
41).
0.0
0.1
0.2
03
0.4
TENPERATURE I T/ TcIm))
05
e l e c t r o n s i s s m a l l e r than a t y p i c a l r i p p l o n wavelength,
j u s t i l ' y i n g the
v a l i d i t y o f our model c a l c u l a t i o n .
6. THE PHASE-DIAGRAM
Very r e c e n t l y , the phase diagram o f e l e c t r o n s on helium f i l m s
has been i n v e s t i g a t e d 3 ' . As we have discussed i n s e c t i o n 2,
f e a t u r e o f the behavior o f these s u r f a c e e l e c t r o n s i s
the
a striking
possibility
t h a t they can undergo a phase t r a n s i t i o n from a l i q u i d phase t o an
or-
dered s t a t e as one increases the densi t y o r lowers the temperature
for
r.
This
a f i x e d density,
i.e.,
f o r l a r g e values o f the plasma parameter
c l a s s i c a l f r e e z i n g t r a n s i t i o n has been observed f o r e l e c t r o n s
on
bulk
h e l ium17. The mechanism o f t h i s t r a n s i t i o n has been proposed i n a semina1 paper o f K o s t e r l i t z and Thouless ( K T ) ~and
~ l a t e r e l a b o r a t e d on by
H a l p e r i n and Nelson ( H N ) ~ This
~ . c l a s s i c a l t r a n s i t i o n occurs by unbinding
o f dislocation pairs a t
rm
t o a new phase, a l i q u i d - c r y s t a l phase, w i t h
no long-range t r a n s l a t i o n a l o r d e r b u t w i t h bond o r i e n t a t i o n a l o r d e r . At
r
=
ri
the system undergoes
another
transi tion
by
unbinding of
Revista Brasileira de Física, Vol. 16, no 2, 1986
d i s c l i n a t i o n p a i r s t o the usual l i q u i d w i t h no long o r d e r . The v a l u e o f
'I m i s given, i n the contex o f KTHN theory as
where c t and cR a r e the t r a n s v e r s a l and l o n g i t u d i n a l sound v e l o c i t y o f
the s o l i d , These constants must be renormalized by t a k i n g
i n t o account
the e f f e c t s o f c r e a t i o n o f d e f e c t s i n the s o l i d . Using zero temperature
r, -
values o f ct and cL, ~ h o u l e s sfound
~ ~ out a transition a t
79. More
r e f i n e d numerical c a l c u l a t i o n s by ~ o r f " , who solved the r e n o r m a l i z a t i o n
group equations o f HN, r a i s e the m e l t i n g t r a n s i t i o n t o
r -
125 which i s
c o n s i s t e n t w i t h t h e experimental r e s u l t .
On t h e o t h e r hand, t h e r e i s a r e g i o n o f the phase diagramkhere
the Fermi energy i s l a r g e r than the thermal energy, EF >> T ,
i.e., a low
temperature phase where the Fermi energy dominates the k i n e t i c s o f
the
e l e c t r o n system.
This r e g i o n cannot be reached e x p e r i m e n t a l l y f o r e l e c t r o n s
b u l k h e l ium, because an dectrohydrodynamic i n s t a b i l i t y o f
on
the charged
surface occurs. T h i s i n s t a b i l i t y a r i s e s when one c o n s i d e r s t h e i n f l u e n c e
of the e l e c t r o n s on the d i s p e r s i o n r e l a t i o n o f r i p p l o n s .
I t i s expected
t h a t the frequency o f r i p p l o n s w i t h a wave v e c t o r around k ( t h e i n v e r s e
C
o f the c a p p i l a r y l e n g t h ) i s considerably lowered as the
e l e c t r o n den-
n > nc ' the frequency becomes imaginary,
For
suggesting t h a t the surface becomes u n s t a b l e a g a i n s t deformations.
s i t y i s increased. lndeed, f o r
b u l k h e i ium nc = (ogp) "*/(2ne2)"*
= 10'
cm-'
. However,
for
elec-
trons on h e l i u m f i l m s , t h e van der Waals helium- substrate i n t e r a c t i o n
i s incorporated i n t h e d i s p e r s i o n r e l a t i o n o f r i p p l o n s and
drastically
increases the c r i t i c a l densi t y nc. Remember t h a t , i n t h i s case
the
g,
g r a v i t a t i o n a l constant, i s replaced by g = 3 d p d 4 . For t h i n f i l m s n
C
given by eq. (5.29).
is
So, the quantum r e g i o n o f the phase diagram can be
achieved f o r e l e c t r o n s on t h i n h e l i u m f i l m s . Indeed, a
recent
imenth6 has demonstrated t h a t e l e c t r o n densi t i e s up t o
10"
exper-
cm-2 above
s a t u r a t e d f i l m s on b o t h i n s u l a t i n g and conducting s u b s t r a t e s a r e s t a b l e .
Revista Brasileira de Física, Vol. 16, n? 2, 1986
Peeters and ~ l a t z m a nhave
~ ~ proposed a phase diagram o f e l e c t r o n s on helium f i l m s by u s i n g a simple dimensional argument
and
KTHN
theory. Accordi ng t o them, the r e l a t i o n
must h o l d i n the gas phase, where
eq. (3.5).
i s the p o t e n t i a l energy g i v e n by
Now, t h e eq. (6. I ) t u r n s o u t t o be
where c t ( d ) and
cR(d) a r e the zero- temperature sound v e l o c i t i e s
in
s o l i d formed by p a r t i c l e s i n t e r a c t i n g v i a a p o t e n t i a l $d, and I'm(-)
a
is
taken as the experimental r e s u l t f o r e l e c t r o n s on b u l k helium.The phase
diagram i s d i s p l a y e d i n f i g . 14. These a r e q u a l i t a t i v e r e s u l t s , because
they must take i n t o account the values o f sound v e l o c i t i e s
temperature, where the r e n o r m a l i z a t i o n e f f e c t s i n these
at
finite
consta~nts a r e
q u i t e important. One important c o n c l u s i o n i s t h a t f o r a metal s u b s t r a t e
one f i n d s a f l u i d - l i k e r e g i o n as n
ness.Athighdensities
-+
O and T
-+
O f o r small
thesolidmeltsagainatadensity
fitin thickwhere
n
t]
the quantum t r a n s i t i o n occurs. These densi t i e s depend upon d ir1 such
way t h a t f o r d
- 70
8,
the s o l i d r e g i o n s h r i n k s t o a p o i n t . T h i s i s
e f f e c t o f t h e d i p o l a r i n t e r a c t i o n a t small f i l m thickness.
It
is
i n t e r e s t i n g t o analyse t h e quantum p r o p e r t i e s o f a d i p o l e gas o f
a
an
very
elec-
t r o n s which, now, has a p h y s i c a l experimental r e a l i z a t i o n . The study o f
t h i s system i s i n p r o g r e s s 4 7 .
Most o f the work d e s c r i bed here has been done i n c01 l a b o r a t i o n
with
G.A. F a r i a s , J.P. Rino, J.R. Drugowich de F e l í c i o and M. Degani.
We would l i k e t o acknowledge them f o r v a l u a b l e discussions and f r i e n d l y
cooperation. We would l i k e t o thank S.A.
J a c k s o n and M .
h e l p f u l discussions on t h e p o l a r o n i c s t a t e and E.A.
Andrei
Sai toh
for
f o r conver-
Revista Brasileira de Flsica, Vol. 16, n? 2, 1986
s a t i o n s a b o u t h e r b e a u t i f u l e x p e r i m e n t . We a l s o thank t o P r o f e s s o r R . C .
T. da Costa f o r a c r i t i c a l r e a d i n g o f t h e m a n u s c r i p t .
( ~ r a z1ii a n Govern-
T h i s work was s u p p o r t e d by FAPESP and CNPq
ment Agencies)
.
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Resumo
Resul tados teÕr i cos de. a1gumas propriedades do s i s tema formado
por e l é t r o n s bidimensionais sobre um f i l m e de h é l i o l í q u i d o a d s o r v i do
em um s u b s t r a t o s ó l i d o são r e v i s t o s neste t r a b a l h o . Descrevemos os estados s u p e r f i c i a i s e l e t r ô n i c o s t a n t o para e l é t r o n s sobre h é l i o volumét r i c o quanto para e l é t r o n s sobre f i l m e s de h é l i o . Propriedades t a i s como o f a t o r de e s t r u t u r a e energia de c o r r e l a ç ã o são determinadas em função da l a r g u r a do f i l m e para d i f e r e n t e s t i p o s de s u b s t r a t o através de
uma Aproximação de Fases A l e a t ó r i a s Generalizada. Descrevemos também as
e x c i taçÕes c o l e t i v a s deste sistema. Os resultados para e l é t r o n s sobre
h é l i o v o l u m é t r i c o e f i l m e s f i n o s são f a c i l m e n t e o b t i d o s .
Examinamos a
i n t e r a ç ã o e n t r e o e l é t r o n e as e x c i tações da superf i c i e do hél i o 1 Í q u i do resultando na formação de um novo estado p o l a r ô n i c o , que f o i observa
do recentemente. A energia do estado fundamental e a massa e f e t i v a dest e polaron são determinadas p e l o formalismo de i n t e g r a l de t r a j e t ó r i a e
método de transformação u n i t á r i a . Discutimos ainda algumas especulações
sobre o diagrama de fase dos e l é t r o n s sobre o f i l m e de h é l i o .