Relativistic ferromagnetic magnon at the zigzag edge of graphene

0.19 MB·English

by  Jhih-Shih You

Relativistic Ferromagnetic Magnon at the Zigzag Edge of Graphene
Jhih-Shih You, Wen-Min Huang and Hsiu-Hau Lin
Department of Physics, National Tsing-Hua University, Hsinchu 30013, Taiwan and
Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan
(Dated: January 28, 2009)
We study the spin-wave excitations near the zigzag edge of graphene. It is rather interesting
thatweobtainasinglebranchofrelativisticferromagneticmagnonduetothepresenceoftheopen
boundary. Note that magnons in antiferomagnets appear in pairs, while the single brach magnon
in ferromagnets does not have relativistic dispersion. Thus, the magnon near the zigzag edge of
grapheneisahybridofboth,signalingitsintrinsicpropertyasaboundaryexcitationthatmustbe
9 embedded in a higher dimensional bulk system.
0
0 PACSnumbers: 82.39.Jn,82.80.Gk,87.16.Xa,87.19.lt
2

n
Graphene is the two-dimensional single-layer graphite
a
J composed of carbon atoms in honeycomb lattice. Re-

8 cently, this novel material was successfully fabricated in
2 laboratories[1, 2, 3] and stimulates intense investigations
on both experimental and theoretical sides. Its surpris-
]
l ing band structure and high mobility seem promising for
l
a potential applications in many aspects[4]. For instance,
h there are proposals[5, 6] that graphene nanoribbons can
-
s be the building block for future quantum computation.
e Since graphene is a low dimensional system, we expect
m
the electron-electron interaction should play some role.
. In a recent paper, Son, Cohen and Louie[7] spotted local FIG.1: Latticestructureforthesemi-inifinitegraphenewith
t
a magnetic moments near the zigzag edges of a graphene zigzagedge. TheNeelstateisshownbythealternativespins
m on different sublattices.
nanoribbon due to the Coulomb interaction. In fact,
- correlation-induced magnetic moment in graphene has
d
n be speculated for quite a while[8, 9, 10, 11, 12] but its we are interested in the spin physics, we propose to con-
o low-energy excitations remain poorly understood at this centrateontheeffectiveHeisenbergexchangecouplingin
c point. the system,
[
1 tioTnhseplnayovaelsimgnaigfinceatnistmrolsehoawtsthteheadtgeelsecotfrgornaicphceonrer.elIan- H =J (cid:88)S(l)·S(l(cid:48)), (1)
v condensedmattersystems,thephysicalpropertiesatthe (cid:104)l,l(cid:48)(cid:105)
7
6 edge are often tied up with related bulk properties. For where l labels all lattice sites and (cid:104)l,l(cid:48)(cid:105) means nearest
5 instance, the Andreev bound state[13] near the edge of a neighbor pairs. The coupling constant is J =4t2/U >0
4 superconductorarerelatedtotheedgetopologyandalso in the strong interaction limit but is viewed as an in-
1. the pairing symmetry in the bulk. The edge states[14] of dependent parameter here. The honeycomb lattice of
0 a quantum Hall liquid are described by the chiral Lut- graphenecanbeseparatedintotwosublatticesAandB.
9 tingerliquidthatcannotbeseparatedfromthebulkdue The outmost edge sites belongs to the sublattice A in
0
to gauge invariance. Therefore, we are inspired to inves- our convention as shown in Fig. 1. Starting from a Neel
:
v tigate the low-energy spin excitations near the edge of a ordering state, it is convenient to rotate the spin on sub-
Xi single-layergraphenehere. Forsimplicity,weconcentrate latticeB bySx →Sx,Sy →−Sy,andSz →−Sz sothat
on a semi-infinite honeycomb lattice with zigzag edge as the semi-classical ground state become uniformly ferro-
r
a shown in Fig. 1. Since the band structure of graphene is magnetic. Following the standard procedures, we can
wellapproximatedbythenearest-neighborhoppingt,the represent the spin on the lattice sites by the Holstein-
tight-binding model is sufficient. Furthermore, if we as- Primakov boson
sume the electron-electron interaction can be capture by
(cid:113)
an effective on-site interaction U, it is natural to model S+(r) = 2S−b†(r)b (r)b (r),
L L L L
the semi-infinite graphene by the Hubbard model at half (cid:113)
filling (cid:104)n(cid:105)=1. S−(r) = b†(r) 2S−b†(r)b (r),
L L L L
Unfortunately, it is known that the two dimensional Sz(r) = S−b†(r)b (r), (2)
L L L
Hubbardmodelcannotbesolvedexactly/reliablyatcur-
rent stage. Thus, approximations are inevitable. Since whereb (r),b†(r)areannihilation/creationoperatorsfor
L L
2
bosons. To make the sublattice dependence explicit, we E k
x
have separate the lattice positions into sublattice index 3
L = A,B and r for the triangular lattice. Expanding
the Heisenberg interaction to the quadratic order in the
bosonic operator, one can obtain the familiar Hamilto- 2! "
nian for spin-wave excitations,
(cid:20)
(cid:88)
H = JS b†(r)b (r)+b† (r+λ)b (r+λ) 1
A A B B
r,λ
(cid:21)
+b†(r)b† (r+λ)+b (r+λ)b (r) . (3) k
A B B A !Π !Π Π Π x
2 2
The summation over λ includes all three nearest neigh-
√ √
bors λ = a(±1/2,1/2 3),a(0,−1/ 3) as shown in FIG. 2: Dispersion for bulk and edge magnons with JS = 1
i
Fig. 1. The presence of open boundary ruins the trans- and a = 1 simplicity. The shaded area represents the bulk
magnons with double degeneracy ∆Sz =±1. The thick blue
lationalinvarianceiny directionandtheaboveHamilto-
line indicates the dispersion for the edge magnon with only
nian cannot be diagonalized by the usual Fourier trans-
one brach ∆Sz =−1.
formation.
However, since the translational invariance in the x
directionisstillvalid,wecanperformthepartialFourier to the boundary. For zigzag edge, there is one missing
transformation bondfromtheoutmostAsitestotheoppositesublattice.
1 (cid:88) The semi-infnite nature of the Hamiltonian gives rise to
bL(x,y)= √ eikyyψL(kx,y), (4) the constraint,
N
x kx
ψ (1)+J ψ (0)=0. (7)
to simplify the Hamiltonian. After some algebra, the A 2 B
spin-wave Hamiltonian can be casted into the form,
Now we are ready to find out the eigenstates by the
(cid:88) (cid:20)h h (cid:21) generalizedBlochtheorem[20]. Notethateigenvalueofa
H = JS Ψ† h11 h12 Ψ, (5) unitarydisplacementoperatorcanbewrittenasz =eik,
21 22
kx wherekisthemomentum. However,thepresenceofopen
boundarychangestothisniceconnectiontothemomen-
where we introduce the two-component spinor Ψ† =
[ψ†(k ,y),ψ (−k ,y)]. The matrix elements in the re- tum. It is easy to check that the displacement operator
A x B x satisfies DD† =1 but not in the inverse order D†D (cid:54)=1.
duced 2 × 2 matrix are semi-infinite matrices: h =
11
2+D†D, h = 3 and h = h† = J +J D, where Thus,itisnolongerunitary. Thus,theeigenstateshould
22 21 12 1 2 be written in the more general form,
J1 = 2coskx2a comes from the tilted bonds and J2 = 1
from the vertical bonds. For anisotropic interaction, J (cid:20) (cid:21) (cid:20) (cid:21)
2 ψ (n) c
would derivate from one. The semi-infinite displacement Ψ= A = A zn. (8)
ψ (n) c
B B
matrix D is
  UponsubstitutionintothecoupledHarperequations,the
0 1 0 0 ···
eigenvaluez ofthedisplacementoperatorsatisfythesim-
 0 0 1 0 ···
  ple algebraic constraint,
D = 0 0 0 1 ···.
 
 0 0 0 0 ··· (cid:18) (cid:19)
1
··· ··· ··· ··· ··· J J z+ +(J2+J2+ε2−9)=0. (9)
1 2 z 1 2
Now the problem in converted into solving the eigen-
Usually, for given momentum k, we look for energy ε
statesforaonedimensionalsemi-infinitesystem. Forno-
of the system. Here it is more convenient to look for
tationclarity,wesuppressthek momentumdependence
x
the complex z for a given energy ε. It is obvious that
in the following. Writing down the matrix elements of
the solutions z,1/z appear in pairs. Thus, the solutions
theHamiltonianexplicitly,itleadstothecoupledHarper
canbeclassifiedintotwotypes: (1)plane-wavesolutions
equations,
withz =e±ik (2)evanescentmodeswithrealz =α,1/α.
3ψ (n)+J ψ (n)+J ψ (n−1) = εψ (n) Sincewerequirethewavefunctiontobefiniteatinfinity,
A 1 B 2 B A
only one of the evanescent solutions survives while, not
−3ψ (n)−J ψ (n)−J ψ (n+1) = εψ (n), (6)
B 1 A 2 A B
surprisingly, both plane-wave solutions are allowed.
where n=1,2,3,... denote the lattice coordinates in the The plane-wave solutions correspond to the bulk
y direction. However,oneneedstopayspecialattentions magnons. Thepresenceoftheopenboundarydoescause
3
(a) (b) (cid:80) ∆Sz(r)=−1, as for those found in conventional fer-
r
romagnets. Thespindensityprofilesoftheedgemagnon
with two different momenta k = 0.1 and k = 0.5 are
x x
shown in Fig. 3. For momentum fairly close to k = 0,
x
the spin density extends into the bulk with long decay
1
length. However, for k =0.5, though not very far away
x
fromtheorigin, thespinprofileisalreadyquitelocalized
to the zigzag edge.
0
So,what’sthesignificanceoftheabovesimplecalcula-
tions? For an ordinary ferromagnet, there is one branch
ofnon-relativisticmagnon. Ontheotherhand,antiferro-
-1 magneticgroundstatesupportstwobranchesofrelativis-
tic magnons. Here, near the zigzag edge of graphene, we
FIG.3: Spindensityprofilefor(a)kx =0.1and(b)kx =0.5 find a hybrid between the two cases – a single branch of
edge magnons. For clarity, the magnitudes on the edge are the relativistic ferromagnetic magnon. This implies that
heldconstant. Themagnitudesofthespindensity(aftersub-
the spin-wave excitations near the edge of the graphene
tracting the background from the ground state) are repre-
cannotbedescribedastheordinaryonedimensionalfer-
sented by the radius and also the color gradients.
romagnet and shall be viewed as an intrinsic boundary
field theory. The missing branch can be understood by
a rather complicated mixing between counter propagat- the following symmetry argument. Although an antifer-
ing modes ±k . However, the energy spectrum remains romagnetic order breaks both time-reversal and sublat-
y
the same as shown in Fig. 2. Therefore, we would only tice exchange symmetries separately, the ground state
concentrate on the more interesting evanescent modes remainsinvariantunderbothtransformationssimultane-
where z is real. Solving the coupled Harper equations ously. Thus, starting from the ∆Sz = −1 branch, one
together with the boundary condition, there exist only can reverse the spin orientations by time-reversal trans-
(cid:112)
one solution ε = 4−J2. Therefore, the dispersion for formation followed by sublattice exchange to construct
1
the single-branch edge magnon is the other ∆Sz =1 branch. This explains why antiferro-
magnetic magnons on bipartite lattices always appear in
E (k )=2JS|sin(k a/2)|, (10)
e x x pairs with opposite spins. However, the sublattice sym-
metry is broken in the presence of the zigzag edge, this
which becomes linear (relativistic) in the small momen-
makes the single-branch magnon possible. In fact, we
tum regime. Note that, as shown in Fig. 2, the velocity
check with the armchair edge which doesn’t destroy the
of the edge magnon is slightly smaller than that for the
sublatticesymmetryandfoundnoevidenceforthenovel
bulk magnon and thus is protected by a gap. In fact,
edge magnon discussed here.
this protection arises from the sharp distinction between
Finally, we address the validity of the approximations
|z| = 1 and |z| < 1 solutions. Furthermore, the wave
inthelinearizedspin-wavetheory. Firstofall,thecharge
function of the edge magnon is
excitations are all left out. Since there is no charge gap
(cid:20) (cid:21)
Ψ= 1/z zn, with z = 1−|sin(kxa)|. (11) in graphene, it is expected that the neutral particle-hole
−1 cos(k a/2) excitations with ∆Sz = −1 will mix up with the edge
x
magnon. However, since the Fermi surface shrinks to
Notethat0≤z <1asrequiredfortheevanescentmodes
twoDiracpoints,thephasespaceislargelyreduced. Fur-
exceptatk =0wheretheedgemagnonmergesintothe
x thermore, since the Fermi velocity v is expected to be
F
bulk with z =1.
faster than the magnon velocity v , the hybridization is
m
The spin density profile of the edge magnon can be
further suppressed by a finite gap ∆ ∼ (v −v )k for
F m x
obtainedbysubtractingthebackgroundformtheground
any finite momentum. This is similar to the protection
state, ∆Sz(r) ≡ (cid:104)1|Sz(r)|1(cid:105)−(cid:104)G|Sz(r)|G(cid:105), where |1(cid:105) is
betweenbulkandedgemagnonswediscussedpreviously.
one magnon state and |G(cid:105) is the ground state. Since the
Another approximation is the existence of the Neel or-
spin density is uniform in the x direction, a factor 1/N
x der. This is the most serious approximation since it is
appears where N is the total transverse length. The
x likely that the antiferromagnetic order may not exist at
spin density can be expressed in terms of the normalized
all in graphene. Though the quadratic fluctuations from
wave functions,
themagnonsdonotdestroytheNeelorderandthespin-
(cid:26)− 1 |ψ (n)|2, r∈A, wave theory remain self-consistent, it remains an open
∆Sz(r)= +N1x|ψA(n)|2, r∈B. (12) questionwhetherourfindingsherearetruebeyondspin-
Nx B wave theory. Given the novelty of the edge magnetism
Summing over all lattice sites, it is easy to check the in graphene, alternative theoretical investigations are in
edge magnon we found carries the total spin ∆Sz = oder. And, as always, more experimental data will help
4
to clear up our understanding of spin-wave excitations
living on the edge.
We acknowledge supports from the National Science
Council in Taiwan through grants NSC-96-2112-M-007-
004 and NSC-97-2112-M-007-022-MY3. Financial sup-
portsandfriendlyenvironmentprovidedbytheNational
CenterforTheoreticalSciencesinTaiwanarealsogreatly
appreciated.
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