More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\\n\\n
Our breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\\n\\n
“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\\n\\n
Additionally, each book published by IntechOpen contains original content and research findings.
\\n\\n
We are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
Simba Information has released its Open Access Book Publishing 2020 - 2024 report and has again identified IntechOpen as the world’s largest Open Access book publisher by title count.
\n\n
Simba Information is a leading provider for market intelligence and forecasts in the media and publishing industry. The report, published every year, provides an overview and financial outlook for the global professional e-book publishing market.
\n\n
IntechOpen, De Gruyter, and Frontiers are the largest OA book publishers by title count, with IntechOpen coming in at first place with 5,101 OA books published, a good 1,782 titles ahead of the nearest competitor.
\n\n
Since the first Open Access Book Publishing report published in 2016, IntechOpen has held the top stop each year.
\n\n\n\n
More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\n\n
Our breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\n\n
“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\n\n
Additionally, each book published by IntechOpen contains original content and research findings.
\n\n
We are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\n\n
\n\n
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"5378",leadTitle:null,fullTitle:"The Epidemiology and Ecology of Leishmaniasis",title:"The Epidemiology and Ecology of Leishmaniasis",subtitle:null,reviewType:"peer-reviewed",abstract:"Leishmaniasis is a vector-borne, parasitic disease with tremendous variety in presentation, biology, and epidemiology. 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\n\t\t\t
1. Introduction
\n\t\t\t
The perennial interest in studying the physical properties of nanofilms has increased substantially over the last few years due to the development of nanotechnologies and the synthesis of new compounds – especially those based on carbon, which are extremely interesting for both fundamental research and potential applications.
\n\t\t\t
An important feature of carbon nanofilms (including those with defects) is a close relation between the electronic and phonon properties, which is exhibited, for example, in the graphene-based systems with superconducting properties [1,2].
\n\t\t\t
It is well known that graphene monolayers cannot exist as planar objects in the free state, because in flat 2D-crystals the mean-square amplitudes of the atoms in the direction normal to the layer plane diverge even at \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t(see, e.g., [3]). So we can study and practically apply only such graphene, which is deposited on a certain substrate providing the stability of the plane carbon nanofilms (see, e.g., [4-6]). Only small flakes can be detached from the substrate and these flakes immediately acquire a corrugated shape [7]. When studying the electronic properties of graphene a dielectric substrate is often used. The presence of the substrate greatly increases the occurrence of various defects in graphene and carbon nanofilms. Our investigations make it possible to predict the general properties of phonon and electron spectra for graphene and bigraphene containing different defects.
\n\t\t\t
This chapter consists of three sections: first section is devoted to the calculation of local discrete levels in the electron spectra of graphene with different defects. In the second section we describe the electronic properties of bilayered graphene and, finally, the third section deals with the influence of defects on electron spectra of bigraphene.
\n\t\t
\n\t\t
\n\t\t\t
2. Impurity levels in the electron spectra of graphene
\n\t\t\t
The exceptionality of graphene is manifested in the phonon and electron properties. Graphene is a semimetal whose valence and conduction bands touch at the points K and K’ of the Brillouin zone [8,9]. In the pure graphene unique electronic properties are manifested by the charge carriers behaving as massless relativistic particles - the dependence of energy on the momentum is linear rather than - as in ordinary solids - quadratic. Thus, the lower-dimensionality affects the formation of phonon localized states [10] and also the formation of localized states in the electronic spectrum. Absent gap between the valence and conduction bands is a consequence of the symmetry between two equivalent sublattices in graphene [11]. Presence of impurities lowers the symmetry. The influence of vacancies placed into one of the graphene sublattices was investigated in [12], where it was shown that the equivalence of the sublattices is broken.
\n\t\t\t
In this section we describe the characteristics of localized and local states present in graphene due to the impurities of nitrogen and boron, respectively. The presence of the substrate greatly increases the possibility to introduce various defects into graphene. For example, in the graphene deposited on silicon, vacancies can occur [13, 14], whereas in graphite (a set of weakly interacting graphene monolayers) vacancies heal and form a stacking fault with local fivefold symmetry axis [15]. Impurity atoms embedded in graphene may lead to the appearance of impurity states outside the band of quasi-continuous spectrum. At low impurity concentrations (when impurity is considered as an isolated defect) these states appear in the form of local discrete levels (LDL).
\n\t\t\t
Although such levels in various quasiparticle spectra have been known and studied over 60 years, an adequate description is still absent, even in the harmonic approximation for sufficiently realistic models of the crystal lattice. The dependence of the appearance conditions and characteristics of LDL on the parameters of a perfect lattice and defect was identified only in the most general terms. However, LDL may be used as an important source of information about the defect structure and force interactions in real crystals. To extract such information it is useful to have analytical expressions that relate main characteristics of LDL to the parameters of both the defect and the host lattice.
\n\t\t\t
Here we present the results of our calculations and analyses of the characteristics of the electronic local discrete levels for substitutional impurities in graphene, especially for a boron substitutional impurity, using an analytical approximation based on the Jacobi matrices method [16, 17].
\n\t\t\t
The fact that the charge carriers in graphene are formally described by the Dirac equation and not by the Schrödinger equation is due to the symmetry of the crystal lattice of graphene, which consists of two equivalent carbon sublattices. Electronic subbands formed by the symmetric and antisymmetric combinations of wave functions in the two sublattices intersect at the edge of the Brillouin zone, which leads to a cone-shaped energy spectrum near the K and K´ points of the first Brillouin zone. The electrons obey the linear dispersion law (in ordinary metals and semiconductors the dispersion law is parabolic).
\n\t\t\t
The electronic spectrum of graphene can be described by a strong coupling approximation, and it is sufficient to consider the interaction between nearest neighbors only (see, e.g., [5,6,18-20]). The corresponding Hamiltonian is
where i and j are the labels of the nodes of the two-dimensional lattice, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the energy of electron at node i, and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the so-called overlap integral.
\n\t\t\t
Curve 1 in Figure 1 shows the density of electronic states of graphene as calculated using the method of Jacobi matrices [16, 17]. In a perfect graphene the local Green\'s function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t coincides with the total Green’s function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tlim\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t〈\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t〉\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t because of the physical equivalence of the atoms of both sublattices. Peculiarity of the density of states at \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (the value \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t corresponds to the Fermi energy \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t in graphene) determines the behavior of the real part of the Green\'s function near \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. For a wide class of perturbations caused by defects we can find, using the Lifshitz equation [21], quasilocalized states in the interval \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (in this model\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t). This equation, which determines the energy of these states, can be written as (see, e.g., [3,17])
where the \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tΛ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t function is determined by the perturbation operator \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΛ\n\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΛ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tare matrix elements of this operator on defined basis).
\n\t\t\t
The local spectral densities \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tlim\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t↓\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tIm\n\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t of impurity atoms are calculated in [6]. For an isolated substitutional impurity with the energy \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t of the impurity node \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and with the overlap integral\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t has the form
\n\t\t\t
Figure 1.
Electronic density of states of perfect graphene (curve 1) and the local density of states for an isolated boron substitutional impurity (curve 2).
For a nitrogen impurity \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t0.525\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (according to [4]). As shown in [22], equation (2) has a solution for both interval \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and interval\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
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The local density of states of the nitrogen substitutional impurity calculated in [11] has quasi-local maxima in both intervals. For an boron substitutional impurity (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) [6], quasi-localized states are absent in the \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t interval [12]. Figure 2 shows the graphical solution of the Lifshitz equation (2) for a given impurity atom. In this case the Lifshitz equation has no solutions in interval \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (corresponding dependences \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are shown as curves 3 in Figure 2). The local Green\'s function of the boron impurity (curve 2 in Figure 1) has two poles outside the band of quasi-continuous spectrum, which are called local discrete levels and which are also solutions of equation (2). As is clearly seen in Figure 1 the area under the curve 2 is smaller (by the sum of the residues at these poles) than the area under the curve 1.
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Local discrete levels can be an important source of information about defective structure and force interactions in real crystals. To extract this useful information we should have analytical expressions that relate the main characteristics of LDL (primarily their energy) to the parameters of the defect and the host lattice.
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Figure 2.
Graphical solution of equation (2) for boron substitutional impurity in graphene. Curve 1 is the electronic density of states of ideal graphene, curve 2 is the corresponding real part of the Green\'s function. Curves \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t are the “approximations of two moments” of these functions. Curve 3 represents the function\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
\n\t\t\t
Such expressions were obtained in [23] for localized vibrations in the phonon spectrum of a three-dimensional crystal. Authors proposed an analytical approximation of the basic characteristics of local vibrations based on the rapid convergence of the real part of the Green\'s function outside the band of quasi-continuous spectrum using the method of Jacobi matrices [16,17].
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Let us, briefly, to the extent necessary to understand the use of the classification of the eigenfunctions of Hamiltonian (1), to present the basics of the method of Jacobi matrices. This method allows, without finding the dispersion laws, to calculate directly the local partial Green\'s functions of the system, corresponding to the perturbation of one or more atoms. This perturbation is described by the so-called generating vector\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where H is the space of electronic excitations of atoms. Its dimension is\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where N is the number of atoms in the system, and q is the dimension of the displacement of a single atom (q = 1, 2, 3). Vectors of the space H are denoted by an arrow above the symbol, and “ordinary” q-dimensional vectors are in bold italics.
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If, using the generating vector \n(p is the number of excited atoms) and the Hamiltonian (1), we construct the sequence\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, then the linear envelope covering the vectors of this sequence forms, in the H space, a cyclic subspace invariant to the operator\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. This subspace contains, within itself, all the atomic displacements generated by the vector\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. The corresponding partial Green\'s function is determined as a matrix element\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t00\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the eigenvalue. Quantity \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tIm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t00\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is called the spectral density generated by the initial displacement\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. In the basis \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\twhich is obtained by the orthonormalization of the sequence\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the operator (1) is represented in the form of a tridiagonal Jacobi matrix (or J-matrix). This matrix has a simple spectrum, what greatly simplifies finding the partial Green\'s functions and spectral densities. As can be seen, this method does not use explicitly the translational symmetry of the crystal, making it extremely effective for treating systems in which such symmetry is broken. The method of Jacobi matrices is particularly effective for treating systems with a simply connected quasi-continuous band of spectrum D. In this case, with increasing rank of the J-matrix (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t), its diagonal elements \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t converge to a corresponding to the middle of the bandwidth D, and nondiagonal elements b\n\t\t\t\t\n\t\t\t\t\tn\n\t\t\t\t converge to b corresponding to the one-quarter of the bandwidth D.
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For the local Green\'s function (LGF), corresponding to the excitations of one or more atoms, which are determined by the generating vector\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, we get following expression using the J-matrix method
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the unit operator and polynomials \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are determined by the following recurrence relations
The initial conditions are\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t,\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t,\n\t\t\t\t\n\t\t\t\t\t\n \n \n Q\n 0\n \n (\n ε\n )≡0\n \n\n\t\t\t\t , and\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t corresponds to the LGF operator, with all elements of its J-matrix being equal to their limit values a and b:
The method of Jacobi matrices can treat as a regular singular perturbation a much larger number of perturbations of the phonon spectrum due to the presence of various crystal defects than the traditional methods [19, 20]. In addition, perturbations do not change the bandwidth of the quasi-continuous spectrum, and consequently, the asymptotic values of the elements of the J-matrix can be regarded as an asymptotically degenerated regular perturbation [23]. This type of perturbations covers virtually all perturbations of the phonon spectrum caused by local defects. The calculation of vibration characteristics of such systems is performed, using the method of J-matrices, with the same accuracy as for the initial ideal system.
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In practice, it is usually possible to calculate the Jacobi matrix of the Hamiltonian of a finite rank. The expression
is called analytical approximation of LGF. All dependences in Figure 1 and curves 1 and 2 in Figure 2 were calculated by the formula (8), using the Jacobi matrix of the Hamiltonian (1) with rank\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t600\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. If we count the energy from the Fermi energy level, then all diagonal elements of Jacobi matrices are zero (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the half-width of the quasi-continuous spectrum). A good accuracy of the approximations shown in figures is confirmed also by the fact that they show the nonanalyticity effects corresponding to the densities of states of systems with the dimension larger than unity (so-called van Hove singularities). In the vicinity of these singularities the expression (8) slowly converges to the true values of the really and imaginary parts of LGF.
\n\t\t\t
Curves \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t in Figure 2 show local density of states and their corresponding real parts of the LGF calculated by formula (8) with n=1. As can be seen in the band of the quasi-continuous spectrum these relationships have very little in common with curves 1 and 2. Thus, the curve \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t does not even hint at the V-shaped “Dirac” singularity at \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, and on the curve \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t in the interval \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t both non-monotonous parts and the logarithmic singularities at the edges of the band of quasi-continuous spectrum, characteristic for the 2D systems are absent. However, outside the band of quasi-continuous spectrum (also in the area of intersection of the real part of LGF with curve (3)) curves 2 and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tpractically coincide, and if we put in the Lifshitz equation (2) instead of LGF its approximation (8) for n=1, the obtained solutions give the energies of LDL with quite high accuracy. Moreover, these solutions can be easily found analytically. The LGF approximation by formula (8) for n=1 was named the approximation of two moments in [23]. Indeed, it follows from the orthonormality of the polynomials defined in [16,17] that
Finding the characteristics of LDL is more convenient without using equation (2), looking for them as the poles of the LGF perturbed Hamiltonian \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tΛ\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. In the approximation of two moments for the subspace generated by the excitation of an impurity atom, we get
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and
Local discrete levels are poles (11), i.e. the roots of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tare
Residues at these poles \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t‵\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are called intensities of LDL and they determine the relative LDL “amplitude” on the impurity atom: \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∫\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tIm\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\td\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. The condition that the intensity differs from zero defines the existence region of LDL. In this case
This implies that the damping of LDL, i.e. the decay of its intensity with the increasing distance from the impurity atom (i.e. with the increase of n) follows the equation \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Using the method of mathematical induction we can prove that
The intensities \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t decay with increasing n according to \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, that is, starting from n=1 they form an infinitely decreasing geometric progression whose denominator
Summing these progressions we see that \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, that is the formation of each LDL is the formation of one quasi-particle outside the band of the quasi-continuous electron spectrum.
\n\t\t\t
Formulas (10, 12-14, 16) give simple analytical expressions of the local conditions of the existence of discrete levels due to the presence of a substitutional impurity in graphene.
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Regions of the LDL existence for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (in this case\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) and for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t(in this case\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) lie above and under curves in Figure 3, as indicated by arrows.
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It is seen that such levels exist in a very wide range of variables \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tη\n\t\t\t\t\t\n\t\t\t\t. The absence of LDL is possible only in a narrow range of values \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t at \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. If \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t at least one local discrete level exists. In fact, the lines delineating the area of existence of LDL in graphene in the plane \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t must pass through the origin of coordinates, since \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRe\n\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. However, since this divergence is logarithmic, for any appreciable splitting of LDL from the boundary of quasi-continuous spectrum there is a certain threshold. Curves 2 and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t in Figure 2 merge at \n\t\t\t\t\t\n \n |ε|>\n ε\n 0\n \n \n \n\n\t\t\t\t.
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Figure 3.
Regions of the existence of discrete levels for substitutional impurities in graphene.
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\n\t\t\t\tFigure 4 shows, for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t0.525\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (for boron, from [6]), the dependences of energies, the LDL intensities at the boron impurity and the damping parameters of the value \n\t\t\t\t\t\n\t\t\t\t\t\tη\n\t\t\t\t\t\n\t\t\t\t that characterizes the change in the overlap integral of the boron impurity atom (10). Solid lines show the characteristics of LDL, calculated using the approximation of the Green\'s function (9), i.e. according to the analytical formulas (13), (14) and (16). Open circles show the results of numerical calculations of dependences \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tη\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, using the Green\'s function in the form of (8), calculated by the Jacobi matrices of the \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t100\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t rank.
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Figure 4.
The basic characteristics of LDL in the presence of a boron substitutional impurity in graphene.
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It is seen that at the threshold values \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t of the LDL formation, the intensities of LDL equal zero and the parameters of damping are equal to unit. Further increase of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis accompanied by the increase of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and by the strengthening of the containment level.
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So, a good agreement between the results of numerical calculation of the LDL characteristics using Jacobi matrices of high rank, and their analytical description by the Green\'s function (9), which relates these characteristics to the parameters of the defect (13) (14), makes it relatively easy to extract the defect parameters from the known characteristics of LDL. Experimental measurement (e.g. by scanning tunneling microscopy) of values \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t should lead, using (10), (12) and (13), to the determination of the parameters \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t˜\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tη\n\t\t\t\t\t\n\t\t\t\t and this might represent a significant advance in creating nanomaterials with predetermined spectral characteristics. As can be seen from Figure 4, with increasing \n\t\t\t\t\t\n\t\t\t\t\t\tη\n\t\t\t\t\t\n\t\t\t\t the intensity of LDL \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tμ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. That is, the impurity levels can not be completely localized on the impurities, but they also appear in the spectra of surrounding carbon atoms. This greatly increases the probability of experimental detection of such levels, even at low concentrations of impurities.
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3. The electronic spectrum of bilayer graphene
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Bilayer graphene is a carbon film consisting of two graphene monolayers, linked together by (as in bulk graphite) van der Waals forces. Since the distance between the layers (film thickness) \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t3.5\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tÅ the bilayer graphene can be considered not a nanofilm but a subnanofilm. The constants of the interatomic interaction of bilayer graphene were determined and its phonon density of states and partial contributions to this quantity from the atomic displacements along different crystallographic directions were calculated [7,24]. On the basis of the analyses of the mean-square amplitudes of atomic displacements calculated using data from the spectral densities, we have shown that the flat shape of a free bilayer graphene remains stable up to the temperatures much higher than the room temperature, which makes this compound promising for nanoelectronics. In this section we calculate and analyze the electronic spectrum of a defect-free bilayer graphene. Naturally of greatest interest is its behavior in the energy range close to \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t where there are characteristic Dirac points on the spectrum of graphene monolayer (whose plane shape is unstable).
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The unit cell of graphene contains two physically equivalent atoms and therefore local Green\'s function and the local density of states (LDOS) of the atoms of different sublattices are identical. On the other hand, bilayer graphene unit cell contains four atoms, and atoms of different sublattices of a single graphene layer interact differently with the atoms of the other layer and their physical equivalence is disrupted (Figure 5a).
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The electronic spectrum of bilayer graphene, as well as the electronic spectrum of graphene can be described in the strong coupling approximation. Corresponding Hamiltonian has form (1). For graphene and bigraphene we assume that the electron hopping within the layer is possible only between nearest neighbors \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t2.8\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\teV (see for example [25]). Electron hopping between layers is also assumed to be possible only between nearest neighbors from different layers, that is, between those which lie at a distance h from each other. Denote the corresponding hopping integral \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Note that only half of the bilayer graphene atoms have such neighbors (sublattice AI and AII, see Figure 1). In sublattices BI and BII no such neighbors exist, since nearest neighbors from different layers are at a distance \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t1.415\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tÅ is the distance between nearest neighbors in the layer plane). Since this distance is only by less than 10% greater than\n\t\t\t\t\t\n\t\t\t\t\t\th\n\t\t\t\t\t\n\t\t\t\t, we can neglect the interaction with the atoms of the sublattices BI and BII and this does not lead to qualitative changes in the behavior of the spectra near \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (see, e.g. [6]). Then the dispersion relation of each of the four branches of the electronic spectrum of bilayer graphene can be written as
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the electronic spectrum of graphene, calculated in the strong coupling approximation:
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are two-dimensional Bravais lattice vectors (see Figure 5b).
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Figure 5.
The structure of bilayer graphene (a); Bravais lattice and the first Brillouin zone (b).
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Dispersion curves along highly symmetric directions ΓK, ΓM and KM for J’ = 0.1J are shown in Figure 6. Region near the K-point \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΚ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t0,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t,0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΚ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\tπ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t,0\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is shown in the inset. The same inset shows also the dispersion curves for graphene (18). We clearly see the quasi-relativistic nature of the electronic spectrum of graphene as well as ordinary quadratic dispersion curves \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1,2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t for bigraphene. Spectral branches \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t3,4\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t∉\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are determined in the \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t interval. Indeed, if k takes value along ΓK, then \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\tcos\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and putting \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tΚ\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tκ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tκ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) we find \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tΚ\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tκ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t∓\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\tκ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, i.e. a linear (relativistic) dispersion relation. For electronic modes of bilayer graphene \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1,2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t near the \n\t\t\t\t\t\n\t\t\t\t\t\tΚ\n\t\t\t\t\t\n\t\t\t\t point, we can write
Dispersion curves along high-symmetry directions of bilayer graphene.
\n\t\t\t
The electron effective mass in considered branches, determined from the relation \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tℏ\n\t\t\t\t\t\t\t\t\t\t\t\t\tκ\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, is equal to
In the case considered above \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0.1\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t the effective mass is \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t2.75\n\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t10\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t32\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\tkg, and if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t it tends to a value close to the free electron mass. Since \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, then by changing the interlayer hopping integral we can change the effective mass of charge carriers.
\n\t\t\t
It should be noted that for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, that is for the transition from bilayer graphene to two non-interacting graphene monolayers the effective mass, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, and formulas (19 - 21) cannot correctly describe this transition, since they were obtained under the assumption\n\t\t\t\t\t\n \n \n ε\n 0\n \n (\n k\n )<<\n J\n ′\n \n \n \n\n\t\t\t\t.
\n\t\t\t
Electron density of states for values of energy near εF is determined by branches of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t only, and it follows from (3) that \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Then
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tΣ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the area of the two-dimensional Bravais cell, and integration is done along a closed isoenergetic line \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1,2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. At \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (Fermi level) the line contracts into a point and near εF the contour of integration is a circle. Taking into account (20) we may write
This means that the electron density of states (DOS) is constant and different from zero. As follows from (20), DOS is an analytical function and has minimum at \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and near \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t the function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
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Total electron DOS can be represented as a mean-arithmetic function of the two LDOS corresponding to atoms of sublattices A and B: \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t For each sublattice with perfect structure the LDOS may be written as
where index s is the designation of sublattice, index α is the designation of branch, and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tκ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tare the eigenwave functions corresponding to atoms from sublattices. LDOS are calculated by the method of Jacobi matrix [16, 17] and are shown in Figure 7.
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In this figure we clearly see two-dimensional van-Hove peculiarities at energy values corresponding to points \n\t\t\t\t\t\n\t\t\t\t\t\tΓ\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tΜ\n\t\t\t\t\t\n\t\t\t\t of the first Brillouin zone ( see Figure 5b). These eigenvalues are given in the top inset in Figure 7. The bottom inset shows LDOS, in enlarged scale, near\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. It is seen that near the Fermi level the local density of states as well the total density of states are analytical and their dependences on energy are essentially nonlinear (for comparison the DOS of graphene is also shown in the same inset). Besides, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tdiffers from both \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and the total density of states and it approaches to zero for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
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Figure 7.
LDOS for atoms of different sublattices of bigraphene.
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Indeed, putting zero eigenvalue in the equation of eigenfunctions of Hamiltonian (1) we get the values \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tκ\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Therefore, near the Fermi level \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t~\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
\n\t\t\t
Peculiarities at both ends, i.e. at \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t0.1\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, originate from the contribution of modes ε3,4 which are not represented on interval \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\t′\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Peculiarities at \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t≈\n\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t0.05\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are due to the fact that beginning from these energies the anisotropy of isofrequency lines becomes essential.
\n\t\t\t
So we can conclude that, in contrary to graphene, the bigraphene has an ordinary non-relativistic form of the electronic dispersion law. The effective mass of electron in the bigraphene strongly depends on the value of integral describing the hopping between two layers, and this value may be changed by external conditions (for example by pressure). Near the Fermi level the LDOS of atoms of different sublattices qualitively differ from each other. If the LDOS for the atoms of sublattice A at the Fermi level equals to zero and it slowly increases near this level (“a quasi-gap” appears), then the LDOS of sublattice B for the same energy values differs from zero and increases very quickly.
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\n\t\t
\n\t\t\t
4. The influence of defects on the electron spectrum of bigraphene
\n\t\t\t
Some peculiarities in the behavior of the bigraphene electron spectrum near \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tΚ\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, that is near the Fermi level, indicate the possibility of a strong influence of various defects [26-28].
\n\t\t\t
Let us first consider point defects [26]. Figure 8 shows the real parts of the local Green\'s functions \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRe\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRe\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t of atoms of the sublattices A and B.
\n\t\t\t
Figure 8.
Real parts of the local Green’s functions for the atoms of the two sublattices of bigraphene. The dashed lines are \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tfunctions correspond to substitutional impurities of Boron (B-curves) and Nitrogen (N-curves).
\n\t\t\t
This figure also shows the dependences S(ε) appearing in the Lifshitz equation (2) and corresponding correspond to the presence of substitutional impurities of boron and nitrogen in bigraphene.
\n\t\t\t
For an isolated substitutional impurity that differs from the atom of the basic lattice by the values of energies at the impurity site and also by the overlap integral, function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t has for each sublattice form (3). Also, as in the case of graphene, for the nitrogen impurity in the interval \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tΜ\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tΜ\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\tequation (2) has a solution and this impurity forms quasilocalized states in this interval. For the boron impurity, equation (2) has two solutions outside the band of quasi-continuous spectrum \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tΜ\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tΜ\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, corresponding to local discrete levels.
\n\t\t\t
LDOS of isolated impurity atom of nitrogen in sublattice A or B are shown in Figure 9 (we remind that atoms AI and AII as well as BI and BII are physically equivalent).
\n\t\t\t
As the inset shows, the nitrogen impurity does not forms a quasilocal maximum on LDOS and substantially changes it near the Fermi level. Figure 10 shows a LDOS of the boron impurity in sublattices A and B. As in the case of boron impurity in graphene (see Figure 1), the area under this curve is less than unity. Outside the band of the quasi-continuous spectrum local discrete energy levels are formed.
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Figure 9.
LDOS of nitrogen impurity in the sublattices of bigraphene (red curve is for A sublattice, purple is for B sublattice). For comparison the figure shows the DOS of graphene (black dashed line) and LDOS (thin solid gray and dashed gray, respectively).
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Figure 10.
LDOS of the boron impurity in different sublattices of bigraphene (designations are the same as in Figure 9).
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The energies of these levels for the considered ratio between the overlap integrals J and J’ are slightly different. As can be seen from Figure 8, the energies of the local discrete levels for the substituted boron atom in sublattice A or sublattice B can be calculated using the two-moment approximation (13). Near the Fermi level the LDOS of the boron atom, substituting an atom of sublattice B, is considerably lower than the LDOS of the carbon atom of this sublattice. That is the boron impurity lowers the conductivity of bigraphene.
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Because in bigraphene atoms of sublattices A and B are physically inequivalent, the influence on their electron spectra by various defects is different. In the first part of this section we have described the influence of substitutional impurities on the electron density of states. However, the influence of vacancies in sublattices on the electron DOS is even more profound. Figures 11 and 12 show the LDOS of neighbors of vacancies in sublattices A (Figure 11) and B (Figure 12).
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Neighboring atoms are in the same layer as a vacancy, either in sublattice B (top) or in sublattice A (bottom). Insets show the arrangement of atoms. Atoms are shown in the same color as the corresponding LDOS.
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Figure 11.
LDOS of bigraphene atoms which are neighbors of a vacancy in the sublattice A. Neighboring atoms are in the same layer as a vacancy, either in sublattice B (top) or in sublattice A (bottom). Insets show the arrangement of atoms. Atoms are shown in the same color as the corresponding LDOS.
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Figure 12.
LDOS of bigrafene atoms which are neighbors of a vacancy in the sublattice B.
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The reason for the specific evolution of the LDOS with increasing distance from the vacancy has been explained in [22]. In bigraphene there is an analogous situation. Electronic spectra of vacancy neighbors belonging to other sublattice also have sharp resonance peaks at ε = εF (see upper parts Figure 11 and 12). So, if the vacancy is in the sublattice A, then the maximum of the LDOS of the sublattice B atoms is sharp, and the maximum on the spectrum the sublattice A atoms is blurred. Therefore we can conclude that the vacancy in bigraphene should have a more pronounced effect.
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5. Conclusion
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Unique properties of both graphene and bigraphene are caused, above all, by an unusual symmetry leading to the absence of a gap between the valence and conduction bands. The quantum states of quasiparticle in these bands are described by the same wave function, i.e. quasiparticle in graphene and bigraphene have the so-called chiral symmetry. In graphene this leads to the fact that dispersion relation of electron spectrum is linear and is described by the Dirac equation, characteristic for massless ultrarelativistic quasiparticle in quantum electrodynamics. On other hand, in bigraphene the presence of chiral symmetry leads at low energies to an ordinary parabolic dispersion relation, i.e. quiasiparticle of a new type appear – massive chiral fermions having no analogy in quantum electrodynamics. At symmetry breaking a gap between the valence and conduction band appears, allowing to tune the conducting properties of these materials.
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In this chapter, using the method of Jacobi matrices, we analyzed how different impurities affect the energy gap width and the local density of states. The method of Jacobi matrices enables to investigate heterogeneous systems and to calculate the densities of states for each atom in different sublattices. Such analysis is necessary for correct comparison with experimental results. As defects we considered the vacancies and the impurity of nitrogen or boron. The presence of nitrogen leads to the formation of sharp resonance peaks (quasilocal states) inside the continuous spectrum; on the other hand, the boron impurity leads to states outside the continuous spectrum (local states). Both quasilocal and local states can be investigated experimentally by, for example, a scanning tunneling microscope. In the presence of vacancies we have analyzed how the density of states in each of the sublattices A or B changes. Different situations were analyzed. For example, if the vacancy in this sublattice vanishes at Fermi level, whereas in sublattice B the density has maximum at this point. We also investigated the conditions for opening the gap and changing its width. Main attention was paid to the analysis of electronic properties of considered systems. Moreover, computational method we used has also been successfully applied to the analysis of vibrational states. For example, increasing the overlap integral between the boron impurity and carbon atom leads to the strengthening of force interaction contacts between them. In addition, boron atom is 16% lighter than carbon atom, i.e. all necessary conditions are sent for the appearance of local vibrations in the phonon spectrum of both graphene and bigraphene with boron impurity. We hope that predicted peculiarities in electronic and vibrational spectra of perfect graphene and bigraphene as well as of graphene and bigraphene with defects will be detected experimentally.
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Acknowledgments
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This work is the result of the project implementation: Research and Education at UPJŠ – Heading towards Excellent European Universities, ITMS project code: 26110230056, supported by the Operational Program Education funded by the European Social Fund (ESF). This work was also supported by the grant of the Ukrainian Academy of Sciences under the contract No 4/10-H and by the grant of the Scientific Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic and Slovak Academy of Sciences under No. 1/0159/09.
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\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/37677.pdf",chapterXML:"https://mts.intechopen.com/source/xml/37677.xml",downloadPdfUrl:"/chapter/pdf-download/37677",previewPdfUrl:"/chapter/pdf-preview/37677",totalDownloads:1973,totalViews:254,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,dateSubmitted:"February 9th 2012",dateReviewed:"June 11th 2012",datePrePublished:null,datePublished:"March 27th 2013",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/37677",risUrl:"/chapter/ris/37677",book:{slug:"new-progress-on-graphene-research"},signatures:"Alexander Feher, Eugen Syrkin, Sergey Feodosyev, Igor Gospodarev, Elena Manzhelii, Alexander Kotlar and Kirill Kravchenko",authors:[{id:"15947",title:"Dr.",name:"Alexander",middleName:null,surname:"Feher",fullName:"Alexander Feher",slug:"alexander-feher",email:"alexander.feher@upjs.sk",position:null,institution:null},{id:"19872",title:"Prof.",name:"Eugen",middleName:null,surname:"Syrkin",fullName:"Eugen Syrkin",slug:"eugen-syrkin",email:"syrkin@ilt.kharkov.ua",position:null,institution:null},{id:"19873",title:"Prof.",name:"Sergey",middleName:null,surname:"Feodosyev",fullName:"Sergey Feodosyev",slug:"sergey-feodosyev",email:"feodosiev@ilt.kharkov.ua",position:null,institution:null},{id:"19874",title:"Dr.",name:"Igor",middleName:null,surname:"Gospodarev",fullName:"Igor Gospodarev",slug:"igor-gospodarev",email:"gospodarev@ilt.kharkov.ua",position:null,institution:null},{id:"62279",title:"Dr.",name:"Kirill",middleName:null,surname:"Kravchenko",fullName:"Kirill Kravchenko",slug:"kirill-kravchenko",email:"kravchenko@ilt.kharkov.ua",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Impurity levels in the electron spectra of graphene",level:"1"},{id:"sec_3",title:"3. The electronic spectrum of bilayer graphene",level:"1"},{id:"sec_4",title:"4. The influence of defects on the electron spectrum of bigraphene",level:"1"},{id:"sec_5",title:"5. Conclusion",level:"1"},{id:"sec_6",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tUchoa\n\t\t\t\t\t\t\tB.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tCastro Neto\n\t\t\t\t\t\t\tA. H.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007\n\t\t\t\t\tSuperconducting States of Pure and Doped Graphene\n\t\t\t\t\tPhysical Review Letters\n\t\t\t\t\t98\n\t\t\t\t\t14681\n\t\t\t\t\t4\n\t\t\t\t\n\t\t\t'},{id:"B2",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMcChesney\n\t\t\t\t\t\t\tJ. 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S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFeodos’ev\n\t\t\t\t\t\t\tS. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2009\n\t\t\t\t\tEffect of Defects on Quasi-particle Spectra of Graphite and Graphene\n\t\t\t\t\tLow Temperature Physics\n\t\t\t\t\t35\n\t\t\t\t\t7\n\t\t\t\t\t679\n\t\t\t\t\t686\n\t\t\t\t\n\t\t\t'},{id:"B15",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChen\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZhang\n\t\t\t\t\t\t\tY.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tShen\n\t\t\t\t\t\t\tY.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007\n\t\t\t\t\tSelf-healing in Defective Carbon Nanotubes: a Molecular Dynamic Study\n\t\t\t\t\tJournal of Physics: Cond. Matter\n\t\t\t\t\t9\n\t\t\t\t\t38612\n\t\t\t\t\t6\n\t\t\t\t\n\t\t\t'},{id:"B16",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPeresada\n\t\t\t\t\t\t\tV. I.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1968\n\t\t\t\t\tNew Calculation Method in the Theory of Crystal Lattice\n\t\t\t\t\tCondensed Matter Physics\n\t\t\t\t\t4\n\t\t\t\t\ted B.I. Verkin (Kharkov: FTINT AN Ukr. SSR)\n\t\t\t\t\t172\n\t\t\t\t\t(in Russian)\n\t\t\t\t\n\t\t\t'},{id:"B17",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPeresada\n\t\t\t\t\t\t\tV. I.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAfanas’ev\n\t\t\t\t\t\t\tV. N.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBorovikov\n\t\t\t\t\t\t\tV. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1975\n\t\t\t\t\tOn Calculation of Density of States of Single-Magnon Perturbations in Ferromagnetics\n\t\t\t\t\tSoviet Low Temperature Physics\n\t\t\t\t\t1\n\t\t\t\t\t4\n\t\t\t\t\t227\n\t\t\t\t\t232\n\t\t\t\t\n\t\t\t'},{id:"B18",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSkrypnyk\n\t\t\t\t\t\t\tYu. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLoktev\n\t\t\t\t\t\t\tV. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2006\n\t\t\t\t\tImpurity Effects in a Two-Dimensional Systems with the Dirac Spectrum\n\t\t\t\t\tPhysical Review B\n\t\t\t\t\t73\n\t\t\t\t\t241402\n\t\t\t\t\t6\n\t\t\t\t\n\t\t\t'},{id:"B19",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSkrypnyk\n\t\t\t\t\t\t\tYu. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLoktev\n\t\t\t\t\t\t\tV. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tSpectral Function of Graphene with Short-Range Imourity Centers\n\t\t\t\t\tLow Temperature Physics\n\t\t\t\t\t34\n\t\t\t\t\t9\n\t\t\t\t\t818\n\t\t\t\t\t825\n\t\t\t\t\n\t\t\t'},{id:"B20",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBena\n\t\t\t\t\t\t\tC.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tKivelson\n\t\t\t\t\t\t\tS. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2005\n\t\t\t\t\tQusiparticle Scattering and Local Density of States in Graphite\n\t\t\t\t\tPhysical Review B\n\t\t\t\t\t72\n\t\t\t\t\t12\n\t\t\t\t\t125432\n\t\t\t\t\t7\n\t\t\t\t\n\t\t\t'},{id:"B21",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLifshits\n\t\t\t\t\t\t\tI. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1945\n\t\t\t\t\tOn the Theory of regular Perturbations Report of AS SSSR (in Russian)\n\t\t\t\t\t48\n\t\t\t\t\t83\n\t\t\t\t\t86\n\t\t\t\t\n\t\t\t'},{id:"B22",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFeher\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSyrkin\n\t\t\t\t\t\t\tE. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFeodosyev\n\t\t\t\t\t\t\tS. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGospodarev\n\t\t\t\t\t\t\tI. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tManzhelii\n\t\t\t\t\t\t\tE. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tKotlyar\n\t\t\t\t\t\t\tA. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tKravchenko\n\t\t\t\t\t\t\tK. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2011\n\t\t\t\t\tThe Features of Low Frequency Atomic Vibrations and Properties of Acoustic Waves in Heterogeneous Systems\n\t\t\t\t\tIn: Vila R. (ed.)\n\t\t\t\t\tWaves in Fluids and Solids\n\t\t\t\t\tRijeka\n\t\t\t\t\tInTech\n\t\t\t\t\t103\n\t\t\t\t\t126\n\t\t\t\t\n\t\t\t'},{id:"B23",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tKotlyar\n\t\t\t\t\t\t\tA. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFeodosyev\n\t\t\t\t\t\t\tS. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2006\n\t\t\t\t\tLocal Vibrational Modes in Crystal Lattices with a Simply Connected Region of the Quasi-continuous Phonon Spectrum\n\t\t\t\t\tLow Temperature Physics\n\t\t\t\t\t32\n\t\t\t\t\t3\n\t\t\t\t\t256\n\t\t\t\t\t269\n\t\t\t\t\n\t\t\t'},{id:"B24",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGospodarev\n\t\t\t\t\t\t\tI. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tEremenko\n\t\t\t\t\t\t\tV. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tKravchenko\n\t\t\t\t\t\t\tK. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSirenko\n\t\t\t\t\t\t\tV. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSyrkin\n\t\t\t\t\t\t\tE. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFeodos’ev\n\t\t\t\t\t\t\tS. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2010\n\t\t\t\t\tVibrational Charakteristics of Niobium Diselenide and Graphite Nanofilms\n\t\t\t\t\tLow Temperature Physics\n\t\t\t\t\t36\n\t\t\t\t\t4\n\t\t\t\t\t344\n\t\t\t\t\t350\n\t\t\t\t\n\t\t\t'},{id:"B25",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNovoselov\n\t\t\t\t\t\t\tK. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2010\n\t\t\t\t\tGrafen: Materialy Flatlandii\n\t\t\t\t\tUspehi Fizicheskih Nauk (in Russian)\n\t\t\t\t\t181\n\t\t\t\t\t12\n\t\t\t\t\t1299\n\t\t\t\t\t1311\n\t\t\t\t\n\t\t\t'},{id:"B26",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDahal Hari\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBalatsky\n\t\t\t\t\t\t\tA. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZhu\n\t\t\t\t\t\t\tJ-X.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tTuning Impurity States in Bilayer Graphene\n\t\t\t\t\tPhysical Review\n\t\t\t\t\t77\n\t\t\t\t\t11\n\t\t\t\t\tB 115114\n\t\t\t\t\t1\n\t\t\t\t\t10\n\t\t\t\t\n\t\t\t'},{id:"B27",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWang\n\t\t\t\t\t\t\tZ. F.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLi\n\t\t\t\t\t\t\tQ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSu\n\t\t\t\t\t\t\tH.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWang\n\t\t\t\t\t\t\tH.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tShi\n\t\t\t\t\t\t\tQ. W.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChen\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tYang\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHou\n\t\t\t\t\t\t\tJ. G.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007\n\t\t\t\t\tElectronic Sructure of Bilayer Graphene; Areal-Space Green’s Function Study\n\t\t\t\t\tPhysical Review\n\t\t\t\t\t78\n\t\t\t\t\t8\n\t\t\t\t\tB 085424\n\t\t\t\t\t1\n\t\t\t\t\t8\n\t\t\t\t\n\t\t\t'},{id:"B28",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tCastro\n\t\t\t\t\t\t\tE. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPeres\n\t\t\t\t\t\t\tN. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLopes dos Santos\n\t\t\t\t\t\t\tJ. M. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tCastro Neto\n\t\t\t\t\t\t\tA. H.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGuinea\n\t\t\t\t\t\t\tF.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tLocalized States at Zigzag Edges of Bilayer Graphene\n\t\t\t\t\tPhysical Review Letters\n\t\t\t\t\t100\n\t\t\t\t\t26\n\t\t\t\t\t026802\n\t\t\t\t\t1\n\t\t\t\t\t4\n\t\t\t\t\n\t\t\t'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Alexander Feher",address:"alexander.feher@upjs.sk",affiliation:'
Institute of Physics, Faculty of Science, P. J. Šafárik University in Kosice, Slovakia
B.I.Verkin Institute for Low Temperature Physics and Engineering NASU, Ukraine
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1. Introduction
Current therapies for cancerous tumors suffer from both toxic secondary effects and the development by the tumor of drug resistance. These effects usually block therapy for metastatic cancers, the cause of 90% of cancer deaths [1, 2, 3, 4, 5]. For solving these problems, our first thesis is that the best strategy is to increase tumor specificity of anticancer drug delivery in several, independent stages. If, for example, three stages are used and each stage is 80% efficient (20% nonefficient) in increasing specificity, then overall efficiency is 99% [100 × (1.0 − 0.23)]. In this case, (1) drug dosages to tumors can be raised 100× without changing toxicity and, therefore, (2) tumor cell evolution of drug resistance is minimized.
The primary alternative is to continue testing chemotherapies [6, 7, 8], immunotherapies [9, 10, 11] and radiotherapies [12, 13, 14] that have tumor-specificity determined at one independent stage. This one stage is often cellular DNA replication, which is more rapid and, therefore, more drug- and radiation-sensitive, in cancerous cells than it is in healthy cells. One-stage strategies are >100 years old for immunotherapy and radiotherapy. Chemotherapeutic agents typically used are over 50 years old [8]. Even major effort has not produced systematic therapies for metastatic cancer. Apparently, new, possibly more biology-based, strategy is needed to counter risk that the above one-stage-based strategies, in general, are not realistic for reaching objectives (reviewed in [1, 15, 16, 17]).
In theory, one implementation of multi-stage strategy starts with drug delivery in a drug delivery vehicle (DDV) that is gated. The gate is opened to load drug, closed in circulation to deliver drug and opened again in tumors to administer drug. Stages of DDV-derived toxicity reduction are the following tumor-specific events: (1) DDV delivery, (2) DDV opening and, (3) drug activation, for masked drugs. We address stages (1) and (2) here.
In this implementation, tumor-specific delivery is achieved via the EPR effect. The EPR effect is the spontaneous accumulation of nanoparticles in tumors, observed for an uncharacterized phage in 1940 [18] (reviewed in [19]). The causes of the EPR effect are (1) porosity of tumor blood vessels and relative tightness of healthy blood vessels, so that nanoparticles enter tumors, but usually not healthy tissue and (2) poor tumor lymphatic drainage, so that nanoparticles remain [20, 21, 22, 23, 24, 25]. The EPR effect is the basis for the use of several FDA-approved, drug-loaded, liposomal DDVs [23, 24, 26].
However, circulating, drug-loaded, liposomal DDVs undergo drug leakage that causes significant toxicity [22, 23, 25, 27]. Also, liposomes are removed from circulation by the macrophage-phagocyte innate immune system. That is to say, liposomes are not very persistent. Chemical solutions to the leakage problem do not exist to our knowledge. Chemical solutions to the persistence problem (e.g., polyethylene glycol derivatization [28, 29]) introduce quality control problems and are not adaptable to future improvements, for example, achieving of tumor-specific unloading. The second thesis is that the optimal solution is linked to finding an appropriate, biologically produced, microbial DDV. Unlike some biology-based anticancer strategies [1, 15, 16, 19], use of a DDV is implementable with nononcolytic viruses and, therefore, avoids the dangers [19] of using oncolytic viruses.
In practice, we have discovered a phage T3 capsid that appears to have DDV-favorable characteristics needed for implementation of our strategy. First, phage T3 (and presumably its capsid) has recently been found to have exceptionally high persistence in mouse blood (3–4 h), unlike the T3 relative, T7 [30]. Second, one empty, but otherwise phage-like T3 capsid, is impermeable (for over 20 years) to the compound, Nycodenz (821 Da molecular weight) (reviewed in [31]). But, when the temperature is raised to 45°C, Nycodenz enters this capsid [32], presumably through a gate that opened. The concept is that, if we adapt this capsid to use as gated DDV, some of the needed engineering has already been done by natural selection.
Phages T3 and T7 are illustrated at the left in Figure 1. The gated capsid is illustrated at the right in Figure 1. This capsid is generated during DNA packaging that had been initiated by a DNA-free procapsid called capsid I (not shown). During packaging, capsid I expands and becomes the more angular and stable capsid (capsid II) illustrated in Figure 1. In nature, the capsid-gate is a ring of 12 gp8 molecules (Figure 1) that acts as entry portal for DNA during DNA packaging [31, 32, 33]. Most T3 and T7 capsid II particles are purified after having detached from DNA during infected cell lysis. The amount is 5–10 mg capsid II per liter of culture. The last purification step is buoyant density centrifugation in a Nycodenz or Metrizamide density gradient. Nycodenz low density (NLD) capsid II has the gp8 “gate” and is impermeable to Nycodenz and Metrizamide. The low density (1.08 g/ml) is caused by high internal hydration, which is caused by Nycodenz impermeability. Nycodenz high density (NHD) capsid II is Nycodenz-permeable (1.28 g/ml) and is completely separated from NLD capsid II during buoyant density centrifugation in a Nycodenz or Metrizamide density gradient [31, 33, 34, 35]. We use NHD capsid II as a control during loading experiments.
Figure 1.
Phages T3 and T7 (left); and T3 and T7 NLD capsid II (right). The graphic legend at the top indicates the various capsid components. A protein is labeled by gp, followed by the number of the encoding gene. Analogous T3 and T7 genes have the same numbers. If NLD capsid II is used as DDV, the perimeter of the DDV is defined by the gp10 shell; the gate is the gp8 portal. All structures are the same for phage T3 as they are for phage T7 (reviewed in [31, 32]).
In the case of T3 NLD capsid II, gated entry of Nycodenz has been observed via raising of temperature. The likely entry channel was the axial hole of the gp8 ring. This conclusion was drawn, in the case of T7 NLD capsid II, from (1) entry kinetics of the fluorescent dye, bis-ANS [1,1′-bi(4-anilino)naphthalene-5,5′-di-sulfonic acid; 673 Da], and (2) covalent cross-linking of bis-ANS to channel proteins [33]. Asymmetric reconstruction-cryo-EM [36] revealed that obstruction of the T7 gp8 channel (and presumably the T3 channel) varies.
In support of working to implement the above gating-based strategy, the following quote from 2005 presents an expert opinion of what is needed for the next generation of anticancer DDVs [27]. “An ideal liposomal anticancer drug would exhibit little or no drug release while in the plasma compartment, thus ensuring limited exposure of the drug to healthy tissue. This feature would also maximize drug delivery to disease sites, as mediated by the movement of the drug-loaded liposomes from the plasma compartment to the extravascular space at disease sites, such as a region of tumor growth. Following localization, however, the drug-loaded liposome must transform itself from a stable carrier to an unstable carrier. This would ensure that the drug, which has localized in the diseased site, is bioavailable.” To our knowledge, no details for such “controlled release” via a liposomal DDV have been published. The system described here is designed to accomplish what is described in the above quote. However, implementation uses gating of a DDV, not programmed instability of a DDV.
To proceed further, we need a procedure for rapidly determining whether a drug is loaded in a capsid. In the current study, we have developed native agarose gel electrophoresis (AGE) for this purpose. The detection is performed via capsid band fluorescence produced by the compound loaded.
2. Results
2.1 Detection of test compounds: GelStar
We tested the loading of two fluorescent compounds. The first was GelStar, a fluorescent nucleic acid stain typically used after AGE. In contrast, we incubated GelStar with our capsid and then performed AGE without further use of GelStar. The second compound was bleomycin, an anticancer drug [37, 38] that is also fluorescent [38]. Neither the manufacturer nor the vendor provided either the structure of GelStar or the concentration of commercial GelStar solutions. GelStar is sold in solution only.
The dominant fluorescence emission of nucleic acid-bound GelStar is in the green range. Apparently not previously documented is that the dominant fluorescence emission of free GelStar is in the orange range, at least when the GelStar is in an agarose gel. Ultraviolet light stimulated GelStar fluorescence emission vs. GelStar dilution is shown in Figure 2. Free GelStar, at several dilutions, had been pipetted in 5 μl amounts onto an agarose gel before ultraviolet light illumination and photography through an orange filter (spots labeled G in Figure 2). The effective volume in μl (dilution, multiplied by 5) of the stock GelStar solution is also indicated. In Figure 2, the color of GelStar spots is orange for all dilutions, as it also is found to be (not shown) with yellow and green emission filters. The orange color is real and is not produced by the emission filter because green Alexa 488 dye fluorescence retains its green color (spots labeled A in Figure 2). The number next to the Alexa 488 spots is the total amount (μg) of Alexa 488, also applied in 5 μl amounts.
Figure 2.
Fluorescence of free GelStar. GelStar and Alexa Fluor 488 were diluted and, then, pipetted onto the surface of an agarose gel. The fluorescence was photographed through the orange filter. The GelStar samples are indicated by G, followed by the effective volume (μl) of the original, undiluted GelStar solution. The Alexa Fluor 488 samples are indicated by A, followed by the amount (μg) of Alexa Fluor 488.
DNA-bound GelStar had the expected green emission at all dilutions, when viewed through the same orange emission filter used for Figure 2 (right side of the right panel of Figure 3). Green emission was also dominant when yellow and green emission filters were used for DNA-bound GelStar (not shown).
Figure 3.
Association of GelStar with T3 NLD capsid II. A commercial GelStar solution was diluted to the extent indicated above a lane and incubated with T3 NLD capsid II. Association of GelStar with the capsid was determined by AGE, followed by, first, photography of fluorescence (right panel, lanes labeled NLD CII) and, then, staining of protein with Coomassie blue (left panel). Also analyzed was purified T3 DNA (right panel, lanes labeled DNA), which does not stain with Coomassie blue. The arrow indicates the direction of electrophoresis; the arrowheads indicate the origins.
2.2 Detection of test compounds: bleomycin
Without fluorescent compound, the background of an agarose gel was blue when emission was photographed without an emission filter (not shown). Thus, not surprising was that optimal detection of bleomycin was not obtained with a blue filter, even though the blue range was where peak emission was previously found for bleomycin [38]. Among the blue, green, yellow and orange filters, optimal detection was obtained with the green filter.
With the green emission filter, the minimal detected bleomycin amount was 0.2–0.4 ng when a bleomycin dilution series like the GelStar dilution series in Figure 2 was photographed (not shown). Contrast enhancement of images was used at these lower amounts.
2.3 Loading of GelStar in NLD capsid II
We succeeded in loading GelStar into NLD capsid II. To achieve loading, 10 μg of NLD capsid II was incubated with GelStar at 45°C. Loading was then assayed by AGE at 10°C. Then, the gel was illuminated with ultraviolet light. The result was a fluorescent band of intensity that monotonically increased with decreasing GelStar dilution (left section of the right panel of Figure 3). The capsid amount was invariant, as judged by Coomassie staining of the same gel (left panel of Figure 3). At GelStar dilutions lower than those in Figure 3, down to 1/10, the band intensity reached a plateau (not shown). The dominant fluorescence, at all dilutions, was green, implying that the GelStar was bound to something capsid associated. GelStar did not detectably associate with NHD capsid II (not shown).
The following data indicated that the GelStar-binding capsid site was not on a DNA molecule associated with the capsid. As the dilution of GelStar decreased, the DNA-bound GelStar fluorescence underwent, first, an increase and then a decrease (Figure 3, right segment of right panel). However, the decrease was not observed for the binding to NLD capsid II. Second, although a minor NLD capsid II fraction has DNA [31], the DNA-containing NLD capsid II had been excluded during purification by selecting the low-density side of the NLD capsid II band after buoyant density centrifugation in a Nycodenz density gradient. Thus, the GelStar was apparently either self-bound or bound to capsid protein.
2.4 Loading of bleomycin in NLD capsid II
Association of bleomycin with T3 NLD capsid II was also achieved. However, the fluorescence signal was relatively weak (Figure 4). The bleomycin fluorescence signal of a NLD capsid II band did not change when the concentration of bleomycin was changed from 2 to 16 mg/ml. A bleomycin-associated NLD capsid II band is shown in Figure 4. Most of the free bleomycin migrated toward the cathode (not shown), i.e., in a direction opposite to the direction of capsid migration.
Figure 4.
Association of bleomycin with T3 NLD capsid II. The experiment of Figure 3 was repeated with bleomycin (8 mg/ml), instead of GelStar. The capsid region of the post-AGE gel is shown. The right (protein) panel has a single band of capsid stained with Coomassie blue. This band marks the position of the capsid-associated bleomycin fluorescence in the left panel. The arrow indicates direction of electrophoresis.
The strength of the signal in Figure 4 was weakened by the blue background and use of a green filter. In addition, a contaminant in the bleomycin preparation migrated close to the capsids, and is seen above the capsid band at the top of the left panel of Figure 4.
Calibration data for bleomycin, like the data for GelStar in Figure 2, were obtained. These data revealed that the amount of bleomycin loaded was 150–300 molecules per capsid.
3. Discussion
In the Introduction, we outlined a strategy that is expected to work, if we can achieve the following objectives: (1) high (~4 h) persistence of NLD capsid II in blood so that the EPR effect has time to work, (2) adequate loading and sealing of NLD capsid II and (3) tumor-specific, controlled release (de-sealing or unloading). Objective #1 is likely already achieved, given the high persistence of T3 phage. That is to say, if one considers this strategy to be engineering based, some of the engineering might already be been done by natural selection.
Concerning adequate loading, the volume of the internal cavity of NLD capsid II = 6.95 × 10−17 ml. For volume occupancy (FV) of 0.5 (equal to the FV of DNA packaged in mature phage [34]), the number of bleomycin molecules (1416 Da; density estimated at 1.6 g/ml as sulfate) per NLD capsid II particle is 2.4 × 104. The recommended dose of DDV-free bleomycin depends on the tumor, but is typically [39, 40] 10–20 units/m2, corresponding roughly to 10–20 mg/m2; 15 mg/m2 is 1.76 × 1018 bleomycin molecules/m2.
To calculate the number (ND) of NLD capsid II particles needed for this dose at FV = 0.5, we initially assume a 25 g mouse, which on average, has 78.6 cm2 surface area [41]. Then, ND is 3.6 × 1011. A 6-liter culture yields 150–300 mouse doses of this size (cost ~ $1500), assuming (1) laboratory-scale production technique, (2) no development of procedures to increase the amount produced per bacterial cell, and (3) no drug-dose reduction caused by improved targeting. That is to say, if we can half-fill the volume of NLD capsid II, we have a viable beginning. However, thus far, we have filled no more than 2% of FV = 0.5, NLD capsid II volume. So, increasing the loading is a major objective for the future.
An apparent obstacle to achieving this goal is the nonincrease in loading as bleomycin concentration increases above 2 mg/ml. At least two possible explanations exist. (1) After passing through an open gate, the bleomycin eventually causes the gate to close. We were hoping to close the gate by lowering temperature. (2) After diffusing through an open gate, the bleomycin is prevented from diffusing in reverse by binding to internal proteins; the internal proteins become saturated as the concentration of bleomycin increases. In either case, increasing the loading is a problem of engineering.
An advantage of using a phage DDV is that the human-design engineering potential is relatively high. First of all, the capsids in a T7 NLD capsid II preparation are structurally uniform enough so that symmetric cryo-EM reconstruction is obtained at 3.5 Å [34] and asymmetric reconstruction, at ~8 Å [36]. Assuming T3 capsids to be comparably homogeneous, use of chemistry to improve gating should produce relatively uniform results.
Second, phages, in general, and phage T3 in particular, can be genetically manipulated, which is not possible with liposomes. Information for determining which nucleotides to change can be obtained from high-resolution cryo-EM structure. Structure of this type is not obtainable with liposomes.
Finally, we note that, as far as we know, the only phages tested for production of an NLD capsid II-like capsid are the related coliphages, T7, T3 and ϕII. All three of these phages produce a NLD capsid II-like capsid [42]. Other phages are potential sources of gated capsids, perhaps with properties even more DDV-favorable.
4. Materials and methods
4.1 T3 bacteriophage, capsids and DNA (nanoparticles)
We obtained bacteriophage T3 and T3 capsid II from 30°C-lysates of host, Escherichia coli BB/1, that had been infected by phage T3 in aerated liquid culture [43]. The growth medium was 2× LB medium: 2.0% Bacto tryptone, 1.0% Bacto yeast in 0.1 M NaCl. We initially purified both phage and capsids by centrifugation through a cesium chloride step gradient, followed by buoyant density centrifugation in a cesium chloride density gradient [43]. The latter fractionation separates capsid I from capsid II.
To separate NLD capsid II from NHD capsid II, we performed buoyant density centrifugation of capsid II in a Nycodenz density gradient, as previously described [32]. The purified NLD and NHD capsid II were dialyzed against 0.1 M NaCl, 0.01 M Tris-Cl, pH 7.4, 0.001 M MgCl2. NLD capsid II, which formed a band near the top of the Nycodenz density gradient, had no detected contamination with NHD capsid II and vice versa, as previously seen by analytical ultracentrifugation [31]. Phage, NLD capsid II and NHD capsid II were dialyzed against the following buffer before use in the experiments described below: 0.2 M NaCl, 0.01 M Tris-Cl, pH 7.4, 0.001 MgCl2.
T3 DNA was obtained from purified T3 phage by phenol extraction. The DNA was dialyzed against and stored in 0.1 M NaCl, 0.01 M Tris-Cl, pH 7.4, 0.001 M EDTA. DNA concentration was obtained from optical density at 260 nm.
4.2 Fluorescent compounds: test of fluorescence emission
GelStar was obtained from Lonza (Basel, Switzerland) in solution. The company recommends dilution by a factor of 1:10,000 for use as a nucleic acid stain after gel electrophoresis. Alexa Fluor 488 succinimidyl ester was obtained from Molecular Probes (Eugene, OR, USA) as a powder.
Bleomycin was obtained from Cayman Chemical Company (Ann Arbor, MI, USA) as a powder. The bleomycin was dissolved in the aqueous buffer indicated and diluted to the concentrations indicated before incubation with capsids and DNA.
Tests of fluorescence emission vs. fluorescent molecule concentration were made by pipetting 5 μl of diluted fluorescent molecule onto the surface of a 0.7% agarose gel (LE agarose, Lonza) that had been cast in a plastic Petri dish in the electrophoresis buffer of Section 4.3. The gel was then photographed by use of the procedures described in Section 4.3.
4.3 Loading experiments: AGE
To test for fluorescent compound/nanoparticle association, fluorescent compounds were mixed with one of the following T3 nanoparticles: NLD capsid II, NHD capsid II, phage, DNA. First, a 12.5 μl amount of fluorescent compound in 0.1 M NaCl, 0.01 M sodium citrate, pH 4.0, 0.001 M MgCl2 (citrate buffer) was added to 4.5 μl of additional citrate buffer. Then, 8.0 μl of a nanoparticle sample was added and mixed (final pH, 4.1). This mixture was incubated at 45.0°C for 2.0 h.
To perform AGE, we added to this mixture 2.5 μl of the following solution: 60% sucrose (to increase the density for layering in sample wells) in the electrophoresis buffer below. This final mixture was layered in a well of a horizontal, submerged, 0.7% agarose gel (LE agarose, Lonza), cast in and submerged under the following electrophoresis buffer: 0.05 M Tris-acetate, pH 8.4, 0.001 M MgCl2. The temperature of the gel and buffer had been pre-adjusted to 10°C in an effort to seal NLD capsid II and, therefore, prevent leakage of fluorescent compounds.
AGE was performed at 1.0 V/cm for 18.0 h with the gel and buffer maintained at 10°C by circulation through a controlled-temperature water bath. After AGE, the gel was soaked in 25% methanol in electrophoresis buffer for 1.5 h at room temperature, to cause leakage of fluorescent compounds from NLD capsid II and, therefore, to prevent auto-quenching.
Finally, the gel was photographed during illumination with a Model TM-36 ultraviolet transilluminator (Ultra Violet Products, Inc.). The camera used was a Canon Power Shot G2, 4.0 Megapixels. The following Tiffin emission filters were used as described in Section 2: Blue, 80A #290513; Green, 11Green 1—#287305; Yellow, Yellow 12—#282224; Orange, Orange 16—#289750. To detect capsid protein, the gel was subsequently stained with Coomassie blue and photographed during illumination with visible light [43].
5. Conclusions
Obtaining an increase in the current tumor-specificity of anticancer drugs should be possible via use of a DDV that implements multiple, independent stages of specificity increase. T3 NLD capsid II is an example of a bio-nanoparticle that has undergone some of the needed DDV-bioengineering via mutation/selection in the environment. Other examples, not yet found, are assumed to exist and potentially have even more favorable characteristics.
Acknowledgments
The authors acknowledge support from the San Antonio Area Foundation and the Morrison Trust.
Conflict of interest
The authors declare no conflict of interest.
\n',keywords:"agarose gel electrophoresis, bacteriophage T3, bleomycin, buoyant density centrifugation, capsid impermeability, GelStar",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/70945.pdf",chapterXML:"https://mts.intechopen.com/source/xml/70945.xml",downloadPdfUrl:"/chapter/pdf-download/70945",previewPdfUrl:"/chapter/pdf-preview/70945",totalDownloads:166,totalViews:0,totalCrossrefCites:0,dateSubmitted:"September 5th 2019",dateReviewed:"January 8th 2020",datePrePublished:"April 20th 2020",datePublished:"May 6th 2020",dateFinished:null,readingETA:"0",abstract:"Over the last 25 years, cancer therapies have improved survivorship. Yet, metastatic cancers remain deadly. Therapies are limited by inadequate targeting. Our goal is to develop a new drug delivery vehicle (DDV)-based strategy that improves targeting of drug delivery to solid tumors. We begin with a capsid nanoparticle derived from bacteriophage (phage) T3, a phage that naturally has high persistence in murine blood. This capsid has gating capacity. For rapidly detecting loading in this capsid, here, we describe procedures of native agarose gel electrophoresis, coupled with fluorescence-based detection of loaded molecules. We observe the loading of two fluorescent compounds: the dye, GelStar, and the anticancer drug, bleomycin. The optimal emission filters were found to be orange and green, respectively. The results constitute a first milestone in developing a drug-loaded DDV that does not leak when in blood, but unloads its cargo when in a tumor.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/70945",risUrl:"/chapter/ris/70945",signatures:"Philip Serwer, Elena T. Wright and Cara B. 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Introduction",level:"1"},{id:"sec_2",title:"2. Results",level:"1"},{id:"sec_2_2",title:"2.1 Detection of test compounds: GelStar",level:"2"},{id:"sec_3_2",title:"2.2 Detection of test compounds: bleomycin",level:"2"},{id:"sec_4_2",title:"2.3 Loading of GelStar in NLD capsid II",level:"2"},{id:"sec_5_2",title:"2.4 Loading of bleomycin in NLD capsid II",level:"2"},{id:"sec_7",title:"3. Discussion",level:"1"},{id:"sec_8",title:"4. Materials and methods",level:"1"},{id:"sec_8_2",title:"4.1 T3 bacteriophage, capsids and DNA (nanoparticles)",level:"2"},{id:"sec_9_2",title:"4.2 Fluorescent compounds: test of fluorescence emission",level:"2"},{id:"sec_10_2",title:"4.3 Loading experiments: AGE",level:"2"},{id:"sec_12",title:"5. Conclusions",level:"1"},{id:"sec_13",title:"Acknowledgments",level:"1"},{id:"sec_16",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'Park GT, Choi KC. 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Nanomedicines for cancer therapy: An update of FDA approved and those under various stages of development. SOJ Pharmacy and Pharmaceutical Sciences. 2014;1:13. DOI: 10.15226/2374-6866/1/2/00109'},{id:"B27",body:'Abraham SA, Waterhouse DN, Mayer LD, Cullis PR, Madden TD, Bally MB. The liposomal formulation of doxorubicin. Methods in Enzymology. 2005;39:71-96. DOI: 10.1016/S0076-6879(05)91004-5'},{id:"B28",body:'Kanwal U, Irfan Bukari N, Ovais M, Abass N, Hussain K, Raza A. Advances in nano-delivery systems for doxorubicin: An updated insight. Journal of Drug Targeting. 2018;26:296-310. DOI: 10.1080/1061186X.2017.1380655'},{id:"B29",body:'Gabizon A, Martin F. Polyethylene glycol-coated (pegylated) liposomal doxorubicin. Rationale for use in solid tumours. Drugs. 1997;54(Suppl. 4):15-21. DOI: 10.2165/00003495-199700544-00005'},{id:"B30",body:'Serwer P, Wright E, Lee JC. High murine blood persistence of phage T3 and suggested strategy for phage therapy. BMC Research Notes. 2019;12:560. DOI: 10.1186/s13104-019-4597-1'},{id:"B31",body:'Serwer P, Wright ET, Demeler B, Jiang W. States of T3/T7 capsids: Buoyant density centrifugation and cryo-EM. Biophysical Reviews. 2018;10:583-596. DOI: 10.1007/s12551-017-0372-5'},{id:"B32",body:'Serwer P, Wright ET, Chang J, Liu X. Enhancing and initiating phage-based therapies. Bacteriophage. 2014;4:e961869. DOI: 10.4161/21597073.2014.961869'},{id:"B33",body:'Khan SA, Griess GA, Serwer P. Assembly-associated structural changes of bacteriophage T7 capsids. Detection by use of a protein-specific probe. Biophysical Journal. 1992;63:1286-1292. DOI: 10.1016/S0006-3495(92)81724-1'},{id:"B34",body:'Guo F, Liu Z, Fang P-A, Zhang Q , Wright E, Wu W, et al. Capsid expansion mechanism of bacteriophage T7 revealed by multistate atomic models derived from cryo-EM reconstructions. Proceedings of the National Academy of Sciences of the United States of America. 2014;111:E4606-E4614. DOI: 10.1073/pnas.1407020111'},{id:"B35",body:'Serwer P. 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DOI: 10.1364/BOE.7.002400'},{id:"B39",body:'Medscape, bleomycin (Rx). 2019. Available from: https://reference.medscape.com/drug/bleomycin-342113'},{id:"B40",body:'Drugs.com. 2019. Available from: https://www.drugs.com/dosage/bleomycin.html'},{id:"B41",body:'Dawson NJ. The surface-area/body-weight relationship in mice. Australian Journal of Biological Sciences. 1967;20:687-690. ISSN: 0004-9417'},{id:"B42",body:'Serwer P, Watson RH, Hayes SJ, Allen JL. Comparison of the physical properties and assembly pathways of the related bacteriophages T7, T3 and phi II. Journal of Molecular Biology. 1983;170:447-469. DOI: 10.1016/s0022-2836(83)80157-0'},{id:"B43",body:'Serwer P, Wright ET, Liu Z, Jiang W. Length quantization of DNA partially expelled from heads of a bacteriophage T3 mutant. Virology. 2014;456-457:157-170. DOI: 10.1016/j.virol.2014.03.016'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Philip Serwer",address:"serwer@uthscsa.edu",affiliation:'
The University of Texas Health Science Center, San Antonio, Texas, USA
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The University of Texas Health Science Center, San Antonio, Texas, USA
'},{corresp:null,contributorFullName:"Cara B. Gonzales",address:null,affiliation:'
The University of Texas Health Science Center, San Antonio, Texas, USA
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General requirements for Open Access to Horizon 2020 research project outputs are found within Guidelines on Open Access to Scientific Publication and Research Data in Horizon 2020. The guidelines, in their simplest form, state that if you are a Horizon 2020 recipient, you must ensure open access to your scientific publications by enabling them to be downloaded, printed and read online. Additionally, said publications must be peer reviewed.
',metaTitle:"Horizon 2020 Compliance",metaDescription:"General requirements for Open Access to Horizon 2020 research project outputs are found within Guidelines on Open Access to Scientific Publication and Research Data in Horizon 2020. The guidelines, in their simplest form, state that if you are a Horizon 2020 recipient, you must ensure open access to your scientific publications by enabling them to be downloaded, printed and read online. Additionally, said publications must be peer reviewed. ",metaKeywords:null,canonicalURL:null,contentRaw:'[{"type":"htmlEditorComponent","content":"
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Metadata for all publications is also automatically deposited in IntechOpen's OAI repository, making them available through the Open Access Infrastructure for Research in Europe's (OpenAIRE) search interface further establishing our compliance.
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In other words, publishing with IntechOpen guarantees compliance.
When choosing a publication, Horizon 2020 grant recipients are encouraged to provide open access to various types of scientific publications including monographs, edited books and conference proceedings.
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Authors requiring additional information are welcome to send their inquiries to funders@intechopen.com
Publishing with IntechOpen means that your scientific publications already meet these basic requirements. It also means that through our utilization of open licensing, our publications are also able to be copied, shared, searched, linked, crawled, and mined for text and data, optimizing our authors' compliance as suggested by the European Commission.
\n\n
Metadata for all publications is also automatically deposited in IntechOpen's OAI repository, making them available through the Open Access Infrastructure for Research in Europe's (OpenAIRE) search interface further establishing our compliance.
\n\n
In other words, publishing with IntechOpen guarantees compliance.
When choosing a publication, Horizon 2020 grant recipients are encouraged to provide open access to various types of scientific publications including monographs, edited books and conference proceedings.
\n\n
IntechOpen publishes all of the aforementioned formats in compliance with the requirements and criteria established by the European Commission for the Horizon 2020 Program.
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Authors requiring additional information are welcome to send their inquiries to funders@intechopen.com
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