Soft polar molecular layers adsorbed on charged nanowire
A selfconsistent theoretical approach is developed for describing an uncompensated charge carrier inside a perfect semiconductor quantum wire covered with a soft molecular layer. Deformation of the molecular layer under inhomogeneous carrier electric field is described in the model of liquid cry...
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Інститут фізики конденсованих систем НАН України
2004
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| Цитувати: | Soft polar molecular layers adsorbed on charged nanowire / V.A. Lykah, E.S. Syrkin // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 805–812. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1190112017-06-04T03:04:16Z Soft polar molecular layers adsorbed on charged nanowire Lykah, V.A. Syrkin, E.S. A selfconsistent theoretical approach is developed for describing an uncompensated charge carrier inside a perfect semiconductor quantum wire covered with a soft molecular layer. Deformation of the molecular layer under inhomogeneous carrier electric field is described in the model of liquid crystal with intrinsic electric dipole. The longitudinal quantization of a charge carrier is reduced to the spectral problem for nonlinear Schrodinger equation which is solved in terms of elliptic functions. The features of behavior of the system are as follows: the higher is the interaction, the higher is the nonlinearity; the lowest quantum levels experience the highest nonlinearity; the effect should be more pronounced for heavier holes. Under the increase of interaction the carrier is localized. The occurrence of localized states could be responsible for the experimentally observed decrease of conductivity in nanowires. Самоузгоджений теоретичний підхід розроблено для опису незкомпенсованого носія заряда всередені напівпровідникового квантового дроту, що вкритий м’яким молекулярним шаром. Деформація молекулярного шару в неоднорідному електричному полі носія описана в моделі рідкого кристала з власним електричним дипольним моментом. Продольне квантування носія заряда зведено до спектральної проблеми для нелінійного рівняння Шредингера. Самоузгоджене рішення спектральної проблеми отримано в термінах еліптичних функцій. Система має таку поведінку: чим вища взаємодія, тим більша нелінійність; найнижчі квантові рівні відчувають найбільшу нелинійність; ефект має бути більш відчутним для більш важких дірок; при зростанні взаємодії енергетичні рівні знижуються і носій локалізується. Поява локалізованих станів може бути відповідальною за зменшення провідності в нанодротах, що наблодають в експерименті. 2004 Article Soft polar molecular layers adsorbed on charged nanowire / V.A. Lykah, E.S. Syrkin // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 805–812. — Бібліогр.: 12 назв. — англ. 1607-324X DOI:10.5488/CMP.7.4.805 PACS: 61.30.Gd, 11.10.Lm, 68.65.La, 67.70.+n http://dspace.nbuv.gov.ua/handle/123456789/119011 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
A selfconsistent theoretical approach is developed for describing an uncompensated
charge carrier inside a perfect semiconductor quantum wire
covered with a soft molecular layer. Deformation of the molecular layer under
inhomogeneous carrier electric field is described in the model of liquid
crystal with intrinsic electric dipole. The longitudinal quantization of a
charge carrier is reduced to the spectral problem for nonlinear Schrodinger
equation which is solved in terms of elliptic functions. The features of behavior
of the system are as follows: the higher is the interaction, the higher
is the nonlinearity; the lowest quantum levels experience the highest nonlinearity;
the effect should be more pronounced for heavier holes. Under the
increase of interaction the carrier is localized. The occurrence of localized
states could be responsible for the experimentally observed decrease of
conductivity in nanowires. |
| format |
Article |
| author |
Lykah, V.A. Syrkin, E.S. |
| spellingShingle |
Lykah, V.A. Syrkin, E.S. Soft polar molecular layers adsorbed on charged nanowire Condensed Matter Physics |
| author_facet |
Lykah, V.A. Syrkin, E.S. |
| author_sort |
Lykah, V.A. |
| title |
Soft polar molecular layers adsorbed on charged nanowire |
| title_short |
Soft polar molecular layers adsorbed on charged nanowire |
| title_full |
Soft polar molecular layers adsorbed on charged nanowire |
| title_fullStr |
Soft polar molecular layers adsorbed on charged nanowire |
| title_full_unstemmed |
Soft polar molecular layers adsorbed on charged nanowire |
| title_sort |
soft polar molecular layers adsorbed on charged nanowire |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119011 |
| citation_txt |
Soft polar molecular layers adsorbed on charged nanowire / V.A. Lykah, E.S. Syrkin // Condensed Matter Physics. — 2004. — Т. 7, № 4(40). — С. 805–812. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT lykahva softpolarmolecularlayersadsorbedonchargednanowire AT syrkines softpolarmolecularlayersadsorbedonchargednanowire |
| first_indexed |
2025-07-08T15:05:09Z |
| last_indexed |
2025-07-08T15:05:09Z |
| _version_ |
1837091641401278464 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 4(40), pp. 805–812
Soft polar molecular layers adsorbed
on charged nanowire
V.A.Lykah 1 ∗, E.S.Syrkin 1,2 †
1 National Technic University “Kharkiv Polytechnic Institute”,
21 Frunze Str., Kharkiv, 61002, Ukraine
2 Institute for Low Temperature Physics and Engineering,
47 Lenin Ave., Kharkiv, 61103, Ukraine
Received August 16, 2004
A selfconsistent theoretical approach is developed for describing an un-
compensated charge carrier inside a perfect semiconductor quantum wire
covered with a soft molecular layer. Deformation of the molecular layer un-
der inhomogeneous carrier electric field is described in the model of liq-
uid crystal with intrinsic electric dipole. The longitudinal quantization of a
charge carrier is reduced to the spectral problem for nonlinear Schrodinger
equation which is solved in terms of elliptic functions. The features of be-
havior of the system are as follows: the higher is the interaction, the higher
is the nonlinearity; the lowest quantum levels experience the highest nonli-
nearity; the effect should be more pronounced for heavier holes. Under the
increase of interaction the carrier is localized. The occurrence of localized
states could be responsible for the experimentally observed decrease of
conductivity in nanowires.
Key words: liquid crystals, nonlinear Schrodinger equation, nanowire
PACS: 61.30.Gd, 11.10.Lm, 68.65.La, 67.70.+n
1. Introduction
The creation of new mesoscopic objects and the prospects of their application
to nanoelectronics generate interest in studying their fundamental properties [1].
Optical and conducting properties of such nanoobjects are conditioned with a set
of energy quantum levels [2]. The effect of a structure of levels onto conductivi-
ty is found in quantum nanowires (an ordinary intrinsic semiconductor or carbon
nanotube). The average free path of the carrier exceeds 10 microns in nanotube
[3] which is important for quantization through the nanotube [4]. The conductivity
of the nanowire is extremely sensitive to the appearance of adsorbed layers of NH3
∗E-mail: lykah@ilt.kharkov.ua, lykah@kpi.kharkov.ua
†E-mail: syrkin@ilt.kharkov.ua
c© V.A.Lykah, E.S.Syrkin 805
V.A.Lykah, E.S.Syrkin
molecules [5] and Langmuir-Blodgett thin film [6]. The contacting of nanotubes with
a complicated organic medium causes new effects: small nanotube components to
nematic liquid crystals lead to giant electromechanical effect [7], mixture of nan-
otubes and DNA results in the formation of a structurally ordered phase with each
nanotube “wrapped” by DNA [8].
The present work shows that an interaction with a soft medium consisting of
organic polar molecules changes the fundamental characteristic of nanowire: the
energy spectrum of the carrier. The possibility of the carrier localization under the
movement along the covered nanowire is shown .
2. Model of the system
The stationary Schrodinger equation for the additional carrier inside the intrinsic
semiconducting quantum wire could be commonly written as [2,9]:
− h̄2
2mef
∆ψ + U(r)ψ = Wψ, (1)
where ψ(r) is the carrier wave function, mef is the carrier effective mass, h̄ is Plank
constant, ∆ is Laplas operator, W is the total energy, U(r) is potential energy,
r(x, y, z) is a radius-vector of the particle. In semiconductors an approximation of
the rectangular well being of the infinite depth is usually applied: U(r) ≡ U0(r)
where U0(r) = 0 inside and U0(r) = ∞ outside the well. An interaction with an
adjacent medium leads to an additional potential Uint(r):
U(r) = U0(r) + Uint(r). (2)
An effect of the charge’s field reaches its maximum if the molecules possess an in-
trinsic electric moment and the molecular system is soft. Electrical potential created
by one dipole d could be written [10] as φi = diR/εR
3, where R = r−r
′ is a radius-
vector which connects the centre of the dipole r
′ and a point r of observation, ε is
relative dielectric permeability of the medium. Summarizing the contributions of all
dipoles of the adsorbed molecules in the layer we get potential of interaction:
U e
int(r) = e
∫
n(r′)φ(r − r
′)dr
′, (3)
where n(r′) is molecule concentration, e is the carrier charge. In its turn, the electric
field intensity E(r′), created by the carrier, determines the potential of interaction
between the carrier and an individual molecular dipole:
Ud
int(r
′) = −d(r′)E(r′). (4)
Generally it is necessary to supplement a set (3), (4) by material equations
n(r′) = n(E(r′)), d(r′) = d(E(r′)). (5)
806
Soft polar molecular layers adsorbed on charged nanowire
Figure 1. Cylindrical quantum wire with adsorbed polar molecules. Empty arrows
indicate the direction of electric dipole vectors. (a) General view. (b) Crossection
along the nanowire axes Ox.
The system has to be convenient for calculations, so the part of degrees of free-
doms have to be soft only. The liquid crystals of the layered smectic A [11] satisfy
these requirements. The system of long linear molecules is rigid in the direction
along the layer (i. e., surface of a nanowire) and is soft in the direction perpendic-
ular to layer. Let us assume a dipole directed along the axes of the molecules and
perpendicular to the layer [11]. See geometry of the system in figure 1.
The Schrodinger equation (1) with potentials (2)–(4) transforms into a nonlinear
integral-differential one. Such equations are solved using approximation methods [9]
only. Let us show that the approximation of a long nanowire reduces the nonlocal
contribution to the local one. Rewrite an integral (3) in cylindrical coordinates
(figure 1) to divide coordinates: ψ(r) = ψ(x)ψ⊥(y, z), W = Wx + W⊥. We also
assume, that all modifications ψ(x) occur on the scales ∼ L � r, R (see photo in
[6]).
In order to analytically evaluate the potential of interaction between the carrier
and a molecular dipole (4), the space integration is substituted by finding a flux:
Ud
int(x, r
′) = −Ed = −2τ(x)d
εr′
, (6)
τ(x) = e|ψ(x)|2F⊥ , F⊥ =
∫
|ψ(y, z)|2dydz. (7)
Here τ(x) is the local linear density of charge and it is taken into account that the
dipole moment is directed to the centre of the nanowire (3). Relative error of the
approximation is ∼ (R/L)2 � 1.
Let us find the material equation (5) for the thin molecular layer. The condition
Fi = −kδ = −∇Ud
int of an equilibrium elastic displacement δ = 2τd/kεr′2 of a dipole
for attraction (retraction) in the region of more intensive fields and the condition of
conservation of a number of the molecules lead to the potential of the carrier in a
self-consistent field under elastic displacement of dipoles:
U e
int = U e0
int −G|ψ(x)|2 , (8)
807
V.A.Lykah, E.S.Syrkin
U e0
int = −4π
ε
dne(R0 − r0), G =
8πn0d
2e2F⊥
kεR3
(R0 − r0). (9)
Then the equation (1) can be written for 1D movement of the carrier inside nanowire:
− h̄2
2mef
∂2ψ(x)
∂x2
+
[
U e0
int −G|ψ(x)|2
]
ψ(x) = Wxψ(x). (10)
Here −L < x < +L , and U e0
int, G are parameters of linear and nonlinear interaction
of the carrier with an elastic molecular subsystem, correspondingly. Signum of the
linear parameter of interaction depends on a charge signum and the direction of
the dipoles. The nonlinear interaction always decreases the system energy (the even
powers of a charge and dipole in parameter G).
3. Analysis of the system’s spectrum
Using the method standard in quantum mechanics [9] we introduce new variables:
k2
p =
(
Wx − U e0
int
) 2mef
h̄2 , 2g = G
2mef
h̄2 , (11)
where kp is a component of the wave vector of the particle along the quantum wire,
g is the renormalized parameter of the interaction. Then equation (10) transforms
into
ψ′′(x) + k2
pψ(x) + 2g|ψ(x)|2ψ(x) = 0. (12)
Multiplying this equation by dψ(x) and integrating we obtain the integral of the
equation. Devision of variables and the further integration results in elliptic integrals
of the first type. The choice of constants yields even and odd solutions of Schrodinger
equations:
ψeven = b cn κx, ψodd = k′mb sd κx, (13)
where km, k′m =
√
1 − k2
m are elliptic moduli, cn y, sn y, sd y = sn y/dn y are
elliptic functions, κ is a constant which is related with km and b by equations:
b2 =
1
2g
(−k2
p + κ2), km =
b
√
g
κ
. (14)
Other two equations which set the relation between km, κ, b are given by boundary
conditions ψ(±L) = 0 and normalization condition
∫ |ψ(x)|2dx = 1. These four
equations form a selfconsistent system for parameters b, κ, km, kp. Excluding b, κ it
is possible to find the following solution for kp:
k2
p =
(
1 − 2k2
m
) K2 (km)
L2
n2. (15)
where n is the number of the level, km is the solution of the equation
2
gL
K (km)
[
E(km) − k
′2
mK(km)
]
=
1
n2
, (16)
808
Soft polar molecular layers adsorbed on charged nanowire
Figure 2. Dependence between dimensionless parameter of interaction gL and
elliptic modulus k
(n)
m for n = 1 (bottom curve), 2, . . . , 6. (b) Renormalized energy
of levels Wn = k2
pL
2 depending on gL according to equation (15).
Parameter k2
p (square of quasimomentum) determines a spectrum of the carrier (11),
which can be investigated experimentally.
Let us graphically analyse the dependence k2
p of the system on parameters. Equa-
tion (16) determines the relation between a parameter km which describes the non-
linearity and the dimensionless parameter of interaction gL. This dependence is
presented in figure 2a which permits to make some conclusions. Namely, the great-
est nonlinearity (k(n)
m ) is observed under the greatest parameter gL and for the lowest
levels. Parameter k2
p (11) which describes the energy spectrum pass to a zero value
under km = 1/
√
2. Figure 2 (b) gives the dependence of energy levels on a parameter
of interaction. The nonlinearity is observed for the first level.
Under extremely weak interaction (gL, km → 0) the nonlinearity disappears and
(15) transforms into the well-known solution [9] for a rectangular potential well of
infinite depth. The extremely strong interactions could be realized not for such a
high value of the parameter g but for a sufficiently long nanowire (Lg → ∞, km → 1).
Here we use an asymptotical behavior of elliptic functions [12]. Then equations (16),
(15) transform into
K(km)n
L
=
g
2n
, k2
p = −
(
g
2n
)2
, (Lg → +∞) (17)
and the carrier is localized (k2
p < 0) into soliton-like states, the energy does not
depend on L. Such localized states could be responsible for the decrease of conduc-
tivity in chemisorption sensors based on nanowires [5] and in nanowires covered with
Langmuir-Blodgett thin film [6]. The carrier wave function and the molecular layer
deformation are shown in figure 3.
809
V.A.Lykah, E.S.Syrkin
Figure 3. Form of the wave function of the carrier in dependence on elliptic
modulus km for the first level (along Ox) according to equations (13). Elliptic
modulus is km = 0 for g = 0 (for comparison) and km = 0.8. Deformation of the
molecular layer in the case of (a) molecules polarization directed to the nanowire
axes and hole carrier; (b) molecules polarization directed to the nanowire axes
and electron carrier; (c) antiferroelectric dipoles ordering and hole carrier.
Let us qualitatively account for the effect of the actual factors, since the elemen-
tary model was considered. The following factors lead to the increase of localization.
1. Strong interaction of the carrier with a thick molecular layer. The nonlinearity
causes the growth of localization, but the analytical description is inconvenient.
2. Displacement of molecules along the nanowire in the direction of stronger
fields.
3. Rotation of the dipole groups increases both nonlinear and linear (which be-
comes nonlinear) interactions and makes them nonlocal ones.
The effect of localization decreases in the cases:
1. the molecules do not have an intrinsic dipole moment (as a rule an induced
moment is smaller than the intrinsic one);
2. the increase of temperature.
Each of the listed factors requires a separate research for the account.
In order to make the comparison with the experimental research of a nanowire
spectrum or conductivity it is necessary to take into account an inertia of molec-
ular cover. As it is seen in figure 2b energy levels are displaced downwards almost
in parallel manner under the increase of the interaction gL. Really the situation
strongly depends on the relation between the lifetime of an excited state τe and the
relaxation time of a molecular system τM. If τM � τe then the molecular layer has
the time to be set up under the modification of an electrical field of the carrier,
and the spectrum looks like the one in figure 2b. If τM � τe the molecular layer
has no time to be set up, the carrier comes on an energy level which is defined
by deformation of the molecular layer in the ground state of the carrier. This also
concerns the behaviour of the carrier which tunnels in the nanowire. If τM ∼ τe, τtunn
the excitation and tunneling of the carrier have to be considered together with the
excitation of the molecular system.
810
Soft polar molecular layers adsorbed on charged nanowire
4. Conclusion
It is shown, that the nonlinear interaction exists for the carrier inside the quan-
tum nanowire being in contact with a soft polar medium. In the approximation of a
long wire and thin molecular layer, the longitudinal quantization for the additional
carrier can be described using nonlinear Schrodinger equations, and the spectral
problem is solved analytically. The effect of nonlinear interaction and possible lo-
calization have to be more pronounced for heavier holes. Linear interaction changes
a signum depending on a signum of the carrier charge and polarization of molecules.
The energy spectrum of the carrier hardly depends on the rigidity of the molecu-
lar system and on the intrinsic dipole moment of molecules. Crystalization or solidi-
fication of a molecular film under a phase transition result in a discontinuous growth
of a rigidity and an appropriate decline of the parameter of nonlinear interaction,
while the localization of longitudinal movement of the carrier disappears. I. e., un-
der such a phase transition in a molecular system the temperature dependence of
conductivity of nanowire should possess discontinuity. Therefore, nanowire can be
used as a sensor of state of the molecular system being in contact. The obtained re-
sults could be used in order to explain the decrease of conductivity in chemisorption
sensors.
References
1. Datta S. Electronic Transport in Mesocopic Systems. Cambridge, 1995.
2. Ferry D.K., Goodnick S.M. Transport in Nanostructures. Cambridge, Cambridge Uni-
versity Press, 1997.
3. Poncharal P., Berger C., Yan Yi et al. // J. Phys. Chem. B, 2002, vol. 106, p. 12104.
4. McEuen P.L.// Phys. World., 2000, vol. 13(6), No. 6, p. 31–36.
5. Dai H.// Phys. World., 2000, vol. 13(6), No. 6, p. 43–47.
6. Armitage N.P., Gabriel J.-C.P., Gruner G. Cond-mat/0307712.
7. Courty S., Mine J., Tajbakhsh A.R., Terentjev E.M. Cond-mat/0309216.
8. Ming Zheng, Jagota A., Strano M.S., et al. // Science, 2003, vol. 302, p. 1545–1548.
9. Landau L.D., Lifshits E.M. Quantum Mechanics. New York, Pergamon Press, 1980.
10. Landau L.D., Lifshits E.M. Theory of Field. New York, Pergamon Press, 1980.
11. Sonin A.S. Introduction in Liquid Crystals Physics, (in Russian). Moscow, Nauka,
1983.
12. Janke E., Emde F., Losch F. Tafeln Hoherer Functionen. Stuttgart, 1960.
811
V.A.Lykah, E.S.Syrkin
М’які полярні молекулярні шари на зарядженому
нанодроті
В.О.Ликах 1 , Є.С.Сиркін 1,2
1 Національний технічний університет “Харківський політехнічний
інститут”, вул. Фрунзе 21, Харків, 61002, Україна
2 Фізико-технічний інститут низьких температур,
пр. Леніна 47, Харків, 61103, Україна
Отримано 16 серпня 2004 р.
Самоузгоджений теоретичний підхід розроблено для опису незком-
пенсованого носія заряда всередені напівпровідникового квантово-
го дроту, що вкритий м’яким молекулярним шаром. Деформація
молекулярного шару в неоднорідному електричному полі носія
описана в моделі рідкого кристала з власним електричним диполь-
ним моментом. Продольне квантування носія заряда зведено
до спектральної проблеми для нелінійного рівняння Шредингера.
Самоузгоджене рішення спектральної проблеми отримано в термі-
нах еліптичних функцій. Система має таку поведінку: чим вища
взаємодія, тим більша нелінійність; найнижчі квантові рівні відчу-
вають найбільшу нелинійність; ефект має бути більш відчутним
для більш важких дірок; при зростанні взаємодії енергетичні рівні
знижуються і носій локалізується. Поява локалізованих станів може
бути відповідальною за зменшення провідності в нанодротах, що
наблодають в експерименті.
Ключові слова: рідкі кристали, нелінійне рівняння Шредингера,
нанодріт
PACS: 61.30.Gd, 11.10.Lm, 68.65.La, 67.70.+n
812
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