Collective excitations in carbon nanotubes

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...

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Дата:2010
Автор: Korostil, A.
Формат: Стаття
Мова:English
Опубліковано: Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України 2010
Назва видання:Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/28299
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати: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, 2010. — Вип. 57. — С. 48-53. — Бібліогр.: 5 назв. — англ..

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-282992016-04-14T15:08:44Z Collective excitations in carbon nanotubes Korostil, A. 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. 2010 Article 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, 2010. — Вип. 57. — С. 48-53. — Бібліогр.: 5 назв. — англ.. XXXX-0067 http://dspace.nbuv.gov.ua/handle/123456789/28299 539 en Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
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.
format Article
author Korostil, A.
spellingShingle Korostil, A.
Collective excitations in carbon nanotubes
Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
author_facet Korostil, A.
author_sort Korostil, A.
title Collective excitations in carbon nanotubes
title_short 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_sort collective excitations in carbon nanotubes
publisher Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
publishDate 2010
url http://dspace.nbuv.gov.ua/handle/123456789/28299
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, 2010. — Вип. 57. — С. 48-53. — Бібліогр.: 5 назв. — англ..
series Збірник наукових праць Інституту проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
work_keys_str_mv AT korostila collectiveexcitationsincarbonnanotubes
first_indexed 2025-07-03T08:21:49Z
last_indexed 2025-07-03T08:21:49Z
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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. 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