Impurity and vacancy effects in graphene
A Green function analysis has been developed for quasiparticle spectrum of a 2D graphene sheet in presence of different types of substitutional disorder, including vacancies. The anomalous character of impurity effects in this system is demonstrated, compared to those in well known doped semicondu...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Impurity and vacancy effects in graphene / V.M. Loktev, Yu.G. Pogorelov // Физика низких температур. — 2012. — Т. 38, № 8. — С. 993-1000. — Бібліогр.: 33 назв. — англ. |
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Loktev, V.M. Pogorelov, Yu.G. 2017-05-23T15:04:34Z 2017-05-23T15:04:34Z 2012 Impurity and vacancy effects in graphene / V.M. Loktev, Yu.G. Pogorelov // Физика низких температур. — 2012. — Т. 38, № 8. — С. 993-1000. — Бібліогр.: 33 назв. — англ. 0132-6414 PACS: 03.65.Pm, 71.30.+h, 71.55.–i https://nasplib.isofts.kiev.ua/handle/123456789/117436 A Green function analysis has been developed for quasiparticle spectrum of a 2D graphene sheet in presence of different types of substitutional disorder, including vacancies. The anomalous character of impurity effects in this system is demonstrated, compared to those in well known doped semiconductors, and explained in terms of conical singularities in the band spectrum of pure graphene. The criteria for appearance of localized states on clusters of impurity scatterers and for qualitative restructuring of band spectrum are established and a possibility for a specific metal/insulator transition at presence of vacancies is indicated. Y.P. thanks V.A. Miransky, J.P. Carbotte and S.G. Sharapov for very useful discussions of disorder effects in graphene, and kind hospitality by the Department of Applied Mathematics of University of Western Ontario during his sabbatical stay there, when this work was initiated. The support from Portuguese Fundação para a Ciência e a Tecnologia is gratefully acknowledged. V.L. thanks the SCOPES grant and also the Special Program of fundamental researches by Physics and Astronomy Department of NAS of Ukraine for the partial support. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Low Temperature Spectroscopy and Radiation Effects Impurity and vacancy effects in graphene Article published earlier |
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Impurity and vacancy effects in graphene Loktev, V.M. Pogorelov, Yu.G. Low Temperature Spectroscopy and Radiation Effects |
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Impurity and vacancy effects in graphene |
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impurity and vacancy effects in graphene |
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Loktev, V.M. Pogorelov, Yu.G. |
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Low Temperature Spectroscopy and Radiation Effects |
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Low Temperature Spectroscopy and Radiation Effects |
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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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A Green function analysis has been developed for quasiparticle spectrum of a 2D graphene sheet in presence
of different types of substitutional disorder, including vacancies. The anomalous character of impurity effects in
this system is demonstrated, compared to those in well known doped semiconductors, and explained in terms of
conical singularities in the band spectrum of pure graphene. The criteria for appearance of localized states on
clusters of impurity scatterers and for qualitative restructuring of band spectrum are established and a possibility
for a specific metal/insulator transition at presence of vacancies is indicated.
|
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0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/117436 |
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Impurity and vacancy effects in graphene / V.M. Loktev, Yu.G. Pogorelov // Физика низких температур. — 2012. — Т. 38, № 8. — С. 993-1000. — Бібліогр.: 33 назв. — англ. |
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2025-11-25T21:20:34Z |
| last_indexed |
2025-11-25T21:20:34Z |
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1850550651630649344 |
| fulltext |
© V.M. Loktev and Yu.G. Pogorelov, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 8, pp. 993–1000
Impurity and vacancy effects in graphene
V.M. Loktev
Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, 14b Metrolohichna Str., Kiev 03680, Ukraine
E-mail: vloktev@bitp.kiev.ua
Yu.G. Pogorelov
IFIMUP/Departamento de Fisica, Universidade do Porto, R. Campo Alegre, 687, Porto, 4169-007, Portugal
E-mail: ypogorel@fc.up.pt
Received May 15, 2012
A Green function analysis has been developed for quasiparticle spectrum of a 2D graphene sheet in presence
of different types of substitutional disorder, including vacancies. The anomalous character of impurity effects in
this system is demonstrated, compared to those in well known doped semiconductors, and explained in terms of
conical singularities in the band spectrum of pure graphene. The criteria for appearance of localized states on
clusters of impurity scatterers and for qualitative restructuring of band spectrum are established and a possibility
for a specific metal/insulator transition at presence of vacancies is indicated.
PACS: 03.65.Pm Relativistic wave equations;
71.30.+h Metal-insulator transitions and other electronic transitions;
71.55.–i Impurity and defect levels.
Keywords: graphene, quasiparticle spectrum, impurity and vacancy effects.
1. Introduction
There is a growing attention to electronic properties of
a single carbon layer known as graphene [1]. Its 2D honey-
comb lattice defines a peculiar band structure [2] with two
nodal points in the Brillouin zone (BZ) where conical
energy surfaces (with zero effective mass) of conduction
and valence bands touch each other. This gives rise to elec-
tronic dynamics of relativistic Dirac type [3], extraordinary
for condensed matter, and generates such unusual pheno-
mena as half-integer Hall effect [4–6] and, possibly, the
magnetic catalysis of an excitonic gap [7,8], ferromagnet-
ism and superconductivity [9]. On the other hand, it is of
interest to examine the effects that various kinds of impuri-
ties can produce on this remarkable material, regarding for
instance a fundamental role of such effects in physics of
common semiconductors (with finite effective mass) [10].
An intriguing situation with impurity levels near conical
singularities was recognized in d-wave superconductors
[11] where theoretical predictions are sometimes contra-
dictory [12] and not fully confirmed by the existing expe-
rimental data. To this time, the disorder effects in graphene
were theoretically studied, searching for weak localization
in this 2D electronic system under weak scattering (Born
limit) [13,14] or for strong localization under infinitely
strong (unitary limit) perturbation [15]. This work is aimed
on a consequent description of restructured electronic spec-
trum, at arbitrary perturbation strength and in a rather
broad range of impurity concentration, and on specifics of
this restructuring for Dirac quasiparticles under realistic
perturbation, compared to more common quasiparticles
with parabolic dispersion. In this respect, our results con-
tribute to the overall topic of the present issue dedicated to
Prof. E.V. Savchenko's jubilee and her notable works on
the electronically induced defects, especially vacancies, in
cryocrystals [16–18].
2. Hamiltonian and Green functions
Let us start from the simplest tight-binding Hamiltonian restricted
to nearest neighbor hopping
†
,
= h.c.,H t a bn n+δ
n δ
+∑ (1)
where t is the hopping amplitude, an and bn δ+ are the
Fermi operators of (spinless) electrons on sites of type 1
and 2, respectively, and the atomic energy on each site is
chosen as zero energy reference. The set of vectors δ
(of length | |= )aδ relates to three nearest neighbors of
a type 1 site (due to lack of central symmetry, it does not
coincide with the set of δ− for nearest neighbors of a type 2
V.M. Loktev and Yu.G. Pogorelov
994 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 8
site, see Fig. 1,a). Passing from site operators to plane
waves: 1/2= e ia N akn
k n n
− −∑ and 1/2= e ib N bkn
k n n
− −∑
(N being the number of cells in the lattice) the Hamilto-
nian, Eq. (1), is splitted in quasimomentum k:
†= ( h.c.),H t a bγk kk
k
+∑ (2)
with the complex function
/2 3
= e = e 2e cos .
2
yiak iaki x x
ak
γ k δ
k
δ
−⋅ +∑
It is then fully diagonalized:
† †= ( ),H ε α α β βk k kk k
k
−∑ (3)
in terms of the normal modes operators:
( )/2 /21= e e ,
2
i ia bϕ ϕα k kk k k
− +
( )/2 /21= e e ,
2
i ib aϕ ϕβ k kk k k
−−
with the eigen-energies = tε γk k . The main specifics of
graphene is that the function γk vanishes near two iso-
lated points in the BZ: = (0, 4 /3 3 )aπK and –K (Fig. 1).
Also the lack of central symmetry for the point group 3C
makes this vanishing linear in a small difference
=q k K− (or =q k K+ ):
3
( ), at 1,
2 y xa q iq aqγ γk q≡ ≈ − + (4)
defining the conical form of isoenergetic surfaces =ε εq± as
q F qε ≈ v
with the Fermi velocity 3
2= / .F tav This conical spec-
trum is commonly compared with the relativistic massless
Dirac fermions. The following analysis of this system is
restricted to the low energy physics, essentially determined
by the vicinities (below also referred to as valleys) of two
nodal points K± .
The electronic quasiparticle states at given k can be
suitably generated by the operator spinors:
= ,
a
b
ψ k
k
k
⎛ ⎞
⎜ ⎟⎝ ⎠
(5)
and the Hamiltonian, Eq. (1), is expressed through such
spinors as:
† ˆ= ,H hε ψ ψk k kk
k
∑ (6)
with the matrix ˆ ˆ ˆ= ( )h t γ τ γ τk k k
∗
+ −+ , and ˆ ˆ ˆ= x yiτ τ τ± ±
involving Pauli matrices ˆiτ . In what follows, the integration
in k over BZ is presented as twice of that in q, done in an
approximate way over a circle of radius = 8 /3 3 /mq aπ
around K (half of the full BZ area). The corresponding ef-
fective bandwidth = 2.87F mW q t≈v , by the approximate
dispersion law ,εq is close to the exact bandwidth 3 ,t by
the exact dispersion law .εk
We describe the dynamics of this system by the (Fouri-
er transformed) two-time Green functions (GF's) [19], here
combined into a 2 × 2 matrix †
,
ˆ = |G ψ ψk k k k′ ′〈〈 〉〉 and
satisfying the equation of motion:
[ ]† †
,| = , | ,Hε ψ ψ δ ψ ψk k k kk k′′ ′〈〈 〉〉 + 〈〈 〉〉 (7)
where the commutator [ ], Hψk is understood as a spinor
whose components are commutators of respective ψk
components with H: [ ] ,, = ,jjH Hψ ψk k⎡ ⎤⎣ ⎦ . For the
unperturbed system with the Hamiltonian, Eq. (5), the ex-
act form of GF is readily obtained from Eq. (6) as
, ' , '
ˆ ˆ=G g δk k k k k where
2 2
ˆ
ˆ = .
h
g
ε
ε ε
k
k
k
+
−
(8)
The observable characteristics of the system follow
from the Green functions in a usual way, for instance, the
density of states (DOS) is obtained as
,
1 ˆ( ) = Im Tr .G
N
ρ ε
π k k
k
∑ (9)
Fig. 1. 2D lattice structure of a graphene sheet. Rhombic primi-
tive cell (shadowed) with two non-equivalent positions for carbon
atoms, 1 (open circles) and 2 (solid circles), and elementary trans-
lation vectors 1a and 2a of length 3a (a). Rhombic Brillouin
zone (shadowed) with two non-equivalent nodal points, K (open)
and –K (solid), and vectors of reciprocal lattice 1b and 2b of
length 4 / ( 3)aπ (b).
a
b
2 1
Impurity and vacancy effects in graphene
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 8 995
Another important quantity, the local density of states
(LDOS) on a site jn of jth type:
( )
,
,
1 ˆ ˆ( ) = Im e ,
i j
jj
G
N
ρ ε τ
π
k k n
n k k
k k
− ′
′
′
∑ (10)
with the projecting matrices ˆ ˆ= 1 ( 1) / 2j
j zτ τ⎡ ⎤+ −⎣ ⎦ . In the
unperturbed system, the exact solution, Eq. (7), leads to
simple linear energy dependencies of DOS: 0 2( ) = / ,Wρ ε ε
and also of LDOS: 0 0( ) = ( ) / 2
j
ρ ε ρ εn . The following
analysis is focused on modifications of these dependencies
under local perturbations by impurities or vacancies.
3. Impurity perturbations
The notable distinction of graphene from genuine rela-
tivistic systems of quantum field theory is the possibility to
study the effects of localized perturbations on its dynamics,
which is our main purpose here. The disordered systems
are specific by presence of localized states whose wave
functions are restricted to atomically short distances
[20,21], they always appear near the limits of non-
perturbed energy spectrum but can either emerge near
some impurity resonance energies [22]. Besides them, ex-
tended (also called Bloch-like) states are present, similar to
common Bloch waves in perfect crystals but with limita-
tion of wave function coherence by certain (long enough)
mean free path .
Here we adopt the Lifshitz model of impurity perturba-
tion, where the perturbation potential is identical for all
impurity sites randomly distributed among the lattice sites
[20]. This model looks more adequate to the case of rare
defects in graphene, than the alternative choice [23] of
Anderson model with random perturbations at each lattice
cite [24]. Within the tight-binding framework, two charac-
teristic types of local perturbation are considered: i) that on
the on-site atomic energy, also referred to as the “diagonal
disorder” [25], here more relevant for substitution impuri-
ties and ii) that on the inter-site hopping, or the “nondia-
gonal disorder”, more relevant for vacancies.
We begin from the simplest point-like diagonal pertur-
bation by a Lifshitz impurity, producing a certain shift V
of the on-site energy at the impurity site. We denote 1p
the defect sites of type 1 with concentration 1c and 2p
those of type 2 with concentration 2c (not necessarily
equal to 1c ), the total impurity concentration being
1 2 = 1c c c+ . Then the perturbation operator in terms of
local Fermi operators appears as
† †
1 21 2
1 2
= ,H V a a b bp pp p
p p
⎛ ⎞
⎜ ⎟+′
⎜ ⎟⎝ ⎠
∑ ∑
and, passing to the spinor representation, Eq. (5), it gets
the form of scattering operator:
( )†
, ,
ˆ= e .
i j
j
j j
VH
N
ψ τ ψ
k k p
kk
k k p
−′
′
′′
′ ∑ ∑ (11)
The equation of motion for the perturbed GF matrix,
Eq. (7) with Hamiltonian H H+ ′ , takes the form:
( )
, , ,
,
ˆ ˆˆ ˆ= e ,
i j
j
j j
VG g G
N
δ τ
k k p
k k k k k k k
k p
−′′
′ ′ ′′ ′
′′
⎡ ⎤
⎢ ⎥+⎢ ⎥
⎢ ⎥⎣ ⎦
∑ ∑ (12)
the sum in the brackets presenting a set of “scattered” GF’s
by impurity perturbations. Then certain iteration routines
on the scattered GF's ,Ĝk k′′ ′ lead to solutions for the ini-
tial (or “master”) GF as certain expansions in groups of
impurity scatterers, called group expansions (GE’s), spe-
cific for band-like or localized states in the disordered sys-
tem [22].
Namely, the band-like states are better described by the
so-called fully renormalized GE, resulting from consecu-
tive iterations of Eq. (12) on each scattered GF except
those already present in the previous iterations. Collecting
the same GF's from all the iterations forms some coeffi-
cients beside them as infinite series of respective scattering
amplitudes by all multiple scatterings. The subsequent al-
gebraic solution of resulting chained equations either for
the master GF and its posterior scattered ones can be
closed so that the master GF gets expanded in terms of
multiple scatterings on a single impurity, on a pair of im-
purities, etc. The full renormalization means that each ele-
ment of such expansion also presents as a similar expan-
sion (but subdued to a particular scattering history of the
master GF), and so on for its own elements. Such hierar-
chical structure is assured by the fact that one can uniquely
trace the scattering history for each element at each level
of the expansion. The most important momentum-diagonal
GF is written as:
( ) 11
,
ˆ ˆ ˆˆ= ,G G g Σk k k k
−−≡ − (13)
that is the inverse GF involves the self-energy matrix in
the GE form:
·
, 0
ˆ ˆ ˆˆˆ ˆ ˆ ˆ ˆ= 1 (e )i
j j j j j j
j j
T c c A A AΣ τ τ τ τk n
n n n
n
−
−′ ′ ′
≠′
⎡
⎢ + + ×
⎢⎣
∑ ∑
( ) 1ˆ ˆˆ ˆ1 .j jA Aτ τn n …
−
−′
⎤
× − + ⎥
⎥⎦
(14)
Here 1ˆˆ = (1 )T V GV −− is the common T-matrix, the pro-
jectors ˆ jτ refer to the types of impurity scatterers and
the matrix of effective interaction:
·1ˆ ˆ ˆ= e .' iA G T
N
k n
n k
k
∑ (15)
in a sequence ˆˆ ˆj jAτ τn ′ describes electron scatterings on
a pair of impurities of types j and j ′ at separation n.
V.M. Loktev and Yu.G. Pogorelov
996 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 8
The primed sum in Eq. (15) is subdued to the history be-
fore this scattering, that is restricted only to momenta non-
coincident with its preceding ones in products like
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆj j j jA A A Aτ τ τ τn n n n…− −′ ′ . Thus the sum next to the uni-
ty in the brackets of Eq. (14) defines the contribution to
GE by impurity pairs and the dropped terms are due to
impurity triples, etc. [22].
Other iteration routine, applied to all the scattered GF’s
with no exception to collect all the terms due to the free
term ,ĝ δk k k ′ at any iteration of Eq. (13), leads to the non-
renormalized representation:
ˆ ˆ ˆ ˆ ˆ= ,G g g gσk k k k+ (16)
more adequate for localized states. The respective GE for
the self-energy σ̂ appears in the GF itself, instead of its
inverse in Eq. (13), and has a similar structure to Eq. (14)
but with the non-renormalized T-matrix ˆ ˆ= / (1 )t V gV− ,
1ˆ ˆ=g N gkk
− ∑ , and pair scattering matrices ˆˆ ˆ=a g tn n
having no restrictions in quasimomentum sums. This re-
presentation is easier for practical evaluation of impurity
effects and can also serve as an approximation for the
(more involved) fully renormalized representation. Gener-
ally, each representation only makes sense until the corres-
ponding GE is convergent that is qualitatively checked
below by comparison of the contributions by T-matrix and
pair terms.
In the low-energy approximation, Eq. (4), one has
( ) 0Fγ εk kk ≈∑ for any regular function F and can neg-
lect the nondiagonal elements in the above local GF matrix
ĝ to present it as a scalar:
2
2 2
ˆ ( ) = ln 1 .Wg g
W
ε
ε
ε
⎛ ⎞
≈ − −⎜ ⎟
⎝ ⎠
(17)
Then the T-matrix also turns a scalar: ˆ / ( )t V D ε≈ , whose
denominator ( ) = 1 ( )D Vgε ε− would define a single-
impurity resonance at the energy res=ε ε such that
resRe ( ) = 0.D ε But, this requires a strong enough per-
turbation: cr| | >V V , and, accordingly to Eq. (17),
2
cr = 1 e / 2 1.45V W W+ ≈ amounts up to cr 10.4V ≈ eV
(with a common choice of 2.5t ≈ eV). Of course, this is
not an easy condition for real impurities like N or B substi-
tutes for C in the graphene plane, as can be seen from
rough estimates of related perturbation parameters, by
the difference of corresponding ionization potentials:
B C B 3V I I≈ − ≈ eV and N C N 3.2V I I≈ − ≈ − eV. Then,
in a realistic situation when cr| |<V V , impurity effects
would mainly affect the conical point in the energy spec-
trum.
The choice between the above two GE forms for given
k is done through the known Ioffe–Regel–Mott (IRM) cri-
terion for band-like states 1k [26,21], which can be
written in more detail as:
,ε Γk k kk ⋅∇ (18)
in terms of the renormalized dispersion law
= Re ( )ε ε Σ εk k k− and its broadening = Im ( )Γ Σ εk k .
Since Re ( )Σ ε is almost constant at low energies, for
k K≈ , the left hand side of Eq. (18) is close to
F K Wv ∼ here, while the right hand side is much small-
er, as shown below. Thus the IRM criterion, in the form
accepted by Eq. (18), is granted and all the states are band-
like in this k-range.
Then, using the renormalized form, Eq. (14), approx-
imated to its first term, we obtain the self-consistency equ-
ation:
( )( ) = ,sc scG gε ε Σ− (19)
with the related self-energy ( ) = / [1 ( )]sc sccV VGΣ ε ε− en-
tering the argument of the g-function, Eq. (17). It admits a
simple solution, revealing the most evident impurity effect as
the shift of the nodal point by cV : ( ) ( )scG gε ε≈ for all
= cVε ε − , except for an exponentially narrow vicinity of
the shifted nodal point, 2 2 2| | = (2 / ) exp ( /2 )cV W W cVε δ −
[27], where it behaves as:
( )
2
2
1( ) .
2sc
WG i
cV
ε ε δ
⎛ ⎞≈ − −⎜ ⎟⎝ ⎠
(20)
By this solution, DOS at the shifted nodal point would
drop to an exponentially low minimum value,
2
,min 2 2exp .
2
sc
W W
cV cV
ρ
π
⎛ ⎞
≈ −⎜ ⎟
⎝ ⎠
However some stronger contributions can come from the
next GE terms as shown below.
Since the T-matrix estimated with use of Eq. (17) is
scalar: ( ) / [1 ( )]T V Vgε ε≈ − , the matrix structure of next
GE terms is essentially defined by the structure of interac-
tion matrix:
( )ˆˆ ˆ= e cos ( · ).i zyA T g fτ θτ nn n n K n+
Here e = ( )/i
x yn in nθn + and the two propagator functions
are approximated at low energies, | | Wε , and not too
short distances, ,n a as:
(2)
02 2
3, .n n
F
i n t ag H f
nW W
πε ε π⎛ ⎞
≈ ≈⎜ ⎟⎝ ⎠v
(21)
involving the zeroth order Hankel function (2)
0H [28].
Calculation of the pair GE term in Eq. (14) is simplified
by the two facts:
i) since the fast oscillating factor has zero average:
1cos ( ) = cos ( ) = 0,
nN
K n K n⋅ ⋅∑
the odd part of the pair term can be neglected beside
the even part,
Impurity and vacancy effects in graphene
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 8 997
ii) the even part is defined by the diagonal form of
the matrix product:
2 2
, ,2 (1 )ˆ ˆ ˆˆ ˆ ˆ ˆ= ,
2
j j j jjj g f
A jA j A j jT
δ δn n
n n n
′ ′′
−
+ −
≈′ (22)
(using the average: 2 ( ) = 1/ 2cos K n⋅ ). The related cor-
rection to Σ̂ is a scalar 2 3 2
2 / ,c V WΣ ∼ that is small
enough beside the T-matrix term cT cV≈ to assure con-
vergence of the renormalized GE. The contribution to 2 ,Σ
non-vanishing at | | 0ε → , is due to interactions between
impurities of different types ( j j≠ ′ ) realized by the func-
tions nf and amounts to:
22 2 3
2 2 ln 1.c V W
VW
π
Σ
π
⎛ ⎞≈ −⎜ ⎟⎝ ⎠ (23)
Note that Im 2Σ by Eq. (23) is non-zero only if the pertur-
bation parameter is not too weak: | | /V W π (though this
restriction is easier than by the above cr .V Otherwise, for
2/ | | /W V Wπ π , the maximum GE correction to ˆIm Σ
would come from impurity quadruples: 4 5 4
4Im / ,c V WΣ ∼
and so on. Anyway, such power low (in c ) contributions
would dominate over the exponentially small ,minscρ in
the minimum DOS under impurity effect. Also, smallness
of these contributions assure the IRM criterion, Eq. (18),
and thus the band-like type of states through entire low
energy range.
4. Vacancy perturbations
The defects most commonly discussed in graphene are
vacancies and the following analysis concentrates on their
effects in this material. Since vacancy perturbation has no
explicit energy parameter, its modeling finds a certain
freedom of choice and the most usual approach in literature
uses the model for vacancy by the before considered local
perturbation, Eq. (11), in the limit of V → ∞ [15]. Such
a strong assumption may look somewhat artificial physi-
cally, so it can be of interest to check its validity by a com-
parison with a more realistic perturbation model. To this
end, we employ here the mechanism of simple exclusion of
the hopping processes from and to the vacancy sites in
the Hamiltonian, Eq. (1), this perturbation being expressed
in terms of the local operators as:
† †
1 21 2, ,1 2
= ( h.c.) ( h.c.) .H t a b b ap δ p δp p
p δ p δ
+ −
⎡ ⎤
⎢ ⎥− + + +′
⎢ ⎥⎣ ⎦
∑ ∑
(24)
Then, using Eq. (5), the perturbation is rewritten in the
spinor form, analogous to Eq. (11):
( ) ( )†
,
, ,
1 ˆ= e ,
2
i jj
j j
H V
N
ψ ψ
k k p
kk k k
k k p
−′
′′
′
′ ∑ ∑ (25)
with the scattering matrices (1)
,
ˆ ˆ ˆ= ( )γ τ γ τV t ∗
− +′′ − +k kk k and
(2)
,
ˆ ˆ ˆ= ( )V t γ τ γ τk kk k
∗
+ −′′ − + . Also we apply below this model
to study the effects of inter-vacancy interactions using
the respective GE’s.
Table 1. Basic elements and their multiplication rules for the
self-energy problem in graphene with vacancies
But the important difference of the vacancy non-local
perturbation, Eq. (25), from the local perturbation,
Eq. (11), is the momentum k dependence of the V̂ ma-
trices. Their treatment is facilitated by the fact that the con-
stituent matrices τ̂± , together with the above defined 1,2τ̂ ,
form a simple algebra with the multiplication rules given
by Table 1. Then, for instance, the fully renormalized GF
matrix in this case is similar to Eq. (13):
1 1ˆ ˆˆ= ( ) .G g Σk k k
− −− (26)
Here the self-energy matrix (1) (2)ˆ ˆ ˆ=Σ Σ Σk k k+ include the
particular terms
( ) ( ) · ( )
0
ˆˆˆ = 1 (e ) ,j j i j
j jc T c AΣ k n
nk k
n
… …−
≠
⎡ ⎤
+ + +⎢ ⎥
⎢ ⎥⎣ ⎦
∑
for jth type vacancies and the interaction matrices
( )( ) 1 ·ˆ ˆ ˆ= eΣ jj ' iA N G T− ′
′ ′
k n
n kk k . The formal structure of the
GE pair term in Eq. (26) is analogous to that in Eq. (14) for
impurities but, in contrast to that case, here the same sym-
metry condition ( ) 0Fγ εΣk k k ≈ makes only effective
interactions to be those between vacancies of the same type
( =j j ′ ).
The above T-matrices can be found as momentum-
diagonal values, ( ) ( )
,
ˆ ˆj jT Tk k k≡ , of more general forms:
( ) ( ) ( )
, , ,11=0 , ,1
ˆ ˆˆ ˆ ˆ=j j j
n n
n
T N V G G Vn
k kk k k k k k
n k k…
…
∞
−
′ ′∑ ∑ (27)
From this definition, a matrix equation is readily obtained:
( ) ( ) ( ) ( )
, , , ,11 1
1
1 ˆˆ ˆ ˆ ˆ= .j j j jT V V G T
N kk k k k k k k k
k
′ ′ ′+ ∑ (28)
and, using the τ-algebra by Table 1, solved in the form:
( )
2
( ) ( )
, ,
1ˆ ˆ ˆ ˆ= 1 .
( )
j j
j j
t
T V
G
γ γ
τ ε τ
ε ε
k k
k k k k
∗
′
′ ′
⎛ ⎞
+ + − −⎜ ⎟⎝ ⎠
(29)
Under the natural assumption that vacancies are equally
present in two sublattices: 1 2= = / 2c c c , and taking ac-
count of identities: 1 2ˆ ˆ = 1τ τ+ and ˆ ˆ ˆ= xτ τ τ+ −+ , the full
T-matrix results as:
2 21ˆ ˆ= Re ,
( ) xT t
G
ε ε
γ τ
ε ε
k
k k
+
− + − (30)
τ̂+ τ̂− 1τ̂ 2τ̂
τ̂+ 0 1τ̂ 0 τ̂+
τ̂− 2τ̂ 0 τ̂− 0
1τ̂ τ̂+ 0 1τ̂ 0
2τ̂ 0 τ̂− 0 2τ̂
V.M. Loktev and Yu.G. Pogorelov
998 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 8
where the first term describes a characteristic zero-energy
resonance by vacancy and just presents the V → ∞ limit
of the above momentum-independent T̂ for impurities
while the momentum-dependent terms are non-resonant
and only weakly renormalize the dispersion law at higher
energies.
It is also natural to use the T-matrix, Eq. (30), “trun-
cated” to its resonant part, in a self-consistent equation for
local GF, as a V → ∞ limit of Eq. (19):
= ,sc
sc
cG g
G
ε
⎛ ⎞
+⎜ ⎟⎝ ⎠
whose solution would, in particular, tend to a finite imagi-
nary limit: ( 0) ln (1/ ) / .scG i c c Wε→ ∼ However ex-
tending of this solution towards zero energy in the case of
vacancies is essentially restricted by the failure of the IRM
criterion in a certain vicinity of the (here unshifted) nodal
point: 0| |ε Δ . Approximating Eq. (30) to its first term
with ( ) ( )scG Gε ε≈ , such vicinity is estimated from
Eq. (18) as:
0 ,
ln(1 / )
c W
c
Δ ∼ (31)
to define the Mott's mobility edges [21] between band-like
and localized states in this disordered system. Also, the
limit of band-like states and hence of the self-consistent
treatment is manifested here by failing convergence of the
renormalized GE. The latter is checked by comparing the
pair term similar to Eq. (23), but with V replaced by
1( )scG ε− , and the T-matrix term written respectively as
/ ( )scc G ε− .
Otherwise, calculation of DOS within validity of the
renormalized GF, Eq. (26), in the approximation of full
T-matrix, Eq. (30), but taking ( ) ( )G gε ε≈ , leads to the
analytic result:
2
2
1 2( ) = Im 1 ln 1
2
T
cW
R RcW ε ε
ε
ρ ε
ε επ
⎧ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎪ ⎢ ⎥− + − +⎨⎜ ⎟ ⎜ ⎟+⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎣ ⎦⎩
2
1 21 ln 1 ,cW
R Rε εε ε
⎫⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎪⎢ ⎥+ − − ⎬⎜ ⎟ ⎜ ⎟−⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎪⎣ ⎦⎭
(32)
where 2= 1 4 4 / [ ( )]R c c gε ε ε+ + . This function within its
validity is close to the “truncated” T-matrix result:
21 ( )( ) = Im ln 1 ,
( ) ( )T
c Wg
g g c
ε
ρ ε ε
π ε ε ε′
⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎢ ⎥+ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ + ⎠⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
(33)
as shown in Fig. 2, and both them deviate down from the
non-perturbed linear 0 ( )ρ ε . But at approaching the mo-
bility edges, 0| |ε Δ∼ , they also notably deviate from the
self-consistency behavior (which should be in principle
more reliable until the renormalized GE converges).
At least, within the range of localized states, 0| |ε Δ ,
the choice should be done for the non-renormalized GF, in
analogy with Eq. (16). Its contribution in the simplest T-
matrix approximation: ˆ / ( )c gσ ε≈ , used in Eq. (16), leads
to the expression for DOS by localized states:
loc 2( ) ,
/ln
c
W
ρ ε
ε ε
≈
that intercepts the before considered scρ at such low ener-
gies. Nevertheless its validity is also limited by the GE
convergence requirement, to some more narrow vicinity of
zero,
1 ,
ln (1 / )
c W
c
Δ ∼
estimated with use of similar expressions as for the above
case of impurities, Eq. (31), but replacing ( )scG ε by ( ),g ε
This vicinity should measure the area of concentration
broadening around the vacancy resonance peak.
A closer insight on the localization process can be ob-
tained, beginning from a single vacancy on site p (say, of
type 1) and constructing a quasiparticle state with energy
ε : (1) † (2) †| = ( ) | 0a bεψ ψ ψn n n nn〉 + 〉∑ , whose amplitudes on
type j sites ( )jψn are found from the Schroedinger equa-
tion, ( ) | = 0'H H εε ψ+ − 〉 . The solutions having central
symmetry with respect to the impurity site p are of
the same form as the above referred propagator functions,
Eq. (21), dominated at 0ε→ by the scale-free nf . The
resulting marginal behavior of zero-energy states by single
defects [15], though does not correspond to their true loca-
lization, facilitates localization of states on defect pairs
(and their higher numbers). For instance, a pair of vacan-
cies at long enough separation r a produces a doublet
Fig. 2. Results of different approximations for DOS in graphene
with vacancies at their concentration = 0.005c in comparison
with the non-perturbed linear law 0( )ρ ε . The thin dashed lines
indicate the estimated limits of validity of renormalized GE ( 0,Δ
the Mott’s mobility edge) and of non-renormalized GE ( 1Δ ).
ρ,
1
/
u
ni
ts
W
ε, unitsW
c = 0.005
ρloc
ρsc
ρT
ρ0
Impurity and vacancy effects in graphene
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 8 999
of localized levels at finite energies rε± where
/ [ ln ( / )],εr Wa r r a∼ both having characteristic radius
pair ln ( / ) ,r r r a r∼ thus yet much bigger than the sepa-
ration between these vacancies. This situation can be seen
in the spirit of general theory of of quasiparticle spectra in
disordered systems [20,22], as presence of a whole series
of defect pairs with separations ranging from a∼ to
/r a c∼ ∼ (the average distance between single defects,
a reasonable biggest size of an isolated pair). Accordingly,
the biggest pair size will define the closest distance, 1,Δ
from a pair level to zero energy, and the effectively smal-
lest size (but not too rare in the system) will define the
mobility edge 0Δ .
Existence of a finite mobility gap 0 0[ , ]Δ Δ− around
zero energy should produce important practical effects on
electron dynamics and transport in the disordered graphene
system. Though vacancies themselves should not perturb
the initial electron-hole symmetry and zero location of the
Fermi energy Fε , it can be modified by some other me-
chanisms like adsorbed impurities or chemical bias. Then,
with its crossing the mobility edge by a related control,
a transition from insulating to metallic state can occur in
this system at zero temperature and a very high tempera-
ture sensitivity of transport in the vicinity of such transition
can be expected.
It is of interest to compare this type of spectrum restruc-
turing with the known examples for non-relativistic spec-
tra. Thus, the low-frequency acoustic resonance
res /D DM Mω ω ω′∼ by impurities with mass
M M′ in a crystal with atomic mass M and Debye
frequency Dω [32] gives rise to splitting of phononic
spectrum near resω with opening of a quasi-gap (seen as
a dip in DOS) of width 2
res/Dcω ω∼ , at surpassing
the characteristic impurity concentration 3
0 res( / )ω ωDc ∼
(the cubic law) [33]. This dip corresponds to repulsion of
band levels from the impurity resonance (in contrast to the
hump in DOS and attraction of band levels to the reson-
ance expected in graphene). Another example is the donor
level locε near a parabolic conductance band [31], which
rapidly expands and merges with this band when the donor
concentration exceeds loc loc / mc Eε∼ (linear law), in the
2D case, or 3/2
loc loc( / ) ,ε mc E∼ in the 3D case. The 3/2
law also defines the characteristic concentration
3/2
0 res( / )c Jω∼ of weakly coupled ( J J′ ) impurity
spins in a Heisenberg ferromagnet when the magnon spec-
trum splits near the resonance frequency res ,ω J J′∼
and the cubic law 3
0 res( / )c Jω∼ defines such effect in
an aniferromagnet [22]. Hence the case of defects in gra-
phene, where the critical concentration appears to be either
impossibly high: 0 1c ∼ (for substitutional impurities), or
completely vanishing: 0 0c → (for vacancies), differs
from all those, even at seemingly identical linear disper-
sion (as for phonons and antiferromagnons).
5. Conclusions
In conclusion, the effects of local perturbation by vari-
ous types of impurities, including vacancies, in single layer
graphene were studied by means of Green function tech-
niques. Namely, restructuring of electronic quasiparticle
spectrum near its relativistic Dirac-type nodal points was
analyzed through application of various types of group
expansions for a Green function itself or for its inverse
(self-energy) in groups of interacting impurity clusters. A
special study for substitutional impurities with realistic
(not too strong compared to the non-perturbed bandwidth)
perturbation potential permitted to conclude on persistence
of band-like states in the whole nodal area, though the ac-
count of non-trivial cluster terms by the group expansion
provides essential corrections (power-law in impurity con-
centration c ) to the finite density of nodal states, besides
an exponentially small (∼ e 1/c− ) value by the common T-
matrix term, and should so essentially modify the related
low-energy physics. Similar group expansions in the case
of vacancies, indicated the conditions for quasiparticle
localization near nodal points and location of the Mott's
mobility edges around them for any small vacancy concen-
tration c . A possibility for specific metal/insulator transi-
tion in a graphene layer at presence of vacancies, under
modulation of the Fermi energy by some additional me-
chanisms, is indicated. The developed group expansion
approach can be useful for other types of Green functions
in the systems with nodal points in quasiparticle spectrum
under disorder effects.
Acknowledgements
Y.P. thanks V.A. Miransky, J.P. Carbotte and S.G. Sha-
rapov for very useful discussions of disorder effects in gra-
phene, and kind hospitality by the Department of Applied
Mathematics of University of Western Ontario during his
sabbatical stay there, when this work was initiated. The sup-
port from Portuguese Fundação para a Ciência e a Tecno-
logia is gratefully acknowledged. V.L. thanks the SCOPES
grant and also the Special Program of fundamental re-
searches by Physics and Astronomy Department of NAS of
Ukraine for the partial support.
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