Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film
The energies of first two subbands are calculated, within a variational approach, for electrons localized over the surface of a liquid-helium film covering a solid substrate. The results are obtained for arbitrary value of the dielectric constant of the solid substrate, covering both the limit of...
Збережено в:
| Опубліковано в: : | Физика низких температур |
|---|---|
| Дата: | 2004 |
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2004
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/119473 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film / S.S. Sokolov // Физика низких температур. — 2004. — Т. 30, № 3. — С. 271-275. — Бібліогр.: 15 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-119473 |
|---|---|
| record_format |
dspace |
| spelling |
Sokolov, S.S. 2017-06-07T04:33:26Z 2017-06-07T04:33:26Z 2004 Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film / S.S. Sokolov // Физика низких температур. — 2004. — Т. 30, № 3. — С. 271-275. — Бібліогр.: 15 назв. — англ. 0132-6414 PACS: 73.10.Di; 73.20.Dx; 73.90.+f https://nasplib.isofts.kiev.ua/handle/123456789/119473 The energies of first two subbands are calculated, within a variational approach, for electrons localized over the surface of a liquid-helium film covering a solid substrate. The results are obtained for arbitrary value of the dielectric constant of the solid substrate, covering both the limit of a substrate with a dielectric constant close to unity (such as a rare gas solid) and a metal. The results for the subband energies for a metallic substrate are compared with those obtained previously by a different method by Gabovich, Ilchenko, and Pashitskii. The agreement is rather good supporting the applicability of the variational method for calculating the energy spectrum of surface electrons in a wide range of substrate parameters. The author is highly indebted to V.E. Sivokon and Ye.V. Syrnikov for assistance in the numerical calculations. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Квантовые жидкости и квантовые кpисталлы Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film |
| spellingShingle |
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film Sokolov, S.S. Квантовые жидкости и квантовые кpисталлы |
| title_short |
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film |
| title_full |
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film |
| title_fullStr |
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film |
| title_full_unstemmed |
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film |
| title_sort |
variational approach to the problem of energy spectrum of surface electrons over liquid-helium film |
| author |
Sokolov, S.S. |
| author_facet |
Sokolov, S.S. |
| topic |
Квантовые жидкости и квантовые кpисталлы |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
| publishDate |
2004 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The energies of first two subbands are calculated, within a variational approach, for electrons
localized over the surface of a liquid-helium film covering a solid substrate. The results are obtained
for arbitrary value of the dielectric constant of the solid substrate, covering both the limit
of a substrate with a dielectric constant close to unity (such as a rare gas solid) and a metal. The
results for the subband energies for a metallic substrate are compared with those obtained previously
by a different method by Gabovich, Ilchenko, and Pashitskii. The agreement is rather good
supporting the applicability of the variational method for calculating the energy spectrum of surface
electrons in a wide range of substrate parameters.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119473 |
| citation_txt |
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film / S.S. Sokolov // Физика низких температур. — 2004. — Т. 30, № 3. — С. 271-275. — Бібліогр.: 15 назв. — англ. |
| work_keys_str_mv |
AT sokolovss variationalapproachtotheproblemofenergyspectrumofsurfaceelectronsoverliquidheliumfilm |
| first_indexed |
2025-11-24T02:24:13Z |
| last_indexed |
2025-11-24T02:24:13Z |
| _version_ |
1850838064803348480 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3, p. 271–275
Variational approach to the problem of energy spectrum
of surface electrons over liquid-helium film
S.S. Sokolov
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy
of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: sokolov@ilt.kharkov.ua
Received July 25, 2003
The energies of first two subbands are calculated, within a variational approach, for electrons
localized over the surface of a liquid-helium film covering a solid substrate. The results are ob-
tained for arbitrary value of the dielectric constant of the solid substrate, covering both the limit
of a substrate with a dielectric constant close to unity (such as a rare gas solid) and a metal. The
results for the subband energies for a metallic substrate are compared with those obtained previ-
ously by a different method by Gabovich, Ilchenko, and Pashitskii. The agreement is rather good
supporting the applicability of the variational method for calculating the energy spectrum of sur-
face electrons in a wide range of substrate parameters.
PACS: 73.10.Di; 73.20.Dx; 73.90.+f
1. Introduction
The properties of surface electrons (SE) localized
over liquid-helium film are essentially more compli-
cated than those over bulk helium. As is known the po-
tential energy of SE in the point z over bulk liquid oc-
cupying the semispace with z � 0 can be written as [1]
U z
z z
eE zb( ) � �
�
� �
�0
0
(1)
where �0
2 1 4 1� � �e /( ) [ ( )]� �He He , E� is the
holding electric field oriented normally to helium sur-
face, e is the electron charge, and �He� 10572. is the
dielectric constant of liquid helium. The parameter
z0 � 101. Å is introduced in Eq. (1) to account the
finitness of the potential barrier V0 � 1 eV on the liq-
uid helium surface, which is an obstacle to electron
penetration inside the liquid phase, and to avoid di-
vergence of the first term of Eq. (1) at z � 0. The
value of z0 is estimated by comparison of the experi-
mental data on the frequencies of spectroscopic tran-
sitions between the SE surface states and theoretical
calculation based on Eq. (1) in the limit of small
holding field. One should note that the approach
V0 � ( )z0 0� gives a SE energy spectrum very
close to that really observed [2], and for this reason it
is widely used in calculations. The applicability of
the approach V0 � is based on the strong inequal-
ity | |
l V�� 0 where
l is the energy of SE states
numbered by l � 1 2, ,...
For a liquid-helium film located at � � �d z 0 over
a solid substrate with dielectric constant � s the SE po-
tential energy can be written as [3]
U z U z U zf b s( ) ( ) ( )� � (2)
where
U z
a
z nds
n
n
( )
( )
� �
�
�
�
�
��1
1
1
,
�1
2 21� � � �e /s s� � � � � �He He He He( ) [( ) ( )], and
a � �( )�He 1 ( ) ( )( )]� � � � �s s/� � �He He He1 . Due
to small difference between �He and unity one can dis-
regard, in the sum of U zs ( ), the terms with n
2 and
write, to a very good accuracy,U z / z ds ( ) ( )� � ��1 .
The additional contribution U zs ( ) to the equation
for SE potential energy, in comparison with that over
bulk liquid, is connected with polarization interaction
between SE and image forces in the solid substrate at
z d� � . This energy influences strongly the properties
of SEs over film changing not only the structure of the
SE energy states, which were first considered by
Shikin and Monarkha [3], but also the Hamiltonian of
electron–ripplon scattering, which determines the
© S.S. Sokolov, 2004
kinetic properties of the SEs under their motion in the
plane of the vapor–liquid phase boundary [3]. Fur-
thermore, one more scattering mechanism by substrate
surface defects can appear contributing to the SE
transport properties [4].
The role ofU zs ( ) is especially well pronounced for
substrates with � s �� 1, such as, for example, some
types of glass, where � s � 7 to say nothing of metals,
where � s � . For a metallic substrate one has �1 �
� �e /2 21� �He He( ) � e /2 4 and the contribution of
U zs ( ) dominates inU zf ( ) of Eq. (2). The Schrödinger
equation for the SE wave functions and energy spec-
trum has been solved, in that approximation, by
Gabovich, Ilchenko, and Pashitskii [5], and the final
expression for the spectrum at E� � 0, in the limit of
V0 � , can be written as
l � � � �
�
�
�
�
�
�
�
e
a
l
d
a
2
0 0
2
32
3
4
1 2
� (3)
which differs essentially from the hydrogen-like spec-
trum
l /l� � 0
2of SEs over bulk helium [3]. Here
a /me0
2 2� � is the Bohr radius,
0
2
0
2 2� � � / m,
�0 0
2� m /� � , and m is the free electron mass. Equation
(3) is valid for 1 4 22
0�� ��( ) | |) ,e / m/ d/al�
which is well satisfied for d
� �5 10 7cm.
For the substrates with relatively small (� s � 1) or
intermediate values of � s the contribution ofU zb( ) to
U zf ( ) can be compatible with that ofU zs ( ). In such a
condition the only possible way to estimate analyti-
cally the SE spectrum over helium film is to apply the
variational approach. The aim of the present work is to
obtain the variational solution for the energies of two
lowest SE subbands l � 1 and l � 2. For the sake of gen-
erality, the consideration is carried out for the substrate
with arbitrary value of � s and the effect of holding
field is also included. The result for a metallic substrate
is obtained under the limiting transition � s � and is
compared with that given by Eq. (3). Such a compari-
son can make clearer the possibilities of applying the
different approaches in the problem of description of
the SE spectrum. In view of the rising interest in inves-
tigating both quasi-two-dimensional and quasi-one-di-
mensional SE properties over helium film in recent
years, the present study seems timely [7–13].
2. Main relations
To calculate the energies
1 and
2 one applies the
orthonormalized trial wave functions [14,15]
f z z z1 1
3 2
12( ) exp( )/� �� � , (4)
f
z
z2
2
1
2
1 2 2
2
1 22 3
1
3
5
2
1
2
�
� �
�
��
�
�
�
�
�
�
�
�
�
�
�
�
� � � �
� �
( )
exp( )��2z ,
(5)
where �1 and �2 are variational parameters. The en-
ergy of lth subband is calculated as
l fl
m
d
dz
U z l� � � � �| ( )|
�
2
2
.
The method of calculation is a generalization of
that developed in Ref. 15 for the microstratified liquid
solution 3He–4He, and the final expressions for the
energies are
� �1
2
1
2
0 1 1 1 12
1 2� � � � �
� �
� � �
m
d[
� � � �4 2 2
3
21
2
1 1
1
( ) ( ) ( )]� � �
�
d d Ei d
eE
exp (6)
and
�
2
2
1
2
1
2
1 2 2
2
1
2
1 2 2
2
0 2
6
7
2
�
� �
� �
�
�
�
�
�
�
�
�
�
� � � � � �
� � � �
� �
m
1
2
1 2 2
2
1
2
1 2 2
2
1 2
1
2
1 2
2 3
2
� �
� �
�
�
�
�
�
�
�
�
�
� �
� � �
� � � �
�
� � � �
�
( 2
2)
� � � � � � �{ ( )( ) [ ( ) ]� � � � � � � � � �1
2
1 2 2
2
2 1 2
2
2
2
1 22 3
2
3
2
3
3d d 2
2 2
2
2 21 2 4 2 2[ ( ) exp( ) ( )]}� � � �� � � �d d d Ei d
�
� �
� �
�
�
�
�
�
�
�
�
�eE
2
5 2 2
2
1
2
1 2 2
2
1
2
1 2 2
2�
� � � �
� � � � (7)
where Ei x( ) is the exponential integral. In the limit-
ing case d � the terms depending on d in Eqs. (6)
and (7) disappear, and we reproduce the values of � l
obtained in Refs. 14,15 for the bulk liquid. It is inter-
esting to note that for d � 0 the terms depending on
d also disappear in Eqs. (6) and (7), which are for-
mally the same as those for d � but now depend on
� � �0 0 1
! � � . One can easily see that we obtain in
this limit, replacing �Heby unity in �0, the energies
of an electron localized over a semi-infinite medium
272 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
S.S. Sokolov
at z � 0 with a dielectric constant � s and without a
helium blanket.
The values of �1 and �2 are calculated numerically
by cumbersome transcedental equations " " �
1 1 0/ �
and " �
2 2 0/"� . By determining the roots of these
equations and replacing the values of �1 and �2 in Eqs.
(6) and (7) by them one calculates the energies
1
and
2.
3. Results and discussion
Here we restrict ourselves to the limit of zero hold-
ing field E� � 0, where the influence of film effects on
the SE energy spectrum is especially pronounced. The
corrections due to finite value of E� can be included
in a straightforward way [5,6].
We start our consideration by calculating the mean
electron distance from the helium surface. Based on
the SE wave functions of Eqs. (4) and (5) one can eas-
ily obtain
� � � �z 1 1
13
2
� and � � �
� �
� �
z 2
1
2
1 2 2
2
2 1
2
1 2 2
2
5 2 2
2
� � � �
� � � � �( )
. (8)
The dependences of � �z 1 and � �z 2 on d are plotted in
Fig. 1 for metallic substrate. As is seen from Fig. 1 the
values of � �z l increase with d. For a small film thickness
of 5 10 7� � cm one has � �z 1 � 29 Å and � �z 2 � 72.5 Å.
At smaller values of d the mean electron distance, calcu-
lated by Eq. (8), tends to the microscopic range, where
the applicability of the above-mentioned approach to
the description of the SE states over helium film van-
ishes. Note that, for the same d , the values of � �z 1 and
� �z 2 are substantially larger for the substrate with
� s � 1. For example, for solid neon (� s � 1.20) we es-
timate � �z 1 � 89 Å and � �z 2 � 288 Å (for comparison,
� �z 1 � 144 Å and � �z 2 � 456 Å for SE over bulk he-
lium [3]). One concludes that the characteristic values
of the mean electron distance from the liquid surface
satisfy the inequality � � ��z zl 0, being substantially
larger than atomic scale # 10 8� cm. For this reason the
microscopic nature of the helium surface, leading, in
particular, to a small incertainty of the position of the
potential barrier V0, cannot influence appreciably the
SE energy properties; this supports the applicability of
the approach V0 � with the boundary condition for
the SE wave function f zl ( ) � 0 at z � 0 [3].
The dependences of the SE energies
1 2and on d
for a solid neon substrate, calculated numerically by
Eqs. (6) and (7), are presented in Fig. 2. One ob-
serves the increase of the energies with d (the decrease
of absolute values of
1 2and ), which is a natural
consequence of the decreasing contribution of
� ��1/ z d( ) with increasing d. As a result, the abso-
lute values of surface level energies decrease, and the
distance between them also decreases, tending, for
zero holding field, to the hydrogen-like values of SE
energies over bulk helium,
l /l� � 0
2, whereas the
roots of the minimization equations " "�
1 1/ and
" "� $
2/ tend to the values � �1 0� and � �2 0 2� / , co-
inciding with the exact result of solving the Schrö-
dinger equation in the limit d � and E� � 0 [3].
For solid neon this asymptotic limit is achieved for
d � 10 5� cm. At the same time, for a metallic sub-
strate, where the value of �1 is much larger than that
over solid neon,
1 and
2 start to practically coin-
cide with those of the hydrogen-like spectrum at sig-
nificantly larger values d � 10 4� cm.
In Fig. 3 the dependences of
1 and
2 on d are de-
picted for a metallic substrate (solid lines). For com-
parison the values of the level energies calculated by
Eq. (3) are also plotted by the dashed lines. It is seen
that the agreement between the energies calculated in
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 273
200 400 600 800 1000
0
5
10
15
20
25
<z>1
<z>2
<
z
>
,
1
0
7
c
m
–
d, 10 8 cm–
Fig. 1. The mean electron distance from the helium sur-
face for the subbands 1 and 2 as a function of film thick-
ness d for metallic substrate.
0
200 400 600 800 1000
5
10
15
20
25
30
Solid Ne
1
E
n
er
g
y,
K
d, 10 8cm–
2–
–
Fig. 2. The energies of the subbands 1 and 2 vs. film
thickness for solid neon substrate.
different ways is reasonable, especially for relatively
small values of d. At the same time, the agreement be-
comes less satisfactory under increase of d. Note that
the contribution � ��0 0/ z z( ) of the polarization of
liquid helium to the SE potential energy was omitted
in Ref. 5 to derive Eq. (3). Obviously, under increas-
ing d and a decreasing contribution � ��1/ z d( ) of the
polarization of the solid substrate to Eq. (2), the role
of ��0/z becomes more essential which can explain
some divergence of the results calculated by Eqs. (3),
(6), and (7) at d � �10 6 cm. To make this point more
clear we have plotted, in Fig. 4, the values of
1 and
2 calculated by Eqs. (6) and (7), where we put
�0 0� . One can see the substantially better agreement
with the results of Eq. (3) than that in Fig. 3, espe-
cially for the level l � 2 with the larger value of � �z 2
and, consequently, with the larger distance from the
helium free surface, leading to a decreasing a contribu-
tion of the SE potential energy, due to media polariza-
tion, to the structure of subband l � 2 in comparison
with that to the ground subband. As a result, the
choice of approach to describe the SE potential energy
becomes less essential for l � 2 than for l � 1.
4. Conclusions
In the present work the energies of the ground and
first excited SE subbands over helium film are esti-
mated within a variational approach. The expressions
for the level energies are estimated for arbitrary value
of substrate dielectric constant � s . The values of the
mean electron distance from the helium surface are es-
timated in the macroscopic range, supporting the ap-
plicability of the approach V0 � where the poten-
tial barrier is supposed to be exactly at the free helium
surface. The SE energies are calculated, as functions
of film thickness, for substrates of solid neon and a
metal. The results for the metallic substrate are com-
pared with those obtained analytically in Ref. 5. The
agreement between the results of present work and
those of Ref. 5 seems rather good, being especially sat-
isfactory for thin helium films with d � �10 6 cm, in
spite of the different methods of calculations in the
present work and in Ref. 5. One should note the appli-
cation of variational approach to obtain the energies
of subbands with l � 2 leads to overcumbersome calcu-
lations with practically intractable results. In such a
situation the results of Ref. 5 (Eq. (3)) are especially
important, giving the only way to describe analyti-
cally the energy spectrum for subbands with l
3 of
SEs localized over a helium film covering a metal.
In closing, the author is highly indebted to V.E.
Sivokon and Ye.V. Syrnikov for assistance in the nu-
merical calculations.
1. O. Hipólito, J.R.D. de Felício, and G.A. Farias, Solid
State Commun. 28, 635 (1978).
2. L. Zipfel, T.R. Brown, and C.C. Grimes, Phys. Rev.
Lett. 37, 1760 (1976).
3. V.B. Shikin and Yu.P. Monarkha, J. Low Temp.
Phys. 16, 193 (1974).
4. S.S. Sokolov and N. Studart, Phys. Rev. B67, 132510
(2003).
5. A.M. Gabovich, L.G. Ilchenko, and E.A. Pashitskii,
Zh. Eksp. Teor. Fiz. 81, 2063 (1981).
6. S.S. Sokolov and N. Studart, J. Phys.: Condens.
Matter 12, 9563 (2000).
7. H. Etz, W. Gombert, W. Idstein, and. P. Leiderer,
Phys. Rev. Lett. 53, 2567 (1984).
8. X.L. Hu and A.J. Dahm, Phys. Rev. B42, 2010
(1990).
9. T. Günzler, B. Bitnar, G. Mistura, S. Neser, and P.
Leiderer, Surf. Sci. 341/342, 831 (1996).
10. G. Mistura, T. Günzler, S. Neser, and P. Leiderer,
Phys. Rev. B56, 8360 (1997).
274 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
S.S. Sokolov
0 200 400 600 800 1000
0
100
200
300
400
500
Metal
2
1
2
1
E
n
e
r
g
y
,
K
8
–
–
–
–
d, 10 cm
–
Fig. 3. The same as in Fig. 2 but for a metallic substrate.
Solid lines are the results of the present work, the dashed
lines are those of Ref. 5.
0 200 400 600 800 1000
0
100
200
300
400
500
2
2
1
1
Metal, �
0
= 0
E
n
e
rg
y,
K
d, 10 8 cm–
–
– –
–
Fig. 4. The same as in Fig. 3 but for �0 0� .
11. R.J.F. van Haren, G. Acres, P. Fozooni, A. Kristensen,
M.J. Lea, P.J. Richardson, A.M.C. Valkering, and
R.W. van der Heijden, Physica B249–251, 656 (1998).
12. S.P. Gladchenko, V.A. Nikolaenko, Yu.Z. Kovdrya,
and S.S. Sokolov, Fiz. Nizk. Temp. 27, 3 (2000) [Low
Temp. Phys. 27, 3 (2000)].
13. P. Glasson, V. Dotsenko, P. Fozooni, M.J. Lea, W.
Bailey, G. Papageorgiou, S.E. Andersen, and A.
Kristensen, Phys. Rev. Lett. 87, 176802 (2001).
14. Yu.P. Monarkha, S.S. Sokolov, and V.B. Shikin,
Solid State Commun. 38, 611 (1981).
15. S.S. Sokolov, Fiz. Nizk. Temp. 11, 875 (1985) [Sov.
J. Low Temp. Phys. 11, 481 (1985)].
Variational approach to the problem of energy spectrum of surface electrons over liquid-helium film
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 275
|