Local vibrational density of states in disordered graphene
Local vibrational density of states for disordered graphene has been calculated via Green’s functions method. Disordered material has been modeled with Bethe lattice. Density of states does not include particularities specific for ideal graphene.
Збережено в:
| Дата: | 2016 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2016
|
| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/121607 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Local vibrational density of states in disordered graphene / D.L. Kardashev, K.D. Kardashev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 3. — С. 315-317. — Бібліогр.: 11 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-121607 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1216072017-06-15T03:05:54Z Local vibrational density of states in disordered graphene Kardashev, D.L. Kardashev, K.D. Local vibrational density of states for disordered graphene has been calculated via Green’s functions method. Disordered material has been modeled with Bethe lattice. Density of states does not include particularities specific for ideal graphene. 2016 Article Local vibrational density of states in disordered graphene / D.L. Kardashev, K.D. Kardashev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 3. — С. 315-317. — Бібліогр.: 11 назв. — англ. 1560-8034 DOI: 10.15407/spqeo19.03.315 PACS 63.22.Rc t http://dspace.nbuv.gov.ua/handle/123456789/121607 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Local vibrational density of states for disordered graphene has been calculated via Green’s functions method. Disordered material has been modeled with Bethe lattice. Density of states does not include particularities specific for ideal graphene. |
| format |
Article |
| author |
Kardashev, D.L. Kardashev, K.D. |
| spellingShingle |
Kardashev, D.L. Kardashev, K.D. Local vibrational density of states in disordered graphene Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Kardashev, D.L. Kardashev, K.D. |
| author_sort |
Kardashev, D.L. |
| title |
Local vibrational density of states in disordered graphene |
| title_short |
Local vibrational density of states in disordered graphene |
| title_full |
Local vibrational density of states in disordered graphene |
| title_fullStr |
Local vibrational density of states in disordered graphene |
| title_full_unstemmed |
Local vibrational density of states in disordered graphene |
| title_sort |
local vibrational density of states in disordered graphene |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121607 |
| citation_txt |
Local vibrational density of states in disordered graphene / D.L. Kardashev, K.D. Kardashev // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2016. — Т. 19, № 3. — С. 315-317. — Бібліогр.: 11 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT kardashevdl localvibrationaldensityofstatesindisorderedgraphene AT kardashevkd localvibrationaldensityofstatesindisorderedgraphene |
| first_indexed |
2025-07-08T20:12:48Z |
| last_indexed |
2025-07-08T20:12:48Z |
| _version_ |
1837110991769305088 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 3. P. 315-317.
doi: 10.15407/spqeo19.03.315
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
315
PACS 63.22.Rc t
Local vibrational density of states in disordered graphene
D.L. Kardashev1, K.D. Kardashev2
1 National University “Odessa Maritime Academy”
8, Didrikhson str., 65029 Odessa, Ukraine; e-mail: dlkardashev@ukr.net
2A.S. Popov Odessa National Academy of Telecommunication
1, Kuznechnaya str., 65029 Odessa, Ukraine; e-mail: konstantinkardashev@gmail.com
Abstract. Local vibrational density of states for disordered graphene has been calculated
via Green’s functions method. Disordered material has been modeled with Bethe lattice.
Density of states does not include particularities specific for ideal graphene.
Keywords: Bethe lattice, Green function, local vibrational density of states.
Manuscript received 06.04.16; revised version received 14.07.16; accepted for
publication 13.09.16; published online 04.10.16.
1. Introduction
The Green functions (GF) method has been successfully
applied for investigation of electronic [1, 2] and dynamic
[3, 4] properties of disordered solids [5, 6]. It is
impossible to use cyclic boundary conditions in these
materials. In a real graphene, there are always some
variations of chemical bounds lengths and valence
angles magnitudes [5].
The most simple model of structurally disordered
graphene is the Bethe lattice [7-9] (Cayley tree). It is an
infinite connected cycle-free dendritic system of atoms
with remaining short-range order. We’ve used lattice
potential model in Born’s approximation
( )[ ]
( ) ( )[ ] ,
4
1
)(
4
1)(
2
1)(
22
,
2
∑∑
∑
×−γ+−×
×β−α+⋅−β=
ij
jiji
ij
iji
ji
jiji
ruuuu
iruurV
rrrrr
rrrr
The sums are on atoms i and its nearest neighbors j,
and rj(i) is the unit vector joining they equilibrium
positions; ui and uj – vectors of displacement of these
atoms.
In order to determine GF, it is necessary to solve
infinite sequence of Dyson’s linear matrix equations:
( ) ∑
νν
νν
νν+=−ω
ji
ji
n ijijijijm
K
K
K GDIGDI
10
2 .
Here, the sums are on all possible paths joining i-th
and j-th atoms, Dij – matrices of force constants.
The local density of states is given by
∑ ω
⋅π⋅
⋅ω⋅
−=
k
kkG
n
mEg )(Im
3
2)( ,
where n is a number of atoms in a system, and Gii (ω) –
diagonal matrix element of Green’s function for i-th
degree of freedom. Each summand is a partial density of
vibrational states, which characterize contribution of a
given degree of freedom with a frequency ω in the total
density of states.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 3. P. 315-317.
doi: 10.15407/spqeo19.03.315
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
316
2. Solution for the Bethe lattice
In order to transform graphene into the Bethe lattice, we
use the following technique: any random atom we
choose as initial and mark him with subscript 0; next
nearest neighbors will have indexes 1, 2, etc. To
investigate dynamics of this lattice, we postulate that
each bond with a nearest neighbor can be characterized
via 3×3 symmetric matrices of force constants Dij:
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
γ
αβ
βα
=
00
0
0
01D ;
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
γ
β⋅
+α
β
−
β
−
β⋅
−α
=
00
0
2
3
2
0
22
3
02D ;
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
γ
β⋅
−α
β
−
β
−
β⋅
+α
=
00
0
2
3
2
0
22
3
03D .
Then, the local GF of 0-th atom can be determined
from an infinite sequence of matrix equations [1, 2]:
( )
( )
( )
⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎨
⎧
⋅
⋅
⋅
+=−ω
+=−ω
+=−ω
∑
∑
∑
≠
≠
≠
2
22112
2
1
11
2
0
2
j
i
j
j
m
m
m
jjii0
jj000i0i0
0j0j000
GDGDGDI
GDGDGDI
GDIGDI
Here, m is the mass of an atom, I – identity matrix,
∑
= ⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
γ
α
α
==
3
1 300
030
003
i
0i0 DD – force constants matrix
joining atom with itself.
This system of equations can be solved, if one
introduces the transfer matrices Фji in the form:
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
⋅
⋅
⋅
=
=
0ii1i
00i0i
GΦG
GΦG
1
0
,
Figure. Local density of vibrational states of graphene. Solid
line – density of states for the Bethe lattice, dashed line –
density of states of ideal graphene [10].
which satisfies equation
( ) ∑
≠
+=−ω
jk
m kjikikijij0 ΦΦDDΦDI2 .
Then, the Green function is determined as
1
2
−
⎥
⎦
⎤
⎢
⎣
⎡
−−ω= ∑
i
m 0i0i000 ΦDDIG .
Matrix elements were determined in an analytical
form (here we does not perform them due to their
massiveness).
The results of calculation of vibrational density of
states for disordered graphene are represented in the
figure. The dashed line represents density of states for
ideal graphene [10]. Values of force constants have been
selected due to the best proximity to X-ray inelastic
scattering experimental data [11].
3. Conclusions
Local vibrational density of states describes well ZO,
ZA, LO, TO modes. But TA and LA peaks are highly
smoothed.
In a presence of structural disorder, the expression
for local density of states must be averaged over
different local configurations.
The Bethe lattice can be used in a cluster modeling
calculations, including presence of cycles of bonds, as
boundary conditions.
References
1. S. Viola Kusminskiy, D.K. Campbell, and
A.H. Castro Neto, Lenosky’s energy and the
phonon dispersion of graphene // Phys. Rev. B, 80,
p. 035401-1–035401-5 (2009).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2016. V. 19, N 3. P. 315-317.
doi: 10.15407/spqeo19.03.315
© 2016, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
317
2. M. Neek-Amal, Graphene nanoribbons subjected to
axial stress // Phys. Rev. B, 82, p. 085432-1–
085432-6 (2010).
3. D. Cheliotis, B. Virág, The spectrum of the random
environment and localization of noise // Probab.
Theory Relat. Fields, 148, p. 141-158 (2010).
4. Z.H. Ni, T. Yu, Y.H. Lu, Y.Y. Wang, Y.P. Feng,
Z.X. Shen, Uniaxial strain on graphene: Raman
spectroscopy study and band-gap opening // ACS
Nano, 2, p. 2301-2305 (2008).
5. S.Yu. Davydov, Model of adsorption on
amorphous graphene // Semiconductors, 50(3),
p. 377-383 (2016).
6. S.Yu. Davydov, On the density of states of
disordered epitaxial graphene // Semiconductors,
49(5), p. 615-620 (2015).
7. Koh Wada, Takehiko Fujita, Takashi Asahi, Lattice
Vibration of the Cayley Tree // Progr. Theor. Phys.
59, No. 4, p. 1101-1114 (1978).
8. H. Böttger, Principles of the Theory of Lattice
Dynamics. Physik-Verlag, 1983.
9. R. Alben, D. Weaire, J.E. Smith, M.H. Brodsky,
Vibrational properties of amorphous Si
and Ge // Phys. Rev. B, 11, No.6, p. 2271-2295
(1975).
10. V. Adamyan, V. Zavalniuk, Phonons in graphene
with defects // J. Phys: Condens. Matter, 24,
015402 (10 p.) (2011).
11. M. Mohr, J. Maultzsch, E. Dobardžić et al., Phonon
dispersion of graphite by inelastic X-ray scattering
// Phys. Rev. B, 76, 035439 (2007).
|