PRL 102, 146806 (2009)PHYSICAL REVIEW LETTERS

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Hamiltonian Decomposition for Bulk and Surface States
Ken-ichi Sasaki,1,* Yuji Shimomura,1 Yositake Takane,1 and Katsunori Wakabayashi1,21Department of Quantum Matter, Graduate School of Advanced Sciences of Matter, Hiroshima University,
Higashi-Hiroshima 739-8530, Japan
2
PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012, Japan(Received 21 November 2008; published 10 April 2009)
We demonstrate that a tight-binding Hamiltonian with nearest- and next-nearest-neighbor hopping
integrals can be decomposed into bulk and boundary parts for honeycomb lattice systems. TheHamiltonian decomposition reveals that next-nearest-neighbor hopping causes sizable changes in theenergy spectrum of surface states even if the correction to the energy spectrum of bulk states is negligible.
By applying the Hamiltonian decomposition to edge states in graphene systems, we show that the nextnearest-neighbor hopping stabilizes the edge states. The application of Hamiltonian decomposition to ageneral lattice system is discussed.
DOI: 10.1103/PhysRevLett.102.146806

PACS numbers: 73.20.-r, 73.21.-b

The energy band structure is of central importance inunderstanding the electronic properties of material. Atight-binding (TB) model is a versatile approach to study
the electronic, magnetic and transport properties of a solid
since the TB model describes the qualitative features of theenergy band structure [1]. The long-range hopping terms
such as next-nearest-neighbor (NNN) hopping are often
added to the TB model with nearest-neighbor (NN) hopping to improve the energy band structure. In many cases,
the NNN correction to the energy band structure is not amatter of particular importance. In this Letter, we showthat the NNN hopping can change appreciably the energy
spectrum of surface states which appear near the boundaryof a system even when the correction to the energy spectrum of bulk states is negligible. We explain this by decomposing the NNN TB Hamiltonian into two parts: bulk andboundary parts. This Hamiltonian decomposition is essential to understanding the stability of surface states.
We use graphene systems to demonstrate theHamiltonian decomposition for the following reasons.
(1) Graphene is known to have both bulk and surface states.
The bulk states exhibit a ‘‘relativistic’’ energy band structure called the Dirac cone [2,3], and the surface statescalled the edge states appear near the zigzag edge [4].
(2) The edge states have been observed by several experimental groups [5–8], and a theoretical understanding of theexperimental results is called for. In fact, the NNN hopping
is important to explain the experimental results [9]. (3) The
edge states have a large density of states (DOS) near theFermi energy, which is responsible for the Fermi instabilities. Since the DOS depends on the energy spectrum or
bandwidth of the edge states, the NNN hopping is important for the appearance of many-body effects of the edgestates [10,11]. Thus, graphene is a good testing system for
the Hamiltonian decomposition not only from a theoreticalbut also from an experimental point of view.
We study the NN (NNN) TB Hamiltonian, HNN (HNNN ),which is defined as
0031-9007=09=102(14)=146806(4)
X yHNNci ½H NN ij cj ;¼
0i;j2allX yHNNNci ½H NNN ij cj ;¼
ni;j2all(1)

where ci (cyi ) is the annihilation (creation) operator of anelectron at the ith site, 0 (n ) is the NN (NNN) hopping
integral, and the matrix element ½H NN ij (½H NNN ij ) is 1
when the ith site is a (next) NN site of jth site and is zerootherwise. In the following, we will show that HNNN can be
decomposed into bulk and boundary (edge) parts for agraphene with zigzag edge.
Suppose that we put an electron on the central sitedenoted by the empty circle in Fig. 1(a). The initial stateis labeled as j0 i ¼ cy0 j0i. We operate on j0 i with
HNN =ð0 Þ, then H NN transfers the electron to three
NN sites denoted by the solid circles in Fig. 1(a). ThisP
state is written as j00 i ¼ i2all ½H NN i0 cyi j0i. The numbers associated with the solid circles in Fig. 1(a) indicate
the matrix element of H NN . The successive operation ofFIG. 1. (a) An electron at the central site, j0 i ¼ cy0 j0i, istransferredP
to three NN sites by HNN =ð0 Þ. The resultant stateis j00 i ¼ i2all ½H NN i0 cyi j0i. (b) H 2NN transfers the electron
to the NNN sites. At the same time, H 2NN returns the electron tothe original site. The matrix element of ½H 2NN ij that returns theelectron to the original site is given by 3 because there are three
NN sites around the central site. (c) The matrix element for thereturn process ½H 2NN ii depends on whether the site is a bulk site(½H 2NN ii ¼ 3) or a zigzag edge site (½H 2NN jj ¼ 2).

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Ó 2009 The American Physical Society

PRL 102, 146806 (2009)

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HNN =ð0 Þ on j00 i givesX
j000 i ¼
½H NN ji ½H NN i0 cyj j0i:

HNNN ¼ n(2)i;j2all

Starting from the initial site, the electron reaches the NNN
sites as shown in Fig. 1(b). Thus, the two successive NN
hopping processes relate to the NNN hopping process. This
indicates that H 2NN includes H NNN . However H 2NN andH NNN are not identical because in H 2NN there is a diagonal matrix element that returns the electron to the originalsite, that is, j000 i of Eq. (2) contains the term with j ¼ 0.
Since there are three NN sites around the original site, theamplitude of this return process is 3 as shown in Fig. 1(b).
H 2NN and H NNN become identical if we subtract thecorresponding diagonal matrix element from H 2NN . Thismatrix is proportional to the unit matrix, ½Iij ¼ ij .
For a periodic system, since the number of bonds ofevery site is three, we have H NNN ¼ H 2NN  3I. Putting
this into Eq. (1), we see that HNNN can be rewritten as
X yHNNN ¼ nci ½H 2NN  3Iij cj :(3)i;j2all

Xi;j2allcyi ½H 2NN ij cj þ nfor jEp i. In Eq. (3), HNNN contains the on-site potentialPpart, 3n i2all cyi ci . This on-site potential can be ignored
since it changes only the origin of the energy band structure, as shown by 3n in Eq. (4) [2]. This statement iscorrect for a periodic system without boundary, but is not
approved for a system with boundary. It is because of thatthe number of bonds of the edge sites is different from thatof a bulk site and the corresponding on-site potentials at theedge sites are different from those at the bulk sites.
To show this explicitly, we put an electron on the zigzag
edge site labeled as 2 in Fig. 1(c). The electron is transferred to the NNN sites by H 2NN . For this time, however,
the matrix element of the on-site potential term that weneed to subtract from H 2NN in order to get H NNN is 2
because the number of bonds is 2 for the edge site. It isdifferent from 3 for a nonedge (bulk) site. Thus, we obtainthe formula for HNNN as
X y
HNNN ¼ nci f½H 2NN ij  gi ½Iij gcj ;(5)i;j2all

where gi is the number of bonds of the ith site. Since wecan shift the origin of the energy by 3n without a loss ofgenerality, Eq. (5) can be written asðgi  3Þn^ i ; (6)i2allwhere n^ i ¼ cyi ci is the number operator of the ith site. The
first term in Eq. (6) [or the second term of Eq. (4)] showsthat HNNN breaks the particle-hole symmetry of HNN because H NNN contains the square of H NN . The secondterm in Eq. (6) represents quantum well potentials at theedge sites with potential depth of n because gi ¼ 2 for azigzag edge site. The quantum well potential appears onlyat an edge site whose number of bonds is different fromthose of a bulk site. Therefore, a surface state appearing
near the boundary is strongly affected by this potential. Forbulk states, only the first term of the right-hand side ofEq. (6) is important. In fact, if a system has no boundary (ifa system is periodic), the second term of Eq. (6) disappears
and the NNN Hamiltonian is given only by the first term.
Therefore, the first term can be considered as the bulk partand the second term is as the edge part of the NNNbulk þ H edge , whereHamiltonian, i.e., HNNN ¼ HNNNNNNX y2
bulkHNNN  nci ½H NN ij cj ;i;j2all

The matrix H NN can be diagonalized by a unitary matrix
as ½UH NN Uy pq ¼ Ep =ð0 Þpq , where Ep is the energy
eigenvalue of an eigenstate jEp i. Then, from Eq. (3) we seethat HNN and HNNN can be diagonalized simultaneously by
the basis of jEp i, and the energy eigenvalue of the total
Hamiltonian, HNN þ HNNN , is given byEp 2
þ 3n ;
(4)
Ep  n0Xedge n
HNNNXðgi  3Þn^ i :(7)i2allThe edge state is labeled by the wave vector along thezigzag edge, k, as jEðkÞi. Here EðkÞ denotes the energy
eigenvalue of HNN . Since EðkÞ of the edge state is very
bulk ,close to zero [4], the energy shift due to HNNN
22ðn =0 ÞEðkÞ , is negligible. Thus, the energy correction
to the edge state arises from HNNN asX
EðkÞ ¼ nhEðkÞjn^ i jEðkÞi:(8)i2edgeIt is only the density at the edge sites that determinesEðkÞ. The energy bandwidth (W) of the edge states canbe calculated in the following way. As k approaches
the Fermi point, i.e., ka ! 2=3 or 4=3 (a is lattice constant), the edge state changes into a bulk statesince
the localization length ðkÞ ! 1 [4,12]. ThenP^ i jEðkÞi is negligible and Eð2=3aÞ ¼
i2edge hEðkÞjnEð4=3aÞ ¼ 0. On the other hand, the k ¼ =a state is
the most localized
P state satisfying ð=aÞ ¼ 0. For thisstate, we have i2edge hEðkÞjn^ i jEðkÞi ¼ 1 and Eð=aÞ ¼
n . It shows that the k ¼ =a state (the Fermi point) islocated at the bottom (top) of the energy band and W ¼ nfor the edge state. This result of W ¼ n reproduces theenergy bandwidth that is numerically calculated for thezigzag edge shown in Fig. 2(a) [9]. The quantum welledge
potential of HNNN
lowers the edge state’s energy if n is
a positive value. We adopt n ¼ 0:3 eV. A similar value isobtained by a first-principles calculation with the local
density approximation [13].
It should be mentioned that Eqs. (5) and (6) include
NNN hopping between sites which are connected by two146806-2PRL 102, 146806 (2009)

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Pi2allpﬃﬃﬃﬃﬃ ii yni e ci j0i, then we havedc ji ¼ hjHNNNnNNNX

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ iði j Þni nj e:(11)i;j2edge

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
This result shows that not only the density ( ni nj ) but also
the relative phase (i  j ) of the localized wave function
is important for the energy shift. This is contrasted to thefact that the quantum well potential couples only to thedensity of a quantum state. If there is a lattice periodicity
along the edge, we can set i ¼ ðkaÞi and nj ¼ ni . Then
Eq. (11) becomes
Xdcji ¼ 2n cosðkaÞni :(12)hjHNNNi2edge

FIG. 2. (a) The energy band structure of graphene system withthe zigzag edge. This plot is obtained by diagonalizing HNN þHNNN  3n numerically. We adopt 0 ¼ 3:0 eV and n ¼
0:3 eV. The horizontal axis is a wave vector along the zigzagedge (k) multiplied by the lattice constant (a). (b) The lattice
structure of the Klein edge. The NNN hopping between NNNedge sites (i and j or i and j0 ) is not represented by the double of
the NN hopping. (c) The energy band structure of a graphenesystem with the Klein edge. (d) A graphene system with aedge
gives an on-site potential
boundary and a lattice vacancy. HNNN
energy shift of n at carbon atoms denoted by solid circles.
There is a disconnected NNN Hamiltonian at the dotted lines
around the lattice vacancy.

successive NN hopping processes. Two successive NN
hopping processes do not include disconnected NNN hopping process. The disconnected NNN process is relevant tothe Klein edge as shown in Fig. 2(b) [14]. Because H 2NN
cannot transfer an electron at the edge site (i) to the NNNedge site (j or j0 ), we have to add H dc
NNN , which represents
the NNN hopping between disconnected sites, to the matrix of the right-hand side of Eq. (5) in order to get acomplete H NNN . Thus we have
X yci f½H 2NN ij  gi ½Iij þ ½H dc
HNNN ¼ nNNN ij gcj :i;j2all(9)
Since ½H dc
NNN ij is not zero only when i and j are both theedge sites, the disconnected NNN Hamiltonian is written
asdc
HNNN nXNNNXci ½H dc
cyi cj :NNN ij cj ¼ ni;j2edgei;j2all
(10)

dc
HNNN

can be classified into the edge part of the NNNdc
is given by the creation andHamiltonian since HNNN
annihilation operators at the edge sites. If we represent
the wave function using the density n and phase  as ji ¼

Since gi ¼ 1 for the Klein edge sites, the edge part of theNNN Hamiltonian is written asedge
HNNN¼ 2nXn^ i  ni2edge

NNNXi;j2edge

cyi cj :(13)edgejEðkÞi) for
Using Eq. (12), we get EðkÞ (  hEðkÞjHNNNthe Klein edge asXEðkÞ ¼ 2n ½1 þ cosðkaÞni ðkÞ;(14)
i2edgewhere ni ðkÞ  hEðkÞjn^ i jEðkÞi. The energy bandwidth for
the Klein edge states is calculated as follows. Near theKlein edges, the edge states appear for 0  k < 2=3a and
4=3a < k  2=a [see Fig. 2(c)]. It can be shown thatthe wave function of the most localized state is given by
k ¼ 0 state, and P
the sum of the densities at the Klein edge
sites is given by i2edge ni ð0Þ ¼ 3=4 [12]. Then, by putting
k ¼ 0 into Eq. (14), we have Eð0Þ ¼ 3n . Thus, W for
the Klein edge is 3n (¼0:9 eV). This result also reproduces the energy bandwidth that is numerically calculated
for the Klein edge shown in Fig. 2(c). Half of W comesfrom the quantum well potential and the other half of W isdue to the disconnected NNN hopping process.
Here let us summarize the formula for the NNN
Hamiltonian: HNNN can be decomposed into bulk andbulk þ H edge withedge parts as HNNN ¼ HNNN
NNNX ybulk
HNNN nci ½H 2NN ij cj ;i;j2alledge
 n
HNNN

Xi2edgeðgi  gÞn^ i  nXi;j2edgecyi ½H dc
NNN ij cj :(15)g is the number of bonds of a bulk site. The first term of theedge
right-hand side of HNNNin Eq. (15) represents quantum
well potentials at the edge sites. It is only the number ofbonds at the edge site which determines the depth of thequantum well potential. As a result, the quantum well146806-3

PRL 102, 146806 (2009)PHYSICAL REVIEW LETTERS

potential appears at the edge sites, regardless of the edge
shape. For example, in the case of graphene, gi Þ gð¼ 3Þ
not only for the zigzag (or Klein) edge sites but also for thearmchair edge sites. Thus, for a finite system of graphene
shown in Fig. 2(d), the quantum well potentials of n are
denoted by the solid circles. The quantum well potential at
the armchair edge may not be of as much importance as thezigzag edge because the edge state of a graphene is absent
from the armchair edge [4–6]. It is also interesting to notethat gi Þ g for the three sites around a lattice vacancy. In
addition to the quantum well potentials at the three sites,
the disconnected NNN hopping appears between them.
We remark on other effects that can modify the energy
spectrum of the edge states. First, it is naively expected thatthe orbital energy at an edge carbon atom is different fromthat at a bulk atom when a functional group attaches to theedge atom. The attachment of a functional group gives riseto an additional change of the energy bandwidth of theedge states. However, this energy shift may be a positive or
negative value depending on the type of functional group.
This can be distinguished from the energy shift due to the
NNN Hamiltonian because it is always a negative value.
Second, W can be modified by the electron-electron orelectron-phonon interactions because they give rise to aself-energy correction to the edge states. A theoretical
calculation of the self energy for the edge states is given
in Refs. [10,15].
We note that Eq. (15) is not restricted to graphenesystems but is applicable to other two-dimensional systems
like the square lattice, and three-dimensional systems. Let
us consider the application of Hamiltonian decomposition to a square lattice system. On a square lattice, twoapplications of a NN hopping produces hopping not justto NNN, but even to the 3rd NN sites. Thus we need tosubtract hopping between 3rd NN sites (H 3rd ) from H 2NNbulk in Eq. (15) asin order to get H NNN . This modifies HNNNbulk nHNNNXi;j2allcyi f½H 2NN ij  ½H 3rd ij gcj :(16)edgedoes not change even for theOn the other hand, HNNN
square lattice system. The NNN Hamiltonian stabilizes
surface states in a general system through the quantum
well potentials. The energy dispersion relation of surface
states is observed below the Fermi level by high-resolution
photoemission studies of the (111) surfaces of copper,
silver, and gold [16]. We speculate that the observed
stability of the surface states is due to the edge part of
the NNN Hamiltonian.

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In conclusion, HNNN can be decomposed into the bulkand edge parts as shown in Eq. (15). If the energy spectrum
bulkof HNN is symmetric with respect to E ¼ 0, then HNNN
breaks this symmetry. If HNN has a localized energy eigenedge
state near the edge of a system, then HNNN
is relevant toshifting the energy eigenvalue through the quantum wellpotential and the disconnected NNN edge Hamiltonian.
The quantum well potential couples only to the density,whereas the disconnected NNN edge Hamiltonian couples
to the phase of the localized wave function. Although theHamiltonian decomposition is proved for two-dimensionalgraphene systems, it works for other systems like a onedimensional chain of atoms and a three-dimensional latticesystem as well.
This work is financially supported by a Grant-in-Aid forSpecially Promoted Research (No. 20001006) fromMEXT.

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