On the density of states of graphene in the nearest-neighbor approximation
We propose an alternative analytical expression for the density of states of a clean graphene in the nearestneighbor approximation. In contrast to the previously known expression, it can be written as a single formula valid for the whole energy range. The correspondence with the previously known ex...
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Інститут фізики конденсованих систем НАН України
2017
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | On the density of states of graphene in the nearest-neighbor approximation / V.O. Ananyev, M.I. Ovchynnikov // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43705: 1–4. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1570312019-06-20T01:27:38Z On the density of states of graphene in the nearest-neighbor approximation Ananyev, V.O. Ovchynnikov, M.I. We propose an alternative analytical expression for the density of states of a clean graphene in the nearestneighbor approximation. In contrast to the previously known expression, it can be written as a single formula valid for the whole energy range. The correspondence with the previously known expression is shown and the limiting cases are analyzed. Ми пропонуємо альтернативний аналiтичний вираз для густини електронних станiв у чистому графенi в наближеннi найближчих сусiдiв. На противагу вже вiдомим виразам, вiн представляє собою єдину формулу, справедливу на всьому iнтервалi енергiй. Також було перевiрено вiдповiднiсть уже вiдомим виразам i дослiджено граничнi випадки. 2017 Article On the density of states of graphene in the nearest-neighbor approximation / V.O. Ananyev, M.I. Ovchynnikov // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43705: 1–4. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 71.20.Tx, 73.22.Pr DOI:10.5488/CMP.20.43705 arXiv:1705.08120 http://dspace.nbuv.gov.ua/handle/123456789/157031 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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We propose an alternative analytical expression for the density of states of a clean graphene in the nearestneighbor approximation. In contrast to the previously known expression, it can be written as a single formula
valid for the whole energy range. The correspondence with the previously known expression is shown and the
limiting cases are analyzed. |
| format |
Article |
| author |
Ananyev, V.O. Ovchynnikov, M.I. |
| spellingShingle |
Ananyev, V.O. Ovchynnikov, M.I. On the density of states of graphene in the nearest-neighbor approximation Condensed Matter Physics |
| author_facet |
Ananyev, V.O. Ovchynnikov, M.I. |
| author_sort |
Ananyev, V.O. |
| title |
On the density of states of graphene in the nearest-neighbor approximation |
| title_short |
On the density of states of graphene in the nearest-neighbor approximation |
| title_full |
On the density of states of graphene in the nearest-neighbor approximation |
| title_fullStr |
On the density of states of graphene in the nearest-neighbor approximation |
| title_full_unstemmed |
On the density of states of graphene in the nearest-neighbor approximation |
| title_sort |
on the density of states of graphene in the nearest-neighbor approximation |
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Інститут фізики конденсованих систем НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/157031 |
| citation_txt |
On the density of states of graphene in the nearest-neighbor approximation / V.O. Ananyev, M.I. Ovchynnikov // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43705: 1–4. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ananyevvo onthedensityofstatesofgrapheneinthenearestneighborapproximation AT ovchynnikovmi onthedensityofstatesofgrapheneinthenearestneighborapproximation |
| first_indexed |
2025-07-14T09:22:26Z |
| last_indexed |
2025-07-14T09:22:26Z |
| _version_ |
1837613655231823872 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 4, 43705: 1–4
DOI: 10.5488/CMP.20.43705
http://www.icmp.lviv.ua/journal
On the density of states of graphene in the
nearest-neighbor approximation
V.O. Ananyev, M.I. Ovchynnikov
Faculty of Physics, Taras Shevchenko National Kyiv University,
6 Academician Glushkov Ave., 03680 Kyiv, Ukraine
Received June 8, 2017, in final form July 21, 2017
We propose an alternative analytical expression for the density of states of a clean graphene in the nearest-
neighbor approximation. In contrast to the previously known expression, it can be written as a single formula
valid for the whole energy range. The correspondence with the previously known expression is shown and the
limiting cases are analyzed.
Key words: graphene, density of states, Van Hove singularity
PACS: 71.20.Tx, 73.22.Pr
1. Introduction
A theoretical background of the electronic applications of graphene is based on the knowledge
of its band structure. The vast majority of theoretical approaches exploit the fact that the low-energy
quasiparticle excitations in graphene have a linear Dirac-like spectrum up to the energies of the order of
0.5 eV. Widely used simple analytical expressions for the density of states (DOS), optical conductivity,
etc., ultimately originate from this spectrum [1–3].
There are very few analytical results describing electronic properties of graphene beyond the con-
tinuum linear approximation. Among them there is an expression for the DOS provided by Hobson and
Nierenberg in 1953 [4] without derivation. It appears in various forms in the modern literature (see e.g.,
review [2] and reference [5]). In particular, in [3] the DOS per unit cell and one spin component reads
D(E) =
|ε |
tπ2
1√
F( |ε |)
K
(
|ε |
F( |ε |)
)
, 0 6 |ε | 6 1,
|ε |
tπ2
1√
|ε |
K
(
F( |ε |)
|ε |
)
, 1 6 |ε | 6 3,
(1.1)
where the energy ε = E/t is measured in units of the nearest-neighbor hopping energy t ≈ 3 eV, the
function g(x) is given by
F(x) =
(1 + x)2
4
−
(x2 − 1)2
16
=
1
16
(x + 1)3(3 − x), (1.2)
and K(m) is an elliptic integral of the first kind,
K(m) =
1∫
0
dx
[
(1 − x2)(1 − mx2)
]−1/2
. (1.3)
We stress that the definition (1.3) corresponds to the notations of Wolfram Mathematica [6]. The
definitions of the complete elliptic integrals, for example, in [7–9] employ the parameter k2 as argument
in place of the modulus m, viz. K(k) = K(k2). The purpose of the present brief report is to propose a
more compact form of the DOS.
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published articleŠs title, journal citation, and DOI.
43705-1
https://doi.org/10.5488/CMP.20.43705
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
V.O. Ananyev, M.I. Ovchynnikov
2. Derivation
For completeness, we recapitulate the main steps of the derivation that lead both to the new and
old expressions for the DOS. We begin with the tight-binding dispersion law of graphene written in the
nearest-neighbor approximation [10]
ε(k) = ±t
√
1 + 4 cos2 kxa
2
+ 4 cos
kxa
2
cos
√
3kya
2
, (2.1)
where k = (kx, ky) is the wave-vector and a =
√
3aCC is the lattice constant with aCC being the distance
between the neighboring carbon atoms.
The DOS can be calculated as a trace of the imaginary part of the corresponding Green’s function [11]
D(E) = −
2E
π
Im[g(E)], (2.2)
with
g(E) = S
∫
BZ
d2k
(2π)2
1
(E + i0)2 − ε2(k)
. (2.3)
The factor of 2 in equation (2.2) originates from the trace over the sublattice degree of freedom and
S =
√
3a2/2 in equation (2.3) is the area of a unit cell. The integration is done over the Brillouin
zone (our notations correspond to [1]). Introducing dimensionless variables and doubling the domain of
integration to make it rectangular, −2π/a 6 kx 6 2π/a and −2π/(a
√
3) 6 ky 6 2π/(a
√
3), one obtains
g(E) =
1
8π2t2
π∫
−π
dx
π∫
−π
dy
1
τ − cos 2x − 2 cos x cos y
(2.4)
with τ = (ε + i0)2/2 − 3/2. Replacing s = tan y/2 we integrate over s and obtain
g(E) =
1
4πt2
π∫
−π
dx√(
τ − cos 2x
)2
− 4 cos2 x
. (2.5)
Equation (2.5) can be expressed in terms of an elliptic integral of the first kind (see equation (3.147.3)
in [7])
g(E) =
2
πt2
√
(ε − 1)3(ε + 3)
K
(√
16ε
(ε − 1)3(ε + 3)
)
, (2.6)
where for the brevity of notations we omitted +i0 in the argument. The argument of the elliptic function
in equation (2.6) is imaginary for 0 6 ε < 1, but it is real and larger than 1 for 1 < ε 6 3 . In the former
case, 0 6 ε < 1, using the imaginary modulus transformation [9]
K(ik) = 1
√
k2 + 1
K
(√
k2
k2 + 1
)
(2.7)
we arrive at the fist line of equation (1.1). In the latter case, we use the relationship (see equation (8.128)
in [7] and [9])
K(k) = 1
k
[
K
(
1
k
)
− iK
(√
1 −
1
k2
)]
, k > 1, (2.8)
where the sign in front of the imaginary term is chosen in accordance to imaginary shift ε + i0. Then,
the last term of equation (2.8) leads us to the second line of equation (1.1).
43705-2
On the DOS of graphene
0.2
0.4
0.6
0.8
1.0
1.2
k
(|
ϵ|
)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
|ϵ|
Figure 1. (Color online) On the plot there is shown a dependence of the argument k(|ε |) = 4
√
|ε |F( |ε |)
[
√
|ε |+
√
F( |ε |)]2
of the elliptic integral in the DOS expression (2.10) on the modulus |ε | = |E |
t (prepared with
SciDraw [13]).
Using Landen’s transformation (see equation (8.126.3) in [7, 9] and [12])
θ(q − k)
q
K
(
k
q
)
+
θ(k − q)
k
K
( q
k
)
=
1
q + k
K
(
2
√
qk
q + k
)
, (2.9)
the DOS (1.1) can be represented in one line expression. Returning back to the Wolfram’s definition of
the elliptic integral (1.3), we arrive at the final result of this report:
D(ε) =
1
tπ2
|ε |θ(3 − |ε |)√
|ε | +
√
F(|ε |)
K
(
4
√
|ε |F(|ε |)[√
|ε | +
√
F(|ε |)
]2
)
. (2.10)
Here, the argument of the elliptic integral is 6 1 (see figure 1).
The unified result (2.10) can be made clearer in the following way. By using (2.6) and (2.7), the
expression (2.2) for the DOS can be converted to the form
D(ε) =
4|ε |/(tπ2)√
(|ε | + 1)3(3 − |ε |)
Re K
(√
16|ε |
(|ε | + 1)3(3 − |ε |)
)
, (2.11)
valid for 0 < |ε | < 3. The result (2.10) then straightforwardly follows from analytical properties of the
Green function; this is due to the identity
Re K(z) = 1
1 + z
K
(
2
√
z
z + 1
)
, 0 < z < ∞ (2.12)
following from (2.9).
3. Conclusions
To conclude, we reproduce limiting cases of the DOS. The low-energy expansion of the DOS (2.10) is
D(ε) =
1
t
[
2|ε |
√
3π
+
2|ε |3
3
√
3π
+O(|ε |5)
]
, (3.1)
where the first term originates from the contribution of K(0) = π/2. One can see that the second term
of the expansion is 100 times smaller than the first one for |E | . 0.17t ∼ 0.5 eV. Assuming that
the Fermi velocity is vF =
√
3ta/(2~), we reproduce the commonly used DOS per unit area and spin
43705-3
V.O. Ananyev, M.I. Ovchynnikov
D(E) = |E |/(π~2v2
F). Finally, near ε = 1, the argument of the elliptic integral in equation (2.10) is
1 − (ε − 1)6/256. Assuming that K(z) = −1/2 ln(1 − z)/16 for z → 1 [6], we reproduce the asymptotic
of the DOS near the van Hove singularity [4] D(E) = −3/(2π2t) ln(|1 − ε |/4).
Acknowledgements
We thank S.G. Sharapov for suggesting to reproduce the result of [4] in his lecture course on graphene
and for providing a great help in writing the article. Also, we would like to thank O.O. Sobol for a helpful
and productive discussion.
References
1. Gusynin V.P., Sharapov S.G., Carbotte J.P., Int. J. Mod. Phys. B, 2007, 21, 4611,
doi:10.1142/S0217979207038022.
2. Castro Neto A.H., Guinea F., Peres N.M.R., Novoselov K.S., Geim A.K., Rev. Mod. Phys., 2009, 81, 109,
doi:10.1103/RevModPhys.81.109.
3. Katsnelson M.I., Graphene: Carbon in Two Dimensions, Cambridge University Press, Cambridge, 2012.
4. Hobson J.P., Nierenberg W.A., Phys. Rev., 1953, 89, 662, doi:10.1103/PhysRev.89.662.
5. Rammal R., J. Phys., 1985, 46, 1345, doi:10.1051/jphys:019850046080134500.
6. URL http://functions.wolfram.com/EllipticIntegrals/EllipticK/.
7. Gradshteyn I.S., Ryzhik I.M., Table of Integrals, Series and Products, Academic Press, New York, 1980.
8. Bateman H., Erdélyi A., Higher Transcendental Functions, McGraw-Hill, New York, 1953.
9. Byrd P., Friedman M., Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, 2013.
10. Wallace P.R., Phys. Rev., 1947, 71, 622, doi:10.1103/PhysRev.71.622.
11. Horiguchi T., J. Math. Phys., 1972, 13, 1411, doi:10.1063/1.1666155.
12. Gamayun O.V., Gorbar E.V., Gusynin V.P., Phys. Rev. B, 2009, 80, 165429, doi:10.1103/PhysRevB.80.165429.
13. Caprio M.A., Comput. Phys. Commun., 2005, 171, 107, doi:10.1016/j.cpc.2005.04.010.
Про вираз для густини станiв у графенi
В.О. Ананьєв,М.Ю. Овчиннiков
Фiзичний факультет, Київський нацiональний унiверситет iменi Тараса Шевченка,
просп. Академiка Глушкова, 6, 03680 Київ, Україна
Ми пропонуємо альтернативний аналiтичний вираз для густини електронних станiв у чистому графе-
нi в наближеннi найближчих сусiдiв. На противагу вже вiдомим виразам, вiн представляє собою єдину
формулу, справедливу на всьому iнтервалi енергiй. Також було перевiрено вiдповiднiсть уже вiдомим
виразам i дослiджено граничнi випадки.
Ключовi слова: графен, густина станiв, сингулярнiсть Ван Гове
43705-4
https://doi.org/10.1142/S0217979207038022
https://doi.org/10.1103/RevModPhys.81.109
https://doi.org/10.1103/PhysRev.89.662
https://doi.org/10.1051/jphys:019850046080134500
http://functions.wolfram.com/EllipticIntegrals/EllipticK/
https://doi.org/10.1103/PhysRev.71.622
https://doi.org/10.1063/1.1666155
https://doi.org/10.1103/PhysRevB.80.165429
https://doi.org/10.1016/j.cpc.2005.04.010
Introduction
Derivation
Conclusions
|