Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra
Electronic spectra in functionalizing semiconducting carbon nanotube (CNT) are researched theoretically. The soft degrees of freedom are considered in the molecular system: radial deformation, misfit dislocations, and conformation. For each model the selfconsistent system of equations is derived. A...
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Інститут хімії поверхні ім. О.О. Чуйка НАН України
2010
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Lykah, V.A. Syrkin, E.S. 2011-11-27T17:17:08Z 2011-11-27T17:17:08Z 2010 Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra / V.A. Lykah, E.S. Syrkin // Хімія, фізика та технологія поверхні. — 2010. — Т. 1, № 3. — С. 296-302. — Бібліогр.: 14 назв. — англ. 2079-1704 PACS: 68.65.La;67.70.+n;79.60.Dp https://nasplib.isofts.kiev.ua/handle/123456789/28996 Electronic spectra in functionalizing semiconducting carbon nanotube (CNT) are researched theoretically. The soft degrees of freedom are considered in the molecular system: radial deformation, misfit dislocations, and conformation. For each model the selfconsistent system of equations is derived. A system describes a charge carrier longitudinal quantization in CNT, interaction of a carrier in CNT and molecular electric dipoles, material equations for a soft degree of freedom and its reconstruction. It is shown that the functionalization reconstructs the CNT electronic spectra and creates different conditions of localization or tunneling for holes and electrons. Теоретично досліджено електронні спектри функціоналізованої напівпровідникової вуглецевої нанотрубки (НТ). В молекулярній системі розглянуто наступні м’які ступені свободи: радіальна деформація, дислокації невідповідності і конформації. Для кожної моделі отримана самоузгоджена система рівнянь. Кожна система описує поздовжне квантування носія заряду в НТ, взаємодію носія в НТ з електричними диполями молекул, матеріальні рівняння для м'якого ступеня свободи і його реконструкції. Показано, що функціоналізація модифікує електронні спектри НТ і створює різні умови для локалізації або тунелювання дірок й електронів. Теоретически исследованы электронные спектры функционализированной полупроводниковой углеродной нанотрубки (НТ). В молекулярной системе рассматриваются следующие мягкие степени свободы: радиальная деформация, дислокации несоответствия и конформации. Для каждой модели получена самосогласованная система уравнений. Каждая система описывает продольное квантование носителя в НТ, взаимодействие носителя в НТ с электрическими диполями молекул, материальные уравнения для мягкой степени свободы и ее реконструкции. Показано, что функционализация модифицирует электронные спектры НТ и создает различные условия для локализации или туннелирования дырок и электронов. en Інститут хімії поверхні ім. О.О. Чуйка НАН України Хімія, фізика та технологія поверхні Неорганічні та вуглецеві наноматеріали і наносистеми Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra Функціоналізовані напівпровідникові вуглецеві нанотрубки: три моделі для спектрів носія Функционализированные полупроводниковые углеродные нанотрубки: три модели для спектров носителя Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra |
| spellingShingle |
Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra Lykah, V.A. Syrkin, E.S. Неорганічні та вуглецеві наноматеріали і наносистеми |
| title_short |
Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra |
| title_full |
Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra |
| title_fullStr |
Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra |
| title_full_unstemmed |
Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra |
| title_sort |
functionalized semiconducting carbon nanotubes: three models for carrier spectra |
| author |
Lykah, V.A. Syrkin, E.S. |
| author_facet |
Lykah, V.A. Syrkin, E.S. |
| topic |
Неорганічні та вуглецеві наноматеріали і наносистеми |
| topic_facet |
Неорганічні та вуглецеві наноматеріали і наносистеми |
| publishDate |
2010 |
| language |
English |
| container_title |
Хімія, фізика та технологія поверхні |
| publisher |
Інститут хімії поверхні ім. О.О. Чуйка НАН України |
| format |
Article |
| title_alt |
Функціоналізовані напівпровідникові вуглецеві нанотрубки: три моделі для спектрів носія Функционализированные полупроводниковые углеродные нанотрубки: три модели для спектров носителя |
| description |
Electronic spectra in functionalizing semiconducting carbon nanotube (CNT) are researched theoretically. The soft degrees of freedom are considered in the molecular system: radial deformation, misfit dislocations, and conformation. For each model the selfconsistent system of equations is derived. A system describes a charge carrier longitudinal quantization in CNT, interaction of a carrier in CNT and molecular electric dipoles, material equations for a soft degree of freedom and its reconstruction. It is shown that the functionalization reconstructs the CNT electronic spectra and creates different conditions of localization or tunneling for holes and electrons.
Теоретично досліджено електронні спектри функціоналізованої напівпровідникової вуглецевої нанотрубки (НТ). В молекулярній системі розглянуто наступні м’які ступені свободи: радіальна деформація, дислокації невідповідності і конформації. Для кожної моделі отримана самоузгоджена система рівнянь. Кожна система описує поздовжне квантування носія заряду в НТ, взаємодію носія в НТ з електричними диполями молекул, матеріальні рівняння для м'якого ступеня свободи і його реконструкції. Показано, що функціоналізація модифікує електронні спектри НТ і створює різні умови для локалізації або тунелювання дірок й електронів.
Теоретически исследованы электронные спектры функционализированной полупроводниковой углеродной нанотрубки (НТ). В молекулярной системе рассматриваются следующие мягкие степени свободы: радиальная деформация, дислокации несоответствия и конформации. Для каждой модели получена самосогласованная система уравнений. Каждая система описывает продольное квантование носителя в НТ, взаимодействие носителя в НТ с электрическими диполями молекул, материальные уравнения для мягкой степени свободы и ее реконструкции. Показано, что функционализация модифицирует электронные спектры НТ и создает различные условия для локализации или туннелирования дырок и электронов.
|
| issn |
2079-1704 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/28996 |
| citation_txt |
Functionalized Semiconducting Carbon Nanotubes: Three Models for Carrier Spectra / V.A. Lykah, E.S. Syrkin // Хімія, фізика та технологія поверхні. — 2010. — Т. 1, № 3. — С. 296-302. — Бібліогр.: 14 назв. — англ. |
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Хімія, фізика та технологія поверхні. 2010. Т. 1. № 3. С. 296–302
_____________________________________________________________________________________________
* Corresponding author lykah@ilt.kharkov.ua.
296 ХФТП 2010. Т. 1. № 3
PACS 68.65.La;67.70.+n;79.60.Dp
FUNCTIONALIZED SEMICONDUCTING CARBON
NANOTUBES: THREE MODELS FOR CARRIER SPECTRA
V.A. Lykah1*, E.S. Syrkin2
1National Technical University "Kharkov Polytechnic Institute"
21 Frunze Street, Kharkov 61002, Ukraine
2Verkin Institute for Low Temperature Physics and Engineering
of National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
Electronic spectra in functionalizing semiconducting carbon nanotube (CNT) are researched theoreti-
cally. The soft degrees of freedom are considered in the molecular system: radial deformation, misfit dis-
locations, and conformation. For each model the selfconsistent system of equations is derived. A system
describes a charge carrier longitudinal quantization in CNT, interaction of a carrier in CNT and molecu-
lar electric dipoles, material equations for a soft degree of freedom and its reconstruction. It is shown that
the functionalization reconstructs the CNT electronic spectra and creates different conditions of localiza-
tion or tunneling for holes and electrons.
INTRODUCTION
Carbon nanotubes (CNTs) are nanomaterials
with a small diameter of about 1 nm, length of
about 1 µm and the mean free path of the charge
carriers that exceeds 10 µm [1] which is
important for quantization along CNT axis [2].
The optical properties and conductance of a
nanoobject are determined by the set of quantum
energy levels of its charge carriers [3] what has
been observed for CNTs [4]. Functionalization is
new powerful method for tuning CNTs quantum
energy levels and a set of their physical properties
[5]. The CNTs novel high-sensitive biosensors,
electronic and optoelectronic devices are created
due to functionalization. Chips based on CNTs
and DNA [6] or surface self-organizing organic
structures [7] are discussed.
Theory of energy spectra tuning in the
semiconductor CNTs as the result of
functionalization by enough thick molecular films
were developed by the authors in [8, 9]. The
spectrum is extremely sensitive to the state of the
molecular subsystem. We considered the effect of
the interaction of the uncompensated charge
carried by an electron or hole in a quantum
nanowire with the neighboring medium that has
low mechanical rigidity and consists of molecules
possessing an intrinsic electric-dipole moment.
The nonlinear nonlocal equations describing the
system were derived. Longitudinal quantization
was reduced to the spectral problem for a
nonlinear Schrodinger equation. The shift of the
charge-carrier energy levels was calculated.
Possibility of carrier localization was shown.
The aim of this research is to develop
theoretical approach to the CNTs spectra tuning
as a result of functionalization. The selfconsistent
system of equations will be derived; it includes (i)
the time-independent Schrodinger equation for a
charge carrier in a semiconductor CNT; (ii)
nonlinear equation for the molecular structure;
(iii) the material equations for interaction of an
extra carrier in CNT and molecular electric
dipoles subsystem.
MODEL OF THE SYSTEM
The time-independent Schrodinger equation
for an extra charge carrier in an intrinsic-
semiconductor nanotube can be written as [3, 10]
2
( ) .
2 ef
U r W
m
ψ ψ ψ∆ + =h r (1)
Here ( ), efr mψ ψ≡ r and r
r
are wave function,
effective mass and the radius vector of the
particle, ∆ is the Laplace operator, W is the total
energy, ( )U r
r is the potential energy.
Approximation of infinite depth well is applied:
0( ) ( )U r U r≡r r where
0( ) 0U r =r inside CNT and
0( )U r = ∞r outside one. Accounting the interaction
Functionalized Semiconducting Carbon Nanotubes
_____________________________________________________________________________________________
ХФТП 2010. Т. 1. № 3 297
with the surrounding medium ( )intU r
r the
potential is given by relation
0 int( ) ( ) ( ).U r U r U r= +r r r
(2)
This interaction can be relatively strong if
functionalizing molecules have an intrinsic
dipole moment d
r
. Summing the contributions
from all the dipoles, one can get the
interaction potential energy
intU eφ= within the
nanotube by integration in respect to external
space 'r
r
. Here e is the carrier charge. In turn,
a charge carrier determines the potential of the
interaction with an individual dipole.
Approximations of a long nanowire, relatively,
thin layer and cylindrical coordinates
( ( ) ( ) ( , ),r x y z xψ ψ ψ ⊥=r is the coordinate along
the wire) allow to reduce the integral
contributions to those of a local nature [8, 9].
The potential energy of the interaction of a
charge carrier with the dipole subsystem is
0
0 04 ( ) 4 . e
int SU n de R r n deπ π= − − → − (3)
Here nS0 is the surface density of the molecules in
the limit case of the thin functionalizing layer.
The local value of the radial field strength is
replaced with a value for an infinitely long wire
taking the local value of the wave function
2
( ) ; ( ) | ( ) | ; | ( , ) | .d
int
d
U x Ed x e x F F y z dydz
r
τ τ ψ ψ
ε
= − = − = =
′
rr
2 2( ) ; ( ) | ( ) | ; | ( , ) | .
d
U x Ed x e x F F y z dydzτ ψ ψ⊥ ⊥= − = − = =
′ ∫ (4)
Here τ(x) is the local linear charge density and ε is
the relative permittivity of the medium. The
positive direction of a dipole moment d
r
coincides with the direction towards the center of
CNT (see Fig. 1a). The set (1)–(4) is to be
completed with the material equations
( ') ( ( ')).n r n E r=
rr r
(5)
Here ( ')n r
r
is the volume number density of
the molecules. The effect of the charge-carrier
field is the strongest in the case when the
molecules possess an intrinsic electric-dipole
moment d
r
. The electric-dipole moment in the
organic molecules exists due to the presence
of atomic groups that break the charge
symmetry [7, 11].
With accounting of relations (4) and (3),
Eq. (1) can be rewritten to describe the 1D
motion of the charge carriers in the CNT as
follows
2 2
2
( )
( ) ( ) ( ).
2 x
ef
x
U x x W x
m x
ψ ψ ψ∂− + =
∂
h (6)
SOFT RADIAL DEGREE OF FREEDOM
The soft molecular layer consisting of a liquid
crystal or other adsorbed molecules with intrinsic
electric dipole moment was considered in [8, 9].
The longitudinal quantization of a charge carrier
is reduced to the spectral problem for nonlinear
Schrodinger equation. The selfconsistent solution
of the spectral problem is obtained in terms of
a
b
c
Fig. 1. (a) Cylindrical CNT with adsorbed polar mole-
cules. Arrows indicate direction of electric po-
larization vector. (b) and (c) form of the carrier
wave function, molecular layer and renormal-
ized energy of levels Wn at different nonlinear
interaction gL according to [8, 9]
V.A. Lykah, E.S. Syrkin
_____________________________________________________________________________________________
298 ХФТП 2010. Т. 1. № 3
elliptic functions. Features of behavior of the
system are following: the higher interaction, the
more nonlinearity; the lowest quantum levels feel
the most nonlinearity; the effect should be more
pronounced for heavier holes; under increasing of
interaction the carriers are localized (see
Fig. 1b,c).
MISFIT DISLOCATIONS
Hamiltonian of molecular chain on a
substrate in continual approximation is [12]
0
2 ( , )
( ( , )) [1 cos( ( , ))]; ( , ) ;s
s
u x t
U x t U x t x t
a
πφ φ φ= − = (7)
where φ is effective phase, u is the displacement
of a molecule from the equilibrium position in the
layer. Then continual equation of motion has
integral E
2
2
02
0 0
2 2
( ) ( cos ); .
2
ma
E
U
κφ φ λ
λ π
′ = − = (8)
Here am is the molecular lattice constant, λ0 is
characteristic length and κ is elastic constant in
the molecular layer. At E>1 continual equation
has periodic solution
0
0
( )
cos( ) ( , ) ; .
2 m
m
x xx
sn k
k
φ σ ξ ξ
λ
−= − = (9)
Here ( , )msn kξ is Jakobi elliptic function with
elliptic module km defined by relation [12]:
2 2 / ( 1); 0 1.m mk E k= + ≤ ≤ The period of ( , )msn kξ
in dependence on ξ is 4 ( )mK k . ( )mK k is full
elliptic integral of the first kind. The space period
on x [12] in (9) is
04 ( ) .x m ml k K k λ= (10)
The periodic misfit dislocations (9) arise in
the molecular layer. Regions of local extension
(kink, σ = +1) or constriction (antikink, σ = –1)
exist. Then deviation of concentration and the
potential 0e
intU (3) are
0
0
0 0
( ) ( , );S s
S S m
m m
n au x
n x n dn k
x k k
σδ
πλ λ
∂= =
∂
(11)
0( ) ( ); ( ) 4 ( ).e e e e
int int int int SU x U U x U x de n xδ δ π δ= + = − (12)
Accounting to relatively large size and
complicated construction of organic molecules,
creating of partial dislocations is very probable.
Renormalization of substrate interaction.
Material equation for renormalized interaction of
the molecular layer with substrate has to account
simultaneously the effect of the substrate
sU and
carrier charge field d
intU (4) potentials. The
amplitude of the substrate potential can be found
as
0( ) [ ( ) ( )] / 2si scU r U r U r= − through the
interaction potentials in incommensurate
fragments
0( , )siU r x and commensurate ones
0( , )scU r x
4 2
0 0 0 0 4 0 4( , ) | ( ) | [ ] [ ] ;;E s sU U r x U G x U Gδ ψ δ→ = − < = − ×
4 2
0 0 0 0 4 0 4 2
0 0 0
1 1
( , ) | ( ) | [ ] [ ] ;
( ) (
;
)E s s
si sc
deF
U U r x U G x U G
k r k r r
δ ψ δ
ε
⊥→ = − < = − ×
′
(13)
The potential near the equilibrium radius 0r
depends on the rigidity 2 2
0( ) ( ) /s sk r U r r= ∂ ∂ .
Commensurate fragments are more rigid
0 0( ) ( )sc sik r k r> . Substitution of (13) into (7) gives
the Frenkel-Kontorova potential renormalized by
a carrier electric field. The terms 2~| ( ) |xψ are
compensated. According to (10, 8)
0 0 0 0~ ~ 1/ ; ;xE E E E xE xl U U U l lλ < > . Due to
nonlinearity of a molecule-substrate interaction,
one has more rigid repulsion and more soft
attraction.
Superlattice potential of the misfit
dislocations. Let us consider an action of the
functionalizing layer on a carrier inside CNT. In
this case the Schrodinger equation (6) can be
rewritten with the potential
0
0( ) ( ) ( ) ( )e e
int intU x U x U x U xδ= + + . The initial potential
equals inside U0(x) = 0 and outside U0(x) = ∞ of
interval –L < x < +L. The potential 0e
intU is given
by relation (3) and is responsible for a
homogeneous shift of the quantum well bottom.
The potential ( )e
intU xδ is given by relations (11, 12)
and is responsible for creation of a superlattice
structure in the case of a periodic dislocation
arrangement. The inequalities
0 xl Lλ << << have
to be satisfied. Band structure modulation of
semiconducting CNT at E>0 and σd > 0 is shown
in Fig. 2. The valence band top has narrow peaks.
Functionalized Semiconducting Carbon Nanotubes
_____________________________________________________________________________________________
ХФТП 2010. Т. 1. № 3 299
Fig. 2. The band structure of semiconducting CNT
modulated according to (12) at E>0 and σd›0.
Quantum levels of electrons in conductivity
band (C) and holes in a valent band (V) are
shown. The molecular layer creates opposite
conditions for localization or tunneling of the
carrier on a superlattice of misfit dislocations
Selfconsistence. Misfit dislocation
rearrangement. Let's consider a turn action of a
carrier in CNT on the molecular layer. After
substitution of (13) into (7) we obtain the
renormalized Frenkel-Kontorova potential. A
carrier attracts (repulses) the molecular layer that
leads to the coherent regions widening and the
characteristic length (8) takes form
2
0 0
0 0
2
( ) [1 ].
2 ( ) 2E
E
a
x
U x U
κλ λ
π
= +≈
4
4
0 0
0 0
| ( ) |
( ) [1 ].
2 ( ) 2
sG x
U x U
δ ψλ λ= +� (14)
In the case of a fast carrier tunneling the
dislocations have no time to move, so the
undisturbed superlattice in Fig. 3a is kept. In
relations (10) the space period of misfit
dislocation depends on the carrier electric field
0( ) ~ ( )xE El x xλ . The higher carrier density inside
CNT creates regions with larger space period of
misfit dislocation. Direct inserting of (14) with
~ cos( / 2 )x Lψ π into (11) yields the picture of the
dislocation rearrangement and band modulation
shown in Fig. 3b The superlattice tunneling is
destroyed and the electron turns to be locked in
one period of the deformed dislocation structure.
The periodic quantum wells for a hole in CNT are
relatively narrow. They are divided by wide
commensurate regions. This can lead to the
carrier localization at a dislocation Fig. 3c.
a
b
c
Fig. 3. Form of the conducting band bottom. (a) Homo-
geneous misfit dislocation distribution along
CNT as in Fig. 2. (b) The first stage of the misfit
dislocation rearrangement in the electron field.
(c) Hole localization about a peak of valence
band top
To estimate a carrier potential change due to
the molecules we use parameters from [9, 11, 12].
From (3) the bottom level of the quantum well in
CNT with functionalizing molecular layer is
0| |~ (1 10)e
intU ÷ eV and the barrier height is
0
0| ( ) |~| | / ~ (01 1)e e
int intU x U a λ∆ ÷ eV.
CONFORMATION TRANSITION
The conformation of a molecule means that
two or more molecular configurations exist. The
conformation transition (CT) means change of the
configuration. In the general case an asymmetric
molecule have different energies of conformation
configurations [14] (the energy difference Wc).
Let's denote the molecular electric dipole moment
d0 in the initial stable conformation and d1 in the
unstable one, the electric dipole moment change
is ∆d=d1 – d0. CT is possible if the carrier electric
field intensity exceeds the critical value Ec
defined by relation that leads (4) to the critical
values of the linear charge density τc and wave
function density |ψc|
2 in CNT:
2; ; | | .
2
c c c
c c c
r r
W W r
E
d d eF
ε ττ ψ
⊥
′
= = =
∆ ∆
(15)
V.A. Lykah, E.S. Syrkin
_____________________________________________________________________________________________
300 ХФТП 2010. Т. 1. № 3
Here dr is the radial projection. CT is possible
only for one mutual orientation of the vectors. A
carrier with another sign of charge does not create
CT. The CT goes as order-disorder phase
transition in the carrier electric field. After CT the
carrier decreases its energy on the value
4 .e
int sU U n e dπ∆ ≡ ∆ = − ∆ (16)
The most probable scenario: a carrier tunnels
into CNT on a high level and jumps to the basic
state then conformation transition goes (the
phonon relaxation is faster then conformation
one). The basic state wave function must exceed
the critical value (15) ψ1(x)≥ ψc. As the result, the
wave function has only one maximum and
conformation domain takes symmetric position.
The potential well of infinite depth with width 2l
arises, it includes symmetrically a potential well
of the finite depth ∆U (16) with width 2a(a<l).
So the carrier potential inside CNT is
; | | ;
( ) 0; | | ;
; | | .
x l
U x a x l
U x a
∞ >
= ≤ ≤
−∆ ≤
(17)
Following to [13] let us introduce the wave
numbers k0 (the conformation transition well
depth), k (the basic energy level height), and κ
(the basic energy level depth):
2 2
2 2 2 2 2
0 0; ; ;
2 2x
ef ef
U k W k k k
m m
κ∆ = = = −h h (18)
Let us write the selfconsistent system of
equations
2 2 2
0
2
2 2
;
1
tan ;
tanh ( )
.
1 1 cos 2 ( )
tanh ( ) [ ];
2 ( ) ( )
cos .
sh sh
c
k k
ka
k l a
ka l a
a l a
A l a l a
A ka
κ
κ
κ
κκ
κ κ κ κ
ψ
= −
=
−
− = + − + −
− −
=
(19)
The first equation of the system (19) defines the
wave numberκ , the second one is the first
derivation the continuity condition in point x=a.
The third equation is the normalization
condition, A is the wave function amplitude. The
fourth equation: the wave function on the
boundary of the conformation domain takes the
critical value ψ(a)=ψc. The last equation directly
defines the selfconsistency of all sizes and
energies of the problem.
The first and the last equations in (19)
allow exclude the variables A andκ . After
introducing the dimensionless variables C, D
and parameters L, Ф the rest of the system can
be transformed to the following selfconsistent
form
2 2
2 2
2 2
2 2 2 2 2
1
tan ;
tanh[ ( 1)]
1
1
tan
1 tan 1 tan 1
{ ( 1)( 1) } 0
1 tan tan
D D C D
L
C D
C
D D
D D L D D
D C D D D C C D C
= −
− −
+ +
− − − − − = + − − Φ
(20)
2
0 0
0
| |
; ; ; ;cC k a D ka L k l
k
ψ= = = Φ = (21)
The system of equations (20) is nonlinear and
transcendental with two unknown variables. The
first equation is generalization of the one
unknown variable equation for the finite depth
quantum well in [13] (there tanh(x) is absent,
parameter C is fixed and ka is unknown). In [13]
the equation is solved by graphic method on a
plane. Here we use the graphic method. However,
for two unknown variables graphic must operate
with 3D space. The system's solution for the base
quantum state can be ambiguous. We take the
minimum energy solution. For dimensionless
parameters values L=10; Ф=0.2 graphic solution
of the system (20) gives the relative width and
depth of the arising quantum well and position of
the basic quantum level (see Fig. 4)
ESTIMATIONS
The molecular electric dipole change is
∆d ~ d ~ |er0| ~ 10-28 Cl·m. The carrier energy
decreasing after conformation transition (16)
is ∆U ~ (10+1–10-1) eV. The linear charge
density (15) and the energy difference between
the conformations are τc ~ e/2l ~ 10-13 Cl/m;
Wc ~ 10-3 eV. The shorter is CNT the more Wc
can be overcame by a carrier.
)22.(50.0;71.0;11.0
2
0
2
00
≈=≈=≈=
k
k
W
W
C
D
k
k
L
C
l
a
c
cc
Functionalized Semiconducting Carbon Nanotubes
_____________________________________________________________________________________________
ХФТП 2010. Т. 1. № 3 301
Fig. 4. The potential well, the basic state electron (or
hole) wave function ψ1(x) shape, and position of
basic energy level in CNT after conformation
transition in the functionalizing molecular layer.
Diamonds show level of the critical value ψc for
a conformation transition
CONCLUSIONS
The energy spectrum of uncompensated
charge carrier in a functionalized CNT is
investigated. The physical mechanism responsible
for this interaction is the molecular dipole
moment interaction with the charge carrier
electric field. A homogeneous distribution of the
molecular dipoles creates homogeneous shift of
electric potential along CNT. Following soft
degrees of freedom in the functionalizing
molecular system are considered: (i) radial elastic
deformation of the molecules; (ii) the periodic
system of misfit dislocations of the molecules
along CNT; (iii) the molecules conformation.
The periodic misfit dislocations with extra or
lack dipoles create a superlattice potential for a
carrier inside a nanotube. The topological
invariant sign(σd) = ±1 defines creating the
relatively narrow peaks or wells for electron
(correspondingly the wells or peaks for hole).
Considerable deviation of the bands edges relief
from a sinusoidal function breaks symmetry of
the spectra. The narrow peaks are easily
penetrated by a carrier that causes the narrow gap
or quasigap arising in corresponding band. In
another band the narrow wells are divided by
wide barriers; tunneling is rather difficult. It
causes very narrow miniband. In its turn the
charge carrier attracts the molecular layer that
makes dislocation out of periodicity and coherent
regions expand. It destroys miniband structure
and can lead to carrier localization faster.
In CNT the spectra of holes and electrons are
symmetric. The layer of the functionalizing
molecules with the intrinsic electric dipoles
breaks this symmetry in dependence on a charge
sign, dipole orientation, and a kind of the
dislocation or conformation. The conformation
transition in the electric field and energy
spectrum modification are possible for one sign
of the carrier charge. The carrier with another
sign of charge feels a homogeneous change of
potential along CNT only. Thus, a functionalized
CNT with a conformation transition can be used
as a semiconductor rectifier.
The functionalized CNT and induced spectra
are extremely sensitive to variation of the
molecule-molecule and molecule-substrate
interaction constants. It may be caused by
temperature, phase transitions, filling of
functionalizing layers and impurity containing.
The charge-carrier energy spectrum depends on
the rigidity of the functionalizing molecular
system. The high rigidity of the functionalizing
molecular system leads to suppression of
localization. The CNT conductivity must grow
significantly in these cases. Thus, a nanowire can
be used as a sensor for the state of the molecular
system. The processes considered should be taken
into account in the design of chips based on CNT
and layered organic surface structures [6, 7].
REFERENCES
1. Poncharal P., Berger C., Yan Yi. et al. Room
temperature ballistic conduction in carbon
nanotubes // J. Phys. Chem. B. – 2002. –
V. 106, N 47. – P. 12104–12118.
2. Dekker C. Carbon nanotubes as molecular
quantum wires // Phys. Today. – 1999. –
V. 52. – P. 22–28.
3. Ferry D.K., Goodnick S.M. Transport in
Nanostructures. – Cambridge: Cambridge
University Press, 1997. – 528 p.
4. Orlikowski D., Mehrez H., Taylor J. et al.
Resonant transmission through finite-sized
carbon nanotubes // Phys. Rev. B. – 2001. –
V. 63. – P. 155412(1–12).
5. Daniel S., Rao T.P., Rao K.S. et al. A review
of DNA functionalized/grafted carbon
nanotubes and their characterization // Sens.
Actuators. B. – 2007. – V. 122, N 2. –
P. 672–682.
V.A. Lykah, E.S. Syrkin
_____________________________________________________________________________________________
302 ХФТП 2010. Т. 1. № 3
6. Buzaneva E., Gorchynskyy A., Popova G.
et al. Nanotechnology of DNA/Nano-Si and
DNA/Carbonnanotubes/Nano-Si Chips //
Frontiers of Multifunctional Nanosystems.
NATO Adv. Ser. II V. 57 / Eds. E.Buzaneva,
P.Scharff. – Dordrecht: Kluwer, 2002. –
P. 191–212.
7. Neilands O. Organic Compounds Capable to
form Intermolecular Hydrogen Bonds for
Nanostructures Created on Solid Surface,
Aimed to Sensor Design // Molecular Low
Dimensional and Nanostructured Materials
for Advanced Applications. NATO Adv.
Ser. II. V. 59. / Eds. A. Graja et al. –
Dordrecht: Kluwer, 2002 – P. 181–190.
8. Lykah V.A., Syrkin E.S. Soft polar molecular
layers on charged nanowire // Condens.
Matter Phys. – 2004. – V. 7, N 4(10). –
P. 805–812.
9. Lykakh V.A., Syrkin E.S. The effect of
adsorbed molecules on the charge-carrier
spectrum in a semiconductor nanowire //
Semiconductors – 2005. – V. 39, N 6. –
P. 679–684.
10. Landau L.D., Lifshits E.M. Quantum
Mechanics, – N.-Y: Pergamon, 1980. – 224 p.
11. Blinc R., Zeks B. Soft Modes in Ferroelectrics
and Antiferroelectrics. – Amsterdam, N.-Y:
North-Holland Publ. Co, 1974. – 317 p.
12. Davydov A.S. Solitons in Molecular Systems. –
Kiev: Naukova Dumka, 1984. – 304 p. (in
Russian).
13. Flugge S. Practical Quantum Mechanics. –
Berlin: Springer-Verlag, 1971. – 620 p.
14. Flygare W.H. Molecular Structure and
Dynamics. – New Jersey: Prentice-Hall Inc.,
1978. – 696 p.
Received 18.05.2010, accepted 17.08.2010
Функціоналізовані напівпровідникові вуглецеві нанотрубки:
три моделі для спектрів носія
В.О. Ликах, Е.С. Сиркін
Національний технічний університет "Харківський політехнічний інститут"
вул. Фрунзе 21, Харків 61002, Україна, lykah@ilt.kharkov.ua
Фізико-технічний інститут низьких температур ім. Б.І. Веркіна Національної академії наук України
пр. Леніна 47, Харків 61103, Україна
Теоретично досліджено електронні спектри функціоналізованої напівпровідникової вуглецевої нанот-
рубки (НТ). В молекулярній системі розглянуто наступні м’які ступені свободи: радіальна деформація,
дислокації невідповідності і конформації. Для кожної моделі отримана самоузгоджена система рівнянь.
Кожна система описує поздовжне квантування носія заряду в НТ, взаємодію носія в НТ з електричними
диполями молекул, матеріальні рівняння для м'якого ступеня свободи і його реконструкції. Показано, що
функціоналізація модифікує електронні спектри НТ і створює різні умови для локалізації або тунелювання
дірок й електронів.
Функционализированные полупроводниковые углеродные нанотрубки:
три модели для спектров носителя
В.А. Лыках, Е.С. Сыркин
Национальный технический университет "Харьковский политехнический институт"
ул. Фрунзе 21, Харьков 61002, Украина, lykah@ilt.kharkov.ua
Физико-технический институт низких температур им. Б.И. Веркина Национальной академии наук Украины
пр. Ленина 47, Харьков 61103, Украина
Теоретически исследованы электронные спектры функционализированной полупроводниковой углерод-
ной нанотрубки (НТ). В молекулярной системе рассматриваются следующие мягкие степени свободы: ра-
диальная деформация, дислокации несоответствия и конформации. Для каждой модели получена самосо-
гласованная система уравнений. Каждая система описывает продольное квантование носителя в НТ, взаи-
модействие носителя в НТ с электрическими диполями молекул, материальные уравнения для мягкой степе-
ни свободы и ее реконструкции. Показано, что функционализация модифицирует электронные спектры НТ
и создает различные условия для локализации или туннелирования дырок и электронов.
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