nanomaterials and nanotechnology

132 Pages·2015·33.18 MB·English

NANOMATERIALS AND
NANOTECHNOLOGY
Selection of Papers I 2015 I ISSN 1847-9804

Nanomaterials and Nanotechnology
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Contents
Perturbations of Dipole Decay Dynamics Induced by Plasmonic Nano-antennas –
A Study within the Discrete Dipole Approximation
Stefania D’Agostino, Fabio Della Sala and Lucio Claudio Andreani
Properties of Aluminosilicate Refractories with Synthesized Boron-Modified TiO2 Nanocrystals
Claudia Carlucci, Francesca Conciauro, Barbara Federica Scremin, Antonio Graziano Antico,
Marco Muscogiuri, Teresa Sibillano, Cinzia Giannini, Emanuela Filippo, Caterina Lorusso, Paolo Maria Congedo
and Giuseppe Ciccarella
Synthesis of Nitrogen-Doped Carbon Nanocoils with Adjustable Morphology using Ni–Fe Layered
Double Hydroxides as Catalyst Precursors
Tomohiro Iwasaki, Masashi Tomisawa, Takuma Yoshimura, Hideya Nakamura, Masao Ohyama, Katsuya Asao
and Satoru Watano
Studies of Electrical and Thermal Conductivities of Sheared Multi-Walled Carbon Nanotube with
Isotactic Polypropylene Polymer Composites
Parvathalu Kalakonda, Yanial Cabrera, Robert Judith, Georgi Y. Georgiev, Peggy Cebe
and Germano S. Iannacchione
Polarimetric Detection of Enantioselective Adsorption by Chiral Au Nanoparticles –
Effects of Temperature, Wavelength and Size
Nisha Shukla, Nathaniel Ondeck, Nathan Khosla, Steven Klara, Alexander Petti and Andrew Gellman
A Flexible Sandwich Nanogenerator for Harvesting Piezoelectric Potential from Single
Crystalline Zinc Oxide Nanowires
E. S. Nour, Azam Khan, Omer Nur and Magnus Willander
Graphene-based Electronically Tuneable Microstrip Attenuator
L. Pierantoni, D. Mencarelli, M. Bozzi, R. Moro and S. Bellucci
Plasmonics in Topological Insulators
Yi-Ping Lai, I-Tan Lin, Kuang-Hsiung Wu and Jia-Ming Liu
Towards the Development of a Novel CNTs-based Flexible Mild Heater for Art Conservation
Tomas Markevicius, Rocco Furferi, Nina Olsson, Helmut Meyer, Lapo Governi, Monica Carfagni, Yary Volpe
and Reto Hegelbach
Influence of the Cobalt Phase on the Highly Efficient Growth of MWNTs
Candida Milone, Elpida Piperopoulos, Maurizio Lanza, Saveria Santangelo, Angela Malara,
Emanuela Mastronardo and Signorino Galvagno
Effect of Dopants on Epitaxial Growth of Silicon Nanowires
Sung Hwan Chung, Sarath Ramadurgam and Chen Yang
Characterization and Comparison of Mesoporous Silica Particles for Optimized Drug Delivery
Xinyue Huang, Neil P Young and Helen E Townley
Photonic Structures in Biology: A Possible Blueprint for Nanotechnology
Frank P. Barrows and Michael H. Bartl

Nanomaterials and Nanotechnology
ARTICLE
Perturbations of Dipole
Decay Dynamics Induced by Plasmonic
Nano-antennas – A Study within
the Discrete Dipole Approximation
Invited Review Article
Stefania D'Agostino1,2*, Fabio Della Sala1,3 and Lucio Claudio Andreani2
1 Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia (IIT), Arnesano, Italy
2 Physics Department, University of Pavia, Pavia, Italy
3 Istituto Nanoscienze-CNR, Euromediterranean Center for Nanomaterial Modelling and Technology (ECMT), Lecce, Italy
* Corresponding author(s) E-mail: [email protected]
Received 03 December 2014; Accepted 25 March 2015
DOI: 10.5772/60566
© 2015 The Author(s). Licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License
(http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited.
Abstract distance between the metal and the source of the radiation
is discussed. Applications span from strong coupling
We report a discrete dipole approximation approach to
studies to time-resolved fluorescence spectroscopy.
analyse the perturbations induced by silver nano-particles
on the decay dynamics of a point-like emitter placed in their Keywords Discrete Dipole Approximation, Optical
proximity. Due to the excitation of localized surface Antennas, Localized Plasmons, Dipole Decay Dynamics
plasmons, metallic nano-particles behave like optical
antennas and are able to convert localized fields into free-
propagating optical radiation, and vice versa. Field
localization and enhancement induce strong changes on 1. Introduction
the decay dynamics of dipoles located in the perturbed
electromagnetic environment, and these can be faithfully Since the pioneering work of Purcell in 1946 [1], it has been
quantified within the framework of classical electromag‐ widely known in the field of fluorescence spectroscopy that
netism in terms of total, radiative and non-radiative decay the rate of the emission process can be modified by placing
rates. The method is tested on benchmark cases, i.e., nano- a quantum emitter (QE), such as an atom or a molecule, in
spheres and nano-shells, and it is then applied to analyti‐ a structured polarizable environment [2 - 5]. Usually, the
cally-unsolvable shapes such as sharp nano-cones and interaction between the emitter and its local optical
oxide-covered small nano-antennas. Numerical results environment is such that only the spontaneous emission
reveal 105-order enhancements in the total decay rate of the rate is perturbed, while the emission frequency remains
dipole when located very near to the sharp tip, both with unaltered. In this regime of light-matter interaction, called
and without a thin AgO layer. Moreover, the counter- a weak coupling regime, a fundamental goal of the research
2
intuitive behaviour of the cone response in relation to the has been to quantify the changes induced by the surround‐
Nanomater Nanotechnol, 2015, 5:11 | doi: 10.5772/60566 1
ing medium on the emitter decay dynamics, while at the approach, the molecule is considered as a classical oscillat‐
same time clarifying the balance between the radiative and ing point dipole, and the metal nano-particle as a continu‐
the nonradiative decay channels [6 - 8]. ous body characterized by its own frequency-dependent
dielectric function [13].
In recent years, the plasmonics community has looked with
increasing interest at the possibility of handling the In the regime of very narrow junctions and sub-nanome‐
modification of spontaneous emissions through the tre metal-molecule distances, other interactions become
conscious and controlled use of localized and delocalized important, and chemical effects must also be included in
surface plasmons, this favouring the growth of radiative the treatment [22 - 24]. These primarily comprise the non-
decay-rate engineering (RDE) [9] which has become a central local screening or the finite spatial profile of the plasmon-
issue in nano-photonics. Metallic nano-structures and induced screening charge, and the spill-out or tunnelling
nano-patterned surfaces provide, in fact, a unique way to of electrons outside the nano-particle surfaces [25, 26].
increase the radiative decay rate of fluorophores, and it has Treating these interactions is sometimes mandatory for a
been shown that unusual effects on fluorophores, such as complete understanding of some molecular plasmonics
increasing or decreasing the rates of radiative decay or phenomena, and as such a more sophisticated method
resonance energy transfer (RET), can be achieved [10]. has been developed to study the coupling. It consists of
Moreover, due to both the extreme tunability of the exploiting the continuous body description of the metal
plasmonic response and the near field enhancement in and of treating the molecule atomistically by standard
metallic nano-particle assemblies or geometries, plasmon‐ electronic structure techniques, such as time-dependent
ics offers unprecedented opportunities for controlling light Hartree-Fock (TDHF) and time-dependent density
at the nano-scale. Thus, it allows us to neatly engineer the functional theory (TD-DFT), which include the electro‐
coupling between the electronic excitations of the emitter magnetic interaction in the molecular Hamiltonian [27,
and the collective motion of the metallic electrons in the 28], and by performing a fully quantum mechanical
plasmons. analysis of both the counterparts (i.e., the QE and the
metal) [25, 26].
This important, plasmon-mediated engineering now
becomes possible thanks to the extremely precise control Here, a classical electrodynamics description of a metal-
that we have over the plasmonic modes supported by emitter system is adopted to quantitatively study the
metallic nano-structures. This control derives either from perturbations induced by nano-particles to the spontane‐
the consolidated understanding of the relationship be‐ ous decay rate of a single emitter taken to be point-like, in
tween the details of the nano-structure and the nature of a regime of weak coupling in which a macroscopic descrip‐
the associated plasmon modes, or from the last experimen‐
tion of metal is assumed to be valid [29].
tal developments of powerful and impressive nano-
fabrication and characterization techniques. The A discrete dipole approximation (DDA) [30, 31] approach
development of nano-optics techniques has, thus, greatly is presented as a useful and accurate tool to investigate
contributed to affirming the importance of exploiting coupling problems involving geometries that are not
plasmonic nano-antennas, like metallic nano-particles or analytically solvable and for which more accurate ap‐
nano-tips, and to modifying the excited-state lifetime [11], proaches (partially or fully ab initio) could prove prohibi‐
the fluorescence intensity [11 - 19], and the radiation tive. The idea at the basis of the analysis involves accurately
distribution [17, 20] of isolated emitters. Due to the highly describing the nano-particle shape to acquire a faithful
confined electromagnetic resonances following from the description of the optical response of the metallic compo‐
response of free electrons [12], metallic nano-particles or nent on the emitter, this being necessary to quantify the
nano-structures are in fact able to strongly perturb the perturbations induced on the decay dynamics of the dipole.
electromagnetic fields in their surroundings and to modify A vast literature on DDA exists - it mostly concerns the
both the excitation and the emission rates of proximate solution of the Maxwell problem for metallic structures
fluorophores, chromophores and QDs [21]. excited by plane waves or radiations coming from infinite‐
ly-distant sources. In studying plasmonic antennas’
Despite these experimental examples, the problem of the
behaviour, generally, we have to deal with strongly
electromagnetic coupling between a plasmonic object and
variable electromagnetic fields emitted by dyes or QDs
a source of radiation located in its proximity continues to
placed near the metal nano-structure. Since, for a first
raise numerous open questions in molecular plasmonics,
approximation, the finite size of such kinds of emitters can
especially for complex nano-particle shapes that do not
be ignored (their dimensions being much smaller than
lend themselves to an analytical treatment.
electromagnetic wavelengths in the VIS-NIR range), and
Theoretically, the investigation of electrodynamic coupling these can be considered as point-like radiating dipoles, a
between molecules and metal nano-particles can be worked very attractive possibility would be to extend the numerical
out with different levels of approximation according to the methods (e.g., DDA, the boundary elements method, the finite
description used for the counterparts, and different options element method, etc.) to include incident fields assuming the
can be found in the literature. According to a widely-used form of dipolar fields.
2 Nanomater Nanotechnol, 2015, 5:11 | doi: 10.5772/60566
ing medium on the emitter decay dynamics, while at the approach, the molecule is considered as a classical oscillat‐ The focus of this review is to summarize the recent devel‐ 2. Dipole Decay Rates: Theory and Method
same time clarifying the balance between the radiative and ing point dipole, and the metal nano-particle as a continu‐ opments in this field regarding the DDA method [32 - 34]
The perturbations on the dipole decay dynamics induced
the nonradiative decay channels [6 - 8]. ous body characterized by its own frequency-dependent and to illustrate the working principles and the potential of
by the plasmonic background are analysed here in the
dielectric function [13]. a new implementation of DDA, including incident fields
In recent years, the plasmonics community has looked with framework of the DDA [30, 31, 57] full-wave simulation
generated by local point-like sources [35]. Despite the vast
increasing interest at the possibility of handling the In the regime of very narrow junctions and sub-nanome‐ method, which describes the scatterer as an array of
literature in the field [8, 36 - 53], to the best of our knowl‐
modification of spontaneous emissions through the tre metal-molecule distances, other interactions become polarizable dipolar elements organized on a regular cubic
edge, a crucial issue remains in demonstrating the reliabil‐
conscious and controlled use of localized and delocalized important, and chemical effects must also be included in grid. The polarization of each element is the result of the
ity of a general purpose numerical approach for calculating
surface plasmons, this favouring the growth of radiative the treatment [22 - 24]. These primarily comprise the non- interaction with the local electromagnetic field produced
the total, radiative and non-radiative decay rates of a metal-
decay-rate engineering (RDE) [9] which has become a central local screening or the finite spatial profile of the plasmon-
by all the other elements plus the external field. This
emitter system.
issue in nano-photonics. Metallic nano-structures and induced screening charge, and the spill-out or tunnelling
method yields solutions for the electromagnetic field in
nano-patterned surfaces provide, in fact, a unique way to of electrons outside the nano-particle surfaces [25, 26]. In this paper, a preliminary check performed in two response to an incident electric field in the frequency
increase the radiative decay rate of fluorophores, and it has Treating these interactions is sometimes mandatory for a different cases for which an analytical solution exists domain, including retardation effects. By simply fixing the
been shown that unusual effects on fluorophores, such as complete understanding of some molecular plasmonics (spheres and nano-shells) shows excellent agreement dipole position in the space, we take care of the antenna
increasing or decreasing the rates of radiative decay or phenomena, and as such a more sophisticated method between exact and numerical results for the decay rate response and of the perturbations induced by it on the local
resonance energy transfer (RET), can be achieved [10]. has been developed to study the coupling. It consists of modification, thus placing the method on a firm ground. field at the dipole position in a rigorous way.
Moreover, due to both the extreme tunability of the exploiting the continuous body description of the metal The analysis reported here aims to shed light on the
plasmonic response and the near field enhancement in and of treating the molecule atomistically by standard perturbations on the dipole decay rates induced by 10 nm- We assume a point-like dipole p˜0 (this, and other complex
metallic nano-particle assemblies or geometries, plasmon‐ electronic structure techniques, such as time-dependent sized Ag nano-particles. In particular, the differences quantities throughout this paper, will be indicated with the
ics offers unprecedented opportunities for controlling light Hartree-Fock (TDHF) and time-dependent density between spherically-shaped nano-particles and conically- grapheme placed at r0 and emitting electromagnetic
at the nano-scale. Thus, it allows us to neatly engineer the functional theory (TD-DFT), which include the electro‐ shaped ones are underlined in the dependence of the radiation with a frequency ω near a metallic nano-particle
coupling between the electronic excitations of the emitter magnetic interaction in the molecular Hamiltonian [27, antennas’ response on the dipole distance and orientation: in a vacuum. This nano-particle reflects and/or scatters-back
and the collective motion of the metallic electrons in the 28], and by performing a fully quantum mechanical numerical results assess the capability of a sharp Ag nano- radiation by generating a response electric field E˜ given
scat
plasmons. analysis of both the counterparts (i.e., the QE and the cone to strongly modify the lifetime of an emitter with high by [57]
metal) [25, 26]. sensitivity on the dipole distance and orientation with
This important, plasmon-mediated engineering now
respect to the tip. Numerical results reveal 105-order
btheacto mwees hpaovsesi bolve etrh athnek sp tloa stmheo nexict rmemoedleys psruepcipsoe rctoedn trboyl Hemeritet,e ra cslyasstseimca l ies leacdtroopdtyedn amtoi cqs udaenstcitraiptitvioenly osf tua dmy ettahle- enhancements in the total decay rate of the dipole when E%scat(r)=G%spcat(r,r0,w)×p%0, (1)
metallic nano-structures. This control derives either from perturbations induced by nano-particles to the spontane‐ located very near to the sharp tip. This is of huge impor‐
the consolidated understanding of the relationship be‐ ous decay rate of a single emitter taken to be point-like, in tance, as an example, for the onset of strong coupling where G˜scat(r,r ,ω) is the dyadic Green function for the
tween the details of the nano-structure and the nature of a regime of weak coupling in which a macroscopic descrip‐ interactions [34, 54]. p 0
scattered field describing the electromagnetic response of
the associated plasmon modes, or from the last experimen‐
tion of metal is assumed to be valid [29]. In addition, we present novel results on oxidated nano-
the whole environment (the vacuum and the nano-particle)
tal developments of powerful and impressive nano-
fabrication and characterization techniques. The A discrete dipole approximation (DDA) [30, 31] approach particles. We included a thin Ag2O layer covering the nano- at point r. Here - and hereafter - we use Gaussian units. The
development of nano-optics techniques has, thus, greatly is presented as a useful and accurate tool to investigate particles with the aim of shedding light on the effects of total electric field is given by
oxidation on the optical behaviour of this kind of Ag nano-
contributed to affirming the importance of exploiting coupling problems involving geometries that are not
particle. The appearance of a few nanometres of native
pnalansom-toipnsic, annadn oto-a mntoedninfyaisn, gli tkhee mexectiatleldic- sntaatneo l-ipfeatritmicele [s1 1o]r, apnroaalycthiceas ll(yp asrotilavlalyb loer afunldly faobr inwithioic)h c omulodr ep raocvceu rpartoeh iabpi‐‐ oxide represents, in fact, a real problem in plasmonics E%tot(r)=E%scat(r)+E%inc(r), (2)
applications for metals, e.g., silver and aluminium, and can
the fluorescence intensity [11 - 19], and the radiation tive. The idea at the basis of the analysis involves accurately
be revealed by SEM images or x-ray diffraction analysis
distribution [17, 20] of isolated emitters. Due to the highly describing the nano-particle shape to acquire a faithful where
after a few minutes of exposure to air [55, 56]. We found
confined electromagnetic resonances following from the description of the optical response of the metallic compo‐
that the oxide layer produces a red-shift of the major
response of free electrons [12], metallic nano-particles or nent on the emitter, this being necessary to quantify the
resonances and increased field localization inside the E% (r)=G% (r,r ,w)×p% , (3)
nano-structures are in fact able to strongly perturb the perturbations induced on the decay dynamics of the dipole. inc 0 0 0
metal.
electromagnetic fields in their surroundings and to modify A vast literature on DDA exists - it mostly concerns the
both the excitation and the emission rates of proximate solution of the Maxwell problem for metallic structures The remaining part of the paper is organized as follows. In and G˜ (r,r ,ω) is the free-space dyadic Green function [57].
0 0
fluorophores, chromophores and QDs [21]. excited by plane waves or radiations coming from infinite‐ Sec. II, we summarize the theoretical and computational
The function G˜scat(r,r ,ω) is related to G˜ (r,r ,ω) by [57]
ly-distant sources. In studying plasmonic antennas’ method by underlining the relationship existing among the p 0 0 0
Despite these experimental examples, the problem of the
behaviour, generally, we have to deal with strongly normalized decay rates (radiative, non-radiative and total).
electromagnetic coupling between a plasmonic object and
a source of radiation located in its proximity continues to variable electromagnetic fields emitted by dyes or QDs The reliability of the method is then shown and discussed G%spcat(r,r0,w)=
placed near the metal nano-structure. Since, for a first in Sec. III by comparing the DDA decay rates for Ag nano- (4)
raise numerous open questions in molecular plasmonics, òG% (r,r,w)T% (r,w)G% (r,r ,w)d3r,
especially for complex nano-particle shapes that do not approximation, the finite size of such kinds of emitters can spheres and nano-shells with the exact electrodynamic P 0 1 p 1 0 1 0 1
be ignored (their dimensions being much smaller than results that are well-known for these geometries. In Sec. IV,
lend themselves to an analytical treatment.
electromagnetic wavelengths in the VIS-NIR range), and we then calculate the dipole total decay rate modifications
where the scattering operator for the nano-particle is [57]
Theoretically, the investigation of electrodynamic coupling these can be considered as point-like radiating dipoles, a in the proximity of variously-shaped small Ag nano-
between molecules and metal nano-particles can be worked very attractive possibility would be to extend the numerical particles (spheres, nano-shells and cones) with and without
out with different levels of approximation according to the methods (e.g., DDA, the boundary elements method, the finite a thin silver oxide layer covering the nano-particle and T% (r,w)=c% (r,w)I+
p p
description used for the counterparts, and different options element method, etc.) to include incident fields assuming the study the dependence of them on the emitter distance and +c% (r,w)òG% (r,r,w)T% (r,w)d3r, (5)
can be found in the literature. According to a widely-used form of dipolar fields. orientation. In Sec. V, conclusions are drawn. p P 0 1 p 1 1
2 Nanomater Nanotechnol, 2015, 5:11 | doi: 10.5772/60566 Stefania D'Agostino, Fabio Della Sala and Lucio Claudio Andreani: 3
Perturbations of Dipole Decay Dynamics Induced by Plasmonic Nano-antennas – A Study within the
Discrete Dipole Approximation
and χ˜p(r,ω) is the (local) dielectric susceptibility of the G =1+3 q0 åN Imé p% ×G% (r,r ,w)×p%*ù .
nreasntroi-cpteadrt ictole . thNeo tnea tnhoa-tp ainrttiecglera lvso ilnu mEeq,s . a(s4 )χ ˜an dv a(n5i)s haeres G0 2k3 p%02 i=1 ë i 0 i 0 0û (9)
p
outside it.
Eq. (9) can be further simplified assuming (as usual) that
Here, attention is focused on the scattered field E˜
scat the external perturbing dipole p˜ =p is real. Thus, from Eq.
0 0
exactly at the dipole source position r . Eqs. (4) and (5)
0 (3) we obtain
are difficult to solve, and different approximations have
been considered in the literature. For example, the nano-
phoarwtiecvlee ri,t stehlifs c dainp oblee- daipppolreo xaipmparotexdim aast aio pno cinant dbiep oalpep [l3ie9d]; GG0 =1+23k3qp002åiN=1Iméë p%i×E%inc(ri)ùû , (10)
only for nano-particle-molecule distances larger than a
few (i.e., two to three) radii of the metal nano-particle,
otherwise the total decay rate is largely underestimated which involves only quantities readily available in DDA.
[58]. Higher multi-pole moments are needed for a correct
It has been largely proved that the change expressed by Eq.
description [58, 59], as is the case in the Gersten and
(7) and known as the ’Purcell effect’ [1] can be ascribed to
Nitzan model [59, 60].
the modifications of both the radiative decay rate due to
The underlying idea of the present work is to numerically photon emission and the non-radiative decay rate due to
approach the exact local field at the dipole position by energy dissipation in the environment [67 - 70]. For emitters
solving the Maxwell’s equations within the DDA frame‐ close to metal surfaces, both rates can be enhanced. If the
work. By solving in a self-consistent manner a system of 3N nano-particle is dissipative, Γ/Γ0 is the sum of the intrinsic
-coupled complex equations, DDA gives N dipoles p˜ non-radiative term Γ /Γ =(1−q), the radiative decay
i NR,0 0 0
describing the polarization of the target so that the scattered enhancement (Γ /Γ) and the quenching rate enhancement
R 0
field experienced by the dipole [61] can be written as induced by the lossy environment (Γ /Γ) [71, 15, 16], i.e.,
NR 0
E%scat(r0)=åiN=1G%0(r0,ri,w)×p%i, (6) GG =(1-q0)+GGR +GGNR . (11)
0 0 0
where p˜ is located at r.
i i According to semi-classical theory [59], the non-radiative
decay rate is Γ =P /ℏω, where P is the (time-aver‐
In the weak-coupling interactions regime, the knowledge NR abs abs
aged) power absorbed by the nano-particle
of the scattered field at r allows us to compute the irrever‐
0
sible enhancement of the total emission rate [62 - 65] or the
total decay rate normalized to the free-space value [8, 57] P =ò wIméc% (r)ùE% (r)2d3r
abs P2 ë p û tot
(12)
G =1+3 q0 Imép%*×E% (r )ù, =w2åNVcIméëc%p(ri)ùûE%tot(ri)2 .
G0 2k3 p%02 ë 0 scat 0 û (7) i
In Eq. (12), we discretized the integral over the nano-
where Γ is the total decay rate of a (classical) dipolar particle: the summation is done on the N dipoles’ contri‐
0
butions, V is the volume of each cubic element, and E˜ (r)
emitter with an intrinsic quantum yield q i.e., correspond‐ c tot i
0
ing to the total radiated power (P ) of the dipole p˜ in a is the total internal electric field at position ri.
0 0
vacuum, normalized to the energy
The normalized non-radiative decay rate Γ thus becomes
NR
G0 =q10 hPw0 = 3q10h p%02k3 . (8) GGN0R = 2k33qp%002æçèåiN=1Iméëc%p(ri)ùûE%tot(ri)2×Vcö÷ø (13)
Note that in the case of a QE, p˜ will be double the transition
0 and by using the relation p˜ =V χ (r)E˜ (r) [31],
i c p i tot i
dipole moment between the ground-state and excited-state
level [57].
G 3q N
Ufusninctgio Enq is. s(6y)m amnedt rtihc e[ 6f6a]c, tw teh acat nt hreew srpiatec eE-qd.y (a8d) iacs Green GN0R = 2p002k3åi=1Iméëp%i×E%t*ot(ri)ùû . (14)
4 Nanomater Nanotechnol, 2015, 5:11 | doi: 10.5772/60566