Collective excitations in carbon nanotubes
The effective action functional has been built by a functional integral method for nanotubes. The closed, self-consistent system of equations of the system is built on the basis of the variational differentiation the effective action on collective variables of an electron-phonon subsystem. A gene...
Saved in:
| Published in: | Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
|---|---|
| Date: | 2011 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2011
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/28165 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Collective excitations in carbon nanotubes / А. Korostil // Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2011. — Вип. 58. — С. 48-53. — Бібліогр.: 5 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859774154886086656 |
|---|---|
| author | Korostil, A. |
| author_facet | Korostil, A. |
| citation_txt | Collective excitations in carbon nanotubes / А. Korostil // Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2011. — Вип. 58. — С. 48-53. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| description | The effective action functional has been built by a functional integral method
for nanotubes. The closed, self-consistent system of equations of the system is built on
the basis of the variational differentiation the effective action on collective variables
of an electron-phonon subsystem. A general expression for a polarization function and
spectrum of the system are considered.
|
| first_indexed | 2025-12-02T08:00:39Z |
| format | Article |
| fulltext |
48 © �. Korostil
��� 539
�. Korostil, Kyiv
COLLECTIVE EXCITATIONS IN CARBON NANOTUBES
The effective action functional has been built by a functional integral method
for nanotubes. The closed, self-consistent system of equations of the system is built on
the basis of the variational differentiation the effective action on collective variables
of an electron-phonon subsystem. A general expression for a polarization function and
spectrum of the system are considered.
1. Introduction
The atomic and electron structure of carbon nanotubes can be represented as,
a two-dimensional carbon hexagonal structure rolling along a given direction and
reconnecting the carbon bonds. Systems of carbon atoms can exist in several
modifications: laminated graphite with a hexagonal structure, nite carbon, crystal
diamond, the fullerenes C60, C70, C78, C8, and carbon nanotubes—two-dimensional
extended structures rolled up in a single- or multiwall tube [1,2]. Carbon nanotubes
were synthesized simultaneously with fullerenes and are more interesting
structures because they model a one-dimensional system. Soliton states are known
to be formed in such systems.
The property of nanotubes to absorb liquid metal, hydrogen, oxygen,
methane, and other gases opens a prospect for constructing strong thin conducting
lines of fuel elements and creating new types of fuel. The discovery of
superconductivity in metal-doped C60 [3] feeds the hope to find the same
phenomenon in nanotubes filled with metal or to modify the superconductivity of
known superconductors by injecting them in a nanotube.
Electron spectrum of such structure is characterized by quantum numbers
including the number of radial ( )n , azimuthal ( m ) and longitudinal ( k ) modes
[4,5]. Its physical properties are considerably related to collective electron-phonon
excitations and oscillations of electron density (plasmons or plasma oscillations).
The equations, describing such excitations, can be obtained on the basis the
functional integral method with help of the variational derivatives of the expression
for the effective action integral. We assume that a such approach allows most
precisely to calculate polarizing function of the carbon nanotube in view of all
features of its atomic structure.
2. The effective action function of the system
The researched system consists of ions with charge Ze and degenerate
electrons. Then the functional integral of the system in terms of spatial coordinates
( , ,x y z ) and imaginary time (� ) can be represented as [4,5]
� �exp [ ]Z D D S� � ��� � , (1)
49
where the action [ ]S � is determined by the expression
0
2
0
0
[ ] ( , ) ( , ) ( , )
( , ) ( ) ( , )
2
( )
( ) ( ) , , .
2
a
a a
a
s s
s
l
l r l
l C
S dr dx x r K x r x r
e dr dxdy x r V x y y r
p r
dr ip r q r a
M
� � �
�
�� �
� � �
�
� � � �� �
� �
�� �
� �
��
(2)
Here s is an electron spin, ( , )s x r� is the two-component wave function of the
nanotube lattice ( ,a b )
( , )
( , )
( , )
as
s
as
x
x
x
� �
� �
� �
� �
� � �
� �
,
al
p ,
al
q and 2 CM are a moment, a coordinate and the mass of an ion in al
sublattice cite, ( ) 1/ | |V x y x y� � � is the operator of the Coulomb interaction.
Beside, ( , )K x r is the operator of kinetic energy of the form
,
, ,
( , ) 0
( , ) , ( , )
0 ( , ) 2
a ba
a b r a b
b
K x r
K x r K x r
K x r m�
�
�� �
� � �� � �� �
� �
,
where /r r� � � � , /(2 )a m� is the kinetic energy for the a th sublattice, a� a
chemical potential of the a th sublattice.
The charge density ( , )x r
is composed of ion ( ( , )q x r
) and electron
( ( , )e x r
) parts and equals ( , ) ( , ) ( , )q ex r x r x r
� � , where
,
, ,
( , ) ( , ) , ( , ) ( , )q q e ex r x r x r x r�� � � ��
� � � �
�
� �� � .
The summation on � and � is carried out over all lattice sites a and b .
In the representation of the functional integral (1) can be rewritten as
� �[ , ] exp [ , ]Z D D D S� � � � �� � � , (3)
where the action function [ , ]S � , which contains an electron influence, the field
and its interaction, has the form
1
0
0 0
2
0
1[ ] ( , ) ( ) ( , )
2
( , ) '( , ) ( , ) ( , ) ( , )
( )
( ) ( ) , , .
2a a
a
q
s s
s
l
l r l
l C
S dr dxdy x V x y y
d dx x r K x r x r ie dr dx x x
p r
dr ip r q r a
M
� � �
� � �
� �
�
�
�
� � � �
� � �
�
� � � �� �
� �
� �
�� � � �
��
50
Here
( , ) ( , ) ( , )
'( , )
( , ) ( , ) ( , )
a
b
K x r ie x ie x
K x r
ie x K x r ie x
� �
� �
� �� �
� � ��� �
.
Integrating in (3) on Fermi fields [4] and using the known Liouville formulae,
� � � �lg det ' Sp (ln ) 'A A� , where A is matrix, a prime denotes a first derivative, we
can transform (3) to the form � �exp [ ]effZ D S � � . Here the effective action
1
0
1[ ] ( , ) ( ) ( , )
2effS dr dxdy x V x y y
� ��� � � �� �
0
2
0
2SplnK'(x, ) + ( , ) ( , )
( )
( ) ( ) , , ,
2a a
a
q
l
l r l
l C
ie d dx x x
p r
dr ip r q r a
M
� �
� �
�
� �
�
� � � �� �
� �
� �
��
(4)
allows to describe the system in collective variables.
The matrix Green function, ,|| ||G G� � of the system is determined by the
equation
'( , ) ( , ; , ) ( ) ( )x y x yK x G x y x y� � � � � � �� � � (5)
At presence only the effective field, effV , of single-electron model potential of
carbon nanotube (see [Ah]) the Green function, 0 0,|| ||G G � , is determined by the
equation
'
0 0( , ) ( , ; , ) ( ) ( )x y x yK x G x y x y� � � � � � �� � � ,
where 0 ( , ) '( , ) |
effx x iVK x K x � � �� .
Using the representation 0 1'( , ) ( , ) ( , )x x xK x K x K x� � �� � , where the function
0 ( , ) ( ( , ) ( )) || c ||, c 1, ( , 1,2)x x eff ik ikK x ie x eV x i k� �� � � � � (5) can be rewritten in
the form
0
0 1
0
( , ; , ) ( , ; , )
( , ; , ) ( , ) ( , ; , ).
x y x y
z x z z z y
G x y G x y
d dzG x z K z G z y
� � � �
� � � � � �
� �
� � �
(6)
The obtained expressions for the effective action function together with the
equation (6) for the Green function permit build the equations determining the field
( , )xx � .
3. The equations for field functions
The equations describing states of the system are obtained by equating to zero
the variational derivation of the effective action function (4) with respect to
generalized coordinates ( , )xx � , lq
�
, lp
�
that give the system
51
[ ] [ ] [ ]
0, 0, 0
( ( , )) ( ( ) ( ( ))
eff eff eff
l l
S S S
x q p
� �
� � �
� � � � � �
� � � .
These three equality result in the system of the three equations
1
0
3
( ) ( , ) ( , )
1 1
2 Sp ( , ; , ) 0,
1 1
( ) ( , ) ( ) 0,
( )
( ) 0.
lim
y x
q
x y
y k
l l
l
l
C
dyV x y y ie x
e G x y
i p ieZ d x x x q
p
i q
M
� �
�
�
� �
�
�
�
�
� �
� � �
�
�
�
!
! �
� � � �
" #� �$ $� �% &� �
$ $� �' (
� � ) � �
� � �
�
�
(7)
From the first equation of the system (7) follows that the field function
� �1 2
0
( , ) ( ) ( , ) 4 ( )
( , ; , ) ( , ; , ) ,lim
y x
q
x y x y
y k
z ie dxV z x x ie dxV z x
G x y G x y
� �
�
�
� � � �
!
! �
� � � � *
* �
� �
(8)
means the electrical field of the electrical potential of ions and electrons. This
quantity completely determines the interaction in the system and its collective
excitations. Taking into account that ( ) 4 ( )V x y x y+�� � � � � , the equation (8) can
be transformed to the form
� �1 2
0
( , ) 4 ( , )
16 ( , ; , ) ( , ; , )lim
y x
q
x y x y
z e x
e G x y G x y
� �
� +
�
+ � � � �
! �
� � �
� � , (9)
that together with the equation (6) consists the closed system. For solving this
system we introduce the new notations the 1 2G G G� � and 0 01 02G G G� � . Then
taking into account that for statical ions
1
1
1
0 1
0
( ) ( , ) 4 ( ) ( , ; , )lim
z z
q
eff z z
z z
iV ie dzV x z z ie dxV z x G z z
� �
� � �
!
! �
� �
� � � � � �
� �
� �
and
0 0 0 0 02 ( ) 2 ( ) ...,G G ieG iV G G ieG iV G � � � , � � �
we can obtain the expression
� �
1
2
1 0 1 1
1 1 0 1 1 0 1 1
( , ) 8 ( ) ( , ; , )
( ( ) ( , ; , ) ( , ; , ) ..., ,
eff z
eff z
z iV e dzdz d V x z G z z
z iV z G z z G z z
� � � �
� � � � � �
� � � *
* � � !
�
which describes plasma oscillations.
52
The second and third equations of the system (7) determine motion of carbon
ions. The obtained self-consistent close system of equations describes the electron
and vibrational subsystems via collective variations.
For calculation the electron density fluctuation induced by plasma vibration
relative to the stationary ion lattice we will enter into (8) the polarization operator
1 1( , ; , )P x z� � which is determined by equality
1 1 1 1 1 1
1 1 1 1 1 0 1 1
( ) ( ) ( , ; ', ')
( , ; , ) ( ) ( , ; ', ').
dz d V x z G z z
dz d P x z G z z
� � � � � �
� � � � � � � �
� � �
� �
�
�
Then the field function can represent in terms of the effective potential effV and
polarization operator P in the form
�
�
1 1 1 1
1 1 0 1 1
( , ) ( , ) 4 ( , ; , )
( ) ( ) ( , ; ', ')
effz iV z ie dz d P x z
V x z G z z
� � � � �
� � � � �
� � �
� � �
�
The Green function obeys the matrix equation
� �2
0 0 08G G e G P V G G� � � ,
whence applying the relation 0V G PG� we can obtain the equation
2
08P V e V G P� � , (10)
determining in the linear approximation the polarization P . The poles of the
Fourier transform of the polarization function P determine plasma oscillations of
the density relative to a ground stationary state.
Applying the Fourier transform to (10) we can obtain in the approximation of
the second order in V the expression
1 1 1 1 1 1
1 1 1
2
2
2
, , ;
, ,
4( , ; ', ') ( ') ( ')
41 ( ) ( ) ( ) ,
2
iqx
nmk n m k nmk n m k
n m k
n m k
P q q q q
q
e dxG x G x e E E
q
+- - � � - -
+ � -�
� � � � *
� �
� �* � � � �� �� �
� �
� �
where q and - are coordinate and frequency components of the Fourier
transform; Energy levels of stationary states of the electron subsystem are denoted
as nmkE (see [1]). The spectrum and intensity of the collective excitations are
described by the diagonal part of ( , ; ', ')P q q- - .
1. Kuzuo R, Terauchi M., Tanaka M., Saito Y., Shinohara H. Phys. Rev. B, 49 5054
(1994).
2. Iijima S. Nature, 354 56 (1991).
3. Schon J. H, Kloc Ch., Batlogg B. Nature, 408 549 (2000).
53 © �.�.��
��
4. ������� �. ., Ì. ����� ����������
�
����
���,, �
�
, �., 1986.
5. R. F. Akhmet’yanov,� V. O. Ponomarev,† O. A. Ponomarev,‡ and E. S. Shikhovtseva,
Theor. Math. Phys., 149, 127 (2006).
��
����� 11.10.2010�.
��� 519.832.4
�.�.��
��
,
.�.�., ���, . � �������
��
��������� �
���
�� ���� �������
�
���
����������� �
� ���� � ������
�� ������ ����� ��
��
There has been defined the set of strictly rational strategies and the set of
nonstrictly rational strategies of a player in the antagonistic game. On the example it
has been shown what advantage a player obtains if it applies the set of strictly rational
strategies by the other player swerve from the set of its optimal strategies.
�!�
���� ��"��� #����� �
�$��
����% #��
���$� �
��"��� ��#�����
�
�$��
����% #��
���$� ��
��� �
��
���$#����$� ��$. &
��*� ���� *��
�
��
*�
!
��, �
� ������ ���� �< ��
����, �
=� �$� ��
���#����< ��"��� #�����
�
�$��
����% #��
���$� *�� �$�#��*$ $�>��� ��
��� �$� ��"��� #��?%
�*��
����% #��
���$�.
�������!" #" $���%&'!"(() *"�"+ ��,&-�/�(()
@�������� �*��
����% �$>��� � � ��
%
��D�$
���% #���
�$� <
��
����� �
*�
����� !�
��=�� !
�
��� #��
#��? *��
�
���?
��
��
� [1]. Q
�
�� �
�% !
�
� D��
�$!����#� �
��������#� !
��*� ����
*
�
�� ����$?
��
���$#�����% $��� [2]. ������
��!�’�!��
���
��
���$#�����% $��� !
#���
�
�
�
#���� � *�����*$ �*��
����#�$, ��
�����
�����
�����$<� <
����*�$� �$����
��, !�$�
� $ !�
%�����#� #$����$
���
� � ��#��%, ��, �!
�
�$
"���, ! $>
��% #��
���$�% [3, 4]. ��������, =�
��
����*�$� �� #�*������� $���?����� �������� *�� ���$��$#��, #�$�
$#��
�
#*�
������$#��. @���� �$�� �,
� ��V��
% [5 — 10] !������� ��
�� �
��
�V#�
����, =� �� �#$ ��� ���� ��"��� �*��
����% #��
���$� ��
��� �
*���$�
��
���$#����$� ��$ < �$���*�
��� � ! ���
� !��� ���
���% �
#�$�
$�
?% ��
���#�
���. X
, � ��V��$ [8] �!�
���� ��"��� #����� �
�$��
����%
��#��% #��
���$� �
��"��� ��#����� �
�$��
����% ��#��% #��
���$� ��
���
�
��
���$#����$� ��$, �� *�
!
��, �
� ������ ���� �< ��
����, �
=� �$�
��
���#����< ��"��� #�����
V� ��#����� �
�$��
����% ��#��% #��
���$�
*�� �$�#��*$ $�>��� ��
��� �$� ��"��� #��?% �*��
����% ��#��% #��
���$�.
&�
��
, �
=�
��
���$#����
��
<
������� �
?? ���� <
����� [8]
|
| id | nasplib_isofts_kiev_ua-123456789-28165 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | XXXX-0067 |
| language | English |
| last_indexed | 2025-12-02T08:00:39Z |
| publishDate | 2011 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Korostil, A. 2011-10-31T20:29:37Z 2011-10-31T20:29:37Z 2011 Collective excitations in carbon nanotubes / А. Korostil // Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2011. — Вип. 58. — С. 48-53. — Бібліогр.: 5 назв. — англ. XXXX-0067 https://nasplib.isofts.kiev.ua/handle/123456789/28165 539 The effective action functional has been built by a functional integral method for nanotubes. The closed, self-consistent system of equations of the system is built on the basis of the variational differentiation the effective action on collective variables of an electron-phonon subsystem. A general expression for a polarization function and spectrum of the system are considered. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Collective excitations in carbon nanotubes Article published earlier |
| spellingShingle | Collective excitations in carbon nanotubes Korostil, A. |
| title | Collective excitations in carbon nanotubes |
| title_full | Collective excitations in carbon nanotubes |
| title_fullStr | Collective excitations in carbon nanotubes |
| title_full_unstemmed | Collective excitations in carbon nanotubes |
| title_short | Collective excitations in carbon nanotubes |
| title_sort | collective excitations in carbon nanotubes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/28165 |
| work_keys_str_mv | AT korostila collectiveexcitationsincarbonnanotubes |