Electronic and Magnetic Properties of Graphite Quantum Dots
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Electronic and Magnetic Properties of Graphite Quantum Dots
Hazem Abdelsalam,
1, 2
T. Espinosa-Ortega, and Igor Lukyanchuk University of Picardie, Laboratory of Condensed Matter Physics, Amiens, 80039, France Department of Theoretical Physics, National Research Center, Cairo,12622, Egypt Division of Physics and Applied Physics, Nanyang Technological University 637371, Singapore
We study the electronic and magnetic properties of multilayer quantum dots (MQDs) of graphitein the nearest-neighbor approximation of tight-binding model. We calculate the electronic density ofstates and orbital susceptibility of the system as function of the Fermi level location. We demonstratethat properties of MQD depend strongly on the shape of the system, on the parity of the layer numberand on the form of the cluster edge. The special emphasis is given to reveal the new propertieswith respect to the monolayer graphene quantum dots (GQD). The most interesting results areobtained for the triangular MQD with zig-zag edge at near-zero energies. The asymmetricallysmeared multi-peak 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 diamagneticorbital susceptibility as function of energy.
I. INTRODUCTION
Rise of graphene certainly revived the interest to theclassical graphite systems, presenting a wealth of notyet well understood electronic and magnetic properties.The challenge is related to the complicate semi-metallicmulti-branch energy spectrum in the vicinity of the half-field Fermi-level, caused by splitting of the Dirac-conegraphene spectrum by the graphite-forming inter-carbon-layer coupling. The point of special interest is thecrossover from graphene to graphite through the mul-tilayer structures with few number of layers. It is wellknown that such systems may exhibit metallic or semi-conductor behavior as a function of the number of layersand the stacking process , characteristic that makesthem highly appealing for gated controlled electronic de-vices.The magnetic properties of few-layer structure are evenmore intriguing . The orbital magnetism of the odd-layer structures is reminiscent to that for the monolayergraphene . In particularly, a characteristic for graphenediamagnetic δ -function singularity of susceptibility appears at the Dirac point at E = 0. For even number oflayers the magnetic properties are more similar to thosefor bilayer graphene and the diamagnetic response hasthe weaker logarithmic divergency. Such odd/even layerdecomposition can give the coexistence of Dirac and nor-mal carriers, observed in the pure graphite .Alongside, flake-like graphene quantum dots (GQD)have captured the substantial attention of nanotech-nology due to their unique optical and magneticproperties . The new element here is the finite-sizeelectron confinement, resulted in opening of the energygap for the bulk delocalized electronic states in the vicin-ity of Dirac point . This gap, however can befilled by the energy level of novel electronic states, lo-calized in the vicinity of the sample boundary . Fornanoscopic and even for mesoscopic clusters these edgestates can play the decisive role in electronic and mag-netic properties of the flakes . The situation can drastically depend on size and shape of the clusters.Even the geometrical structure of the edges (armchair vszigzag) plays the important role . In general twotypes of edge states, located nearby the Dirac point canbe discerned, the zero energy states (ZES) that are de-generate and located exactly at E = 0 and the dispersededge states (DES) that fill the low-energy energy-spectradomain within the gap and are symmetrically distributedwith respect to E = 0 In this paper, we consider the electronic and magneticproperties of finite-size multilayer quantum dots (MQD)that should generalize the principal features of GQD inthat sense as the described above extended multilayersystems grasp the properties of graphene. In particularwe show that the edge state located close to E = 0 areagain responsible for the principal electronic properties,but the level arrangement inside the gap is more diverseas in GQD. To stress the most prominent aspects we con-sider the characteristics examples of MQD of hexagonaland of triangular shape having zigzag edges. For calcula-tions we use the approach of Tight Binding (TB) modelin the nearest neighbor (NN) approximation. For mag-netic properties we are mostly concentrated on the orbitaldiamagnetic effects. The role of the spin-paramagneticproperties is briefly discussed in the conclusion and willbe studied elsewhere. II. THE MODEL
Graphene is formed by a two-dimensional honeycomblattice of carbon atoms in which the conducting π -bandelectrons can be described within the TB model as H = X i ε i c † i c i + X h ij i ( t ij c † i c j + t ji c † j c i ) , (1)where c † i and c i are the creation and annihilation electronoperators, t ij are the inter-site electron hopping elementsand ε i is the on-site electron energy. The Hamiltonian(1) can be extended for the multilayer systems by tak-ing into account the NN hopping between the adjacentlayers. Five interlayer coupling parameters γ . . . γ wereintroduced by Slonczewski, Weiss and McClure asthe hopping parameters for the graphite structure (Fig.1a). Parameter γ ≃ .
39 eV represents the coupling be-tween the vertically aligned B N and A N +1 atoms (sub-script index means the number of plane, shift N → N + 1permutes A and B), parameter γ ≃ .
315 eV describesthe coupling between the shifted A N and B N +1 atomsand parameter γ ≃ .
044 eV corresponds to the cou-pling between the A N and A N +1 and between the B N and B N +1 atoms. Another two parameters, γ and γ represent the coupling between the next-nearest neigh-boring (NNN) layers. Parameter γ ≃ .
04 eV connectsatoms B N and B N +2 belonging to the same vertical lineas atoms connected by parameter γ whereas parameter γ ≃ − .
02 eV corresponds to another vertically alignedatoms A N and A N +2 , with no intermediate atom be-tween them. In addition, the on-site electron energies ǫ i of A and B atoms in layers of MQD become differentand described by the gap parameters ∆ = ± .
047 eV alternately. Γ ABA Γ Γ Γ a) Γ Γ B N A N A N + B N + b) FIG. 1. Coupling parameters for multilayer ABA carbonstacking (a). Top view of the multilayer structure (b). Thecarbon atoms belonging to sublattices A N and B N are shownby blue and red colors. We use the TB Hamiltonian (1) to study the MQDof triangular and hexagonal shape with zig-zag termina-tion. We assume that graphene layers are arranged inthe graphite-type ABA stacking as shown in Fig.1. Theelectronic energy levels of MQD, E n and correspondingDOS are found from the TB Hamiltonian (1) with inter-layer hopping. In current article we consider mostly theNN layer coupling, neglecting the effects of γ , γ and∆. The effect of a c -directed magnetic field is accountedby using the Peierls substitution for the hopping matrixelements t ij between atomic sites r i and r j t ij → t Pij = t ij exp (cid:26) e ~ c Z r j r i A · d l (cid:27) . (2)Here A = (0 , Bx,
0) is the vector potential of the mag-netic field.
TRI N=1 a) b) gapc)
FIG. 2. Large-energy scale of DOS of triangular GQD (a),ZES levels (b) and zoom of DOS(c) within the gap region.
Direct numerical diagonalization of Hamiltonian (1)gives the field-dependent energy levels E n ( B ) of elec-tronic states and corresponding on-site amplitudes of thewave function, ϕ n,i . The orbital magnetic energy of theelectronic state at T = 0 can be found as function of thechemical potential µ and magnetic field B by assump-tion that all the energy levels below µ are double-filledby spin-up and spin-down electrons: U ( B, µ ) = 2 E n <µ X n E n ( B ) , (3)The corresponding orbital susceptibility per unit areaand per one layer is calculated as χ ( µ ) = − N σ (cid:20) ∂ U ( B, ǫ ) ∂B (cid:21) B =0 , (4)where N is the total number of layers and σ = √ a n/ n carbon atoms. III. GRAPHENE QUANTUM DOTS
Before consider multi-layer clusters we describe theprincipal electronic and magnetic properties of single-layer clusters with zig-zag edges, studied in .The DOSs of triangular and hexagonal GQDs with to-tal number of atoms n = 526 and 1014, obtained by di-agonalization of TB Hamiltonian (1) are shown in Fig.2and Fig.3. In general, they repeat the DOS of the infi-nite graphene layer smeared by the finite-quantizationnoise, that vanishes when size of the cluster increases.The particle-hole symmetry of DOS, D ( E ) = D ( − E )is conserved.Fig.4 shows the orbital magnetic susceptibility, ob-tained from the magnetic field variation of the energylevels by the method, described in Sect. II. The mag-netic field was varied between 0 T and 4 T where the sus-ceptibility was checked to be almost field-independent.Again, at large energy scale both the dependencies χ ( E )are similar and are represented by series of jumps be-tween paramagnetic and diamagnetic values, provided bythe almost-equiprobable up- and downward displacementof the size-quantized states as function of magnetic field.As cluster size increases, the magnetic susceptibility forall nanostructures, disregarding their shape and edge-termination tends to the bulk limit characterized by adiamagnetic δ -function singularity at E = 0. . HEX N=1 a) b)c)
FIG. 3. DOS of hexagonal GQD (a), the energy levels (b)and the corresponding DOS in the near-zero energy region, E ∼ The most important details, distinguishing triangularand hexagonal GQDs are concentrated at nearly-zero en-ergies when orbital susceptibility is diamagnetic.The DOS of triangular
GQDs (Fig.2) reveals the re-markable feature: the large number of degenerate statesis observed exactly at E = 0 and is manifested by thehuge central peak of zero energy states (ZES) located in-side the energy gap. This property was shown to be ex-plained by the considerable imbalance of atoms in clustersublattices A and B that leads to degeneracy η = √ n + 3 − . (5)The wave functions of ZES are localized mainly at theedges of the flake .Absence of electronic states inside the near-zero energygap results in the field-independent diamagnetic plato in χ ( E ) at E ∼ E = 0when magnetic field is applied. TRI, N=1 a) HEX, N=1 b) FIG. 4. Orbital magnetic susceptibility of triangular GQD(a) and of hexagonal GQD (b).
The level distribution at E ∼ hexagonal GQDsis qualitatively different (see Fig.3). The localized edge −10 −5 0 5 1000.20.4 D O S ( / γ ) −10 −5 0 5 1000.20.4 D O S ( / γ ) −10 −5 0 5 1000.20.4 D O S ( / γ ) −10 −5 0 5 1000.20.4 D O S ( / γ ) Energy (eV) −1 −0.5 0 0.5 100.20.4−1 −0.5 0 0.5 100.20.4−1 −0.5 0 0.5 100.20.4−1 −0.5 0 0.5 100.20.4 Energy (eV) −0.1 0 0.101
TRI,N=2TRI,N=3TRI,N=4TRI,N=5 γ =.044(a) (b) γ =0 FIG. 5. DOS of triangular MQD with N = 2 ÷ γ = 0. states are not gathered exactly at E = 0, but are mostlydistributed nearby, inside the band, corresponded to thegap for triangular clusters. In strike contrast to the tri-angular case, these dispersed states give the considerablecontribution to the orbital diamagnetism , demon-strating the broad diamagnetic peak at E ∼ IV. ELECTRONIC PROPERTIES OFMULTILAYER QUANTUM DOTS
We turn now to multilayer clusters with layer number N = 2 ÷
5. The DOSs of triangular
MQDs with zig-zagedges are shown in Fig.5. Similar to the single-layer casethe energy gap with the interior central peak in DOS isobserved. The difference however is that, these states arenot located exactly at the Dirac point E = 0 but smearedaround it with formation of N peaks of approximately thesame amplitude (Fig.5b). The total number of the near-zero energy states (NZES) is just the multiple of ZES ineach graphene layer, η N = N η .To reveal the detailed structure of NZES we plot theeigenstate index vs its energy for different N (Fig.6) anddo observe the energy eigen-level cumulation at the loca-tions of the split peaks. Importantly, they are not com-pletely degenerate except the states appeared in the odd-layer clusters exactly at E = 0. Another new property isthe electron-hole asymmetry of DOS with respect to theDirac point, D N ( E ) = D N ( − E ). To examine the origin −0.2 −0.1 0 0.1 0.2 E i gen s t a t e i nde x Energy (eV) −0.2 −0.1 0 0.1 0.2 E i gen s t a t e i nde x Energy (eV)−0.2 −0.1 0 0.1 0.2 E i gen s t a t e i nde x Energy (eV) −0.2 −0.1 0 0.1 0.2 E i gen s t a t e i nde x Energy (eV)TRI,N=4 TRI,N=5TRI,N=3TRI,N=2
FIG. 6. Energy levels for triangular MQD with N = 2 ÷ of these new features we tested the variation of DOS un-der consecutive variation of the coupling parameters γ i and found that this is the coupling γ which is responsi-ble for the both effects. Inset to Fig.5(b) shows that thecentral peak is unsplit at γ = 0.Note that D N ( E ) for MQDs with N > D ( E ) is known as function of thecoupling parameters γ , γ and γ . Generalizing the banddecomposition method, proposed for infinite systems in we present D N ( E ) as D N ( E ) = 12 X m D ( E, λ
N,m γ , , ) , (6)with m = − ( N − , ( N − , . . . , N + 1 and with γ -renormalizing scaling factors λ N,m = 2 sin | m | π N + 1) (7)Importantly, the term with m = 0 and λ N, = 0 ex-ists only for the odd number of layers. It correspondsto the contribution from the uncoupled graphene layer,that provides the degenerate ZES observed in Fig.6. Thisresidual degeneracy however is removed when the NNNcouplings γ and γ are taken into account. For evennumber of layers only two-layers states contribute to D N ( E ) and no peaks in DOS appear within the size-quantization gap that vanishes with increasing of thecluster size . Hexagonal
MQDs, in contrast to triangular MQDsdemonstrate practically the same structure of DOS as thesingle-layer GQD with near-zero-energy dispersed elec-tronic states (Fig.7). The only tiny difference is theelectron-hole asymmetry, provided by the NN couplingparameter γ . −1 −0.5 0 0.5 100.010.020.030.040.05 Energy (eV) D O S ( / γ ) −1 −0.5 0 0.5 100.010.020.030.040.05 Energy (eV) D O S ( / γ ) −1 −0.5 0 0.5 100.010.020.030.040.05 Energy (eV) D O S ( / γ ) −1 −0.5 0 0.5 100.010.020.030.040.05 Energy (eV) D O S ( / γ ) HEX,N=2HEX,N=4 HEX,N=3HEX,N=5
FIG. 7. DOS of hexagonal MQD with N = 2 ÷ −2 −1 0 1 2−202468 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) −2 −1 0 1 2−50510 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) −2 −1 0 1 2−202468 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) −2 −1 0 1 2−50510 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) −0.2 0 0.2−1.5−1.3 −0.2 0 0.2−3−2.8−0.2 0 0.2−2.6−2.2 −0.2 0 0.2−4.2−3.8 TRI,N=4TRI,N=2 TRI,N=3TRI,N=5
FIG. 8. Susceptibility of triangular MQD with N = 2 ÷ V. MAGNETIC PROPERTIES OFMULTILAYER QUANTUM DOTS.
The large-energy scale plot of magnetic susceptibilityfor triangular
MQDs with N = 2 ... . However at E ∼ −2 −1 0 1 2−10−50510 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) −2 −1 0 1 2−10−50510 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) −2 −1 0 1 2−10−50510 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) −2 −1 0 1 2−10−50510 Energy (eV) S u sc ep t i b ili t y ( γ χ / α ) HEX,N=4 HEX,N=3HEX,N=5HEX,N=2
FIG. 9. Susceptibility of hexagonal MQD with N = 2 ÷ region, where for the N = 1 case only the flat energy-independent plato was observed. (Fig.4a). This featureis provided by the field-dependent splitting of the centralpeak due γ -coupling.The structure of χ ( E ) for hexagonal MQD is approx-imately the same as for hexagonal GQD, albeit someasymmetry of the broad diamagnetic peak at E ∼ VI. DISCUSSION
In this paper we studied the electronic and magneticproperties of MQD with zig-zag edges in NN TB ap-proximation as function of the Fermi energy and theirrelation with similar properties of GQD. The behaviorof electronic DOS and of orbital magnetic susceptibil-ity in the near-zero energy region in vicinity of Diracpoint is found to be provided by the edge-localized elec-tronic states. The details substantially depend on shapeof MQD and on parity of layer number. In hexagonalMQD the situation is practically the same as in GQD, previously studied in : the quasi-continuum distributionof edge-localized levels is observed at E ∼ γ onto the narrow multi-peak band. This gives the nontypical splash-wavelet feature in the orbital diamagneticsusceptibility at Dirac point, absent for GQD.In our work we were focused on susceptibility aris-ing from the orbital electronic properties whereas thespin-paramagnetic effects were not taken into account.Meanwhile their role can be decisive in case of highly-degenerate electronic states at E = 0 in the half-field tri-angular GQD with zig-zag edges. The smearing of ZESinto near-zero-energy band in MQD removes such degen-eracy and one can assume that the spin-paramagneticeffects will be less pronounced there. Meanwhile, thisquestion is less trivial when the Hubbard-U Coulomb in-teraction and temperature-induced intra-band electronjumps are properly taken into account. Therefore studyof the competition between the temperature-independentorbital-diamagnetic and temperature-dependent spin-paramagnetic properties in MQD posses the challengingproblem for many-body statistical physics. These effectscan be discerned experimentally, basing on the tempera-ture dependence of susceptibility.Another interesting property that we observed is theelectron-hole asymmetry with respect to the level with E = 0, provided by the same interlayer coupling param-eter γ . Being of the same origin as asymmetric semi-metallic multi-branch spectrum of electron-hole carriersin graphite this feature can result in the non-zero loca-tion of Fermi-level in the half-field MQD. Oscillation offinite DOS at Fermi-level as function of magnetic fieldcan give the quasi- de Haas van Alphen oscillations simi-lar to those observed in the bulk graphite. Study of theircharacter and comparison with graphite presents anotherchallenging 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. S. Latil and L. Henrard, Phys. Rev. Lett. , 036803(2006). B. Partoens and F. M. Peeters, Phys. Rev. B, , 193402(2007). J. M. B. Lopes dos Santos, N. M. R. Peres and A. H. CastroNeto, Phys. Rev. Lett. , 256802 (2007). D. Finkenstadt, G. Pennington, and M. J. Mehl, Phys.Rev. B , 121405(R), (2008). M. Koshino, New J. Phys., , 015010 (2013). M. Koshino and T. Ando, Phys. Rev. B , , 085425(2007). S. A. Safran, Phys. Rev. B , 421 (1984). R. Saito and H. Kamimura, Phys. Rev. B , 7218 (1986). M. Nakamura and L. Hirasawa, Phys. Rev. B , 045429(2008). A. H. Castro Neto, F. Guinea, N. M. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod.Phys. , 109(2009). J. W. McClure, Phys Rev. , 606 (1960). J. C. Slonczewski and P. R. Weiss, Phys. Rev. , 272(1958). I. Luk’yanchuk and Y. Kopelevich, Phys. Rev. Lett., , I. Lukyanchuk and Y. Kopelevich, Phys. Rev. Lett., ,256801 (2006) I. Luk’yanchuk, Y. Kopelevich, and M. El Marssi, PhysicaB-Cond. Matt., , 404 (2009) M. Ezawa, Phys. Rev. B , 245415 (2007). Z. Z. Zhang, K. Chang, and F. M. Peeters, Phys. Rev. B , 235411 (2008). J. Fernandez-Rossier and J. J. Palacios, Phys. Rev. Lett. , 177204 (2007). W. L. Wang, S. Meng, and E. Kaxiras, Nano Lett. , 241(2008). W. L. Wang, O. V. Yazyev, S. Meng, and E. Kaxiras, Phys.Rev. Lett. , 157201 (2009). P. Potasz, A. D. Guclu, and P. Hawrylak, Phys.Rev. B ,033403(2010). D. P. Kosimov, A. A. Dzhurakhalov, and F. M. Peeters, Phys. Rev.B , 195414 (2010). T. Espinosa-Ortega, I. A. Luk’yanchuk, and Y. G.Rubo,Superlattices Microstruct. , 283 (2011). M. Zarenia, A. Chaves, G. A. Farias, and F. M. Peeters,Phys. Rev.B , 245403 (2011). H. P. Heiskanen, M. Manninen, and J. Akola, New J. Phys. , 103015 (2008). T. Espinosa-Ortega, I. A. Luk’yanchuk, and Y. G. Rubo,Phys. Rev.B , 205434 (2013). J. Liu, Z. Ma, A. R. Wright, and C. Zhang, J. Appl. Phys. ,103711 (2008). Y. Ominato and M. Koshino, Phys. Rev. B , 165454(2012). Y. Ominato and M. Koshino, Phys. Rev. B , 115433(2013). J. P. Hobson, W. A. Nierenberg, Phys. Rev. , 662 (1953). D. R. da Costa, M. Zarenia, Andrey Chaves, G. A. Farias,and F. M. Peters, Carbon78