Single-Valley Engineering in Graphene Superlattices

0.93 MB·English

by  Yafei Ren

Single-Valley Engineering in Graphene Superlattices
Yafei Ren,1,2 Xinzhou Deng,1,2 Changsheng Li,3 Jeil Jung,4 Changgan
Zeng,1,2 Zhenyu Zhang,1 Qian Niu,5,6 and Zhenhua Qiao1,2,
∗
1International Centre for Quantum Design of Functional Materials,
Hefei National Laboratory for Physical Sciences at Microscale,
and Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
2Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
5 3Department of Physics, Hunan University of Arts and Science, Changde, Hunan 415000, China
1 4Graphene Research Centre and Department of Physics,
0
National University of Singapore, 2 Science Drive 3, Singapore 117551, Singapore
2
5Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

n 6International Center for Quantum Materials and Collaborative Innovation
a Center of Quantum Matter, Peking University, Beijing 100871, China
J (Dated: January 23, 2015)

2
2 The two inequivalent valleys in graphene preclude the protection against inter-valley scattering
offered by an odd-number of Dirac cones characteristic of Z2 topological insulator phases. Here

] we propose a way to engineer a chiral single-valley metallic phase with quadratic crossover in a
i honeycomb lattice through tailored √3N √3N or 3N 3N superlattices. The possibility of
c × ×
s tuning valley-polarization via pseudo-Zeeman field and the emergence of Dresselhaus-type valley-
- orbit coupling are proposed in adatom decorated graphenesuperlattices. Such valley manipulation
l
r mechanisms and metallic phase can also find applications in honeycomb photonic crystals.
t
m
PACSnumbers: 68.65.Cd71.10.Pm73.22.Pr73.43.Cd,
.
t
a
m Introduction—. HoneycombDiracmaterialshavetwo-
- fold degenerate band structures with inequivalent KK′
d
valleys [1–5], whose origin can be traced back to the bi-
n
partitenatureofhoneycomblattices(AandBtriangular
o
c sublattices). This binary valley degree of freedom has
[ led to proposals of valleytronics applications [6–11] that

leverage the valley pseudospins in a manner analogous
1
v to electron spins in spintronics applications. A distinct
3 scenario is that of single (odd-number) Dirac-cone in Z
2
5 topological insulators [12] where their surface states are
5
effectivelydecoupledfromeachotherduetotheirdistant
5
spatial separation. Therefore, a single Dirac-cone struc-
0
. tureisdesirablewhenwerequireaHamiltonianthatem- FIG. 1: (color online) Schematic representation of inter-
1
bodies the chiral anomaly of Dirac fermions [13] and at valleycouplingadatomsuperlatticesandtheirrespectiveBril-
0
5 thesametimeisprotectedagainstinter-valleyscattering. louin zones. (a) and (b) are respectively primitive and recip-
rocallattices forthetopadsorption in√3 √3graphenesu-
1 In this Letter, we propose to engineer a single valley percells. TheredlinesrepresenttheBrillou×inzoneofpristine
:
v phase in2D honeycombDirac materialsthrough√3N graphene.
i √3N or 3N 3N superlattices that fold and couple th×e
X
×
inequivalentKK valleysintothesameΓpoint. Weshow
r ′
a thatthecorrespondingeffectiveHamiltoniansfortop-site valley metallic phase with quadratic band crossover. We
adsorbed superlattices exhibit uniform inter-valley cou- also propose that such inter-valley coupling mechanism
plingandvalley-orbitcouplingmechanismsthatresemble and metallic phase can be observedin photonic crystals.
the conventionalin-plane Zeeman fields and Dresselhaus Inter-Valley Coupling—. When the √3N √3N or
×
spin-orbitcouplingoftheelectronspins[1,2,14–19]. The 3N 3N supercells are tailoredon a honeycomb lattice,
×
pseudo-Zeemanfieldandpseudospin-orbitcouplingallow KK valleys couple and fold into the Γ point, as illus-
′
to controlvalley polarizationcoherently, while the latter trated schematically in Fig. 1(b) showing the reciprocal
one further indicates the possibility of controlling valley lattices for both 1 1 (red) and √3 √3 (black) super-
× ×
polarization via electric fields. Moreover, together with cells. For definiteness, here we only focus onthe top-site
the coexisting sublattice potentials, we find that inter- adsorptionasshowninFig.1(a)andleavethediscussion
valley coupling can drive a topological phase transition of the effective Hamiltonians for bridge- and hollow-site
from a quantum valley-Hall phase into a chiral single- adsorption in the Supplemental Material. For top-site
2
adsorptionin a √3 √3 supercell, the six atoms in each due tothe influence ofthe adatoms,the real-spacetight-
×
primitive cell can be classified into three different cate- binding Hamiltonian in Eq. (6) acquires an additional
gories: (i)oneattheadatomsite,(ii)threeatthenearest term H′ = ′i,j δt(a†ibj +h.c.) where the index i runs
neighborsites,and(iii)twoatnext-nearestsites. Werep- over “A” sitPeshrigiht underneath the adatoms and the j
resent the corresponding site energies as u1, u2, and u3, sitesrepresentthethreenearest“B”sites[20]. Themod-
andsetu3 =0asthereferencevalue. Assumingthatthe ified effective Hamiltonian becomes:
adsorption sites belong to sublattice “A”, the real-space
tight-binding Hamiltonian can be written as: Hteff′(k) = U0′ +vF′ (σx1τkx+σyτzky)+∆′1σz1τ
∆
Ht =H0+u1 ′a†iai+u2 ′ b†i+δbi+δ, (1) + 2′2(1σ+σz)τx+vδσx(τxkx−τyky), (4)
Xi Xi Xδ
where(U ,∆ )havesameformsas(U ,∆ )bychanging
where i′ runs over all adatom sites. Here H0 = ∆ tobe0′∆ =′1 3u t2/(t+2t )2,andth0eFe1rmivelocityis
2 ′2 1 0 0
−t0 <Pij>(a†ibj +h.c.) is the band Hamiltonian with t0 modified to be vF′ =vF(2t+t0)/(t+2t0). The lastterm
beinPg the nearest-neighbor hopping energy, and a†i (b†i) inEq. (4) canbe identifiedas aDresselhaus-typevalley-
is the creation operator of an electron at i-th A(B) site. orbit interaction of strength v = v (t t )/(t + 2t )
δ F 0 0
−
The Brillouin zone of pristine graphene can be rep- coupled with a sublattice-flip potential. This term also
resented through three copies of √3 √3 graphene couplesdifferentvalleysandimpliesthepossibilityofma-
×
supercell’s Brillouin zone as displayed in Fig. 1(b). nipulating the valley degree of freedom by external elec-
By denoting j-th (j=1-3) center as Kj, the operator tricfieldinamanneranalogoustothecontrolofelectron
ai can be expanded in momentum space as: ai = spin by electrical means via spin-orbit coupling.
√1N0 k jexp[−i(Kj+k)·Ri]aj,k,whereN0 isanor- Single-Valley Metallic Phase—. Adatom superlattices
malizPatioPn factor, and k runs over the Brillouin zone of lead to both inter-valley coupling and inversion symme-
√3 √3graphenesupercell. TheHamiltonianofEq.(6) try breaking potentials, and it is easy to understand
×
in momentum space is: that each term can independently contribute in open-
ing a Dirac point gap when they are viewed as uniform
u u
Ht(k)=H0(k)+ [ 31a†j,kaj′,k+ 32ξjj′b†j′,kbj,k], (2) in-plane xy and z contributions to the pseudospin fields
Xj,j′ in the Dirac Hamiltonian [21], where the former shifts
the positionof the Dirac points in momentumspace and
where H0(k) = −t0 j(χjka†j,kbj,k + h.c.) describes thelatterintroducesaninversionsymmetrybreakinggap
the kinetic energy ofPpristine graphene with χjk = in the Dirac cone. Here we show that when those ef-
δe−i(Kj+k)·δ, and ξjj′ = δei(Kj−Kj′)·δ. The Kj fects are present in a superlattice, a topologically dis-
(Pj=1-3) are respectively wavPevectors of K, K′, and Γ tinct single-valley phase can be engineered. We begin
points. The last two terms give sublattice potentials
considering for sake of clarity the top-adsorption config-
whenj =j whicharedifferentforABsublatticesdueto
′ urations neglecting the modification of the hopping en-
inversionsymmetrybreaking. Whenj =j ,theygiverise
6 ′ ergy in the band Hamiltonian and setting the site en-
to inter-valley coupling through a finite u contribution
1 ergies at all “B” sublattices to assume a constant value
while u contribution vanishes due to the phase interfer-
2 (i.e., U =u <0). Whenu =0,thesiteenergiesatall
B 2 1
ence(ξKK′=0). Byblockdiagonalization,thelow-energy “A” sublattices are identical, i.e., U =0. This leads to
A
effective Hamiltonian can be further obtained:
vanishinginter-valleyscatteringandtheimbalancedsub-
Heff = U +v (k σ +τ k σ )+∆ σ (3) lattice potentials open a quantum valley-Hall gap at the
t 0 F x x z y y 1 z Diracpoints,wherethedoubly-degeneratemassiveDirac
∆
+ 2(1+σ )τ , conesarefoldedasasinglevalleyaroundtheΓpointbut
z x
2
remaindistinguishable[SeeFig.2(a)]. Whenweallowu
1
where U =(∆ +u )/2 and ∆ =(∆ u )/2 with ∆ = totakenegativevalues,wefindagradualdecreaseofthe
0 2 2 1 2 2 2
−
u1/3. The third term reflects the effective potential im- inversionsymmetrybreakinginducedgap|∆1|andanin-
balance through a mass term of magnitude ∆1 and the creaseofinter-valleycouplingstrength|∆2|thatlifts the
last term describes inter-valley coupling through the τ degeneracyoftheconductionbandssplittingbyamagni-
x
operator. We note that the coupling between K and K′ tudeof2∆2[SeeFig.2(b)]. Thesimultaneouspresenceof
valleys only occurs at “A” sublattice with the coupling both terms breaks the particle-hole symmetry and leads
amplitude ∆2 depending on u1 linearly. Such an inter- to a smaller bulk gap ∆′ =|2∆1+∆2|.
valley coupling acts onthe valley pseudospin asan effec- When u is even further decreased and reaches a crit-
1
tive Zeeman field that can be used to control the valley ical value of u = 3u /2, the bulk gap ∆ completely
1 2 ′
polarization coherently in valleytronics devices. closes. As shown in Fig. 2(c), we achieve a single band
Whenthe nearestneighborhopping termsofsuperlat- touching point at Γ formed by a Dirac-cone centered at
tice Hamiltonians are allowed to change by δt = t t the edge of the parabolic valence band. In this limit
0
−
3
TABLE I: Inter-valley coupling mechanisms for different ad-
sorption geometries.
Adsorption Site Symmetry Inter-valley Coupling
Top C3v (1+σz)τx
Hollow C6v τxσy
Bridge C2v τx1σ
andC symmetries,thebandstructuresaswellasinter-
2v
valley coupling mechanisms become completely different
as listed in Table I [20].
The main difference of this band crossover from the
FIG.2: (coloronline)Topological transitionfrom aquantum
caseofbilayergrapheneisthatherewehaveonlyasingle
valley-Hall insulator to a single-valley phase as a function of
Dirac parabolic dispersion. This is of interest, because
the parameters u1. Here, we set u2 to be fixed with u2 < 0
and u3 = 0. 2∆2 corresponds to the local band gap from it provides an ideal platform to study the single Dirac-
theinter-valley−scattering. ∆′ measuresthebulk(local)band cone transport phenomena of Z2 topological insulators
gap from the competition between inter-valley coupling and and allows to explore the chiral anomaly of single val-
sublattice potentials. The progressive decrease of u1 leads to ley physics that is not compensated by its time-reversal
a complete closure of the quantum valley-Hall gap and then
counterpart. For example, if broken symmetry gapped
transitions to thesingle valley phase by reversing ∆′.
phases are developedin the presence of electron-electron
interactions [22–24], a mass sign dependent spontaneous
orbital moments will develop per spin-valley [24–26]. In
where the bulk gap is closed, the valley-Halleffect is ab-
oursinglevalleyphase,itisexpectedthatwhentheFermi
sentandthe valleysarenolongerdistinguishable. When
surface lies at the crossing point, a quantum anomalous
we allow even smaller values of u , the inter-valley cou-
1
Hall ground state will develop when both spin compo-
pling strength ∆ further increases, while the magni-
2
| | nents have the same mass, or alternatively a quantum
tude of the staggered sublattice potentials ∆ first de-
1
| | spin-Hall state will be present when the masses for each
creasestozerothenincreasesagain[20]. Whentheinter-
spin term have opposite signs [27]. Besides, a supercon-
valley coupling is strong enough, a valley-mixed metal-
ducting phase can also be expected when the Fermi sur-
lic phase with quadratic band crossover is engineered as
faceisshiftedawayfromthecrossingpoint[28]. Whereas
displayed in Fig. 2(d). In this limit the edges of the
the energetically favored ground state depends on de-
lower energy bands are distant from the corssing point
tails of the band Hamiltonian and the models for the
by ∆ = 2∆ ∆ [see Supplemental Material for
′ (cid:12) | 1|−| 2|(cid:12) electron-electroninteraction, further control of quantum
details of(cid:12)the tight-bind(cid:12)ing band structure]. phasetransitionsshouldbeachievablebymeansofexter-
(cid:12) (cid:12)
Adetaildanalysisofthelowenergybandsrevealsthat
nalmagneticfieldscouplingwiththespontaneousorbital
the parabolicdispersionatΓ pointis chiralandformally
moments. Furthermore, in bilayer graphene, the magni-
similartothebanddispersionnearK/K valleyinBernal-
′ tudeofthegapspredictedinaHartree-Focktheorywith-
stacking bilayer graphene. When ∆ ∆ , the low-
2 1 out dynamical screening is on the order of a few tens of
| | ≫ | |
energy Hamiltonianof the quadratictouching bands can
meV[24]whereasexperimentalgapsturnedoutto bean
be further simplified as:
order of magnitude smaller 2 meV [29] due to the ex-
∼
ponentially increasing screening feedback when the gaps
0 (π )2
Hteff′′(k) = U0′′+αk2−β(cid:20)π2 0† (cid:21), (5) are small. Thus, it is expected that substantially larger
gaps can develop, if flatter bands can be tailored when
which is represented on the basis of “B” sublattice the leadingparabolicdispersioncoefficients canbe made
from both K and K valleys. Here, we define U = smallerthantheoneusedinbilayergraphene. Moreover,
′ 0′′
U ∆ , k2 = k2 + k2, α = ∆ v2/(∆2 ∆2), and in the presence of strong magnetic field, the anomalous
β0=−∆ v12/(∆2 ∆x2). Tyhe last ter1mFcoup2le−s sta1tes be- Landau-level quantization can also be expected as that
tween v2alFleys K2−and1K with π =k +ik , and gives rise in bilayer graphene case [30].
′ x y
to the quadratically dispersing Fermi point band struc- Photonic-crystal bands—. Experimental realizations
ture. Such a two-fold degeneracy at the crossing point, of periodic graphene superlattices could take advantage
whichalsoappearsin Figs.2(a)-2(c), is protectedby the of substrates that can generate the 3 3- or √3 √3-
× ×
C symmetry, since these two basis functions form a typesuperstructure,likeEuO(111)[31]andAg(111)sub-
3v
two-dimensional irreducible representation of the corre- strates [32]. There are also other methods for engi-
sponding point group. For other adsorption geometries, neering such kind of superstructures, e.g., silicene on
e.g., hollow-andbridge-adsorptionwithrespectivelyC Ag(111) substrate [33], InSb(111) surface [34], artificial
6v
4
structures for different r = 0.18a (a), 0.23a (b), 0.28a
1
(c), and0.32a(d) atfixedr =0.25aalonghigh symme-
2
try lines. One can observea topologicalphase transition
from an insulator to a single-valley metallic phase when
r is progressivelyincreased[See the highlightedregions]
1
in a way closely similar to the behavior of the electronic
band structure shown in Fig. 2.
DiscussionsandConclusions—. Wepresentedthethe-
ory for the inter-valley coupling mechanisms in √3 √3
×
graphene supercells that act as in-plane pseudo-Zeeman
fields or pseudospin-orbit coupling. Both contributions
can be used to tailor valley pseudospins of honeycomb
lattices. Especially, the Dresselhauls-type valley-orbit
coupling makes it possible to control the valley polar-
ization via electric means. These valley coupling mecha-
nismshaveimportantimplicationsinvalleytronics,where
the coherent control of valley polarization is yet a grand
FIG. 3: (color online) Upper panel: Schematic representa-
tion of honeycomb photonic crystals with a √3 √3 peri- challenge due to the missing counterpart mechanisms
odicity. α~1 and α~2 denote the primitive vectors×. The dis- of spin-orbit couplings or magnetic fields for spintron-
tance between nearest columns is set to be a and the slab ics. Moreover, our theory also suggests strategies for
width is d = 0.1a. ri (i = 1-3) label the radii of differ- engineering single-valley electronic structure in conven-
ent columns. In our simulation, each column is chosen to tional Dirac materials with two inequivalent degener-
be infinitely long. Lower panel: Photonic band structures of
ate valleys by folding them together. The single-valley
transverse-magneticmodesalonghigh-symmetrylinesfordif-
phasecanbemanipulatedbycombininginter-valleycou-
ferentradiir1 =0.18a(a),0.23a(b),0.28a(c),and0.32a(d),
plings and imbalanced sublattice potentials originated
respectively. Here, we set r2 =0.25a and r3=0.18a.
fromtheinversion-symmetrybreaking. Byincreasingthe
strength of inter-valley coupling from zero, a topological
phasetransitioncantakeplacefromthequantumvalley-
organic molecular lattice [35], or patterned two dimen-
Hall phase to a chiral single-valley metallic phase with
sional electron gas with well-established experimental
quadratic band crossover that resemble the electronic
technique [36]. The applicability of our theory depends
structure of a half Bernal-stacked bilayer graphene. A
on the degree of the achievable commensurability with
concrete proposal for such a single-valley phase is pre-
the crystal structure of honeycomb lattices. It is note-
sented in honeycombphotonic crystals. We verified that
worthy that, since our model is spin independent, it can
allthesefindingscanalsobe realizedin3 3honeycomb
alsoapplytoBosonicsystemslikecoldatoms[37]orpho- ×
supercells.
tonic crystals [7] in honeycomb superlattices. One pos-
Acknowledgements—. We acknowledge financial sup-
sibility is to use honeycomb photonic crystals made of
port from 100 Talents Program of Chinese Academy of
silicon columns linked by thin silicon slabs as shown in
Sciences, NNSFC (11474265, 11104069 and 11034006),
the upper panelofFig.3,anduse electromagneticwaves
DOE (DE-FG03-02ER45958)and Welch Foundation (F-
with transverse-magnetic modes in the xy plane. The
1255). J.J. thanks the support from National Re-
corresponding site potentials and hopping energies for
search Foundation of Singapore under its Fellowship
the photonic crystalsetup canbe controlledthroughthe
program (NRF-NRFF2012-01). Q.N. is also supported
column radius r and the link width d. The confinement
by NBRPC (2012CB921300 and 2013CB921900), and
radii allow to tune the concentration of electrical-field
NNSFC(91121004)duringhisleaveatPekingUniversity.
energy of the harmonic modes.
ThesupercomputingcenterofUSTCisacknowledgedfor
If the columns’ radii are identical and the connecting computing assistance.
slabs have the same width, the two dimensional pho-
tonic band structure for transverse-magnetic modes [20]
obtained from the finite elements method [5, 6] shows
two linearly dispersing Dirac cones, closely resembling
∗
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[35]1Z1.0,F1.9W68a0n1g,(2Z0h1e3n)g. Liu, and Feng Liu, Phys. Rev. Lett. Ht =H0+u1Xi ′a†iai+u2Xi ′Xδ b†i+δbi+δ, (6)
6
FIG. 6: (color online) Schematic of the staggered sublattice
FIG. 4: (color online) (a): Primitive cell for the √3 √3 potential2∆1 andinter-valleyscattering ∆2 asfunctionof
graphenesupercellwithtop-siteadsorption. ui (i=1-3)×rep- u1. | | | |
resentthesiteenergiesofthethreetypeatoms. (b): Recipro-
callatticesfor1 1(inred)and√3 √3(inblack)graphene
× × Hamiltonianinmomentumspacecanbeexpressedasfol-
supercells.
lows:
u u
Ht(k)=H0(k)+ [ 31a†j,kaj′,k+ 32ξjj′b†j′,kbj,k]. (8)
Xj,j′
where H0(k) = −t0 j(χjka†j,kbj,k +h.c.) is the kinetic
energy of pristine grPaphene with χjk = δe−i(Kj+k)·δ,
andξjj′ = δei(Kj−Kj′)·δ. ThelasttwoPtermsgivesub-
lattice potePntials when j =j′ which are different for AB
sublattices due to inversion symmetry breaking. When
j = j , they furnish inter-valley coupling where u con-
′ 1
6
tribution is finite while u contribution vanishes due to
2
the phase interference (ξKK′=0).
FIG. 5: (color online) Evolution of tight-bindingband struc- By exactly diagonalizing Eq. (8), the tight-binding
turesforthe√3 √3graphenesupercellfor differentu1 =0 bandstructuresfor√3 √3supercellswithanegativeu
× 2
(a), 3u2/4 (b), 3u2/2 (c), 7u2/2 (d) at fixed u2 < 0. Note: is plotted as shown in F×ig. S5 for different u . One finds
Bands’ partner-switching happens in the last panel after the 1
thatwhenu decreasesfromzero,thebandgapfromim-
topological phase transition. 1
balanced sublattice potentials [See Fig. S5(a)] gradually
decreases due to the emergence of inter-valley coupling
where i′ runs over all adatom sites. Here H0 = [See Fig. S5(b)]. When u1 further decreases to a crit-
−t0 <Pij>(a†ibj +h.c.) is the band Hamiltonian with t0 ical point, the bulk band gap completely closes and a
linearly dispersed Dirac-cone is formed centered at the
beinPg the nearest-neighbor hopping energy, and a†i (b†i) edge of the parabolic dispersed valence band. For even
is the creation operator of an electron at i-th A(B) site.
smaller u , however, no bulk band gap reopens while a
The second term is the site energy induced by adatoms 1
localbandgapopensduetotheincreasingofinter-valley
possessing the translation symmetry of √3 √3 super-
× coupling. Such a localgaplifts the three-folddegenerate
cell.
at Γ point and form a chiral single-valley metallic phase
In the corresponding momentum space, the Brillouin
with quadratic band crossover. Interestingly, along with
zone ofpristine graphene (redlines)is three times larger
the phase transition, the third and fourth energy bands
thanthatofthe √3 √3supercell(blacklines)areplot-
× represented by green and purple lines switch partners.
tedinFig.S4(b). Thus,onecandividetheBrillouinzone
of pristine graphene into three parts with centers K , To understand the phase transition process more
1
K , and K being wavevectors of K, K, and Γ points clearly, we first obtain the effective Hamiltonian by uti-
2 3 ′
lizing Lo¨dwin block diagonalization method. In this
of graphene, respectively. Each part is the same as the
method, we firstrearrangethe HamiltonianinEq.(8)to
Brillouinzoneof√3 √3graphenesupercell. Therefore,
× divide it into two blocks: one 4 4 matrix representing
the operator ai can be expanded as: ×
low energy bands contributed from valleys KK’, and the
ai = 1 exp[ i(Kj+k) Ri]aj,k, (7) other2×2matrixrepresentinghighenergybandsfromΓ
√N − · valley of graphene. Their coupling is much smaller than
0 Xk Xj
the hopping energy t and thus, the influence of high
0
wherekrunsovertheBrillouinzoneof√3 √3supercell. energy bands on the low energy ones can be obtained
×
By substituting Eq. (7) into Eq. (6), the tight-binding by perturbation method. Through a block diagonaliza-
7
FIG. 7: (color online) Left panel: Energy levels at Γ point FIG. 8: (color online) Left panel: Schematic of honeycomb-
for u1 = 0. Both energy levels are double degenerate. Right structured photonic crystals. The distance between nearest
panel: Energy levels at Γ point for u1 = 3u2/4. The degen- columns is set to be a and the slab width is d = 0.1a. ri
erateoftheupperenergylevelinleft panelislifted whilethe (i = 1-3) label the radii of different columns. Right panel:
lower one remains degenerate. The photonic band structure with uniform column radii and
theslabwidthswhereDiracconesarepresentaroundKpoint
′
andK pointandthelater oneisnot shown forclarity ofthe
tion [9], the effective Hamiltonian can be obtained as figure.
below:
the left panel of Fig. S8. The columns are infinite in z-
Heff = U +v (p σ +τ p σ )+∆ σ (9)
t 0 F x x z y y 1 z direction and their radii are r (i=1-3). In our calcula-
i
∆
+ 2(1+σ )τ , tion, the electromagnetic wave propagateswithin the xy
z x
2 plane, i.e., the wavevector component along z-direction
isk =0. Variousnumericalmethodscanbeusedtocal-
whereU =(∆ +u )/2with∆ =u /3and∆ =(∆ z
0 2 2 2 1 1 2
− culate the photonic band structure, such as plane wave
u )/2. The third term reflects the imbalancedsublattice
2
method (PWE), finite difference time domain (FDTD)
potentials, while the last term describes the inter-valley
method, and finite elements method (FEM) [5, 6]. Here,
coupling occurring at sublattice “A”.
we used FEM, because it is much efficient in calculat-
To show the dependences of inter-valley coupling and
ing structures with extremely small domains needing to
imbalanced sublattice potentials on u , the magnitudes
1
be meshed. The photonic band structure with the uni-
of 2∆ and ∆ are plotted in Fig. S6. It shows that
1 2
form column radii and the slabs widths are calculated
both terms depend on u linearly. When the inter-valley
1
as shown in the right panel of Fig. S8, where two Dirac
coupling term is zero (i.e., ∆ = 0), there are two dou-
2
cones in K and K’ points for transverse-magneticmodes
ble degenerate energy levels at Γ point [See left panel of
[7, 8] are formed, resembling the linear-dispersed Dirac
Fig.S7]. Thestatesattheupper(lower)energylevelare
cones of pristine graphene.
contributedfrom“A(B)”sublattices. Whenu decreases
1
from zero, the inter-valley coupling ∆ increases from
2
| |
zero,whilethesublatticepotentialsdecreaseasshownin HOLLOW AND BRIDGE ADSORPTION
Fig. S6. As a result, the upper energy level split with a
gapof2∆ duetotheinter-valleyscattering,whereasthe
2 In above, we study the top-site adsorption case where
loweroneenergylevelisunchangedsincethe inter-valley
the inversionsymmetryisbrokenandstaggeredABsub-
scatteringonlyoccursatsublattices“A”asshowninthe
latticepotentialisinduced. Inthefollowing,weshowthe
right panel of Fig. S7. As the u further decrease to
1 inter-valley coupling mechanisms for hollow and bridge
thecriticalpoint,anintersectionbetween 2∆ and ∆
| 1| | 2| adsorption cases with inversion symmetry in 3 3 hon-
indicates that the lower energy level splitting from sub- ×
eycomb supercells. We first consider hollow adsorption
lattice “A” reaches the two-folddegenerateenergy levels
asshowninFig.S9(a)withprimitive vectorsdenotedby
of sublattice “B”, for which the bulk band gap closes.
α~ and α~ in black. By considering only the site ener-
Whenu becomesevensmaller,the inter-valleycoupling 1 2
1 giessurroundingthe adatoms,the π-orbitalinreal-space
isalwaysdominantandthebandgapcannotreopenagain
tight-binding Hamiltonian is written as:
and a chiral single-valley metallic phase is formed.
Hh =H0+u1 ′(a†iai+b†ibi), (10)
METHOD FOR CALCULATING BAND Xi
STRUCTURE OF PHOTONIC CRYSTAL
where ′i runs over six atoms nearest to adatoms with
siteenePrgyofu1. NotethatABsublatticesareequivalent
For the simulation of band structure in photonic crys- here because of the inversion symmetry.
tals,weconsiderahoneycomblatticewith√3 √3peri- The reciprocal lattices for both 1 1 (in red) and
× ×
odicity,whichiscomprisedofsiliconcolumnslinkedwith 3 3 (in black) graphene supercells are displayed in
×
thin silicon slabs in the vacuum backgroundas shown in Fig. S9(b). The Brillouin zone of pristine graphene can
8
where τ and σ are valley and sublattice Pauli matrices,
respectively. The first term is an energy shift relative
to the charge neutrality point, and the second term de-
scribes the kinetic energy with v being the Fermi ve-
F
locity. The last term couples valleys K and K, where τ
′ x
implies a pseudo-Zeeman field in x-direction to induce a
processionofvalleypolarization. Moreover,the coupling
only occursbetween differentsublattices, andthe result-
ingbandgap2u2/9t indicatesasecond-ordercorrection
1 0
from site energy u .
1
Then we study the bridge adsorptioncase as shownin
Fig. S9(c). Assuming that the adatom only influences
the site energiesu of the nearesttwo carbonatoms and
1
neglecting the high-order contribution from Γ valley of
graphene, the continuum effective Hamiltonian for four
lower bands can be expressed as follows:
u u
FIG. 9: (color online) Primitive cells for hollow- (a) and Heff = 1 +v (k σ +τ k σ )+ 1τ 1 . (13)
bridge-site (c) adsorption. α~i indicates the primitive lattice b 9 F x x z y y 9 x σ
vectors for 1 1 (in red) and 3 3 (in black) graphene su-
× ×
percell. u1 denotes the site-energy induced by the adatoms.
(b): Reciprocal latticesfor1 1(inred)and3 3(inblack) The first and second terms are the energy shift and ki-
× ×
graphene supercell. (d) and (e): Low energy band structures netic energy respectively, whereas the third term rep-
for hollow- andbridge-siteadsorptions. ∆indicates thelocal resents the first-order inter-valley coupling contributed
gap from pseudo-Zeeman field in bridge adsorption.
from the on-site energy which also acts as a pseudo-
magneticfieldinx-directionyetwithoutasublatticeflip-
ping. This term shifts the two degenerate Dirac cones
be divided into nine copies of that of 3 3 graphene su-
percell. By denoting the j-th (j = 1-×9) center as Kj, oaft gkra=phe0neasansdhoowpnenisnaFliogc.aSl9e(nee)r.gyTghaepg∆ap=is2c|ulo1s|/ed9
the operatora canbe expandedinmomentum spaceas:
i at ( u /9v ,0) due to the dispersion of energy bands
ai = √1N0 k jexp[−i(Kj +k)·Ri]aj,k,whereN0isa whe±re|a1n|otheFr two Dirac cones are formed. All these re-
normalizaPtionPfactor, and k runs over the Brillouin zone
sults indicate that the inter-valley coupling mechanisms
of 3 3 graphene supercell. Therefore, the Hamiltonian
× are sensitive to the geometry of adsorption.
of Eq. (10) in momentum space can be expressed as:
u
Hh(k)=H0(k)+ 91ξjj′(a†j,kaj′,k+b†j′,kbj,k). (11)
Xj,j′
∗ Correspondence author: [email protected]
Inthesecondterm,j =j′ givesequivalentABsublattice [1] P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994).
potentials,whilej =j′ couplesdifferentparts. Although [2] G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15
6
the direct coupling between valleys KK vanishes due to (1996).
′
phase interference, i.e., ξKK′ = δei(K−K′)·δ = 0, a [3] G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169
(1996).
band gap opens at Γ point with foPur lower energy bands
[4] G. Kresseand D.Joubert, Phys. Rev. B 59, 1758 (1999).
aroundthegapmainlycontributedfromeigenstatesnear
[5] J.Jin,Thefiniteelementmethodinelectromagnetics, 2nd
valleys KK. As shown in Fig. S9(d). This suggests that
′ ed. (Wiley,2002).
the gap is induced by inter-valley coupling from higher- [6] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R.
order effects. In below, we demonstrate the physical ori- D. Meade, Photonic Crystals: Molding the Flow of Light
gin of inter-valley coupling for the hollow adsorption. (Princeton Univ.Press, 2008).
Bydoingablockdiagonalization[9]similartothetop- [7] J.-Y. Yea, V. Mizeikisb, Y. Xua, S. Matsuoa, and H. Mi-
sawa, Optics Communications 211, 205 (2002).
site adsorption case, a low-energy effective Hamiltonian
[8] C. Ouyang, Z. Xiong, F. Zhao, B. Dong, X. Hu, X. Liu,
at second-order approximationcan be reached:
and J. Zi, Phys.Rev. A 84, 015801 (2011).
[9] R.Winkler,Spin-orbit couplingeffects intwo-dimensional
u u2
Heff = 1 +v (k σ +τ k σ )+ 1 τ σ , (12) electron and hole systems (Springer,Berlin, 2003).
h 3 F x x z y y 9t x y
0