Size quantization in metal films
Quantum size effect, predicted by I.M. Lifshits and A.M. Kosevich [Izv. Akad. Nauk SSSR, seriya fiz. 19, 395 (1955)], was investigates in a many works. In the basis of analysis of quantum size oscillations of thermodynamics and kinetic characteristics of metal films lies the quasiclassical quantiz...
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irk-123456789-1209362017-06-14T03:07:10Z Size quantization in metal films Nedorezov, S.S. International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" Quantum size effect, predicted by I.M. Lifshits and A.M. Kosevich [Izv. Akad. Nauk SSSR, seriya fiz. 19, 395 (1955)], was investigates in a many works. In the basis of analysis of quantum size oscillations of thermodynamics and kinetic characteristics of metal films lies the quasiclassical quantization of component of momentum for isotropy model and quantization [S.S. Nedorezov, Zh. Eksp. Teor. Fiz. 51, 868 (1966) [Sov. Phys.-JETF 24, 578 (1967)]] of chord of constant-energy surface in the case of anisotropy energetic spectrum. In the given work the research of quantum size levels of energy of electrons in metal films is carried out by the method of J.M. Luttinger–W.Kohn. The exact conditions of size quantization are got. 2007 Article Size quantization in metal films / S.S. Nedorezov // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1032–1035. — Бібліогр.: 11 назв. — англ. 0132-6414 PACS: 71.61.–r, 71.15.Nc, 73.21.Fg http://dspace.nbuv.gov.ua/handle/123456789/120936 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
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International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" Nedorezov, S.S. Size quantization in metal films Физика низких температур |
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Quantum size effect, predicted by I.M. Lifshits and A.M. Kosevich [Izv. Akad. Nauk SSSR, seriya fiz. 19,
395 (1955)], was investigates in a many works. In the basis of analysis of quantum size oscillations of thermodynamics
and kinetic characteristics of metal films lies the quasiclassical quantization of component of
momentum for isotropy model and quantization [S.S. Nedorezov, Zh. Eksp. Teor. Fiz. 51, 868 (1966) [Sov.
Phys.-JETF 24, 578 (1967)]] of chord of constant-energy surface in the case of anisotropy energetic spectrum.
In the given work the research of quantum size levels of energy of electrons in metal films is carried
out by the method of J.M. Luttinger–W.Kohn. The exact conditions of size quantization are got. |
| format |
Article |
| author |
Nedorezov, S.S. |
| author_facet |
Nedorezov, S.S. |
| author_sort |
Nedorezov, S.S. |
| title |
Size quantization in metal films |
| title_short |
Size quantization in metal films |
| title_full |
Size quantization in metal films |
| title_fullStr |
Size quantization in metal films |
| title_full_unstemmed |
Size quantization in metal films |
| title_sort |
size quantization in metal films |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120936 |
| citation_txt |
Size quantization in metal films / S.S. Nedorezov // Физика низких температур. — 2007. — Т. 33, № 09. — С. 1032–1035. — Бібліогр.: 11 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT nedorezovss sizequantizationinmetalfilms |
| first_indexed |
2025-07-08T18:53:39Z |
| last_indexed |
2025-07-08T18:53:39Z |
| _version_ |
1837106014295425024 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9, p. 1032–1035
Size quantization in metal films
S.S. Nedorezov
Kharkov National Automobile-road University, 25 Petrovskogo Str., Kharkov 610002, Ukraine
E-mail:Eme-Tatyana@yandex.ru
Receive January 29, 2007
Quantum size effect, predicted by I.M. Lifshits and A.M. Kosevich [Izv. Akad. Nauk SSSR, seriya fiz. 19,
395 (1955)], was investigates in a many works. In the basis of analysis of quantum size oscillations of ther-
modynamics and kinetic characteristics of metal films lies the quasiclassical quantization of component of
momentum for isotropy model and quantization [S.S. Nedorezov, Zh. Eksp. Teor. Fiz. 51, 868 (1966) [Sov.
Phys.-JETF 24, 578 (1967)]] of chord of constant-energy surface in the case of anisotropy energetic spec-
trum. In the given work the research of quantum size levels of energy of electrons in metal films is carried
out by the method of J.M. Luttinger–W.Kohn. The exact conditions of size quantization are got.
PACS: 71.61.–r Electrical properties of specific thin films;
71.15.Nc Total energy and cohesive energy calculations;
73.21.Fg Quantum wells.
Keywords: band, chord, dispersion equation, oscillations.
1. The electronic energy spectrum of metallic film is
determined by the Schrödinger equation of the electron in
the periodic potential of crystal taking into account poten-
tial U z( ), created by the border surface of film. On the
condition
a l L�� �� , (1)
where a — distance between atoms, l — characteristic
distance, on which the potential U z( ) changes substan-
tially, L — thickness of a film with normal along the axis
of z, the decision of given equation can be found by the
method of Luttinger–Kohn [1] and approximate potential
U z( ) by the infinitely deep potential pit. A similar method
was used by an author at the analysis of size-quantum lev-
els of energy in semiconductor films [2].
We suppose the decision of the Schrödinger equation
in a kind
�( , ) ( ) ( , )|| || ||x p x x|| z
i
F z u zs s
s
� �
�
�
�
�
exp
�
0 , (2)
where x p|| ||( , ), ( , )� �x y p px y , � ( , / )||p p� �i d dz� , s —
labeling the bands, u s0 — the Bloch function at the bot-
tom of band Es in a point p � 0. Here the functions F zs( )
satisfy zero border condition and are the decision of the
system of differential equations
E
p p
m m
d
dz
F zs
x y
s�
�
�
�
�
�
�
�
�
�
2 2 2 2
22 2
�
( )
�
�
��
� �
1
m
F z F z
s s
ss s s( � ) ( ) ( )p K � , (3)
where
K
r
rss s su i u d� �� �
�
�
�
�
�
�
�
���
( ) *2 3
0 0
3��
�
�
�
(4)
is the momentum matrix elements, � is the volume of the
unit cell.
The given decision is written down in a kind
F z A
i
p zs sj zj
j
N
( ) exp� �
�
�
�
�
�
�
1
2
, (5)
where N is number of the bands, p zj is roots of dispersion
equation
det | |( )||E
m
p
m
ps zj ss� � � ��
1
2
1
2
2 2� �
� � � �� � �( )( ) | ||| ||p K ss zj zss ssp K 1 0� , (6)
which can be by imaginaries. Amplitudes Asj are deter-
mined by the homogeneous system of algebraic equations
© S.S. Nedorezov, 2007
( )||E
m
p
m
p As zj sj� � � �
1
2
1
2
2 2�
� � �
��
� � �
( )|| ||p K
s s
N
ss zj zss s jp K A 0 . (7)
The decision of equations (7) contains 2N arbitrary con-
stants, s =1, 2, …, N; j =1, 2, …, 2N.
Taking (5) into zero border condition , we get
Blj
j
N
�
�
1
2
0, l N�1 2 2, , ..., ; (8)
where
B
A l N
A
i
p L N l N
lj
lj
l N j zj
�
� �
�
�
�
�
�
� � �
�
�
�
�
� �
, ,
exp , .,
1
1 2
�
�
�
�
�
�
. (9)
From (8) we get dispersion equation
det| | | |Blj � 0, (10)
which determines the quantum size levels of energy
(subbands) � n ( )||p .
2. We will consider a model of two bands more in de-
tail. In this case energy bands E p z1 2, ||( , )p have a kind
E p
E E p p
m
z
z
1 2
1 2
2 2
2 2
, ||
||
( , )p �
�
�
�
�
�
E E
m
Kp z
2 1
2
2
2
1��
�
�
�
�
� �( )|| ||K p , (11)
where K K|| || ,� 1 2, K K z� 1 2, . From (10), supposing N � 2,
we get dispersion equation
sin ( ) sin ( )
L
p p
L
p p
2 2
2 1 4 3
� �
� � �
� � �W
L
p p
L
p psin ( ) sin ( )
2 2
3 2 4 1
� �
, (12)
where p j is roots of equation
E p j1 2, ||( , )p � �, j = 1, 2, 3, 4; (13)
W �
� �
� �
( )( )
( )( )
� � � �
� � � �
4 3 2 1
4 1 3 2
, �
�
j
j
j
m E p p
Kp
�
� � �
�
2 1
2 2( ) ||
|| ||K p
.
(14)
In the interval of energies e e2 3� �� and � � e4 (see
Fig. 1) all roots have the material value.
If | |W � 1 from dispersion equation (13) the condi-
tions of quantization follow
p p
L
n2 1 1
2
� � �
�
�
( ), p p
L
n4 3 2
2
� � �
�
�
( ) , (15)
where
�1
4 1 3 11 2 2
2
�
� �
arctg
W L p p L p p
L
sin ( / )( ) sin ( / )( )
sin ( /
� �
� � �)( ) sin ( / )( ) cos ( / )( )p p W L p p L p p4 3 4 1 3 12 2� � � �
,
�2
3 2 3 11 2 2
2
�
� �
arctg
W L p p L p p
L
sin ( / )( ) sin ( / )( )
sin ( /
� �
� � �)( ) sin ( / )( ) cos ( / )( )p p W L p p L p p2 1 3 2 3 12 2� � � �
.
If | |W � 1 we have
p p
L
n3 2 3
2
� � �
�
�
( ), p p
L
n4 1 4
2
� � �
�
�
( ) , (16)
where
�3
2 1 4 11 2 2
2
�
� �
arctg
sin ( / )( ) sin ( / )( )
sin ( /
L p p L p p
W L
� �
� � �)( ) sin ( / )( ) cos ( / )( )p p L p p L p p4 1 2 1 4 22 2� � � �
,
�4
2 1 3 11 2 2
�
� �
�
arctg
sin ( / )( ) sin ( / )( )
sin ( /
L p p L p p
W L
� �
2 2 23 2 2 1 3 1� � �)( ) sin ( / )( ) cos ( / )( )p p L p p L p p� � � �
.
The within the framework of considered model of two
bands the got the conditions of quantization are exact and
are written down in a kind comfortable for the analysis of
quantum size oscillations of thermodynamics and kinetic
Size quantization in metal films
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1033
p1
e1
p2
e2
p3
e3
p4p4
e4
p�
E
�
Fig. 1. The dependence of the energy bands E pz1 2, ||( , )p on
component of the momentum pz.
quantities. At n �� 1 these conditions of quantization
within sizes coincide with the quasiclassical condition
of size quantization, got in [3]
*
.
In the interval of energies e e3 4� �� (see Fig. 1) roots
p2 and p3 are complex numbers
p p iq2 0� � , p p iq3 0� � . (17)
In this case the condition of quantization is determined by
formulas
p p
L
n4 1 0
2
� � �
�
�
( ), (18)
where
�0
0 1
0
1
2
�
� �
�
arctg
ch
sh
( / ) cos ( / )( )
( / ) sin (
Lq L p p
W Lq
� �
� L p p/ )( )� 0 1�
,
W0
2
4 1
4 12
�
� � �
�
! � � � �
! � �
( )( )
( )
.
Here, meaning P p0 0� ( , )|| p , we have
�
�
�
� � �
�
( )( ( ) )
( ) ( )
K P P
K P
12 0 1 0
2 2
12 0
2 2
2m E q
Kq
,
!
�
�
� � � �
�
q p K m E q
Kq
[ ( ) ( ( ) )]
( ) ( )
2 20 12 0 1 0
2 2
12 0
2 2
K P P
K P
.
In the interval of energies e e1 2� �� the roots p3 and
p4 are complex numbers. In this case the condition of
quantization is determined by formulas similar to the for-
mulas (17), (18).
3. The quantum oscillations of thermodynamics [4]
and kinetic [5] quantities in dependence on thickness of
film are due to size quantization of energy of conductive
electrons. This was discovered experimentally on films of
bismuth [6] and antimony [7]. In obedience to the general
theory of oscillatory effects [4], a period and amplitude
C of quantum size oscillations depend on the extreme
values of function n n p pe xe ye( ) ( , , )� �� and its derivative
dn de / � a t � �� F , w h e r e �F i s t h e F e r mi l e v e l ,
� � �n pe x/ ,0 � � �n pe y/ 0. From the received higher the
conditions of quantization we find the function
n p p
L
D Lx y( , , ) ( , ) ( , , )|| ||�
�
� �� �
2 �
p p , (19)
satisfying to equality � �n ( )||p � . HereD is a chord of con-
stant-energy surface, a function is limited (| | � 1) and
substantially ( � L) changes with the change �.
Taking
L
De
2
1
��
� into account, from (19) for the pe-
riod of the oscillations we get
"L
D
�
2��
extr
. (20)
Here Dextr is the extreme chord of the Fermi surface (FS).
The amplitude C of considered oscillations is propor-
tional � "[ / ( ]2 2� �#kT , where
"�
�
�
�
�
�
�
�
�
�
2
2
�
�
L D
L
p p Lxe ye| ( ) ( , , , ) |/
extr
, (21)
�( ) / ( )x x x� sh , T is temperature, k = 1, 2, 3…
At an edge of band it is �F $ e1, e2, e3, e4 (see
Fig. 1), | |W << 1 or | |W >> 1, then from (21) follows
" 0
2
�
�
�
�
�
�
L D| ( )|extr
, (22)
that corresponds to the results got from the quasiclassical
condition of the quantization (see [3,8]).
With leaving from the edge of band approach (22) be-
comes inapplicable. The quantity "� is calculated on a
formula (21), where along with D F�( )� it is necessary to
take into account
�
��
�( , , , )p p Lxe ye F . The periods of
quantum size oscillations are determined (see (20)) by the
extreme chords SF. Thus, the presence of the second band
a near Fermi level, not changing a period, substantially
changes amplitude of quantum size oscillations. The
value "� can turn out more the value " 0� that will result in
more slow decrease of oscillations with the increase of
temperature.
Experimental investigation [6,7,9,10] of quantum size
effects in semimetal films show clearly the presence of
periodic oscillations of physical quantities not only at low
temperatures, but also at temperatures T � 100 K. It does
not consent with estimation � "( / ( )2 2
0� �kT of the am-
plitude of oscillations, got from quasiclassical quantiz-
ation. It is necessary to take into account the closeness of
two bands in the energy spectrum of electrons in bismuth.
It is possible to do within the framework of foregoing
theory.
The exact conditions of size quantization got in this
work allow to find the quantum sizes levels of energy of
electrons of conductivity in a film of metal. It can serve
by the basis of research of the effects conditioned dis-
creteness of energy spectrum of electrons in a film.
More than 50 years passed from the prediction [4] of
quantum size effects. The interest to these effects does not
weaken (see, for example, [11]).
1034 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
S.S. Nedorezov
* Work [3] is executed by an author under the guidance and at the direct participation of I.M. Lifshits.
1. J.M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).
2. S.S. Nedorezov, Fiz. Tverd. Tela 12, 2269 (1970) [Sov.
Phys.-Solid State 12, 1814 (1971)].
3. S.S. Nedorezov, Zh. Eksp. Teor. Fiz. 51, 868 (1966) [Sov.
Phys.-JETF 24, 578 (1967)].
4. I.M. Lifshits and A.M. Kosevich, Izv. Akad. Nauk SSSR,
seriya fiz. 19, 395 (1955).
5. V.B. Sandomirski�, Zh. Eksp. Teor. Fiz. 52, 158 (1967).
6. Yu.F. Ogrin, V.N. Lutski�, and M.I. Elinson, Pis’ma Zh.
Eksp.Teor. Fiz. 3, 114 (1966).
7. Yu.F. Komnik and E.I. Bukhshtab, Pis’ma Zh. Eksp.Teor.
Fiz. 6, 536 (1967).
8. I.O. Kulik, Pis’ma Zh. Eksp.Teor. Fiz. 6, 652 (1967).
9. Yu.F. Ogrin, V.N. Lutski�, M.U. Arifova, V.I. Kovalev,
V.B. Sandomirski�, and M.I. Elinson, Zh. Eksp. Teor. Fiz.
53,1218 (1967).
10. H.A. Combet and J.Y. Le Traon, Solid State Commun. 6,
85 (1968).
11. E. Ogando, N. Zabala, E.V. Chulkov and M.J. Puska. Phys.
Rev. B69, 153410 (2004).
Size quantization in metal films
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9 1035
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