Berry Curvature and Nonlocal Transport Characteristics of Antidot Graphene

17 Pages·2017·4.11 MB·English

PHYSICAL REVIEW X 7, 031043 (2017)
Berry Curvature and Nonlocal Transport Characteristics of Antidot Graphene
Jie Pan,1 Ting Zhang,1,2 Haijing Zhang,1,2 Bing Zhang,1 Zhen Dong,1 and Ping Sheng1,2,*
1Department of Physics, Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, China
2Institute for Advanced Study, Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, China
(Received 20 January 2017; revised manuscript received 2 June 2017; published 12 September 2017)
Antidot graphene denotes a monolayer of graphene structured by a periodic array of holes. Its energy
dispersionisknowntodisplayagapattheDiracpoint.However,sincethedegeneracybetweentheAandB
sitesispreserved,antidotgraphenecannotbedescribedbythe2DmassiveDiracequation,whichissuitable
forsystemswithaninherentA=Basymmetry.Frominversionandtime-reversal-symmetryconsiderations,
antidotgrapheneshouldthereforehavezeroBerrycurvature.Inthiswork,wederivetheeffectiveHamiltonian
ofantidotgraphenefromitstight-bindingwavefunctions.TheresultingHamiltonianisa4×4matrixwitha
nonzerointervalleyscatteringterm,whichisresponsibleforthegapattheDiracpoint.Furthermore,nonzero
Berry curvature is obtained from the effective Hamiltonian, owing to the double degeneracy of the
eigenfunctions. The topological manifestation is shown to be robust against randomness perturbations.
SincetheBerrycurvatureisexpectedtoinduceatransverseconductance,wehaveexperimentallyverifiedthis
feature through nonlocal transport measurements, by fabricating three antidot graphene samples with a
triangulararrayofholes,afixedperiodicityof150nm,andholediametersof100,80,and60nm.Allthree
samplesdisplaytopologicalnonlocalconductance,withexcellentagreementwiththetheorypredictions.
DOI: 10.1103/PhysRevX.7.031043 SubjectAreas: Graphene,Mesoscopics
I. INTRODUCTION Creating sublattice asymmetry is not the only way to
Theadventofgraphene[1–3]hasattractedmuchattention openagapingraphene.Toopenagap,onecanfabricatea
narrow graphene nanoribbon [15–18] or make a periodic
because of its novel electronic properties, such as tunable
carrierdensity[1–4],lineardispersion[2],valleytronics[5], array of holes, known as antidot graphene [19,20]. The
bandstructure[19–26]andtransportproperties[27–34]of
etc. The additional valley degree of freedom has been
antidot graphene were extensively investigated; however,
experimentally investigated by breaking the sublattice
symmetry, such as placing graphene on top of hexagonal thetopologicalproperties,suchasBerrycurvature[35],are
boronnitride(h-BN)substrate[6]orgatingbilayergraphene unknown. Onemight expect its Berrycurvatureto bezero
[7,8]. Sublattice asymmetry can be modeled by adding a since creating holes does not break either sublattice
masstermmσ totheWeylHamiltonianℏν σ·k,whereℏ symmetry(inversionsymmetry)ortime-reversalsymmetry.
z F
denotesthePlanck’sconstant,σthe2×2Paulimatrices,ν However, in this work, we show both theoretically and
F
the Fermi velocity, and k the electron wave vector. experimentally that for antidot graphene, the Berry curva-
This mass term opens a gap with magnitude 2m at the ture can be nonzero while preserving inversion and time-
DiracpointandinducesanonzeroBerrycurvature[5,9,10]. reversal symmetries. This is due to the doubly degenerate
Recently, the nonzero Berry curvaturewas experimentally eigenfunctions of antidot graphene.
investigatedbynonlocalmeasurementsofelectricalcurrent To clarify the topological properties of antidot gra-
[6–8,10–12]preciselyonasystemthatcanbedescribedby phene, the first step is to obtain its effective Hamiltonian.
themassive2DDiracHamiltonian.Thenonlocalmeasure- Usually, the effective Hamiltonian is ℏνFσ·kþmσz
ment techniquewas originally proposed for measuring the [36–39], which gives the hyperbolic dispersion relation
spinHall effect[11]andhasbeenappliedforstudyingthe that can fit the band structure around the Dirac point
valleytronics in 2D system [6–8,12–14] like graphene. well. Such a Hamiltonian is a 2×2 matrix, indicating
that there is no valley mixing. The mσ term models the
z
potential difference between the two sublattice sites
*[email protected] [39,40]. Since there is no such sublattice asymmetry in
antidot graphene, it follows that the massive 2D Dirac
Published by the American Physical Society under the terms of Hamiltonian ℏν σ·kþmσ cannot be the effective
F z
the Creative Commons Attribution 4.0 International license.
Hamiltonian for antidot graphene. In the present work,
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation, we exploited the tight-binding wave functions and band
and DOI. structures to derive the effective Hamiltonian for antidot
2160-3308=17=7(3)=031043(17) 031043-1 Published by the American Physical Society
PAN, ZHANG, ZHANG, ZHANG, DONG, and SHENG PHYS. REV. X 7, 031043 (2017)
graphene in the vicinity of the Dirac point. The new by directly diagonalizing the Hamiltonian represented by
Hamiltonian turns out to be a 4×4 matrix that preserves Eq.(1).Theobtainedbandstructureisshownbytheblack
the sublattice symmetry as expected. A similar m term, circles in Fig. 1(c) for various γ with a fixed periodicity
responsible for opening a gap, appears, which describes L¼13 nm. The band structure is plotted as a function of
the intervalley scattering strength. Since the eigenfunc- theBlochwavevectork,whichrangesfromKtoΓ,thento
tions are doubly degenerate, nonzero Berry curvature can M,andfinallybacktoK.Here,theK,Γ,andMpointsare
exist in this system while preserving both time-reversal fortheholereciprocallattices.Astheantidotsarearranged
and inversion symmetries. in the triangular lattice structure, its first Brillouin zone is
A nonzero Berry curvature is expected to induce a hexagonal in shape, the same as the first Brillouin zone of
transverse conductance [5,9,41,42]. We have experimen- the pristine graphene lattice. The Dirac point, i.e., the K
tally verified the existenceofthis nonzero Berrycurvature pointofthepristinegraphene’sreciprocallattice,foldsonto
by measuring the nonlocal transverse current [6,11] and theΓpointoftheholereciprocallattice.Wecanseethata
compared the results with the theory prediction. Excellent gap opens at the Dirac point. Both the conduction and
agreement is found. valence bands are doubly degenerate because of the
A very important point about the opening of a gap existence of K and K0 valleys. We have chosen to ignore
in the dispersion relation and the accompanying topo- some flat bands [43] since they correspond to localized
logical properties of antidot graphene is that they are states and do not contribute to the transport properties.
robust against randomness perturbations. This accounts At the same time, we can also obtain the wave functions
for their experimental observations in plasma-etched ψ ¼ðφ ;φ ÞT,whichcanbeanalyzedinthekdomainby
A B
samples. carrying out the Fourier transform.
In what follows, derivation of the effective antidot
graphene Hamiltonian is described in Sec. II, followed
B. Effective Hamiltonian
by the evaluation of the Berry curvature in Sec. III.
The measurements of nonlocal transverse electrical resis- Since we wish to focus on the low-energy region, the
tance arising from the Berry curvature, attendant with the Fourier coefficients for both φA and φB are largely
simulation of the theory prediction, are presented in determined by two components, i.e., around the K and
Sec. IV. Comparison between the measured data on three K0. In other words,
fabricated samples and the theory predictions is given in
Sec. V. Section VI concludes with a brief summary of φ ≈a ðkÞeiðKþkÞ·rþa ðkÞeiðK0þkÞ·r; ð2aÞ
A K K0
results.
and similarly for the wave function on the B site,
II. EFFECTIVE HAMILTONIAN
OF ANTIDOT GRAPHENE
φ ≈b ðkÞeiðKþkÞ·rþb ðkÞeiðK0þkÞ·r: ð2bÞ
B K K0
A. Tight-binding model
Consider a triangular lattice of holes (antidots) in Therefore, the wave function can be represented by a
monolayer graphene with lattice constant L as shown in four-dimensional vector ða ;a ;b ;b ÞT, with bases
K K0 K K0
Fig.1(a).Theunitcellishexagonalinshapewithaholeat eiðKþkÞ·r andeiðK0þkÞ·r forbothAandBatoms,respectively.
its center. To model antidot graphene, we define a dimen-
It follows that if we define
sionless geometrical factor γ ¼d=L, where d denotes the
0 1
holediameter.Therefore,(L,γ)canbeusedtodescribethe
a1 a2 a3 a4
antidotunitcell.Thebandstructureandwavefunctioncan B K K K K C
be calculated by using the tight-binding model, where the BBa1 a2 a3 a4 CC
Hamiltonian is given by Ψ¼ðψ1 ψ2 ψ3 ψ4Þ¼BB@bK1K0 bK2K0 bK3K0 bK4K0CCA;
X b1 b2 b3 b4
H ¼−t ða†ibiþδþH:c:Þ: ð1Þ K0 K0 K0 K0
i;δ ð3Þ
† where i¼1 or 2 denotes the two conduction-band wave
Herea denotesthecreationoperatorontheithAsublattice
i functions, and i¼3 or 4 denotes the two valence-band
site,biþδ denotesthedestructionoperatorontheiþδthB wave functions; then the effective Hamiltonian H should
sublatticesite,andδincludesthethreenearestneighborsto
satisfy the following equation:
sitei.InEq.(1),thesummationoverigoesoverthewhole
(cid:2) (cid:3)
unit cell. We apply the Bloch boundary condition at the
E I 0
edge of the hexagons. The band structure and wave HΨ¼Ψ þ ; ð4Þ
function of the antidot graphene system can be obtained 0 E−I
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BERRY CURVATURE AND NONLOCALTRANSPORT … PHYS. REV. X 7, 031043 (2017)
FIG.1. (a)Antidotgrapheneinthetight-bindingmodel.RedandblackdotsrepresentthecarbonatomsAandB.Here,Ldenotesthe
periodicityoftheantidotlattice,andddenotesthediameterofthehole.(b)ThematrixelementmagnitudeplottedasafunctionofBloch
wavevectork.ThetermjH12j(blacksolidsquares)isfoundtovarylinearlywithk¼jkj,whiletheintervalleyscatteringtermjH14j(red
solidcircles)isalmostaconstant.(c)BandstructuresforvariousγwithafixedperiodicityL¼13 nm.Theblackcirclesrepresentthe
tight-bindingresults,andredcurvesstandforthedispersionrelationgivenbytheeffectiveHamiltonian.TheBlochwavevectorkranges
from K to Γ, then to M, and finally back to K [here, K, Γ, and M points are for the antidot (hole) reciprocal lattice].
0 1
where Eþ and E− denote the (degenerate) eigenvalues 0 H12 0 H14
of the corresponding eigenfunctions and I is the BBH21 0 H23 0 CC
2×2 identity matrix. The effective Hamiltonian can be H ¼B C: ð6Þ
@ 0 H 0 H A
obtained by 32 34
H 0 H 0
41 43
(cid:2) (cid:3)
E I 0
H ¼Ψ þ Ψ†: ð5Þ We have found that jH12j¼jH21j¼jH34j¼jH43j and
0 E−I jH14j¼jH23j¼jH32j¼jH41j.InFig.1(b),wepresentthe
resultsofthematrixelementasafunctionofkaroundtheΓ
With the eigenvalues and eigenfunctions obtained pointoftheantidotlattice.ItisseenthatjH12jislinearink,
from the tight-binding calculation, we have numerically while jH14j is almost a constant. The details of H12’s k
evaluated the effective Hamiltonian, which is in the dependence(i.e.,asafunctionofk andk )canbeobtained
x y
form of a 4×4 matrix with only eight nonzero matrix bysettingkalongthexandydirections.Itisfoundthatthe
elements: Hamiltonian can be expressed as
0 1
0 ℏvðk −ik Þ 0 m
x y
B C
Bℏvðk þik Þ 0 m 0 C
H ¼B x y C: ð7Þ
@ 0 m 0 ℏvð−k −ik ÞA
x y
m 0 ℏvð−k þik Þ 0
x y
Therefore,theeffectiveHamiltonianconsistsoftwo2× graphene. Note that the velocity v here is not necessarily
2 matrices, ℏvσ·p and −ℏvσ(cid:2)·p, along the diagonal and the same as v (c=300) in pristine graphene. In antidot
F
fourantidiagonalmterms[44].Thetwodiagonalmatrices graphene,itisafunctionofthegeometricfactorγ.Thefour
are seen to originate from the pristine graphene antidiagonal m terms couple the K and K0 valleys; hence,
Hamiltonian, which can be easily verified by doing theyimplyintervalleyscatterings,whichcanarisefromthe
Taylor expansion around K and K0 points of pristine atomically sharp edges of the antidots. We find the
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PAN, ZHANG, ZHANG, ZHANG, DONG, and SHENG PHYS. REV. X 7, 031043 (2017)
C. Uniqueness of the effective Hamiltonian
Since the bands are doubly degenerate, the two wave
functions ðψ1 ψ2Þ at the same energy can form a linear
combination ðψ0 ψ0 Þ, which would be an equally valid
1 2
choice. The question is whether different choices of
the wave functions can result in a different effective
Hamiltonian. To prove the uniqueness of the effective
Hamiltonian,werecognizethefactthatthewavefunctions
must transform into each other via the unitary matrix U
(cid:3)
forboththeconductionandvalencebands.Inotherwords,
(cid:5)
mFIGa.re2.ploEtftfeedctiavseavelfoucnictytiovnanodf itnhteergvaelolmeyetsrcicattfearcintogrstγr.enTghthe ðψ01 ψ02Þ¼ðψ1 ψ2ÞUþ: ð9Þ
calculatedresultsonvelocityv(inunitsofvF)areshownasblack ðψ03 ψ04Þ ¼ðψ3 ψ4ÞU−
solidsquares,andthoseform(inunitsoft)areshownasredopen
circles. These parameters are obtained for a fixed periodicity Let H0 be the alternative effective Hamiltonian and let
L¼13 nm. The dashed curves are linear fittings. Ψ0 ¼ðψ0 ψ0 ψ0 ψ0 Þ. Based on Eq. (5), we have
1 2 3 4
(cid:2) (cid:3)
E I 0
intervalley scattering strength to have a very weak k2 H0 ¼Ψ0 þ Ψ0†: ð10Þ
dependence, which is ignored here since we focus only 0 E−I
(cid:6) (cid:7)
on the low-energy regime. Hence, the intervalley U 0
Here, Ψ0 ¼Ψ þ . We can simplify H0 as
scattering strength is characterized by a constant m. 0 U
−
The efpfefficffiffitffiiffivffiffiffieffiffiffiffiffiHffiffiffiaffiffimffiffiffiffiiffilffiffitonian yields a dispersion relation (cid:2)U 0 (cid:3)(cid:2)E I 0 (cid:3)(cid:2)U 0 (cid:3)†
E¼(cid:3) ℏ2v2k2þm2 ¼(cid:3)ε, where both the conduction H0 ¼Ψ þ þ þ Ψ†
and valence bands are doubly degenerate. This dispersion 0 U− 0 E−I 0 U−
(cid:2) (cid:3)(cid:2) (cid:3)(cid:2) (cid:3)
relation, which is shown as red curves in Fig. 1(c), can fit E I 0 U 0 U 0 †
thenumericallyobtainedbandstructuresverywell,thereby ¼Ψ 0þ E I 0þ U 0þ U Ψ†
− − −
confirming this effective Hamiltonian to be relatively
accurate for modeling the antidot graphene system in the ¼H: ð11Þ
vicinity of the Γ point (of the antidot lattice).
The values of the velocity, v and m, depend on the Therefore, we can conclude that the effective
geometry of antidot graphene. The values evaluated from Hamiltonianobtainedaboveisunique;i.e.,itwouldremain
the tight-binding calculations are summarized in Fig. 2. the same regardless of the choice of wave functions.
For a fixed periodicity L, the effective velocity v (black
solid squares) decreases with increasing γ, while the gap
III. NONZERO BERRY CURVATURE
(2m) increases with increasing γ. A decrease of the Fermi
velocity associated with the gap opening can also be A. Symmetry analysis
found in Ref. [45]. It can be plausibly understood as due The nonzero term m mixes the K and K0 valleys whose
to the narrowing of the “neck” region of the passage Berry curvatures have opposite signs. Hence, one may
channels with increasing γ. We can linearly fit m and v wonder whether there is net nonzero Berry curvature with
as a function of γ, which are shown as dashed curves in this valley mixing. It is noted that both time-reversal and
Fig. 2: inversion symmetries are preserved in this system; how-
(cid:5) ever, that does not guarantee a zero Berry curvature
v¼ð1−0.67γÞv
F [9,41,42,47,48]. We start the symmetry analysis with the
: ð8Þ
m¼0.056γt·13nm eigenvectorsoftheconductionband,obtaineddirectlyfrom
L
the effective Hamiltonian, Eq. (7),
nw8va=uer1Tmy5hhienear(givstcaeeamralmasprelmf1esi=xuBsLelctd),sa,laewL[1nsh9d¼wi,c42ih1t6=h5]5.i0tsh(FensoiamnirmnvoapaeuclnrecrsdoeCerγoxd)f.pa¼tneThrch2eieme=p3ewcenori(itrtosahraldemsispcpapirotmelynev,pdiiolAi.eneus)g.s,,, 8<:ψψ12ððkkÞÞ¼¼(cid:6)(cid:6)−pkeffiεffipþ−ffikffiieffiεffimθffiffi−−ffiffiffiimffiθffiffip−ffiεffiffiþffipffiffiffiffimffiεffiffiffiffiffi−ffiffiffiffipmffiffiffiffiffiεffiffiþffipffiffiffiffiεffimffiffiffiffi−ffiffiffiffiffi−mffiffiffiffipke−ffiεffiþ−ffiffiipffimθkffiffieffiε(cid:7)ffi−−ffiffiiTmffiθffiffi=(cid:7)2Tp=ffiε2ffiffipffiεffiffi;
gap(2m) is around 20meV, and the effectivevelocity v is ð12Þ
about 0.55v ; hence, m=ℏv¼0.03 nm−1 for sample A.
F
Values of m and v for samples B and C can be similarly where θ ¼arctanðk =k Þ denotes the direction of the
y x
deduced from Eq. (8). wave vector k. Since there is double degeneracy for the
031043-4
BERRY CURVATURE AND NONLOCALTRANSPORT … PHYS. REV. X 7, 031043 (2017)
conduction band, any linear combination of ψ1 and ψ2 is Here,theBerryconnectionAα isamatrix,whosematrix
also an eigenwavefunction of the effective Hamiltonian, elements are defined as ðAαÞmn ¼ihumj∂kαjuni, where
Eq. (7). In Eq. (12), the eigenwavefunctions are chosen u denotes the periodic part of the wave function. We
mðnÞ
so that the Berry curvature matrix is diagonalized, as can see that the first two terms originate from the non-
shown in Sec. IIIB. Under time-reversal transformation degenerate case, while the last term is the correction term
(see Appendix A), due to degeneracy [9,42,47–49]. If there is no degeneracy,
Tψ2ðkÞ¼ψ2ð−kÞ(cid:2) ¼eiθψ1ðkÞ; ð13Þ AαTihsea ncuamlcubleartiionnsteaodf otfhea mBaetrrirxy, mcoakninnegct½iAonx;Arye(cid:4)qzueirreos.
information from the periodic parts of the wave function
from which we conclude that the Berry curvature must
denoted by u . We can derive the expression for u
mðnÞ
obey (see Appendix B) based on Eqs. (2a) and (2b), i.e., u≈ða ðkÞeiK·rþ
K
a ðkÞeiK0·rb ðkÞeiK·rþb ðkÞeiK0·rÞT. Since the integra-
Ω2ð−kÞ¼−Ω1ðkÞ; ð14Þ tiKon0 of exp½KiðK−K0Þ·r(cid:4)K0over the unit cell is zero,
where the subscript denotes the respective wave function we obtain ðAαÞmn ¼ihumj∂kαjuni¼ihψmj∂kαjψni, where
ψ are the eigenvectors of the effective Hamiltonian
[Eq. (12)] from which the Berry curvature is evaluated. mðnÞ
Similarly, for the inversion symmetry, we have Ω1ð−kÞ¼ [Eq.(12)].Hence,bysubstitutingEq.(12)intoEq.(18),we
Ω1ðkÞ and Ω2ð−kÞ¼Ω2ðkÞ. Combining both the time- find that the Berry curvature is given by
reversal and inversion symmetries, we find that the sym- 0 1
m=ℏv
metry conditions only require 0
Ω¼BB@2½ðm=ℏvÞ2þk2(cid:4)3=2 CCA: ð19Þ
Ω1ðkÞþΩ2ðkÞ¼0. ð15Þ 0 − m=ℏv
2½ðm=ℏvÞ2þk2(cid:4)3=2
Hence, the Berry curvature for each band is not neces-
sarily zero. Below, we can see that, for our degenerate ItisseenthattheBerrycurvaturematrixisnonzeroonly
system, the Berry curvature is indeed nonzero. for the diagonal terms. It is also noted that the matrix
If we focus on the region near the Γ point (of the hole element Ω11 is the same as the Berry curvature of the
lattices),i.e.,jkj∼0,theeigenvectorsfromEq.(12)canbe massive Dirac Hamiltonian [5], where m characterizes
simplified as the local potential difference between the carbon atoms
(cid:5) pffiffiffi A and B. The Berry curvature is determined solely by one
ψ1≈ð0 1 1 0ÞT= 2p¼ffiffiffijK0iAþjKiB : ð16Þ parameter, i.e., m=ℏv. In Fig. 3, we summarize the Berry
ψ2≈ð−1 0 0 −1ÞT= 2¼jKiAþjK0iB cku¼rva0tuarnedfdoercdayifsfearsenktdgeevoiamteestrfircompar0a.mTehteerBseγr.ryItcpuervaaktsuraet
Equation (16) clearly shows the mixing behavior can also be obtained through tight-binding calculations
between the K and K0 valleys. For instance, for ψ1, the [48]. We found that near the Γ point of the antidot lattice,
wave function of atom A comes from the K0 valley, while
for atom B, it comes from the K valley. However, if we
assume that there is a long-range potential V (such as the
Coulombpotential of the chargedimpurity), the scattering
matrix element should be
hψ1jVjψ2i¼ðhKjVjK0iÞAþðhK0jVjKiÞB∼0. ð17Þ
In other words, the long-range disorders cannot scatter
ψ1 to ψ2. Note that the intervalley scattering is already
takenintoaccountintheHamiltoniananditsrelevantwave
functions.
B. Calculation of the Berry curvature
Since the 4×4 matrix describes a doubly degenerate FIG. 3. Berry curvature matrix element Ω11 calculated as a
system, the Berry curvature for either the conduction or function of wave vector k for various values of the geometric
valencebandisessentiallya2×2matrix,whichisdefined factorγ[basedonEq.(19)].Theparameterm=ℏvcanbeobtained
as [9,41,42,47–49] from the tight-binding calculations, where the periodicity L is
fixed at 13 nm. Here a denotes the carbon-carbon atomic
Ω¼∂ A −∂ A −i½A ;A (cid:4): ð18Þ separation (¼1.42Å).
k y k x x y
x y
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PAN, ZHANG, ZHANG, ZHANG, DONG, and SHENG PHYS. REV. X 7, 031043 (2017)
Eq. (19) agrees well with the Berry curvature obtained of the hole reciprocal lattice; i.e., a gap is expected.
throughtight-bindingcalculations.AwayfromtheΓpoint, Here,theperiodicityisfixedatL¼24a,withγ ¼0.2.
there can be differences between the two. It is also to be (b) Lattice B—This is the “rotated triangular lattice
notedthatbyintegratingtheBerrycurvaturecalculatedby structure” [22], which is predicted to have nopgaffiffipffi .
the tight-binding model over the first Brillouin zone, we The periodicity in this case is fixed at L¼14 3a
obtain a Berry phase of 2π. Therefore, the effective with the same γ ¼p0ffi.ffiffi2 as in (a). The value of the
Hamiltonian, Eq. (7), and the Berry curvature, Eq. (19), periodicity,L¼14 3að≈24.2aÞ,ischosensothatK
can well describe the antidot graphene system only at the and K0 cannot be folded onto the same Γ point. A
low-Fermi-energy region. gapless system is expected; i.e., the conductance
should exhibit no zero-conductance region.
(c) LatticeC—ThisisthesameasLatticeBbutmodified
C. Robustness against randomness perturbations
withatomic-scalerandomnessattheedgesoftheholes.
From the derived effective Hamiltonian and the related We assume that the hole radius is angular dependent;
analysis,itshouldbeclearthatthegapopeningintheantidot i.e.,r ðθÞ¼r·(1þf ðθÞ),wherenreferstothenth
n n
grapheneisduetothefoldingoftheKandK0points[26]of hPole. The term fnðθÞ is defined as fnðθÞ≡
the reciprocal carbon lattice, onto the Γ point of the hole a cosðmθþφ Þ, where the summation of m
m nm nm
reciprocallattice.Specifpicaffiffiffilly,K andK0 (ofcarbonlattice) goesfrom1to5.Inthesimulations,wehaverandomly
are located at ((cid:3)4π=ð3 3aÞ;0), where a is the carbon- chosen a ∈ð0;0.06Þ, φ ∈ð0;2πÞ for each hole.
nm nm
carbonatomicseparation.Iftheperiodicityofholesisgiven Thewhole system for transport simulations is shown in
by3Na,whereN isaninteger,thepnffitffiffiheprimitivevepctffiffioffirof Fig. 4(a). We can see that there are 10 holes in the width
thereciprocalholelatticeis4π=ð3 3NaÞ·ð1=2;(cid:3) 3=2Þ.
We can see that K and K0 can be folded onto the Γ point.
In the above context of a perfect antidot lattice, it was
pointedoutinRef.[22]thatiftheholelatticeisrotatedby
π=6, which is denoted the “rotated triangular lattipceffiffiffi,” and,
inaddition,iftheperiodicityoftheholelatticeis 3Naso
that the primpitiffivffiffie vector of the reciprocal hole lattice is
4π=ð3NaÞ·ð 3=2;(cid:3)1=2Þ, with N ≠3n (n being an inte-
ger),thenKandK0valleysarenotcoupled;therefore,there
can be no band gap. These conclusions were numerically
verified in Ref. [22], where the results were explained by
usingtheClarsextettheory.Below,weshowthatsuchcases
are unstable against small-scale randomness perturbations
ontheedgeoftheholes.Suchrandomnesswouldbedifficult
to avoid in the oxygen etching of the antidot pattern. It is
demonstratedthatoncethereisatomic-scalerandomnesson
theedgesoftheholes,thenthefullbandgapisrestored;the
magnitudeofthebandgapinsuchcasesisfairlyaccurately
predicted by Eq. (8) from the geometric parameter γ.
The conclusion from such randomness perturbation
analysis, shown below, is that the results obtain from
FIG.4. (a)Schematicillustrationofanantidotgraphenelattice
Sec. II are generally applicable to the experimentally
(LatticeB)forcalculatingtheconductanceasafunctionofFermi
fabricated antidot samples. energy. The system contains 20 holes in the x direction and 10
To determine whether the antidot sample is gapped or holesintheydirection.Theinsetisazoom-inimageoftheantidot
gapless,wecalculateitsconductanceasafunctionofFermi latticewithahole.(b)Calculatedconductanceplottedasafunction
energy.Ifthereisazeroconductanceplateau,thenthereisa of Fermi energy. The black curverepresents theconductance of
bandgap;otherwise,itisagaplesssystem.Theconductance LatticeA,wherethezero-conductanceplateauindicatestheband
oftheantidotgraphenesystemcanbewellsimulatedbythe gapofthissystem.ForLatticeB,theredcurveexhibitsnozero-
conductanceregions,implyingthatthereisnobandgap.LatticeC
open-source packageKWANT[50], whichcancalculatethe
differsfromLatticeBonlyattheedgesoftheholes,wherewehave
electronic transport property based on the tight-binding
added atomic-scale randomness. Ten different randomness pro-
models. We have carried out simulations for the three
fileswerecalculated,andthebluecurverepresentstheensemble-
different cases as follows.
averagedconductance.Itisseenthatabandgapreappears,andthe
(a) Lattice A—This is the “triangular lattice structure,” magnitude of this band gap is the same as that of Lattice A,
which is what we have focused on in Sec. II. In this implyingthatthetransportgapinLatticeChasthesameoriginas
case,K andK0 canalways befoldedontotheΓ point thatinLatticeA,i.e.,fromK toK0 intervalleyscatterings.
031043-6
BERRY CURVATURE AND NONLOCALTRANSPORT … PHYS. REV. X 7, 031043 (2017)
direction and 20 holes in the length direction. The inset
imageisazoom-inforaholeinLatticeB,i.e.,“therotated
triangularlatticestructure.”Theconductancesasafunction
of Fermi energy of the three cases are summarized in
Fig. 4(b). For Lattice A, the conductance is plotted as the
blackcurve,whereazeroconductanceplateauisobserved
as expected. For Lattice B, the conductance does not drop
tozeroatzeroFermienergy,shownbytheredcurve.This
clearly implies that there is no band gap for Lattice B, as
predicted. For the third case, we introduced atomic-scale
randomnessintoLatticeB.Inthiscase,wehavecalculated
10 different randomness profiles, and the blue curve in
Fig.4(b)istheaveragedconductance.Itisclearlyseenthat
with randomness, the fullband gapreappears.Themagni-
tudeofthebandgapinthethirdcaseisexactlythesameas
that of Lattice A, around 0.08t. With the appropriate
parametervaluesofL¼24aandγ ¼0.2,Eq.(8)predicts
a band gap around 0.084t, which is consistent with the
conductance simulation results.
Thereisasimpleexplanationforthereappearanceofthe
gap when the atomic-scale perturbations were added. In
this case, small-scale randomness in real space implies
scatterers with large momentum transfers in the reciprocal
space, i.e., short-range scatterers. That means intervalley
FIG. 5. Supercells and the simulated band structures. Each
scatterings are the inevitable consequences of introducing
atomic-scalerandomness,andtheopeningofthegapisthe supercell comppffiffirffiises nine unit cells, whose hole periodicity is
given by 19 3a. In diagram (a), we have an antidot lattice
result.Thereisalsoasimpleheuristicexplanationforwhy
withoutanyrandomness;i.e.,periodicityisaccuratedowntothe
the obtained gap size depends on the geometric parameter
atomicscale.Wecanseethatthereisnobandgap,aspredicted.
γ=L¼d=L2 as given by Eq. (8). The K to K0 scattering (b) Antidot latticewith randomness. For this case, the bandgap
strength is given by hKjVsrjK0i, where Vsr denotes the reappears, accompanied by a small energysplitting for both the
effective short-range disorder potential. Here, V must be conduction and valence bands. In other words, the states in the
sr
proportionaltothenumberofsitesattheedgesoftheholes, conduction and valence bands are quasidegenerate.
which is proportional to the circumference of the hole;
hence,Vsr∼nd,wherenisthenumberofholes.Thewave We have calculated the band structure of antidot gra-
amplitudesofjKiandjK0imusteachbenormalizedinthe phenewithrandomnessbyusingthesupercellapproachin
sample area S, whpicffiffihffi means jKi and jK0i must each be which each supercell comprised several unit cells with
proportional to 1= S. Combining these two facts, we get randomness introduced at the hole edges. The Bloch
hKjV jK0i∝nd=S∼d=L2,asS∝nL2.However,suchan boundary condition was imposed at the boundary of
sr
argumentcannotyieldtheaccuratevalueofthegap,which suppeffirffifficells. The periodicity of the unit cell is set to
must be determined numerically. 19 3a so that there is no K to K0 coupling in the perfect
From this perspective, the existence of a gap in antidot antidot lattice case. For the supercell, we consider nine
graphene is almost inevitable for the experimentally fab- hoplesffiffiffiasshowninFig.5.Theperiodicityofthesupercellis
ricated antidot samples by using plasma etching. The no- 19 3a×3.Fortheband-structuredatapresentedinFig.5,
gapstate is very precarioussince its existencerequires the we have removed the flat bands. If fðθÞ¼0, i.e., no
very precise periodicity, with no atomic-scale randomness
randomness,wecanseethatthereisnobandgap,asshown
allowed.
inFig.5(a).ForFig.5(b),weconsiderthesamecaseasin
Fig.5(a)butwithatomic-scalerandomness.Itisnotedthat
D. Energy splitting of the eigenfunctions
there is a tiny energy splitting of the originally doubly
Accompanying the introduction of edge randomness degenerate eigenfunctions, accompanying the appearance
into the antidot graphene is the splitting of the originally of the band gap. Hence, the conduction (valence) bands
doubly degenerate eigenfunctions as in the case of Lattice becomequasidegenerate.InFig.5(b),thesizeofthegapis
A above. Below, we first quantify such splittings and around 0.05t, which is close to the prediction by Eq. (8),
then show in the subsequent section that the topological which gives a gap around 0.052t.
manifestation of antidot graphene is not affected by such In the presence of the randomness perturbation, the
splittings. effective Hamiltonian can be numerically evaluated to be
031043-7
PAN, ZHANG, ZHANG, ZHANG, DONG, and SHENG PHYS. REV. X 7, 031043 (2017)
(cid:2) (cid:3)
ℏνσ·ðkþcÞ mσ
x
; ð20Þ
mσ −ℏνσ(cid:2)·ðk−cÞ
x
where c≈c0þc1kþc2k2þOðk3Þ. This nonzero
c turns out to be responsible for the energy splitting of
the eigenfunctions in the conduction (valence) band.
In fact, one can easily evaluate the dispersion relation
froqmffiffiffiffitffihffiffiffieffiffiffiffiaffiffibffiffiffioffiffivffiffiffieffiffiffiffieffiffifffiffifffieffiffifficffiffitffiiffivffiffieffiffiffiffiffiHffiffiffiffiaffiffimffiffiffiffiiffilffitffioffiffiffinffiffiiffiffiaffiffinffiffiffiffiffitffioffiffiffiffiffigffiffiffiiffivffiffieffiffiffiffiffiEffiffiffiðffiffikffiffiÞffiffiffiffiffi¼ffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(cid:3) ℏ2ν2k2 þm2 þℏ2ν2jcj2 (cid:3)2ℏν m2jcj2 þðℏνk·cÞ2,
whichindicatesanenergysplittingintheeigenfunctionsof
boththeconductionandvalencebands.Thissplittingturns
outtobeontheorderof0.1m.Hence,boththeconduction
and valence bands can be regarded as quasidegenerate.
One may wonder what effect this splitting might have FIG. 6. Cartoon image illustrating different wave states’ evo-
on the topological characteristics of antidot graphene. lution under an external electric field under the influence of the
Below, we show that the topological manifestation is not Berrycurvature.ThevalueoftheBerrycurvatureisshowninthe
changedfromthecasewhentheeigenfunctionsaredoubly color scale, plotted in the first Brillouin zone of the antidot
degenerate. graphene.Thebottomleft(right)showsthevalueofthediagonal
matrixtermΩ11 (Ω22),withthepeakvalueattheΓpointofthe
antidot reciprocal lattice. The red wave represents the case of
E. Effect of the Berry curvature under
jη0j2 >1=2; it travels to the left as v <0. The blue wave
an applied electric field 1 x
represents the case of jη0j2 <1=2; it travels to the right.
1
Berry curvature is well known to affect the electronic
transport in the presence of an electric field. This is a
values are very close to those shown in Eq. (19). From
physical manifestation of Berry curvature. In the present
case, the Berry curvature Ω is a 2×2 matrix since Eq.(21b),wehavevx ¼ðeEy=ℏÞ·ðjη01j2−jη02j2ÞΩ11.This
the conduction and valence bands are (quasi-) double result means that when jη01j2 >1=2, the electronic wave
degenerate. Let ðφ1;φ2Þ denote the two quasidegenerate tends to go left as Ey is negative (applied in the minus y
states; an arbitrary initial state ψ is expressible as a linear direction), as illustrated by the red wave in Fig. 6. If
superpositionofðφ1;φ2Þ,i.e.,ψ ¼η1φ1þη2φ2,whereη1;2 we focus on the states that flow leftwards, the average
arethecoefficientssatisfyingjη1j2þjη2j2 ¼1.Itwouldbe transversevelocity is given by
convenient to express ψ in a vector form η¼ðη1 η2ÞT. P
Ivbneylto[h4cei7t]ypretrsaennscveerosfeatnoathpeplieeldecetrliecctfriiecldfiedlidre,cthtieoneliesctgroivneinc v¯x ¼eℏEy· Pjηj01ηj201>j2>1=12=2ðηð0ηÞ0†ÞΩ†η00η0; ð22Þ
v ¼ðeE =ℏÞ·η†Ωη; ð21aÞ where the denominator is the normalization factor. The
x y
summationcanbecarriedoutbpyiffiffinffiffitffieffiffigffiffiffirffiaffiffitffiffiiffioffin.Byassuming
where Ey is the external electric field. Based on the tight- thaptηffiffiffi01¼jη01jexpðiβ1Þandη02 ¼ 1−jη01j2expPðiβ2Þ,where
bmthinaattdrtiihnxegboymffuo-sddieinalgg[o4thn8ea],lstwuepremehrscaevoleflΩwnuiatmhreerraninocdanlozlmyernocae.lsFcsou.rlIatstitemudrpntlhsiceoiutΩyt 1R1=1=pffi22ffij≤η01jηj01·jd<jη101,j0R02≤πdββ1;12R<02π2dπβ,2i.tfEolqlouwatsiotnhat(22)jη01jc2>an1=2b¼e
in evaluating the effect of the Berry curvature under an easily evaluated, and the average velocity turns out to be
electricfield,wechoosetodiagonalizeΩbychangingtoa v¯x ¼ðeEy=ℏÞ·ðΩ11=2Þ. For the range jη01j2 <1=2, the
new basis. If the diagonal(cid:6)ization matr(cid:7)ix is U, we have average transverse velocity is v¯x ¼−ðeEy=ℏÞ·ðΩ11=2Þ.
Ω0 ¼UΩU†, where Ω0 ¼ Ω11 0 . It follows that Hence, the effective Berry curvature for inducing the
0 Ω transverse current is half of the diagonal matrix element,
22
Eq. (21a) can be simplified as i.e., (cid:3)Ω11=2.
Intheabove,itcanbeseenthatindependentofwhether
v ¼ðeE =ℏÞ·ðη0Þ†Ω0η0; ð21bÞ the states are doubly degenerate or quasidegenerate, the
x y
Berry curvature in antidot graphene has the same physical
where η0≡Uη¼ðη0 η0 ÞT and jη0j2þjη0j2 ¼1. The manifestation.Theatomic-scalerandomnessattheedgeof
1 2 1 2
values of Ω11 and Ω22 in the first Brillouin zone are theholes,infact,makesthetopologicaleffectmorerobust,
summarized in the inset of Fig. 6. We can see that Ω11 i.e.,insensitivetotherelativerotationoftheholelatticevs
and Ω22 have opposite signs, i.e., Ω22 ¼−Ω11, and their the underlying graphene lattice, as well as to the precise
031043-8
BERRY CURVATURE AND NONLOCALTRANSPORT … PHYS. REV. X 7, 031043 (2017)
values of the hole lattice constant. This predicted “topo- holediametersweredesignedtobed¼100(sampleA),80
logical Hall effect” is interesting, and we show experi- (sample B), and 60 (sample C) nm. Scanning-electron-
mentally, in the following section, by fabricating and microscope images are shown in the inset of Fig. 7(c) for
measuring three samples with different geometric param- sample A, and in Fig. 14 for sample B (upper panel) and
eter values, that the topological current indeed exists in sample C (bottom panel). All measurements were con-
antidot graphene. ducted in PPMS (Quantum Design) after in situ annealing
at390Kfor2hours.Wefocusonanalyzingthedatafrom
sample A. Similar results for samples B and C are
IV. MEASUREMENT OF NONLOCAL
summarized in Fig. 14.
RESISTANCE IN ANTIDOT GRAPHENE
A. Sample and experimental setup B. Nonlocal measurements
To experimentally check the nonzero Berry curvature, ThenonzeroBerrycurvatureinantidotgraphenecanbe
we have fabricated the antidot graphene samples and experimentallyverifiedbyapplyingthenonlocalmeasure-
studied their electronic transport properties by exploiting ments, and the setup is shown in Fig. 7(a). By applying a
thenonlocalmeasurements.Thesingle-layergraphenewas currentI throughonepairofHallelectrodesandrecording
obtained by mechanical exfoliation and transferred onto a the voltage drop V between another nearby pair of
nl
siliconwaferwithaSiO2thicknessof285nm.Theperiodic electrodes, the nonlocal resistance is defined as
antidots are patterned by e-beam lithography, followed by
oxygenplasmaetching.Thewholesamplewasetchedinto R ¼V =I: ð23Þ
nl nl
standardHallbargeometry,shownasthecartoonimagein
Figs. 7(a) and 7(b), where the blue region is the antidot The nonlocal measurement setup differs from the
graphene, while the grey parts denote the electrodes traditional four-probe local measurement, as shown in
(10 nmTi=60 nmAu). The geometric parameters are set Fig. 7(b). The traditional four-probe local measurement
asd1 ¼2 μm,d2 ¼4 μm,andW ¼2 μm.Fortheantidot givesthelongitudinalresistivityρoftheantidotgraphene;
lattice, we have fabricated three different samples. All its dependence on gatevoltage for various temperatures is
samples have periodicity fixed at L¼150 nm, while the shown in Fig. 7(c). We can see that the charge neutrality
FIG.7. Measurementsetupfor(a)nonlocalmeasurementsand(b)localmeasurements.(c)Antidotgrapheneresistivityisplottedasa
function ofgatevoltageforvarioustemperatures (from10to300K).Inset: SEMimagefortheantidotgraphenesample.(d)Room-
temperatureconductivity as a function of gatevoltageis shownto bewell fitted by the Boltzmann transport theory, which givesthe
mobility around 1500 cm2=ðVsÞ and the residue carrier density n0 ¼2×1011=cm2.
031043-9
PAN, ZHANG, ZHANG, ZHANG, DONG, and SHENG PHYS. REV. X 7, 031043 (2017)
point (CNP) is located around 7.5 V. The longitudinal whether ξ is larger or smaller than the sample width W,
resistivity has a strong temperature dependence, which different behaviors of α can be derived.
can be well described by a variable range hopping The diffusion equationbelow wasoriginally formulated
model (VRH) [29,51]. The mobility of antidot graphene todescribespindiffusion,butitcanbeadaptedtodescribe
is usually low compared to pristine graphene [29,52]. Berry curvature diffusion [11]:
The resistivity (conductivity) as a(cid:6)fqunffifficffiffitffiiffiffioffiffinffiffiffiffioffiffifffiffiffigffiffiffiaffiffitffieffiffiffiffivffiffiolt(cid:7)age
can be described by ρ¼ρ þ ðCV =eÞ2þn2eμ −1 ξ2∂2sðx;yÞ−sðx;yÞ¼Ξðx;yÞ; ð26aÞ
s g 0
[29,52–54], where ρs comes from the short-range scatter- wheresisthe“Berrycurvature”(spin)density.Thesource
ings, μ is the mobility controlled by long-range (charged- termΞðx;yÞisdeterminedbytheexternalelectricfield,with
impurity) scatterings, C is the capacitance per unit a form given by [11]
area of SiO2, and n0 is the residue carrier density due (cid:8) (cid:9)
to the electron-hole puddle effect [55–57]. These param- τ Iσ =Wσ
Ξðx;yÞ¼− s xy δðy−W=2ÞþδðyþW=2Þ :
eterscanbeobtainedbyfittingtheroom-temperaturedata. sinhðπx=WÞ
The fitting details are shown in Fig. 7(d), which gives
mobility μ¼1500 cm2=ðVsÞ and residue carrier density ð26bÞ
n0 ¼2×1011=cm2. HereIdenotesthecurrent,andσ ,σstandforthetransverse
xy
andlongitudinalconductivities,respectively;τ istherelax-
s
C. Theory on nonlocal transverse ation time, which is related to ξ by τ ∼ξ2. The two delta
s
conductance and simulations functionsinEq.(26b)indicatethatthesourcetermisnonzero
Thenonlocalresistanceisdeterminedbythetopological onlyatthetwoedges(y¼(cid:3)W=2);hence,Eq.(26b)serves
currentthatflowsinthesamplechannel.Sincethestatehas as a boundary condition for sðx;yÞ. The induced Berry
a nonzero Berry curvature, it will result in a transverse curvature currentR along the x direction is given by
conductance σ , given by [5,6] J ¼−ðξ2=τ Þ∂ W=2 dy·sðx;yÞ¼α·Iσ =σ. Here, the
xy s s x −W=2 xy
Z quantityξ2=τ istheintrinsicdiffusionconstant;itistreated
e2 1 s
σxy ¼4ð2πÞ2ℏ·2 Ω11ðkÞfðkÞdk; ð24aÞ athseacBoenrsrtyanctuorfvtahteurperocbulrermen.tB,awseedcoannthoebatbaoinvethdeefinnoitnioloncoafl
resistance,givenbyEq.(25).InRef.[11],itisshownthatin
wherek¼jkj,f (k)istheFermidistributionfunction,and the range W ≪ξ, the coefficient α can be analytically
theFermilevelisdeterminedbythecarrierdensityn,which expressed as α¼Wexpð−d2=ξÞ=2ξ. However, in the
isrelatedtok bytherelationn¼k2=π.Here,thefactor1=2 regimeW ≥ξ,thecoefficientαhastobesolvednumerically.
F F
comes from the evaluation of effective Berry curvature as Thecaseofsmallξ(ascomparedwithW)iscrucialhere,as
showninSec.IIIE.SubstitutingEq.(19)intoEq.(24a),we the decay length must be quite small in antidot graphene
obtain sincethemobilityislow.Below,weshowthatbyfittingthe
experimentaldata,thedecaylengthwasfoundtobeonthe
e2 m=ℏv orderof450–550nmforourthreeantidotgraphenesamples.
σ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð24bÞ
xy h ðm=ℏvÞ2þπn Theyareconsistentlysmallerthanthedecaylength(2 μm)
reported in graphene placed ontopof h-BN [6].
The nonlocal resistance is strongly related to the trans- Following the work in Ref. [11], we have obtained the
verse conductance and may be expressed as [6,11] coefficient α by numerically solving the differential dif-
fusion equation [Eq. (26a)] with its boundary conditions
e4 ðm=ℏvÞ2 [Eq.(26b)],byusingthecommercialsoftwareCOMSOL[58]
Rnl ¼ρ3σ2xy·α¼ρ3αh2ðm=ℏvÞ2þπn; ð25Þ package. The Berry curvature density sðx;yÞ can be
obtained numerically. One simulated example with ξ¼
where α is a coefficient that is independent of the Fermi W is shown in Fig. 8(a). The color scale indicates the
magnitude of the Berry curvature density; its magnitude
energybutdependsonthesamplegeometricparameters,W
is seen to decay to zero as x deviates from x¼0. The
and d , where W denotes the sample width and d the
2 2
Berry curvature current density, defined as J ðxÞ¼
separation between the electrodes as shown in Fig. 7. R s
Ultimately, the coefficient α is also related to the decay −ðξ2=τsÞ∂x −WW=2=2dy·sðx;yÞ, is summarized in Fig. 8(b)
length ξ of the Berry curvature density, as shown below. for various values of ξ. Here, J ðxÞ is seen to decay
s
An accurate value of α is important for the comparison exponentially with x. Hence, the coefficient α should
between the theory prediction and experimental data. To behave as α≡fexpð−x=λÞ, where f and λ are dependent
find the behavior of coefficient α, a differential diffusion onξ.TheirfunctionaldependenciesareshowninFig.9(a).
equationmustbesolved.Weshowbelowthatdependingon Intherangeξ≫W,λ¼ξandf ¼W=2ξ,oursimulations
031043-10