Tight-binding study of nonmagnetic-defect-induced magnetism in graphene
This review describes a study of nonmagnetic-defect-induced magnetism of graphene on the basis of a tight-binding model. A vacancy induces a quasilocalized impurity state at the chemical potential around itself, and it leads to formation of local magnetic moments. Connection between a quasilocaliz...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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Kumazaki, H. Hirashima, D.S. 2017-05-24T18:12:47Z 2017-05-24T18:12:47Z 2008 Tight-binding study of nonmagnetic-defect-induced magnetism in graphene / H. Kumazaki, D.S. Hirashima // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1025-1032. — Бібліогр.: 43 назв. — англ. 0132-6414 PACS: 75.75.+a;73.20.At;75.70.Ak https://nasplib.isofts.kiev.ua/handle/123456789/117559 This review describes a study of nonmagnetic-defect-induced magnetism of graphene on the basis of a tight-binding model. A vacancy induces a quasilocalized impurity state at the chemical potential around itself, and it leads to formation of local magnetic moments. Connection between a quasilocalized state around a vacancy and the edge localized states near a zigzag edge is studied in detail. Magnetism associated with many vacancies and edge structures is also reviewed. Some new results of magnetism associated with many vacancies are presented. We would like to thank K. Wakabayashi, S. Okada and O. V. Yazyev for useful discussions and correspondence. This work was partially supported by Research Foundation for the Electrotechnology of Chubu. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Graphene and graphite multilayers Tight-binding study of nonmagnetic-defect-induced magnetism in graphene Article published earlier |
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Tight-binding study of nonmagnetic-defect-induced magnetism in graphene |
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Tight-binding study of nonmagnetic-defect-induced magnetism in graphene Kumazaki, H. Hirashima, D.S. Graphene and graphite multilayers |
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Tight-binding study of nonmagnetic-defect-induced magnetism in graphene |
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Tight-binding study of nonmagnetic-defect-induced magnetism in graphene |
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Tight-binding study of nonmagnetic-defect-induced magnetism in graphene |
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Tight-binding study of nonmagnetic-defect-induced magnetism in graphene |
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tight-binding study of nonmagnetic-defect-induced magnetism in graphene |
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Kumazaki, H. Hirashima, D.S. |
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Kumazaki, H. Hirashima, D.S. |
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Graphene and graphite multilayers |
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Graphene and graphite multilayers |
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2008 |
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English |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
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This review describes a study of nonmagnetic-defect-induced magnetism of graphene on the basis of a
tight-binding model. A vacancy induces a quasilocalized impurity state at the chemical potential around itself,
and it leads to formation of local magnetic moments. Connection between a quasilocalized state around
a vacancy and the edge localized states near a zigzag edge is studied in detail. Magnetism associated with
many vacancies and edge structures is also reviewed. Some new results of magnetism associated with many
vacancies are presented.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117559 |
| citation_txt |
Tight-binding study of nonmagnetic-defect-induced magnetism in graphene / H. Kumazaki, D.S. Hirashima // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1025-1032. — Бібліогр.: 43 назв. — англ. |
| work_keys_str_mv |
AT kumazakih tightbindingstudyofnonmagneticdefectinducedmagnetismingraphene AT hirashimads tightbindingstudyofnonmagneticdefectinducedmagnetismingraphene |
| first_indexed |
2025-11-25T20:40:24Z |
| last_indexed |
2025-11-25T20:40:24Z |
| _version_ |
1850530815308464128 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1025–1032
Tight-binding study of nonmagnetic-defect-induced
magnetism in graphene
H. Kumazaki and D.S. Hirashima
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
E-mail: dai@slab.phys.nagoya-u.ac.jp
Received March 3, 2008
This review describes a study of nonmagnetic-defect-induced magnetism of graphene on the basis of a
tight-binding model. A vacancy induces a quasilocalized impurity state at the chemical potential around it-
self, and it leads to formation of local magnetic moments. Connection between a quasilocalized state around
a vacancy and the edge localized states near a zigzag edge is studied in detail. Magnetism associated with
many vacancies and edge structures is also reviewed. Some new results of magnetism associated with many
vacancies are presented.
PACS: 75.75.+a Magnetic properties of nanostructures;
73.20.At Surface states, band structure, electron density of states;
75.70.Ak Magnetic properties of monolayers and thin films.
Keywords: graphene, magnetism, defects, edges.
1. Introduction
The recent discovery of monolayer graphite [1] —
monolayer graphene — has triggered a surge of intensive
research of this unique material. In addition to the unique
physics of massless Dirac fermions realized in graphene,
the potentiality in future nanoelectronics applications is
the driving force of the research effort [2]. Applications
in electronic devices taking advantage of spin degrees
of freedom — spintronics applications — are also dis-
cussed [3].
Magnetism in carbon allotropes has indeed been a fun-
damental and also controversial problem for a long time
[4–7]. Apart from magnetic impurities, the defect-in-
duced mechanism is the most probable mechanism of
magnetism in carbon-based materials. Indeed, defect-in-
duced magnetism has been theoretically studied in vari-
ous carbon allotropes extensively [8–34]. The simplest
mechanism is the one caused by free dangling bonds asso-
ciated with vacancies. A dangling bond has a net magnetic
moment, which may induce a magnetic state. Even if a de-
fect itself is nonmagnetic, it can induce local magnetism
of π electrons around itself. A well-known example is the
localized edge magnetism associated with a (hydrogen-
terminated) zigzag edge of graphene [8] or a carbon nano-
tube [12].
Graphene is the building block of carbon allotropes,
graphite, carbon nanotubes, and fullerenes. Therefore, it
is important to study the possible magnetism of graphene
to understand magnetism in other carbon allotropes. Non-
magnetic-defect-induced magnetism in graphene has also
been reported using first-principles calculations [16–24].
We have also studied the problem using a tight-binding
model, and also found nonmagnetic-defect-induced mag-
netism in graphene [22,26–28,31]. Here, we review these
studies, and show that the origin of the local magnetism is
the impurity state induced by a vacancy introduced on a
honeycomb lattice [22,35–37] and that the sublattice con-
figuration of vacancies matters when more than one va-
cancies are introduced. The latter fact is a manifestation
of Lieb's theorem on magnetism on a bipartite lattice [38].
We then argue that the edge magnetism on a zigzag edge
of graphene [8] can be naturally understood as accumula-
tion of vacancy-induced local magnetism.
In the next section, we introduce a tight-binding model
of graphene and discuss an impurity state induced by a va-
cancy (nonmagnetic defect). Adding the Coulomb inter-
action to the Hamiltonian, we review the possible local
magnetism induced by a vacancy (vacancies) in Sec. 3. In
Sec. 3, we also present some new results of local moment
formation induced by extended vacancies. We then brief-
ly review the magnetism on edges with various structures
� H. Kumazaki and D.S. Hirashima, 2008
in Sec. 4, and the last section is devoted to a summary of
the paper.
2. Quasilocalized states induced by vacancies
We consider π electrons of graphene, which can be de-
scribed by a tight-binging model on a honeycomb lattice
(Fig. 1),
�0 = − + =
=
′
′∑
∑
t c c
c c
B
B[ ]
[ ( )
†
( , ),
,
†
n
n n
n
k
k
k
A
A
A
h.c.σ
σ
σ
σ
σε kBσ + h.c.] ,
,
(1)
where we consider only the electron transfer t between
the nearest neighbor site pairs ( , )n nA ′ B ; n and n′ repre-
sent the positions of unit cells, and A and B stand for the
sublattice indices. cnασ (cnασ
†
) is an annihilation (cre-
ation) operator of a π electron with spin σ at the site ( )nα ,
and ckασ (c
kασ
†
) is its Fourier transformation. Using the
vectors τ j ( j = 1 2 3, , ) connecting a site on sublattice A and
the three nearest neighbor sites on sublattice B, we can
write
ε τ
( )k
k= −
=
⋅∑t
j
i j
1
3
e . (2)
The band dispersion is given by E E( ) | ( ) | ( )k k k= = ±± ε .
It vanishes at points K( / , / )2 3 2 3π πa a and K′ ( / , )4 3 0π a ,
and has a linear dispersion around those points (a mass-
less Dirac fermion),
E k± = ±( ) ,k γδ (3)
where δk is the deviation of k from K (or K′), γ = 3 2ta / ,
and a is the lattice constant, a = 2 46. �.
We denote the number of unit cells by N u ; N Lu = 2
where L is the linear size of the lattice. Without vacancies,
the number of the lattice points N s is N Ns u= 2 . We re-
strict ourselves to the half-filled case where the electron
number N e is equal to the site number N s , N Ne s= , even
in the presence of vacancies.
In this paper, we implicitly assume that all dangling
bonds are terminated appropriately, e.g., by hydrogen
atoms. We consider magnetism caused by π electrons.
Now we introduce impurity potentials. A short-range
impurity potential at ( )nα is expressed by
�( ) .
†
n n nα ασ
σ
ασ= ∑u c c (4)
The limit of u → ∞ simulates a strong impurity such as a
vacancy, although u = ∞ may be unrealistic for actual va-
cancies. The opposite limit u → − ∞ may simulate the ef-
fect of a hydrogen chemisorption site. In both limits,
N N Ns u i= −2 , where N i is the number of impurities.
First, we briefly study the case with a single impurity
[35]. For the moment, we suppress the spin suffixes. Ex-
panding a single electron wave function Φ( )r with the
Wannier functions φn rα ( ), Φ( ) ( )
,
r rn
n
n= φ∑C α
α
α , we can
write down the Schr�dinger equation in the presence of
an impurity at r r= 0 on sublattice A,
E
E
C
C
u
N
A
B
i
u
i
−
−
⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟
=
=
− ⋅ ′⋅
ε
ε
( )
( ) *
k
k
k
k
k r k re e0 0 0
0 0
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝⎜
⎞
⎠⎟
′
′
′
∑
k
k
k
C
C
A
B
, (5)
from which we obtain the eigenvalue equation for a bo-
und state,
u
N
E
Eu
2 2
1
−
=∑
| ( )|ε kk
. (6)
In the limit of u → ∞, we have E u uc c� ε ε2 0/ [ ln ( / )] → ,
where ε c is a cutoff energy, that is, the eigenenergy van-
ishes. This means that an impurity state is induced at the
chemical potential μ = 0 by a strong impurity potential.
Putting E = 0, we readily have C Ak = 0, which also means
that C An = 0. The amplitude of this wave function van-
ishes on sublattice A, the same sublattice where a vacancy
is. On the other hand, we have
C
N
B
i
u
k
k r
k
=
− ⋅
�
e 0
ε( )
, (7)
where � = −
→ ∞
lim
u
AuC 0 is a normalization constant. We
then obtain
C
N
C DB
u
i
B n
n
n
k r
k
k r r= = −⋅∑1
0 1e � ( , )τ , (8)
with
D
N u
i
( , )
( )
,
( )
r
k
k r
k
τ
ε
τ
=
⋅ +
∑1 e
(9)
1026 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
H. Kumazaki and D.S. Hirashima
y
x
e1
e2
aτ1
τ3τ2
Fig. 1. Honeycomb lattice. The primitive vectors are e1 and e2,
whose magnitude is a, | | | |e e1 2= = a. Three vectors connecting
the nearest neighbor sites are denoted by τ j ( j = 1 2, , and 3),
| | /τ j a= 3.
where τ is the vector connecting the two lattice points in a
unit cell.
We can show that D r( , ) /r τ ∝ 1 at r a>> . This means
that the wave function is localized around the impurity
site, but not strongly enough to make the normalization
constant � independent of the system size. This is why
we call this state a quasilocalized state. The normalization
constant � ∝ 1/ ln L, if we normalize the wave function
so that d∫ =r rΦ( ) 2 1.
In the limit of u → ∞, the number of states in the va-
lence band (E < 0) is N u −1 and that in the conduction
band (E > 0) is also N u −1. The strong impurity potential
induces a quasilocalized state just at E = 0 as shown
above, in addition to pushing a state to positive infinity.
Then, in the ground state at the half-filling N Ne L= =
= −2 1N u , either up-spin or a down-spin occupies the
quasilocalized state at E = 0, generating (quasi)localized
magnetic moments around a vacancy even in the absence
of the Coulomb interaction. It is simply natural that the
ground state has a net magnetic moment, because the
number of electrons N e is odd. The point is that the mag-
netic moments are not spread over the system, but are
(quasi)localized around a vacancy, which is specific for
the honeycomb lattice.
Next, we consider cases with two impurities [26].
First, the case with two impurities at r r= 0 and r1 on the
same sublattice A is considered. An eigenvalue equation
for a bound state is also easily found, and is given by
u
N
E
E
C
Cu
A
A
i c
2 2
1
0
1 10 1
−
+⎛
⎝
⎜
⎞
⎠
⎟ =∑ ⋅ −
| ( )|
( )
ε kk
k r r
e , (10)
from which we readily see that the eigenvalue E vanishes
as | |u → ∞. The wave function also vanishes on sublattice
A, C CA Ak n= = 0, and
C D DB n nn r r r r= − + −� �0 0 1 1 1 1( , ) ( , ),τ τ (11)
where � i is a normalization constant. This means that a
quasilocalized state associated with one vacancy is not af-
fected by the other vacancy, the wave function is simply
an arbitrary linear combination of two quasilocalized
states associated with two vacancies, and the two states
remain degenerate at E = 0.
At the half-filling N N Ne L u= = −2 2, two electrons
occupy those two degenerate states. Without the Coulomb
interaction, a singlet state and triplet states are degener-
ate, and Lieb's theorem guarantees that the triplet states
become the ground state in the presence of the Hubbard
U ( )> 0 [38]. As we consider only the nearest neighbor
electron transfer integral, the lattice is bipartite. The rea-
lization of triplet states is also consistent with Hund's
rule.
When two impurities are on the different sublattices,
the situation is different. We can see that the eigenvalue E
does not vanish in general. It vanishes only when the dis-
tance r between two vacancies goes to infinity (Fig. 2);
E r∝ 1/ [26]. This is simply because there occurs a «con-
flict» between the two vacancies: One vacancy on
sublattice A requires the wave function to be finite only
on sublattice B, and the other vacancy on sublattice B re-
quires the opposite. This leads to the effective repulsion
between the two quasilocalized states associated with va-
cancies on different sublattices, and they are no more at
E = 0.
The reduction in the density of states due to the repul-
sion between the quasilocalized states is unfavorable for
local moment formation. Lieb's theorem states that the
ground state must be singlet (at a finite U ) in the present
case, where the number of lattice points on sublattice A is
equal to that on sublattice B [38]. We denote the number
of vacancies on sublattice A by N A , and that on sublattice
B by N B . When N NA B= , the number of lattice points on
sublattice A is equal to that on sublattice B.
When N NA B= and the ground state is a singlet one,
there are still two possibilities [28,29]: One is that no mo-
ment is formed at any sites. The other possibility is that a
magnetic moment is locally generated around each va-
cancy and the two local moments are coupled to form a
spin singlet state. Which is realized depends on the con-
figuration of two vacancies, the strength in interaction,
etc, which will be studied in the next section.
The results obtained for two vacancies can be straight-
forwardly extended to the case with more vacancies. For
example, N vacancies on the same sublattice generate
N -fold degenerate (quasilocalized) states at E = 0. It is in-
teresting to study the case where vacancies are in a row on
the same sublattice, say sublattice A (Fig. 3). These va-
cancies generate a zigzag edge and a Klein edge. Assum-
ing the periodic boundary condition along the x-direc-
Tight-binding study of nonmagnetic-defect-induced magnetism in graphene
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1027
r/a
|E
|/
t
0.10
0.05
0 5 10 15
Fig. 2. Energy E of a vacancy state induced when two vacan-
cies are introduced on the different sublattices as a function of
the distance r between the two vacancies.
tion, the direction along the edges, the wavevector p
(along the x-direction) is a good quantum number. The
wave function C pBn ( ) is then expressed as
C p D d yB
ip a
n
ipx
p n
n
n r r( ) ( , ) ( ),∝ − ∝∑e e�
�
� τ1
(12)
where
r rn n nx y x a= = =( , ), ( , ) ( , ) ,� � �0 0 (13)
and
d y
p k
p n
ik a
yk
ik y
y
y
y n( )
( , )
;
/
= ∑ e
e
3
ε (14)
the region of the summation of k y is shown in Fig. 4. As
the summand e
ik a
y
y / p k
/
( , )
3 ε is a periodic function of
k y (of period 4 3π / ( a)), d yp n( ) decreases faster than
any power of 1/ yn as yn → ∞, except for at p a= 2 3π / ( )
(or 4 3π / ( )a ), where ε( , )p k y vanishes at k ay = 2 3π / ( )
(or 0). This is indeed what is found by Fujita et al. (see
also below) [8].
Although each vacancy generates a quasilocalized
state whose amplitude decreases proportionally to the in-
verse of the distance from the position of the vacancy,
those vacancies in a row on the same sublattice generate
exponentially localized states (plus extended states at
p a= ± 2 3π / ( )) as a results of interference. Note also that
those extended states may be important for coupling be-
tween magnetic moments on both edges of a graphene
ribbon with zigzag edges [25].
Strictly speaking, what we have shown is not exactly
the same as the localized states found by Fujita et al. They
studied the case with a graphene ribbon with two zigzag
edges. In this case, the edge localized states exist only at
2 3 4 3π π/ /< <pa (i.e., the number of the localized states
is one third of the number of vacancies). The result ob-
tained above corresponds to a graphene ribbon with a zig-
zag edge and a Klein edge, where it was shown that the
edge localized states exist for any p (i.e., the number of
the localized states is the same as that of vacancies) [39].
3. Vacancy-induced magnetism
Now we proceed to study magnetism induced by non-
magnetic defects. To do so, we have to introduce the Cou-
lomb interaction between π electrons. We consider the lo-
cal Coulomb interaction U ,
� = ↑ ↓∑U n nn
n
nα
α σ
α
,
. (15)
We thus consider a Hubbard model on the honeycomb lat-
tice to study magnetism of graphene. We resort to a mean
field approximation to treat �,
� → 〈 〉∑ −U n n
n
n n
, ,
.
α σ
α σ ασ (16)
By diagonalizing the mean-field approximated hamiltoni-
an on a finite size lattice (L L× ), we can self-consistently
determine the charge density n n nn n nα α α= 〈 + 〉↑ ↓ and
the spin density m n nn n nα α α= 〈 − 〉↑ ↓ at each lattice po-
int. In actual calculations, we use U t= 1 2. unless other-
wise stated. In the mean field approximation, the ground
state of the half-filled honeycomb lattice is an antiferro-
magnetic state at U U t> =cr 2 23. .
It was shown in a previous study that the ferrimagnetic
order is induced around the vacancy [27]. WithoutU , only
1028 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
H. Kumazaki and D.S. Hirashima
x
y
Fig. 3. Removing carbon atoms (open dots) in a row from the
same sublattice generates a zigzag edge and a Klein edge.
Edge sites on both edges are represented by closed dots.
2 (0, 2/3π 1/2
)/a
2 ( , /3π 1 –1
1/2
)/a
p
K
K'
ky
Γ
Fig. 4. The first Brillouin zone of a honeycomb lattice (the
rhombus), and the thick dashed line shows the integral region
of k y in Eq. (14). When p a= 2 3π / ( ) or 4 3π / ( )a , the line passes
over K( / ( ), / ( )2 3 2 3π πa a or K'( / ( ), )4 3 0π a , where ε( , )p k y vani-
shes.
the density of one spin component (up or down) is finite
because the wave function of the quasilocalized state is fi-
nite only on one sublattice. The interaction U makes the
minor spin component develop, i.e., it enhances the stag-
gered spin moment. At U t= 1 2. , the total staggered mo-
ment
M m ms A B; | ( )|tot = −∑ n
n
n
is M s; .tot � 2 2 (for L = 32). The total moment
M m mA Btot = +∑| ( )|n
n
n
is evidently M tot = 1 (S z = 1 2/ ). The nonmagnetic-de-
fect-induced magnetic moments were also obtained by
first-principles calculations [16,24].
As mentioned in the previous chapter, the ground state
is a singlet one when two vacancies are introduced on the
different sublattices, A and B. However, it is still possible
that local magnetic moments are induced around each va-
cancy [28,29]. Figure 5 shows the dependence of the total
sublattice magnetization M s;tot on the distance r between
two vacancies put on the different sublattices. When the
vacancies are close to each other, no local magnetic mo-
ments are induced. When they separate apart, local mo-
ments develop around each vacancy. The magnitude of
the total staggered moment approaches to 4.4, twice the
staggered moment in the case with a single vacancy. The
moments couple antiferromagnetically, and the ground
state is a singlet one, but the coupling weakens as they
separate apart.
When the two vacancies are introduced on the same
sublattice, the quasilocalized states remain at the chemi-
cal potential, and local moments are induced around both
vacancies. They couple ferromagnetically, and the
ground state is a triplet one in harmony with Lieb's
theorem [38].
The results for two vacancies can be straightforwardly
extended to more-than-two vacancies [28]. The point is,
as is clear from the above argument, Lieb's theorem. If
N NA B≠ , local magnetic moments are always induced.
When N NA B= , the ground state must be a singlet one. If
vacancies are close to each other, no local moments is
likely to be induced, but if they are wide apart, local mo-
ments develop around each vacancy and couple
antiferromagnetically with each other. Examples where
moments are induced around extended vacancies with
N NA B= are shown below.
Cases with N NA B= are shown in Fig. 6. In Fig. 6,a,
N NA B= = 3, and in Fig. 6,b, N NA B= = 6. In both
cases, no local moments are formed around the «void» at
U t= 1 2. . Note that these extended vacancies are ring-cen-
tered. The spin density in cases with | |N NA B− ≠ 0 is
shown in Fig. 7. In Fig. 7,a, N A = 7 and N B = 6, and in
Fig. 7,b, N A = 10 and N B = 6. The result for the case with
N A = 1 and N B = 3 was also previously reported [28].
Tight-binding study of nonmagnetic-defect-induced magnetism in graphene
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1029
0 5 10 15
2
4
r/a
M
s
to
t
Fig. 5. Dependence of the total sublattice magnetization on the
distance r between two vacancies introduced on the different
sublattices.
a b
Fig. 6. Examples of extended vacancies inducing no moments
around themselves at U t= 1 2. . N NA B= = 3 (a), and
N NA B= = 6 (b).
a b
Fig. 7. Spin density around extended vacancies at U t= 1 2. .
N A = 7 and NB = 6 (a), and N A = 10 and NB = 6 (b). Radius ri
of each is dot given by r m ai i= | | , where mi is the spin density
at each lattice point, m n ni i i= 〈 − 〉↑ ↓ . The closed dots represent
positive spin density (mi > 0), and open ones represent negative
density (mi < 0).
These extended vacancies are site-centered. It can be seen
that local ferrimagnetic order is induced around the void.
Even when N NA B= , it is still possible that local
magnetic moments are induced. Examples are shown in
Fig. 8, where large local moments are induced at Klein
edge sites. Here, it is implicitly assumed that the Klein
edge site are doubly hydrogenated when the crosses in
Fig. 8 stand for vacancies or that the crosses stand for hy-
drogen chemisorption sites. In Fig. 8,a, antiferromagnetic
coupling works between one Klein edge site and the
other, but it is so weak because no direct exchange inter-
action exists between these two sites. That is why local
moments survive even though N NA B= . The same argu-
ment also applies to the case in Fig. 8,b.
Other examples where large moments are induced
even though N NA B= holds are shown in Fig. 9, where
n p pairs of carbon atoms on sublattice A and sublattice B
are removed. AtU t= 1 2. , we find no local moments when
n p = 1, 2, 3, but local moments are found to develop on
both sides of the «slit» when n p ≥ 4. Moments at the edge
sites on one side couple ferromagnetically because they
are all on the same sublattice, and moments at edge site on
different sides couple antiferromagnetically because they
belong to the different sublattices. Since there is no direct
exchange coupling between the moments on both sides of
the slit, local moments remain (The coupling is only
through edges of the slit). The true ground state must be a
singlet one even in these cases, but the gap between the
ground state and the excited state must become small as
n p becomes large.
We can also consider a slit consisting vacancies with
N NA B≠ . An example with N A = 7 and N B = 6 is shown
in Fig. 10. The ground state must be a triplet one in this
case.
We also studied the effect of the next-nearest electron
transfer t′. This transfer directly connects the lattice po-
ints on the same sublattice, and the lattice is not bipartite
anymore with t′. At t t′ �0 2. , local moments induced on a
zigzag edge start decreasing considerably and diminish at
t t′ � 0 4. . The effect of t′ on other magnetic states is quite
similar. This is understood as a result of (1) breaking of
particle-hole symmetry resulting in a shift of the energy
of quasilocalized states from zero and (2) antiferromag-
netic exchange coupling between the next-nearest lattice
points caused by t′. As local moment formation is also
obtained with various first-principles calculations, how-
ever, magnitude of t′ is not so large to affect moment
formation appreciably.
The charge density is found to be uniform irrespective
of the local magnetic moment formation. The nonuniform
contribution from the quasilocalized states is cancelled
by the contribution from the other occupied extended
states [26]. However, this result is a consequence of the
neglect of the long-range part of the Coulomb interaction.
Adding the repulsive interaction V between the nearest-
1030 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
H. Kumazaki and D.S. Hirashima
a b
Fig. 8. Spin density around extended vacancies at U t= 1 2. .
N A = 2 and NB = 2 (a), and N A = 3 and NB = 3 (b). Radius ri
of each is dot given by r m ai i= | | , where mi is the spin density
at each lattice point.
a
b
Fig. 9. Spin density around a «slit» with N N nA B p= = = 3 at
U t= 1 6. (a), and with N N nA B p= = = 6 at U t= 1 2. (b). Radius
ri of each is dot given by r m ai i= | | , where mi is the spin den-
sity at each lattice point.
Fig. 10. Spin density around a «slit» at U t= 12. . N A = 7 and
NB = 6. Radius ri of each is dot given by r m ai i= | | , where mi
is the spin density at each lattice point.
neighbor sites to the Hamiltonian and treating it with the
mean field theory, we find that the charge density be-
comes nonuniform around an impurity site. The charge
distribution depends on the sign of u, the impurity poten-
tial. If it is positive, the charge density increases around
the impurity site, and it decreases if u is negative. The
spin density is not affected substantially by the introduc-
tion of the nearesr-neighbor Coulomb interaction as far as
V U�0 4. . It was also argued that the nearest-neighbor
Coulomb interaction could cause charge order in gra-
phene [14].
4. Magnetism induced on edges of graphene
In Sec. 2 we saw that a zigzag edge is obtained by in-
troducing vacancies on the same sublattice. This implies
that localized edge states are associated with a zigzag
edge, which favors local moment formation. Indeed, it
was found by Fujita et al. that the local ferrimagnetic or-
der is formed near a zigzag edge [8]. Note also that the
«slit» introduced in the previous section generates inter-
nal zigzag edges in graphene. We can also see that an arm-
chair edge is obtained by removing atoms in a row alter-
natively from sublattices A and B. This implies that no
localized states at the chemical potential, and therefore no
local magnetic moments, are present near an armchair
edge.
A realistic edge is in general a mixture of zigzag and
armchair structures [40–43], because it is rather difficult
to prepare a graphene sheet with pure zigzag or armchair
edges. We then studied the possible magnetism near an
edge with structures [31], and found that a zigzag part
longer than 3a is sufficient for the formation of local mag-
netic moments [30,31]. This means, on one hand, that it
is rather difficult to suppress magnetism completely on
edges of actual graphene sheets. One has to prepare a
graphene sheet with a pure armchair edge to make it com-
pletely nonmagnetic. On the other hand, it also means that
it is fairly easy to make graphene at least locally mag-
netic.
Summary
Possible nonmagnetic-defect-induced magnetism in
graphene is studied with a tight-binding model. A va-
cancy (or a hydrogen chemisorption site) is found to in-
duce local magnetic moments around it. In cases of many
nonmagnetic defects, the sublattice configuration is es-
sentially important. When the number of vacancies on
one sublattice is different from that on the other sub-
lattice, local magnetic moments are necessarily induced.
Even when they are equal, it is still possible that local
moments are induced.
Our results imply that it is difficult to suppress local
magnetism in graphene even when it is completely free
from magnetic impurity atoms. On the other hand, the
present results also imply that one can control magnetic
properties of graphene by controlling defects or impu-
rities. This may be useful in the future application of
graphene in spintronic devices. Our results may also be
relevant for magnetism of bulk graphite [4–7]. Local mo-
ments induced by nonmagnetic defects can lead to bulk
ferromagnetism through exchange coupling in 3 dimen-
sions.
The stability of magnetic states or magnetic correla-
tion lenght at finite temperatures [32] is important for its
realization. We found that the energy gain due to the local
moment formation around a vacancy was approximately
0 01. t (∼ 300 K) atU t= 1 0. [27]. As the size of vacancies in-
creases, the energy gain increases more rapidly than lin-
early. For example, the energy gain of the configuration
shown in Fig. 8,a is 0 14. t (0 035. t per vacancy), and that
shown in Fig. 9,b is 0 27. t (0 023. t per vacancy). These re-
sults imply that the local magnetic moments persist even
at room temperatures.
We used a simple tight-binding model. We always as-
sumed that dangling bonds are terminated by hydrogen
atoms, and did not consider the possible lattice distor-
tion induced by vacancies (or chemisorption). Electronic
states may depend crucially on the method of termination.
Moreover, introduction of defects usually induces lattice
distortion. As the size of extended vacancies becomes
larger, the effect of lattice distortion will be more impor-
tant. To study those effects, we have to resort to a first-
principles calculation. Our results will serve as useful
references in those studies.
Acknowledgments
We would like to thank K. Wakabayashi, S. Okada and
O. V. Yazyev for useful discussions and correspondence.
This work was partially supported by Research Founda-
tion for the Electrotechnology of Chubu.
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