In this paper we investigate the influence of the next-nearest-neighbor
coupling of tight-binding model of graphene on the spectrum of plasmon
excitations. The nearest-neighbor tight-binding model was previously used to
calculate plasmon spectrum in the next paper [1]. We expand the previous
results of the paper by the next-nearest-neighbor tight-binding model. Both
methods are based on the numerical calculation of the dielectric function of
graphene and loss function. Here we compare plasmon spectrum of the
next-nearest and nearest-neighbor tight-binding models and find differences
between plasmon dispersion of two models.Comment: LaTeX, 4 pages, 4 Fig

Kadirko, V.

Kogan, E.

Ziegler, K.

English

arXiv

Kadirko, Vladimir

Ziegler, Klaus G.

Kogan, Eugene

OPUS Augsburg

Graphene, 2013, 2, 97-101 http://dx.doi.org/10.4236/graphene.2013.23014 Published Online July 2013 (http://www.scirp.org/journal/graphene) Next-Nearest-Neighbor Tight-Binding Model of  Plasmons in Graphene Vladimir Kadirko1, Klaus Ziegler2, Eugene Kogan1 1Department of Physics, Bar-Ilan University, Ramat Gan, Israel 2Institut für Physik, Universität Augsburg, Augsburg, Germany Email: Eugene.Kogan@biu.ac.il  Received May 8, 2013; revised June 7, 2013; accepted July 3, 2013  Copyright © 2013 Vladimir Kadirko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper we investigate the influence of the next-nearest-neighbor coupling on the spectrum of plasmon excitations in graphene. The nearest-neighbor tight-binding model was previously considered to calculate the plasmon spectrum in graphene [1]. We extend these results to the next-nearest-neighbor tight-binding model. As in the calculation of the nearest-neighbor model, our approach is based on the numerical calculation of the dielectric function and the loss func- tion. We compare the plasmon spectrum of the two models and discuss the differences in the dispersion.  Keywords: Graphene; Plasmon; Tight-Binding Model 1. Introduction Graphene, a single layer of carbon atoms arranged as a honeycomb lattice, is a semimetal with remarkable phy- sical properties [2,3]. This is due to the band structure of the material which consists of two bands touching each other at two nodes. The electronic spectrum around these two nodes is linear and can be approximated by Dirac cones. However, calculations of many physical proper- ties demand the knowledge of the full electron dispersion in the entire Brillouin zone, not only in the vicinity of the nodes. This statement becomes particularly relevant when we take into account the fact that graphene can be gated or doped, such that the Fermi energy can be freely tuned. One of the main open issues in the physics of graphene is the role played by electron-electron interaction. In doped graphene long range Coulomb interaction leads to a gapless plasmon mode which can be described theo- retically within the random phase approximation (RPA). Although this is a standard problem in semiconductor physics, it was studied initially in the case of graphene only in the Dirac approximation around the nodes [4-6]. The linear approximation leads to a frequency of the plasmon that is proportional to the square root of the wavevector. Later the plasmon dispersion law was also calculated for the more realistic band structure, obtained in the  framework of the tight-binding model with nearest-nei- ghbor hopping [1,7]. This model is characterized by two symmetric bands, which implies a chiral symmetry. The latter connects the eigenstates of energy E directly with eigenstates of energy −E by a linear transformation. This symmetry, which also realizes a particle-hole symmetry, is broken by a next-nearest-neighbor hopping term. Usu- ally, the physical properties change qualitatively under symmetry breaking. Here we would like to study the ef- fect of particle-hole symmetry breaking due to next-near- est-neighbor hopping on the plasmon dispersion. For this purpose we extend the nearest-neighbor hopping ap- proximation used in [1] by taking into account the next- nearest-neighbor hopping. We consider an electron gas which is subject to an electromagnetic potential  ,iV  q . The response of the electron gas is to create a screening potential  ,sV  q  which is caused by the rearrangement of the electrons due to the external potential. Therefore, the total potential, acting on the electrons, is [8]     , ,i sV V V ,  q q q          (1) can be evaluated self-consistently [9] and is expressed via the dielectric function. Then the total potential reads [8]     1,, iV V ,  qqq          (2) Copyright © 2013 SciRes.                                                                             Graphene V. KADIRKO  ET  AL. 98 2. Nearest and Next-Nearest Hopping Model The tight-binding Hamiltonian for electrons in graphene with both nearest and next-nearest-neighbor hopping has the form [2] (we use units such that h = 1)  †, ,,† †, , ,, ,..i ji ji j ji jH t a b H ct a a b b H       c      (3) where  annihilates (creates) an electron with spin  on site Ri on sublattice A (an equiva- lent definition is used for sublattice B),  is the nearest-neighbor hopping energy (hopping between different sublattices), and  †, ,i ia a  ,   2.8 eVt t  is the next nearest-nei- ghbor hopping integral (hopping in the same sublattice). The value of  is not well known but ab initio calcula- tions find  depending on the tight-bin- ding parametrization [2]. t0.02 0.2t t  tThe matrix representation of the Hamiltonian is 01 2 0        ih h ihHh ih h   2             (4) The non-diagonal terms in the Hamiltonian correspond to the nearest-neighbor hopping [1]:   3 31 21 1cos , sinj jj jh t h t     b k b k     (5) where b1,2,3 are the nearest-neighbor vectors on the ho- neycomb lattice:      1 2 31 2, 3 2 , 1 2, 3 2 , 1,0d d   b b b d  and is the lattice constant ( Å) the diagonal terms correspond to next-nearest-neighbor hopping: 1.42601cos jjh t   a k                (6) where,          1 23 45 62 3, 3 2 3 2, 3 23 2, 3 2 3 2, 3 20, 3 0, 3d dd dd       a aa aa a，， ，， the energy bands derived from this Hamiltonian have the form [2]         32cos 33 34cos cos2 2yy xE t f t ff k ak a k a               k k kk       (7)  where the plus sign applies to the upper (π or conduction) and the minus sign the lower (π* or valence) band. It should be noticed that the presence of  shifts the posi-  ttion of the Dirac point in energy and it breaks elec- tron-hole symmetry. In both cases, nearest-neighbor and next nearest-neighbor hopping, the electronic dispersion is an even function [1] ,lE E ,lk -k                  (8) The dispersion law for next nearest-neighbor hopping is presented on Figure 1, and the eigenvectors of the Hamiltonian read 1 2 1 22 2 2 21 2 1 21 1,2 21 1h ih h ihh h h h                     (9) where the first eigenvector is for the upper band and the second eigenvector for the lower band. The Hamiltonian H in Equation (4) has a chiral sym- metry for h0 = 0: 3 331 0,0 1e He H              (10)  which connects eigenstates of energy −E with eigenstates of energy E by 3E E                 (11) this is not the case after we have broken the chiral sym- metry by the next-nearest-neighbor hopping term h0. 3. Dielectric Function The longitudinal dielectric function in calculated in RPA [9,10]:   22π, 1 ,eq   q q         (12)  where κ is a dielectric constant and χ is a polarizability. For polarizability we used the Lindhard formula [10], which in our case after some straightforward calculations can be reduced to the expression     1 2, , ,      q q q        (13) with the intraband contribution                *10 , 11 221 2 2 21, lim 14 k ks t BZk kf sE Ed ks E E E E t i              qk kk q k qq  (14) and the interband contribution                2*20 , 11 221 2 2 21, lim 14 k ks t BZk kf sE Ed ks E E E E t i             qk kk q k qq  (15) Copyright © 2013 SciRes.                                                                             Graphene V. KADIRKO  ET  AL. 99 Figure 1. Energy dispersion of graphene with The figure shows a broken particle-hole symmetry. The Dirac nodes are shifted up by  The energy is  in the units of the electronic ba ith   where = 0.2t' t .  measured3t' .ndw  (Δ = 3t ).         1 21,cos cos cos ,x y xh h Eh t k d d k d      k k  (16)  23sin cos 3 sin ;x y xkh t k d k d k d        1 exp 1f E E      tribution function,  is the Fermi-Dirac dis- 1 ,Bk T defined as , is a chemical potential. The energies are            1 2, , ,1 2 2, 1 22,lE t kif we take    2 2 0E E k k +q  ,1 3 ,l l lllE E EE h h t ff      k k kk kk    (17) the integral yields the same polarizability form  found in the nearest-nei- ghbor model’s polarizability [1,7].  4. Plasmons in Graphene  proximation, ula as thatIn a first ap we can consider plasmons as collective excitations of electrons, where the dielectric function vanishes [8]:  , 0  q               (18)  c function is complex c function vanishes: 10] in general, however, the dielectridue to poles in the integrals (14) and (15). This implies that (18) has no solution, unless we only request that the real part of the dielectri Re ,   q 0             (19) assuming a real function   q  as the plasmon disp- ersion. For a numerical evaluation of the integrals it is more convenient to consider the loss function [6,8,        2 2Re , Im ,   1ImIm ,    ,        q qq    (20) qwhose broadened peak indicates the plasmon. Here a complex solution   q  and the gives both the dispersion from the real part decay of the plasmons imaginary part. In the present paper the polarizability of grapheme χ is evaluated numerically and the corresponding dielectric nfrom the function is obtained from Equation (12) for different val- ues of the real freque cy ω, the wave vector q  and chemical potential (Fermi energy) μ. Moreover, we as- sume κ = 4. The chemical potential level μ is selected to be relative to Dirac points whose existence is not affected by a variation of the parameter t  but are shifted by 3t  ([11]), as shown in Figure 2. Our results for plasmon dispersion law are sho n in Figures 3 and 4. For each figure we have selected two values for tw , namely 0t   and 0.2t t  . The original chemical potentials μ that appear in Figures 3 and 4 are taken from the previous paper [1] and are modified by the va  tlue0.2t  . The influence of next-nearest hopping parameter t  is insignificant for the plasmon dispersion law when the chemical otential is a irac point, as depicted in Figures 3 and 4. The shape of the plasmon dispersion law in Figure 3 does not change significantly by a varia- p bove D  Figure 2. The scheme shows that chemical potential μ is taken relative to Dirac cone. Dirac point’s level is relative to case and the level equals to  = 0t'  3t' . Copyright © 2013 SciRes.                                                                             Graphene V. KADIRKO  ET  AL. Copyright © 2013 SciRes.                                                                             Graphene 100     (a)                                                         (b) Figure 3. Plasmon dispersion for . (a) Plasmon dispersion for= 0.4μ t  = 0.4μ t ,  = 0t   (qx = 0).and different values of qy component (qx = 0); (b) Plasmon dispersion for and different values of q   = 0μ .4t , = 0.t' 2t  y component  (a)                                                         (b) Figure 4. Plasmon dispersion for  (a) Plasmon dispersion for and different values of qy compo- nent (qx = 0); (b) Plasmon dispersion for and differen  of onent (qx = 0).  tion of the parameter  On the other hand, the result is quite differe  the chemical potential is below the Dirac point. shows that for different values of hopping pa  and for a negative chemical potential the sha dispersion law changes strongly and the dispe e is much sharper when the value of the hoppi ter is larger. In general, our calculations of the persion law show that there is almost no chan on dispersion with when chemical pote he Dirac point. In conclusion we have investigated the 2D tight-binding  hamiltonian model under the influence of the next near- est-neighbour coupling (constant) and we theoretically obtained analytic expression for improved graphene po- larizability expression. Our work is extension to previous results obtained by [1] where only nearest-neighbor con- stant model is used. This work improves the previous results for graphene plasmon’s dispersion law. The research of next-nearest hopping tight-binding model gave the possibility to investigate the plasmon’s t in the case of low values e to Dirac point, by using al calcul ons apersion’s laws in two cases (nearest-neighbor and next- = -2.7μ t .=μ = -2.7μ tt values, = 0t'  qy comp-2.7t , = 0.2t' t  0.2t t  .nt whenFigure 4 rameter pe of rsion curvng paramplasmonge of thntial is above tte dise plasmt  5. Conclusions analytic ati nd numerically to show that dis- dispersion law near Dirac poinof chemical potential relativV. KADIRKO  ET  AL. 101nearest tight binding model) are almost the esam  as pre- icted theoretically.  ZdREFERENCES [1] A. Hill, S. A. Mikhailov and K. iegler, “Dielectric Func- tion and Plasmons in Graphene,” EPL (Europhysics Let- ters), Vol. 87, No. 2, 2009, Article ID: 27005. [2] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. No- voselov and A. K. 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Next-nearest-neighbor Tight-binding Model of Plasmons in Graphene

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