Energy gap opening in submonolayer lithium on graphene: Local density functional and tight-binding calculations

0.22 MB·English

by  M. Farjam

Energy gap opening in submonolayer lithium on graphene: Local density functional
and tight-binding calculations
M. Farjam1 and H. Rafii-Tabar1,2
1Department of Nano-Science, Computational Physical Sciences Laboratory,
Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
2Department of Medical Physics and Biomedical Engineering,
and Research Centre for Medical Nanotechnology and Tissue Engineering,
Shahid Beheshti University of Medical Sciences, Evin, Tehran 19839, Iran
9 (Dated: January 27, 2009)
0
The adsorption of an alkali-metal submonolayer on graphene occupying every third hexagon of
0 the honeycomb lattice in a commensurate (√3 √3)R30◦ arrangement induces an energy gap
2 ×
in the spectrum of graphene. To exemplify this type of band gap, we present ab initio density

n functional theory calculations of the electronic band structure of C6Li. An examination of the
a lattice geometry of the compound system shows the possibility that the nearest-neighbor hopping
J
amplitudes have alternating values constructed in a Kekul´e-type structure. The band structure of

7 the textured tight-binding model is calculated and shown to reproduce the expected band gap as
2 well as other characteristic degeneracy removals in the spectrum of graphene induced by lithium
adsorption. More generally we also deduce the possibility of energy gap opening in periodic metal
i] on graphene compounds CxM if x is a multipleof 3.
c
s PACSnumbers: 73.22.−f,73.20.At,81.05.Uw
-
l
r
t
m I. INTRODUCTION One of the routes toward tailoring the electronic
properties of graphene is through the adsorption of
.
t metals.10,11 Alkali metals, in particular, are donors of
a The isolation of graphene, a honeycomb lattice of car-
m bon atoms, and the observation of the electric-field ef- electrons and can be used for the purpose of doping
graphene to change its carrier concentration. The re-
- fectinthenanostructuredsamples,depositedonoxidized
d lated problem of adsorptionof alkali metals on graphite,
silicon surface, have renewed interest in the electronic
n propertiesofthistwo-dimensionalcarbonmaterial.1 The which consists of weakly linked layers of graphene, has
o been studied extensively in the past.12 In a quite recent
breakthrough discovery made possible the experimen-
c
study,abandgapwasexperimentallyobserved,andthe-
[ tal observation of theoretically predicted exotic physics
oretically confirmed, in the surface electronic structure
of graphene, including an anomalous half-integer quan-
2 of graphite, induced by the adsorption of sodium in a
tum Hall effect, non-zero Berry’s phase, and minimum
0v conductivity.2,3 (fo5r×th5e)(m4od4e)l.m13oTdehle.1s4imDiulaertoefftehcethheaasvyaldsoopbienegn,hreopwoervteerd,
1 The exceptional properties of graphene, two- ×
theFermilevelisdisplacedintothegrapheneconduction
3 dimensional structure and room-temperature high
0 mobility, make it an ideal material for carbon-based π∗ band so that the band gaplies in the occupied levels.
. Thegap-inducingmechanisminthesecasesinvolvesdop-
1 nanoelectronics envisaged in the future nanotechnolo-
ing as well as interaction between the graphene layers in
0 gies. Various applications depend on the possibility
the graphite surface.
9 of fabricating a graphene-based field effect transistor
0
(FET), including transistors to be used in nanoscale In this paper, we study the gap-inducing effect of
:
v electronic circuits, light emitters and detectors, and alkali-metal adsorption on single layer graphene, which
i ultra-sensitive chemical sensors and biosensors.4 Being representsadifferentproblemthangraphitesurfacesince
X
a zero-gap semiconductor, however, graphene cannot the gap cannot be due to interlayer interactions. For in-
r
a be used directly as the conducting channel in FETs. stance, first-principles calculations of metal adsorption
Discovering how to generate band gaps is therefore on graphene in the (4 4) model15 do not show a gap
×
a question of fundamental and applied significance. at the Dirac point in the density of states (DOS). We
Several possibilities exist. Lateral confinement of the present the case of lithium on graphene compound C Li
6
charge carriers in graphene nanoribbons (GNRs),5 and in detail for several reasons. This is a prototypical sys-
different gate voltages applied to the two layers of teminwhichthegapopeningeffectoccurs,itisasimple
bilayer graphene are promising methods of engineering system to study theoretically, and it may be accessible
band gaps in graphene.6 Recently, a band gap has been experimentally. The crucial property here is the struc-
discoveredinepitaxialgraphenederivedfromgraphitiza- tureofthesystem,sothatotheralkaliandalkaline-earth
tionofsiliconcarbide.7 Theeffectofasubstrate-induced metal compounds C M can exhibit the same effect. For
6
band gap has also been investigated experimentally instance, a similar band gap has been calculated for cal-
in epitaxial graphene on nickel,8 and theoretically in cium graphite intercalated compound,16 as well as for
graphene on boron nitride.9 calcium on graphene, both as C Ca.17 Due to doping,
6
2
lithium or another alkali metal on graphene results in a (a) (b)
4 4
metal and is not directly useful for semiconductor ap-
plications. It is, however, important to understand the 3 3
mechanism responsible for the gap opening effect.
2 2
An interesting question is which coveragesof an alkali
metalongraphenecaninduceabandgapinitsspectrum. 1 1
V)
Our first-principles density functional theory (DFT) cal- e 0 0
culationsindicate theabsenceofagapin(1 1),(2 2), gy (
× × er −1 −1
and (4 4) models, corresponding to C Li, C Li, and n
2 8 E
×
C Li,respectively,butitspresenceinthe(√3 √3)R30◦ −2 −2
32
×
structure, which includes C3Li and C6Li, as well as in −3 −3
(3 3)coverageC Li. Explainingtheunderlyingreason
18
× −4 −4
is the subject of the following sections.
Our paper is organized as follows. Section II presents −5 −5
K G M K K G M K
the results of DFT calculations. Section III develops the
tight-binding(TB)model. Itincludesadescriptionofthe
FIG.1: (a)BandstructureofC6 ,i.e.,graphenedescribedby
lattice geometry, from which we clarify the gap mech- theunitcellof(√3 √3)R30◦ s∗tructure. (b)Bandstructure
anism within the TB model. We conclude here that, ×
of C6Li. A band gap has opened at the neutrality point,
when lithium atoms are arranged above the hollow sites the degenerate bands along the Γ-K direction are split, and
of graphene honeycomb lattice in a periodic commensu- a small band gap has opened at K near the bottom of the
rate structure to form C6Li, the nearest-neighbor hop- figure. Horizontal dashed lines indicate the Fermi levels of
ping amplitudes acquire alternating values constructed therespective systems.
in a Kekul´e-type structure. This section also includes a
discussion of the TB Hamiltonian and numerical results
for the Kekul´e textured model with two hopping ampli- in the bond lengths has an important implication for
tudes. Section IV contains a summary and conclusion. opening a band gap, we then used the code to relax the
The Appendix develops the solution of the TB model. positions of the carbon atoms as well as the position of
lithiumatoms. ThisindeedresultedinatinyKekul´edis-
tortion of C-C bond lengths, so that the near bonds to
II. DFT CALCULATIONS Li were contracted by 2.8 10−4 ˚A, and the far bonds
to Li were stretched by 5×.6 10−4 ˚A. We then made
×
Quantitative data for the electronic band struc- band structure and DOS calculations for C6Li. For re-
ture of C Li were obtained from DFT calculations, peated images of the systems, we used supercells, with
6
using a plane wave basis set and pseudopotentials. c=12˚Aforgraphene,andc=15˚AforC6Li. Akinetic-
We used the Quantum-ESPRESSO code and norm- energycutoffof70Rywasneededfortotal-energyconver-
conserving pseudopotentials,18 and performed the cal- gence. Brillouin-zone (BZ) integrations were made with
culations with local-density approximation (LDA) and a 6 6 2 Monkhorst-Packsampling of k-space,21 with
× ×
Perdew-Zunger parametrization19 of the Ceperley-Alder Gaussiansmearingof 0.05Ry. For the calculationof the
correlation functional.20 We used a lattice parameter DOS we used the tetrahedron method22 with a finer k-
of a = 2.46 ˚A for graphene, corresponding to a C-C point mesh of 36 36 1 grid.
0 × ×
bond length of 1.42 ˚A. Other theoretical calculations15 The band structures of C Li, with unrelaxed carbon
6
have shown that alkali metals generally prefer to ad- atom positions, and C , i.e., graphene described with
6
∗
sorb on the hollow sites of graphene rather than the the unit cell of C Li, are compared in Fig. 1. We note
6
bridge or top sites. We positioned the lithium atoms the fourfold degeneracy of the bands at Γ, at the Fermi
over the hollow sites of graphene in a (√3 √3)R30◦ level (Dirac or neutrality point), in Fig. 1(a), which is
commensurate superstructure. We remark×, however, due to the folding of K and K′ points of the graphene
that lithium has not been shown to coat graphene or BZonto Γ inthe BZ forC structure. We alsonote the
6
graphite in this manner.12 As a first approximation and linear dispersion of the en∗ergy bands in the neighbor-
in view of the strength of the graphene bonds, initially hood of the Dirac point. In Fig. 1(b), we see a number
we used the DFT code to only relax the distance of ofchangesthatappearinthebandstructureofC Li. Of
6
lithium atoms from the graphene plane, keeping the co- interestforourstudyisthe gapopeningofE =0.39eV
g
ordinates of carbon atoms fixed. This yielded a distance at the neutrality point. For the C Li with fully relaxed
6
of1.79˚Abetweenthe lithium layerandgraphene,which positions,includingthoseofcarbonatoms,thegapopen-
is a somewhat smaller value, as expected for the LDA, ing is E = 0.41 eV, and otherwise the band structure
g
thanthatobtainedinothertheoreticalcalculationsbased is quite similar to the unrelaxed case. This means that
on generalized-gradient approximation (GGA).14 In our the change in C-C bond lengths is responsible for only
tight-binding model presented in the Sec. III we use the about 5% of the band gap. We also note the lifting
dataobtainedforthisstructure. However,sinceachange of other degeneracies, at K and along the Γ-K direc-
3
tion. Other changes that can be readily noted, as a re-
sult of lithium adsorption,include the appearanceof the
parabolic-shaped band of lithium, with its minimum lo-
cated at Γ, at 0.92 eV above the neutrality point. The τ τ
3 2
lithium band weakly hybridizes with the graphene con- τ d
1 3 5 2
duction bands, resulting in some shifts and kinks in the 6
bands of graphene. There is a substantial charge trans- 214 b2 K’ Kb1 d
ferof&0.2eperLiatomfromlithiumlayertographene, M 1
witheverycarbonatomreceivingthesamechargeofone- a2 c2 a1 Γ
sixth of this value, i.e., & 0.03e. As a result, the Fermi
c
levelisraisedby1.5eVrelativetotheneutralitypointof 1
graphene. A rigid band model is clearly not a complete
description, but it can be used to describe the charge
transfer,which is another important aspectof this prob-
FIG. 2: (Color online) Lattice structure of lithium on
lem.
graphene C6Li. Carbon atoms on sublattices A and B are
depicted as filled and open black circles. Lithium atoms, in
magenta,occupypositionsabovethehollowsites. Thedashed
III. TIGHT-BINDING MODEL
hexagon is the Wigner-Seitz primitive unit cell of C6Li, con-
taining six numbered C atoms and one Li atom. Two types
A. Lattice geometry ofbonds,two-thirdsredandone-thirdblue,aredistinguished
duetothepresenceofLiatoms. Onthelowerleft,therhom-
ThebandgapofC Liisaconsequenceofthespecialre- bus defined by c1 and c2 and enclosing two carbon atoms is
6 theprimitiveunitcellofgraphene. The30◦ rotatedrhombus,
lationshipbetweenitsstructureandthatofthegraphene
defined by a1 and a2 and having √3 times larger side, is the
substrate. The Bravais lattice of C Li is hexagonal, as
6 unitcellofC6Li. Intheupperleftpartareshownthevectors
it is for graphene, but with a √3 times larger lattice pa- pointingfromacarbonatomonsublatticeAtoitsthreenear-
rameter and rotated by 30◦, as shown in Fig. 2. It is estneighborsonsublatticeB. Theinsetshowsthereciprocal
convenient to describe the base vectors of the hexago- space. Thevectorsb1,2 and d1,2 are setsof reciprocal lattice
nal lattice in a standard way, with the lattice parameter vectors of C6Li and graphene, respectively. The hexagons in
used as unit of length, so for C Li we use the coordinate theinsetaretheBrillouinzones. TheregiondefinedbyΓKM
6
system with its x-axis taken along a in Fig. 2, is theirreducible wedge of graphene Brillouin zone.
1
1 √3
a1 =(1,0)a, a2 = , a, (1) of relationship is the basic reason why a gap opens in
−2 2 ! C M if x is a multiple of 3, since in these structures the
x
K pointsoftheunderlyinggraphenebecomecoupledand
where a = a √3. [Thus the same base vectors [Eq. (1)]
0 theperturbationcausedbytheforeignatomsremovesthe
describe graphene, i.e., c and c in Fig. 2, with a = a
1 2 0 degeneracy. Thewavefunctionswithwavevectorscorre-
and with the x axis taken along c1.] The corresponding sponding to K andK′ of graphene mix to formdifferent
reciprocal-lattice vectors of the hexagonal lattice are
standing waves. One standing wave piles up electronic
charge in hexagonal cells occupied by Li, and depletes
1 2π 2 2π
b = 1, , b = 0, . (2) it from the hexagonal cells devoid of Li, and the other
1 2
√3 a √3 a
(cid:18) (cid:19) (cid:18) (cid:19) standing wave does the opposite. (The commensurate
structure of C Li breaks a Z symmetry.) These stand-
We denote the reciprocal-lattice vectors of graphene, 6 3
ingwavestatesthenexperiencedifferentpotentialswhich
which are also given by Eq. (2) relative to its own sys-
tem, by d . The two sets of vectors b and d are qualitatively explains the origin of the gap.
1,2 1,2 1,2
EachcarbonatomofsublatticeAisconnectedtothree
related by a 30◦ rotation and a √3 change in scale, i.e.,
carbonatoms of sublattice B by the τ vectors,as shown
b = d/√3.
| |As s|h|own in Fig. 2, the K points of graphene are lo- inFig.2. ThesevectorsinthecoordinatesystemofC6Li,
needed in our tight-binding calculations, are given by
cated at
K : 31d1+ 13d2 ≡b1, K′ : −13d1+ 23d2 ≡b2, (3) τ1,3 = −61,∓2√13 a, τ2 = 13,0 a. (4)
(cid:18) (cid:19) (cid:18) (cid:19)
i.e., the K points of graphene coincide with reciprocal- Wenowexaminethe systemfromtheviewpointofthe
lattice points of C . It also follows from Eq. (3) that tight-binding model. The presence of the lithium atoms,
6
∗
the K points ofgraphenebelong to the reciprocallattice in C Li, above the hollow hexagonal sites can be imag-
6
of the (3 3) structure. Evidently, (n√3 n√3)R30◦ ined to have two effects, within a simple tight-binding
× ×
and (3n 3n) constructions, with n a positive integer, model: (a) a contribution (Hartree potential) to the on-
×
will also share this property. The existence of this kind siteenergiesofthe Catoms,and(b)apossiblechangein
4
the hoppingamplitudes depictedasC-C bondsinFig.2. (a) (b)
FromaninspectionoftheWigner-SeitzcellinFig.2,the 3 3
C atoms are seento occupy equivalentpositions with re-
spect to the metalatoms since eachis ata vertexshared 2 2
by one metal-filled and two empty hexagons of the hon-
eycomb lattice. Therefore all C atoms receive the same 1 1
chargetransferfromLiatoms,aswehavealsoseeninour
DFTcalculations,andthesymmetrybreakingcannotbe E/t 0 0
attributed to on-site energies. This reduces the number
of parameters and makes a TB model description much −1 −1
simpler for C Li than for more dispersed compounds.
6
−2 −2
The bonds, however, occupy two kinds of positions in
a Kekul´e construction, two-thirds of them (colored red)
−3 −3
between a filled and an empty hexagon, and one-third
K G M K K G M K
(colored blue) between two empty hexagons. We con-
clude that there are two different hopping amplitudes
FIG.3: BandstructuresofgrapheneintheBrillouin zonesof
corresponding to the two kinds of bond positions. Our
(a)the(1 1)structure,and(b)the(√3 √3)R30◦structure.
DFT calculations showed that changes in bond-length × ×
account for 5% of the energy gap. The modulation in
≃
hopping amplitudes, which are matrix elements of the
Hamiltonian between carbon 2pz orbitals, is therefore energy can be written as
caused mainly by different potential energies and elec-
3
tronconcentrationsinthe regionsofredandbluebonds, E (k)= t eik·τi , (6)
with a small contribution due to bond-length distortion. ± ± (cid:12) (cid:12)
Of course, both variations in potential energy and the (cid:12)Xi=1 (cid:12)
(cid:12) (cid:12)
distortion of bond lengths are caused by the presence of where the vectors τi are de(cid:12)fined in F(cid:12)ig. 2. When we
(cid:12) (cid:12)
lithium ions, and their effects on the band gap add with use the unit cell of the (√3 √3)R30◦ superlattice to
×
each other. describe graphene, we will have six π bands since there
are six carbon atoms within this unit cell. The addi-
tionalbands arederivedfromEq.(6) asE (k b )and
± 1
B. TB Hamiltonian E±(k b2),withthevectorsb1,2 givenbyEq.(−2),which
result−inthe foldingofK andK′ pointsontothe Γpoint
of BZ, because of relation (3). The energy bands from
TheπbandsofgraphenearedescribedwellbyaHamil-
these analytic formulas are plotted in Fig. 3. It is seen
tonian of spinless electrons hopping on the honeycomb
that some degeneracies are present at the new K points
lattice of graphene,
[Fig.3(b)]aswellasalongΓK andMK lines. Thelinear
H = ǫ c†c t c†c . (5) dispersionneartheDiracpointisshownintheAppendix
i i i− ij i j to be E±(k) = ~vF k, where, vF = √3a0t/2~ is the
i ij ± | |
X X Fermi velocity, t is the nearest-neighborTB hopping pa-
rameter, and a is the graphene lattice parameter.
Here c (c†) annihilates (creates) an electron at site i, ǫ 0
i i i
areon-siteenergies,whichareallequalandthereforecan
be set to zero, and t are the hopping amplitudes.
ij C. Numerical results
Thenearest-neighborTBmodelofgraphenehasagap-
lessbandstructure,butaKekul´emodulationofthehop-
We must verify that the band gap of C Li can be de-
ping amplitudes couples the K and K′ points, leading 6
scribed by a textured TB model with two different hop-
to a mixing of degenerate states and opening of an en-
pingamplitudes,t =t . Inthiscase,weobtaintheband
ergy band gap.23 Following our discussion above, we al- 1 6 2
structure by numerical diagonalization of the Hamilto-
low the hopping amplitudes t to have two values, t
ij 1 nian [Eq. (5)] in momentum space. The method is de-
for two-thirds of the bonds on the hexagonal cells be-
scribedintheAppendix,wherewealsoderiveananalytic
neath a metal atom, and t for the remaining one-third
2 expressionfor the energydispersioninthe neighborhood
of the bonds. Here we are interested in describing the
of the gap as
gap, and therefore we exclude the metal atoms from our
TBmodel,exceptfortheireffectonmodulatingthehop- 1/2
E2
pingamplitudes. Theycanbeincludedinamoregeneral E = ~2v 2k2+ g , (7)
± F
treatment,24 butthisisnotneededfordescribingthegap ± 4 !
of C Li.
6 where
For clean graphene t = t = t. There are two π
1 2
bandscorrespondingtothetwo-atombasis,andtheband E =2t t (8)
g 2 1
| − |
5
for the textured TB model. The Fermi velocity is given (a) (b)
interms ofthe TBparameterasbefore,exceptthatnow 8 8
an average value,
6 6
2t +t
1 2 4 4
t= , (9)
3
V) 2 2
must be used. y (e 0 0
From the slope of the linear dispersion near the neu- erg
n
trality point of Fig. 1(a), we can find the Fermi velocity E −2 −2
and then the TB parameter t = 2.72 eV, in good agree-
−4 −4
mentwiththecommonlyusedrangeforthisparameter.25
From Fig. 2(b), we can find the renormalized Fermi ve- −6 −6
locity v and thus the hopping amplitude t by fitting
F −8 −8
Eq. (7) to the band-structure data in the vicinity of the K G M K K G M K
gap. Wefoundthatthevalueoftisslightlylargerforthe
upperbranchthanthatforthelowerbranch,by 0.1eV,
∼ FIG.4: BandstructuresofgraphenefromtheTBmodelwith
but thatthe changein the averagevalue is muchsmaller two hopping amplitudes: (a) t1 = 2.79 eV, t2 = 2.59 eV,
thanthisamount. Inthefollowing,weneglecttherenor- and (b) t1 =2.59 eV, t2 =2.79 eV. Note that the three-fold
malizationoftheFermivelocityandtheaveragehopping degeneraciesatK,e.g.,thelowerleftone,areliftedsothatin
amplitude. (a)wehavefirstanon-degeneratelevelandaboveitadoubly
Giventhevaluesoft=2.72eVandE =0.39eV,from degenerate level, and in (b) we have just the opposite order
g
first-principles calculations, we can determine t and t of the degenerate and nondegenerate levels.
1 2
fromEqs.(8)and(9). Thuswefindt =2.79eVandt =
1 2
2.59 eV. Interchanging t and t is also a solution since
1 2
E depends on the absolute value of their difference. To
g
distinguishbetweenthe two,weplotthe bandstructures
corresponding to both results in Fig. 4. The band gaps
are identical in Figs. 4(a) and 4(b), but a characteristic IV. SUMMARY
differencecanbeseeninthewaythethreefolddegeneracy
is lifted at K. Comparing with Fig. 1(b), it is seen that
Fig. 4(a) has the same qualitative feature, i.e., the same We have studied the opening of a band gap induced
shape of band gap, and the lifting of degeneracies along in single layer graphene by the adsorption of a sub-
ΓK and at the K point. This shows the validity of the monolayer of an alkali metal. A band gap induced by
Kekul´e textured model for describing lithium adsorbed the adsorption of alkali metals was previously observed
graphene, as well as showing that for C6Li, t1 > t2. In in graphite, and its origin was explained to be due to
fact, Fig. 4(b) corresponds to our DFT calculations of charge transfer to substrate and interlayer interaction of
the band structure of C3Li (not shown), which has the graphene layers. In single layer graphene, where inter-
same lattice structure as C6Li, and is its complement. layer interaction does not exist, the origin of an alkali-
A useful quantity that can be calculated from energy metal induced band gap is in the coupling of K and K′
bandsobtainedisthedensityofstatesperunitcellgiven points which is only possible for appropriate superstruc-
by tures. The required property exists in the superstruc-
tures(3n 3n)and(n√3 n√3)R30◦,withnasapositive
g(E)=2A 6 dk δ[E E (k)], (10) integer. F×or other cover×ages on graphene we expect the
(2π)2 − n bandstoremaindegenerateattheK pointsofgraphene,
n=1ZBZ
X resulting in a gapless spectrum. A tight-binding model
whereAistheareaoftheC unitcell. FortheTBDOS withalternatinghopping amplitudes formingthe Kekul´e
6
weuseda60 60k-pointmesh∗andthetrianglemethod,26 constructionwasshowntodescribethecharacteristicfea-
×
which is the two-dimensional version of the tetrahedron turesofthebandstructureofC Li. FromourDFTcalcu-
6
method22 thatwealsousedfortheab initio calculations. lation,we concluded thatthe modulationof the hopping
The density of states calculated by first-principles DFT amplitudes is mainly due to variations in charge density
and our TB model compare favorably in the vicinity of in the graphene plane due to the attraction caused by
the gap,asshowninFig.5. The extradensities ofstates the Li ions. A tiny distortion of bond lengths additively
in the energy range above the gap in Fig. 5(a) are due contributes 5% to the band gap and, thus, enhances
∼
tothe bandofthelithiumatoms,whicharenotincluded the modulation in hopping amplitudes. In conclusion,
in our tight-binding model. The other main difference we showed that the band gap opening in the lithium on
between the two results is the positions of the Van Hove graphene compound C Li is consistent with the nearest-
6
singularitieswhichareclosertothegapinfirstprinciples neighbor tight-binding model of graphene with Kekul´e
calculations. modulated hopping amplitudes.
6
(IPM). The financial support of the Iranian Nanotech-
(a) nology Initiative (INI) is gratefully acknowledged.
3 EF
2
APPENDIX: SOLUTION OF TB MODEL
1) 1
−
ell
1c 0 In this appendix, we summarize our method of solv-
−
V −3 −2 −1 0 1 2 3 ing the Kekul´e textured nearest-neighbor tight-binding
e
S ( model[Eq.5]. First,weexpandtheeigenfunctionsofour
O
D (b) generalperiodicsystem,characterizedbycrystalmomen-
3 tumk, intermsofthe2p orbitals, nj ,localizedonthe
z
| i
jth atom in the nth unit cell, as
2
1 1 N M
k = A eik·rnj nj . (A.1)
j
| i √N | i
0 n=1j=1
XX
−3 −2 −1 0 1 2 3
E (eV) HereN isthe numberofunitcells,andM isthenumber
of carbon atoms in the basis, r are position vectors of
nj
FIG. 5: (Color online) (a) DOS of C6Li calculated by DFT the atoms, and the amplitudes A form the components
j
code. TheverticalredlineindicatesthepositionoftheFermi oftheeigenvectorstobedeterminedbydiagonalizingthe
level. (b) The corresponding DOS for graphene calculated
Hamiltonian. ForgrapheneM =2,andtheHamiltonian
within the tight-binding model. The position of Van Hove
matrix in momentum space is given by
singularities are displaced relative to the DOSof (a) and the
extradensitiescorrespondingtoLiatomsarenotincludedin
(b). H(k)= t 0 3i=1eik·τi , (A.2)
− 3 e−ik·τi 0
(cid:18) i=1 P (cid:19)
P
which is readily diagonalized to give Eq. (6).
Acknowledgments For the textured model, M = 6. Referring to Fig. 2
for numbering of the basis atoms, and the τ vectors,
i
M. F. thanks S. Farjam for graphical assistance and we can write the 6 6 Hamiltonian matrix. For exam-
×
A. Saffarzadeh for useful discussions. First principles ple, to obtain the first row we observe that atom 1 has
computations were performed with the resources of the nearestneighbors 2, 4, and 6 (due to periodicity), and is
Computational Nanotechnology Supercomputer Centre connected to them by τ , τ , and τ , respectively, with
3 2 1
at the Institute for Research in Fundamental Sciences hopping amplitudes t , t , and t . Thus we find
1 1 2
0 t eik·τ3 0 t eik·τ2 0 t eik·τ1
1 1 2
t e−ik·τ3 0 t e−ik·τ1 0 t e−ik·τ2 0
1 1 2
 0 t eik·τ1 0 t eik·τ3 0 t eik·τ2
H(k)=−t1e−ik·τ2 1 0 t2e−ik·τ3 2 0 t1e−ik·τ1 1 0 . (A.3)
 0 t2eik·τ2 0 t1eik·τ1 0 t1eik·τ3
t2e−ik·τ1 0 t1e−ik·τ2 0 t1e−ik·τ3 0 
 
 
For generalk BZ, we diagonalizeEq. (A.3) numerically, but we obtainan analytic expressionfor the neighborhood
∈
of the gap using perturbation theory.
At k = 0, the unperturbed Hamiltonian, case of t = t = t, has four of its eigenvalues equal to 0, with the other
1 2
two being 3t. The correspondingeigenvectorsareeasily written downby noting that they are relatedto the K,K′,
±
and Γ points of graphene described by its primitive unit cell,
1 0 0 1 1
0 ω ω∗ 0 1
1 ω 1  0  1  0  1 ω∗ 1 ±1 
, , , , , (A.4)
√3 0  √3ω∗ √3ω √3 0  √6 1
ω∗  0   0  ω ±1 
         
 0   1   1   0   1
        ± 
         
7
where ω =ei2π/3.
Next we calculate the Hamiltonian matrix in the fourfold degenerate subspace of the zero eigenvalue, spanned by
the first four vectors of Eq. (A.4), and expand the result near k=0 to obtain (cf. Ref. 23)
0 ~v (k ik ) t t 0
F x y 2 1
~v (k +ik ) 0− −0 t t
H(k)= Ft2x t1 y 0 0 ~vF2(k−x 1 iky), (A.5)
−0 t t ~v (k +ik ) − 0 −
 2− 1 − F x y 
 
where v = √3a t/2~ as for unperturbed graphene, but with t = (2t +t )/3. Diagonalizing Eq. (A.5), we find
F 0 1 2
Eq. (7).
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