Electronic and magnetic properties of graphite quantum dots

We study the electronic and magnetic properties of multilayer quantum dots (MQDs) of graphite in the nearest-neighbor approximation of tight-binding model. We calculate the electronic density of states and orbital susceptibility of the system as function of the Fermi level location. We demonstrate t...

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Datum:	2015
Hauptverfasser:	Abdelsalam, H., Espinosa-Ortega, T., Luk’yanchuk, I.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2015
Schriftenreihe:	Физика низких температур
Schlagworte:	К 70-летию со дня рождения В. М. Локтева
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spelling	nasplib_isofts_kiev_ua-123456789-1220842025-02-09T09:50:36Z Electronic and magnetic properties of graphite quantum dots Abdelsalam, H. Espinosa-Ortega, T. Luk’yanchuk, I. К 70-летию со дня рождения В. М. Локтева We study the electronic and magnetic properties of multilayer quantum dots (MQDs) of graphite in the nearest-neighbor approximation of tight-binding model. We calculate the electronic density of states and orbital susceptibility of the system as function of the Fermi level location. We demonstrate that properties of MQD depend strongly on the shape of the system, on the parity of the layer number and on the form of the cluster edge. The special emphasis is given to reveal the new properties with respect to the monolayer graphene quantum dots. The most interesting results are obtained for the triangular MQD with zig-zag edge at near-zero energies. The asymmetrically smeared multipeak feature is observed at Dirac point within the size-quantized energy gap region, where monolayer graphene flakes demonstrate the highly-degenerate zero-energy state. This feature, provided by the edge-localized electronic states results in the splash-wavelet behavior in diamagnetic orbital susceptibility as function of energy. This work was supported by the Egyptian mission sector and by the European mobility FP7 Marie Curie programs IRSES-SIMTECH and ITN-NOTEDEV. 2015 Article Electronic and magnetic properties of graphite quantum dots / H. Abdelsalam, T. Espinosa-Ortega, I. Luk’yanchuk // Физика низких температур. — 2015. — Т. 41, № 5. — С. 508-513. — Бібліогр.: 31 назв. — англ. 0132-6414 PACS: 75.75.–c, 73.21.La https://nasplib.isofts.kiev.ua/handle/123456789/122084 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	К 70-летию со дня рождения В. М. Локтева К 70-летию со дня рождения В. М. Локтева
spellingShingle	К 70-летию со дня рождения В. М. Локтева К 70-летию со дня рождения В. М. Локтева Abdelsalam, H. Espinosa-Ortega, T. Luk’yanchuk, I. Electronic and magnetic properties of graphite quantum dots Физика низких температур
description	We study the electronic and magnetic properties of multilayer quantum dots (MQDs) of graphite in the nearest-neighbor approximation of tight-binding model. We calculate the electronic density of states and orbital susceptibility of the system as function of the Fermi level location. We demonstrate that properties of MQD depend strongly on the shape of the system, on the parity of the layer number and on the form of the cluster edge. The special emphasis is given to reveal the new properties with respect to the monolayer graphene quantum dots. The most interesting results are obtained for the triangular MQD with zig-zag edge at near-zero energies. The asymmetrically smeared multipeak feature is observed at Dirac point within the size-quantized energy gap region, where monolayer graphene flakes demonstrate the highly-degenerate zero-energy state. This feature, provided by the edge-localized electronic states results in the splash-wavelet behavior in diamagnetic orbital susceptibility as function of energy.
format	Article
author	Abdelsalam, H. Espinosa-Ortega, T. Luk’yanchuk, I.
author_facet	Abdelsalam, H. Espinosa-Ortega, T. Luk’yanchuk, I.
author_sort	Abdelsalam, H.
title	Electronic and magnetic properties of graphite quantum dots
title_short	Electronic and magnetic properties of graphite quantum dots
title_full	Electronic and magnetic properties of graphite quantum dots
title_fullStr	Electronic and magnetic properties of graphite quantum dots
title_full_unstemmed	Electronic and magnetic properties of graphite quantum dots
title_sort	electronic and magnetic properties of graphite quantum dots
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2015
topic_facet	К 70-летию со дня рождения В. М. Локтева
url	https://nasplib.isofts.kiev.ua/handle/123456789/122084
citation_txt	Electronic and magnetic properties of graphite quantum dots / H. Abdelsalam, T. Espinosa-Ortega, I. Luk’yanchuk // Физика низких температур. — 2015. — Т. 41, № 5. — С. 508-513. — Бібліогр.: 31 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT abdelsalamh electronicandmagneticpropertiesofgraphitequantumdots AT espinosaortegat electronicandmagneticpropertiesofgraphitequantumdots AT lukyanchuki electronicandmagneticpropertiesofgraphitequantumdots
first_indexed	2025-11-25T12:26:50Z
last_indexed	2025-11-25T12:26:50Z
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fulltext	© Hazem Abdelsalam, T. Espinosa-Ortega, and Igor Luk’yanchuk, 2015 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5, pp. 508–513 Electronic and magnetic properties of graphite quantum dots Hazem Abdelsalam 1,2 , T. Espinosa-Ortega 3 , and Igor Luk’yanchuk 1 1 University of Picardie, Laboratory of Condensed Matter Physics, Amiens 80039, France E-mail: lukyanc@ferroix.net 2 Department of Theoretical Physics, National Research Center, Cairo 12622, Egypt 3 Division of Physics and Applied Physics, Nanyang Technological University, Nanyang 637371, Singapore Received January 15, 2015, published online March 23, 2015 We study the electronic and magnetic properties of multilayer quantum dots (MQDs) of graphite in the near- est-neighbor approximation of tight-binding model. We calculate the electronic density of states and orbital sus- ceptibility of the system as function of the Fermi level location. We demonstrate that properties of MQD depend strongly on the shape of the system, on the parity of the layer number and on the form of the cluster edge. The special emphasis is given to reveal the new properties with respect to the monolayer graphene quantum dots. The most interesting results are obtained for the triangular MQD with zig-zag edge at near-zero energies. The asym- metrically smeared multipeak feature is observed at Dirac point within the size-quantized energy gap region, where monolayer graphene flakes demonstrate the highly-degenerate zero-energy state. This feature, provided by the edge-localized electronic states results in the splash-wavelet behavior in diamagnetic orbital susceptibility as function of energy. PACS: 75.75.–c Magnetic properties of quantum dots; 73.21.La Electron states and collective excitation in quantum dots. Keywords: multilayer quantum dots, tight-binding model, graphite. 1. Introduction Rise of graphene certainly revived the interest to the classical graphite systems, presenting a wealth of not yet well understood electronic and magnetic properties. The challenge is related to the complicate semimetallic multibranch energy spectrum in the vicinity of the half- field Fermi level, caused by splitting of the Dirac-cone graphene spectrum by the graphite-forming intercarbon- layer coupling. The point of special interest is the crosso- ver from graphene to graphite through the multilayer struc- tures with few number of layers. It is well known that such systems may exhibit metallic or semiconductor behavior as a function of the number of layers and the stacking process [1–5], characteristic that makes them highly appealing for gated controlled electronic devices. The magnetic properties of few-layer structure are even more intriguing [6–10]. The orbital magnetism of the odd- layer structures is reminiscent to that for the monolayer grapheme [6]. In particularly, a characteristic for graphene diamagnetic -function singularity of susceptibility [11,12] appears at the Dirac point at E = 0. For even number of layers the magnetic properties are more similar to those for bilayer graphene [6] and the diamagnetic response has the weaker logarithmic divergency. Such odd/even layer de- composition can give the coexistence of Dirac and normal carriers, observed in the pure graphite [13–15]. Alongside, flake-like graphene quantum dots (GQD) have captured the substantial attention of nanotechnolo- gy due to their unique optical and magnetic properties [16–29]. The new element here is the finite-size electron confinement, resulted in opening of the energy gap for the bulk delocalized electronic states in the vicinity of Dirac point [16–18,21,22,25]. This gap, however can be filled by the energy level of novel electronic states, localized in the vicinity of the sample boundary [16–18]. For nanoscopic and even for mesoscopic clusters these edge states can play the decisive role in electronic and magnetic proper- ties of the flakes [18–20,28,29]. The situation can drasti- cally depend on size and shape of the clusters. Even the geometrical structure of the edges (armchair vs zig-zag) plays the important role [16,17,25]. In general two types of edge states, located nearby the Dirac point can be dis- cerned, the zero-energy states (ZES) that are degenerate and located exactly at E = 0 and the dispersed edge states (DES) that fill the low-energy spectra domain within the gap and are symmetrically distributed with respect to E = 0 [23–26]. mailto:lukyanc@ferroix.net Electronic and magnetic properties of graphite quantum dots Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 509 In this paper, we consider the electronic and magnetic properties of finite-size multilayer quantum dots (MQD) that should generalize the principal features of GQD in that sense as the described above extended multilayer systems grasp the properties of graphene. In particular we show that the edge state located close to E = 0 are again respon- sible for the principal electronic properties, but the level arrangement inside the gap is more diverse as in GQD. To stress the most prominent aspects we consider the charac- teristics examples of MQD of hexagonal and of triangular shape having zig-zag edges. For calculations we use the approach of tight-binding (TB) model in the nearest neigh- bor (nn) approximation. For magnetic properties we are mostly concentrated on the orbital diamagnetic effects. The role of the spin-paramagnetic properties is briefly dis- cussed in the conclusion and will be studied elsewhere. 2. The model Graphene is formed by a two-dimensional honeycomb lattice of carbon atoms in which the conducting -band electrons can be described within the TB model as † † † = ( )i i ij j ji ii i j i ij H c c t c c t c c , (1) where † ic and ic are the creation and annihilation electron operators, ijt are the intersite electron hopping elements and i is the on-site electron energy. The Hamiltonian (1) can be extended for the multilayer systems by taking into account the nn hopping between the adjacent layers. Five interlayer coupling parameters 1 5 were introduced by Slon- czewski, Weiss and McClure [11,12] as the hopping param- eters for the graphite structure (Fig. 1(a)). Parameter 1  0.39 eV represents the coupling between the vertically aligned BN and AN+1 atoms (subscript index means the number of plane, shift 1N N permutes A and B), pa- rameter 3 0.315 eV describes the coupling between the shifted AN and BN+1 atoms and parameter 4 0.44 eV corresponds to the coupling between the AN and AN+1 and between the BN and BN+1 atoms. Another two parameters, 2 and 5 represent the coupling between the next-nearest neighboring (nnn) layers. Parameter 5 0.04 eV connects atoms BN and BN+2 belonging to the same vertical line as atoms connected by parameter 1 whereas parameter 2   –0.02 eV corresponds to another vertically aligned atoms AN and AN+2, with no intermediate atom between them. In addition, the on-site electron energies i of A and B atoms in layers of MQD become different and described by the gap parameters i where i = 0 for AN and BN+1 atoms, i =1 for AN+1 and BN atoms and = 0.047 eV [5] alternately. We use the TB Hamiltonian (1) to study the MQD of triangular and hexagonal shape with zig-zag termination. We assume that graphene layers are arranged in the graph- ite-type ABA stacking as shown in Fig. 1. The electronic energy levels of MQD, ,nE and corresponding DOS are found from the TB Hamiltonian (1) with interlayer hop- ping. In current article we consider mostly the nn layer coupling, neglecting the effects of 2 , 5 and . The effect of a c-directed magnetic field is accounted by using the Peierls substitution for the hopping matrix elements ijt between atomic sites ir and :jr = exp j P ij ij ij i e t t t d c r r A l . (2) Here = (0, ,0)BxA is the vector potential of the magnetic field. Direct numerical diagonalization of Hamiltonian (1) gives the field-dependent energy levels ( )nE B of electron- ic states and corresponding on-site amplitudes of the wave function , .n i The orbital magnetic energy of the electron- ic state at = 0T can be found as function of the chemical potential and magnetic field B by assumption that all the energy levels below are double-filled by spin-up and spin-down electrons: < ( , ) = 2 ( ) En n n U B E B . (3) The corresponding orbital susceptibility per unit area and per one layer is calculated as 2 2 =0 1 ( , ) ( ) = , B U B N B (4) where N is the total number of layers and 2= 3 /4a n is the area of a flake containing n carbon atoms. 3. Graphene quantum dots Before consider multilayer clusters we describe the principal electronic and magnetic properties of single-layer clusters with zig-zag edges, studied in [16–29]. The DOSs of triangular and hexagonal GQDs with total number of atoms n = 526 and 1014, obtained by diago- Fig. 1. (Color online) Coupling parameters for multilayer ABA carbon stacking (a). Top view of the multilayer structure (b). The carbon atoms belonging to sublattices AN and BN are shown by blue and red colors. Hazem Abdelsalam, T. Espinosa-Ortega, and Igor Luk’yanchuk 510 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 nalization of TB Hamiltonian (1) are shown in Figs. 2 and 3. In general, they repeat the DOS of the infinite graphene layer [30] smeared by the finite-quantization noise, that vanishes when size of the cluster increases. The particle-hole symmetry of DOS, 1 1( ) = ( )D E D E is con- served. Figure 4 shows the orbital magnetic susceptibility, ob- tained from the magnetic field variation of the energy lev- els by the method, described in Sec. 2. The magnetic field was varied between 0 and 4 T where the susceptibility was checked to be almost field-independent. Again, at large energy scale both the dependencies ( )E are similar and are represented by series of jumps between paramagnetic and diamagnetic values, provided by the almost-equi- probable up- and downward displacement of the size- quantized states as function of magnetic field. As cluster size increases, the magnetic susceptibility for all nano- structures, disregarding their shape and edge-termination tends to the bulk limit characterized by a diamagnetic -function singularity at = 0E [11]. The most important details, distinguishing triangular and hexagonal GQDs are concentrated at nearly-zero ener- gies when orbital susceptibility is diamagnetic. The DOS of triangular GQDs (Fig. 2) reveals the re- markable feature: a large number of degenerate states is observed exactly at E = 0 and is manifested by the huge central peak of zero-energy states located inside the energy gap. This property is explained by the considerable imbal- ance of atoms in cluster sublattices A and B [18,21] that leads to degeneracy 1 = 3 3.n (5) The wave functions of ZES are localized mainly at the edges of the flake [26]. Absence of electronic states inside the near-zero energy gap results in the field-independent diamagnetic plato in ( )E at E 0 (Fig. 4(a)). Importantly, the degenerate states from the central peak do not contribute to suscepti- bility since, they don’t move from their location at = 0E when magnetic field is applied. The level distribution at E = 0 in hexagonal GQDs is qualitatively different (see Fig. 3). The localized edge states are not gathered exactly at E = 0, but are mostly dis- tributed nearby, inside the band, corresponded to the gap for triangular clusters. In strike contrast to the triangular case, these dispersed states give the considerable contribu- tion to the orbital diamagnetism [26,29], demonstrating the broad diamagnetic peak at E 0 in the orbital diamagnetic susceptibility (Fig. 4(b)). In general, the diamagnetic re- sponse is larger in hexagonal GQDs as compare to triangu- lar GQDs. 4. Electronic properties of multilayer quantum dots We turn now to multilayer clusters with layer number N = 2–5. The DOSs of triangular MQDs with zig-zag edges are shown in Fig. 5. Similar to the single-layer case the energy gap with the interior central peak in DOS is observed. The difference however is that, these states are not located exactly at the Dirac point = 0E but smeared around it with formation of N peaks of approximately the same amplitude (Fig. 5(b)). The total number of the near- zero energy states (NZES) is just the multiple of ZES in each graphene layer, 1= .N N To reveal the detailed structure of NZES we plot the eigenstate index vs its energy for different N (Fig. 6) and do observe the energy eigenlevel cumulation at the loca- tions of the split peaks. Importantly, they are not complete- ly degenerate except the states appeared in the odd-layer clusters exactly at E = 0. Another new property is the elec- tron–hole asymmetry of DOS with respect to the Dirac Fig. 2. (Color online) Large-energy scale of DOS of triangular GQD (a), ZES levels (b) and zoom of DOS (c) within the gap region. Fig. 3. (Color online) DOS of hexagonal GQD (a), the energy levels (b) and the corresponding DOS in the near-zero-energy re- gion, E 0 (c). The levels corresponded to localized edge-states are noted by red color. Fig. 4. Orbital magnetic susceptibility of triangular GQD (a) and of hexagonal GQD (b). Electronic and magnetic properties of graphite quantum dots Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 511 point, ( ) ( ).N ND E D E To examine the origin of these new features we tested the variation of DOS under conse- cutive variation of the coupling parameters i and found that this is the coupling 4 which is responsible for both effects. Inset to Fig. 5(b) shows that the central peak is unsplit at 4 = 0. Note that ( )ND E for MQDs with > 2N can be recon- structed from DOSs of bilayer clusters of the same shape if the dependence of 2 ( )D E is known as function of the coupling parameters 1, 3 and 4 . Generalizing the band decomposition method, proposed for infinite systems in [6] we present ( )ND E as 2 , 1,3,4 1 ( ) = ( , ), 2 N N m m D E D E (6) with = ( 1), ( 2), , 1m N N N and with -renor- malizing scaling factors , \| \| = 2sin 2( 1) N m m N . (7) Importantly, the term with = 0m and ,0 = 0N exists only for the odd number of layers. It corresponds to the contribution from the uncoupled graphene layer, that pro- vides the degenerate ZES observed in Fig. 6. This residual degeneracy however is removed when the nnn couplings 2 and 5 are taken into account. For even number of layers only two-layers states contribute to ( )ND E and no peaks in DOS appear within the size-quantization gap that vanishes with increasing of the cluster size [31]. Hexagonal MQDs, in contrast to triangular MQDs demonstrate practically the same structure of DOS as the single-layer GQD with near-zero-energy dispersed elec- tronic states (Fig. 7). The only tiny difference is the elect- ron–hole asymmetry, provided by the nn coupling para- meter 4. 5. Magnetic properties of multilayer quantum dots The large-energy scale plot of magnetic susceptibility for triangular MQDs with N = 2–5 demonstrates the ran- dom oscillations between diamagnetic and paramagnetic states (Fig. 8), similarly to what was observed for the sin- gle-layer case. Typically, the absolute values of suscepti- bility for the odd-layer MQD are always higher than those Fig. 6. Energy levels for triangular MQD with N = 2–5 layers inside the gap region. Fig. 7. DOS of hexagonal MQD with N = 2–5 layers in the near- zero-energy region. Fig. 5. DOS of triangular MQD with N = 2–5 layers (a) and its zoom at the near-zero energies (b). The nn coupling parameters are indicated in the text. The inset shows the unsplit central peak at 4 = 0. Hazem Abdelsalam, T. Espinosa-Ortega, and Igor Luk’yanchuk 512 Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 for the even-layer MQDs that can be explained by contri- bution of one decoupled single-layer, whose susceptibility is remarkably higher [6]. However at E 0 the prominent asymmetric splash is revealed in the gap region, where for the =1N case only the flat energy-independent plato was observed (Fig. 4(a)). This feature is provided by the field- dependent splitting of the central peak due 4-coupling. The structure of ( )E for hexagonal MQD (Fig. 9) is approximately the same as for hexagonal GQD, albeit some asymmetry of the broad diamagnetic peak at E 0 appears. 6. Discussion In this paper we studied the electronic and magnetic properties of MQD with zig-zag edges in nn TB approxima- tion as function of the Fermi energy and their relation with similar properties of GQD. The behavior of electronic DOS and of orbital magnetic susceptibility in the near-zero- energy region in vicinity of Dirac point is found to be pro- vided by the edge-localized electronic states. The details substantially depend on shape of MQD and on parity of lay- er number. In hexagonal MQD the situation is practically the same as in GQD, previously studied in [26]: the quasi- continuum distribution of edge-localized levels is observed at E 0 that provides the broad diamagnetic peak in the orbital susceptibility. In contrast, in triangular MQD the qualitatively new feature appears. The highly-degenerate ZES states, centered in the near-zero-energy gap region of triangular GQD, are split by the interlayer coupling parame- ter 4 onto the narrow multipeak band. This gives the nontypical splash-wavelet feature in the orbital diamagnetic susceptibility at Dirac point, absent for GQD. In our work we were focused on susceptibility arising from the orbital electronic properties whereas the spin- paramagnetic effects were not taken into account. Mean- while their role can be decisive in case of highly- degenerate electronic states at = 0E in the half-field tri- angular GQD with zig-zag edges. The smearing of ZES into near-zero-energy band in MQD removes such degen- eracy and one can assume that the spin-paramagnetic ef- fects will be less pronounced there. Meanwhile, this ques- tion is less trivial when the Hubbard-U Coulomb interaction and temperature-induced intraband electron jumps are properly taken into account. Therefore study of the competition between the temperature-independent or- bital-diamagnetic and temperature-dependent spin-para- magnetic properties in MQD posses the challenging prob- lem for many-body statistical physics. These effects can be discerned experimentally, basing on the temperature de- pendence of susceptibility. Another interesting property that we observed is the electron–hole asymmetry with respect to the level with E = 0, provided by the same interlayer coupling parame- ter 4. Being of the same origin as asymmetric semi- metallic multibranch spectrum of electron–hole carriers in graphite this feature can result in the non-zero location of Fermi level in the half-field MQD. Oscillation of finite DOS at Fermi level as function of magnetic field can give the quasi-de Haas–van Alphen oscillations similar to those observed in the bulk graphite. Study of their charac- ter and comparison with graphite presents another chal- lenging problem. This work was supported by the Egyptian mission sec- tor and by the European mobility FP7 Marie Curie pro- grams IRSES-SIMTECH and ITN-NOTEDEV. 1. S. Latil and L. Henrard, Phys. Rev. Lett. 97, 036803 (2006). 2. B. Partoens and F.M. Peeters, Phys. Rev. B 75, 193402 (2007). 3. J.M.B. Lopes dos Santos, N.M.R. Peres, and A.H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007). Fig. 8. Susceptibility of triangular MQD with N = 2–5 layers. Insets show the zoom of asymmetric splash-wavelet feature in the near-zero-energy region. Fig. 9. Susceptibility of hexagonal MQD with N = 2–5 layers. Electronic and magnetic properties of graphite quantum dots Low Temperature Physics/Fizika Nizkikh Temperatur, 2015, v. 41, No. 5 513 4. D. 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Electronic and magnetic properties of graphite quantum dots

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