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National Academy of Sciences of Ukraine, Chuiko Institute of Surface Chemistry, Ukraine

- *Corresponding Author:
- Pokutnyi S

National Academy of Sciences of Ukraine

Chuiko Institute of Surface Chemistry

17 General Naumov Str, Kyiv, 03164, Ukraine

**Tel:**+38044 424 12 35

**E-mail:**[email protected], [email protected]

**Received Date**: May 10, 2017; **Accepted Date:** May 31, 2017; **Published Date**: June 31, 2017

**Citation: **Pokutnyi S (2017) Strongly Optical Absorbing Nanostructures Containing
Metal Quantum Dots: Theory. J Laser Opt Photonics 4: 156. doi: 10.4172/2469-410X.1000156

**Copyright:** © 2017 Pokutnyi S. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Lasers, Optics & Photonics

In framework of dipole approximation it is shown, that the oscillator strengths of transitions as well as the dipole moments for transitions for one-particle electron Coulomb states emerging above the spherical surface quantum dot of metal assume giant values considerably (by two orders of magnitude) exceeding the typical values of the corresponding quantities for dielectrics. It has been established that the giant values of the light absorption cross section in the nanosystems under investigation make it possible to use such nanosystems as strongly absorbing nanomaterials in a wide range of infrared waves with a wavelength that can be varied in a wide range depending in the type of contacting materials.

One-particle electron Coulomb states; Quantum dots; Light absorption

At present, the optical and electro optical [1-6] properties of quasizero-
dimensional structures are extensively studied. Such structures
commonly consist of spherical semiconductor, metal and insulator
nanocrystals (the so-called quantum dots (QDs)) with a radius a ≈ 1-10^{2} grown in dielectric (or semiconductor) matrices. The studies in this
field are motivated by the fact that such nanoheterosystems represent
new promising materials for the development of new elements of nanooptoelectronics
to be used, specifically, for controlling optical signals in
optical computers or for manufacturing active layers of optical lasers [1]
as well as new strongly absorbing nanomaterials [7]. Special attention is
paid to analysis of optical properties of such nanosystems in view of its
unique photoluminescence properties and the ability to effectively emit
light in the visible or near infrared ranges at room temperatures [1].

Previously [8], the conditions for the localization of charge carriers
near the spherical interface between the two dielectric media were
analyzed. In this case, the polarization interaction of a charge carrier
with the surface charge induced at the spherical interface, U(r,a)
depends on the relative permittivity ε=(ε_{1}/ε_{2}). Here, r is the spacing
between the charge carrier and the center of the dielectric QD; a is the
radius of the QD; and ε_{1} and ε are the permittivities of the surrounding
medium and of the dielectric QD embedded in the medium,
respectively. For the charge carriers in motion near the dielectric QD,
there are two possibilities: due to the polarization interaction U(r,a),
the carriers can be attracted to the QD surface (to the outer or inner
surface at ε<1or ε>1, respectively), with the formation of outer [9] or
inner surface states [9].

It has been show [8,9] that the formation of the above-mentioned
local states is of a threshold – type nature and is possible if the radius of
the dielectric QD a is large than a certain critical radius a_{c}:

where (1)

(2)

is the Bohr radius of a charge carrier (mi is the effective mass of the
charge carrier) in a medium with the permittivity; ε_{i} (i=1,2) is the
average distance from a charge carrier localized over the planar interface
in the ground state to this surface, parameter β=(ε_{2} – ε_{1})/(ε_{1} +ε_{2}). The
interaction of the electromagnetic field with one-particle localized states of charge carriers emerging near the spherical QD–matrix
interface [9,10] was studied in [7,8]. It was shown that localization
of charge carriers on a spherical surface and in the bulk of QDs was
manifested in different ways in the size and frequency dependences of
light absorption and scattering. This paved new ways for spectroscopic
investigations of such localized states in nanosystems [2-5].

Investigations in the theory of absorption and scattering of light at outer surface Coulomb states in nanosystems have not been performed as of yet; to fill this gap, a theory of interaction of the electromagnetic field with the Coulomb states of charge carriers emerging in nanosystems on the outer surface of metal QDs is developed in this study. In present work in framework of dipole approximation it is shown, that the oscillator strengths of transitions as well as the dipole moments for transitions for one-particle electron Coulomb states emerging above the spherical surface QDs of metal assume giant values considerably (by two orders of magnitude) exceeding the typical values of the corresponding quantities for dielectrics. It has been established that the giant values of the light absorption cross section in the nanosystems under investigation make it possible to use such nanosystems as strongly absorbing nanomaterials in a wide range of infrared waves with a.

Let us consider model of a quasi-zero-dimensional system,
viz., a neutral spherical insulator QD of radius a with permittivity
ε2 surrounded by a medium with permittivity ε1 (such that relative
permittivity is ε=(ε_{1}/ε_{2}) << 1). An electron (e) with an effective mass
m1 is localized over a spherical interface (QD-dielectric matrix) (the
electron moves in a dielectric matrix with permittivity ε_{1}). The fact that all characteristic sizes of the problem (a and b_{1}) are considerably larger
than atomic spacing a_{0} makes it possible to consider the motion of
quasi-particles in the nanoparticle in the effective mass approximation
[8]. In ref. [8], the energy spectrum of outer surface states of a quasiparticle,
which appear over a spherical interface (QD-dielectric matrix)
(for ε<<1) was investigated, as well as its dependence on radius a of
the QD under the conditions when the polarization interaction of the
charge carrier with the spherical interface between the two media plays
the leading role. It was shown that the spectrum of the outer surface
states of the quasi-particle upon an increase in nanoparticle radius a
such that

(3)

is transformed into the spectrum of the Coulomb form

(4)

where n and l are principal and orbital quantum numbers, L^{2}=l (l +1).
Here, we are using the energy units (Ry/36)=(/2(h^{2}/2 m_{1} b_{1}^{2}).

In the frequency range corresponding to
Coulomb states (n, l) (4) an electron of a QD localized above the surface
in the QD of radius S (3), the wavelength of the light wave considerably
exceeds the sizes of these states (≈b_{1} (1)). Therefore, the behavior of such
Coulomb states in the electromagnetic field is successfully described by
the dipole approximation [7]. In this case, the dipole moment operator
for a charge carrier in the QD has the form [10]

(5)

where r is the radius vector determining the distance between the charge carrier and the center of the QD.

To estimate the dipole moment D_{2,1,1,0}(a) it is sufficient to consider
the transition between the lowermost Coulomb states (4) (e.g., between
Coulomb ground state 1s⟩=(n=1, l=0) and 2p⟩=(n=2, l=1) Coulomb
state). A transition between such states is allowed by the selection rules
in the Coulomb field (in this case, the principal quantum number n
changes arbitrarily, while the orbital quantum number l changes by
unity) [7].

Using relations (5), we can write the expression of the dipole moment of the transition [7]:

(6)

in QDs with radii S satisfying inequality (3).

The oscillator strength of the transition of a charge carrier with
effective mass m_{1} from ground state 1s to state 2p assumes the form
[7,8]

(7)

where are the energies of Coulomb levels 2p and 1s, respectively. With allowance for formulas (4) and (6), we can write the oscillator strength (7) of the transition in the form

(8)

The cross section of light absorption on the spherical surface of a QD of radius a can be expressed in terms of its polarizability A″(ω,a) [8]:

(9)

where ω is the frequency of the external electromagnetic field and c is the speed of light in vacuum. At temperatures

(10)

lower than the binding energy Eb (S)=Enl=(S) (4) of the Coulomb states (n, l) (4) (where k is the Boltzmann constant), the polarizability of a charged QD can be determined if we treat the QD as a giant ion [7]. The main contribution to polarizability A″(ω,a) Ain this case comes from transitions in the discrete spectrum of such Coulomb states. Separating in polarizability A″(ω,a) the contribution from only one resonant term corresponding to the transition between the ground 1s and 2p Coulomb states, we can write polarizability A″(ω,a) of the QD in the form [7]

(11)

where Ґ_{2,1}(a) is the width of the Coulomb 2p level.

Assuming that frequency ω of the light wave differs significantly
from resonance frequency ω_{2,1}(a) (4) of the Coulomb 2p state and that
broadening Ґ_{2,1}(a) of level 2p is small Ґ_{2,1}(a)/ω_{2,1}(a)<< 1 ([8]), we obtain
the following expression for the qualitative estimate of polarizability
A″(ω,a) (11) of the QD with allowance for eqn.(4).

(12)

We can now write the expression for the cross section of elastic scattering of an electromagnetic wave of frequency an QD of radius S (3) [8]:

(13)

The outer surface Coulomb states of electrons under investigation,
which are localized over a spherical of metal QDs of radii a (3), can be
studied in the processes of absorption (and emission) on transitions with frequencies , which lie
in the infrared spectral region in accordance with (4). Let us estimate
absorption cross sections σ_{abs}(ω,a) (9) and scattering cross sections
σ_{SC} (ω,a) (13) for light at the above-mentioned Coulomb states of the
electron localized over a spherical surface QDs of metal of radii a (3),
in the case of singled-out transition . The estimates of
oscillator strength of the transition f_{2,1;1,0}(S) (8), dipole moment of the
transition D_{2,1;1,0}(S) (6), polarizability A’’(ω, a) (11), and cross section
σabs(ω,a) (9) of absorption of a light wave with frequency ω (in this case,
ratio (ω/ω_{2,1}(S))2 =9 10-2 and the wave frequency ω lies in the infrared
region) at the above Coulomb states of the electron appearing over a
spherical surface (QD of metal – matrix silicate glass) are given in the
table. If we take into account the fact (see the table) that the oscillator
strength f_{2,1;1,0} ≈ 0.4 and the dipole moment D_{2,1;1,0} ≈ 1.85 (where
(D0 = e Ǻ), (Debye)) of the transition over a spherical surface QDs
of metal of radii a=10 nm assume giant values (exceeding the typical
values of oscillator strength and dipole moments in matrix silicate glass
by two orders of magnitude [1-5] and dipole transitions between the
nearest Coulomb levels Enl (a) (4) in QDs in the electromagnetic field
are allowed by the selection rules with a change (or preservation) of
principal quantum number n and with a change in orbital quantum
number l by unity [7], the quasi-zero-dimensional nanosystems under
investigation are obviously strongly absorbing nanostructures for
infrared radiation.

The estimates given in the table lead to the conclusion that the cross sections of light absorption in QD of radius a=10 nm attains
giant values σ_{abs}(,a) ≈ 10-17cm^{2}. This value of σ_{abs}(ω,a) (9) is seven
orders of magnitude higher than typical values of atomic absorption
cross sections [7,8]. Since the scattering cross section σ_{abs}(,a) (13) is
negligibly small as compared to the corresponding value of absorption
cross section σ_{abs}(ω,a) (9) ((σ_{cs}/_{abs}) ≈ 10^{-12}), the value of σ_{cs}(ω,a) is not
given in the table.

Thus, we have shown using the dipole approximation that the oscillator strengths of transitions as well as the dipole moments for transitions for one-particle electron Coulomb states emerging in over a spherical surface (QD of metal-matrix silicate glass) assume giant values considerably (by two orders of magnitude) exceeding the typical values of the corresponding quantities for dielectric matrices. It has been established that the giant values of the light absorption cross section in the nanosystems under investigation make it possible to use such nanosystems as strongly absorbing nanomaterials in a wide range of infrared waves with a wavelength that can be varied in a wide range depending in the type of contacting materials.

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