Pati S.K., Enoki T., Rao C.N. R. (eds.) Graphene and Its Fascinating Attributes
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74 A. K. Manna and S. K. Pati
4. K.S.Novoselov,A.K.Geim,S.V.Morozov,D.Jiang,M.I.Katsnelson,
I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Nature, 438, 197, (2005).
5. Y. Zhang, Y. W. Tan, H. L. Stormer, P. Kim, Nature, 438, 201, (2005).
6. S. Lakshmi, S. Dutta, S. K. Pati, J. Phys. Chem. C, 112, 14718, (2008).
7. A. K. Geim, K. S. Novoselov, Nat. Mater., 6, 183, (2007).
8. R. Saito, G. Dresselhaus, M. S. Dresselhaus, Physical Properties of Carbon
Nanotubes, Imperial College Press, London, (1998).
9. P. M. Ajayan, Chem. Rev., 99, 1787, (1999).
10. M. S. Dresselhaus, Carbon, 33, 871, (1995).
11. T. W. Ebbesen, Annul R ev. of Materials Sci., 24, 235, (1994).
12. J. C. Charlier, X. Blase, S. Roche, Rev. Mod. Phys., 79, 677, (2007).
13. R. S. Baughman, A. A. ZAkhizov, W. A. de Heer, Science, 297, 787, (2002).
14. S. J. Tans, A. R. M. Verschueren, C. Deeker, Nature, 393, 49, (1998).
15. D. M. Guldi, M. Prato, Ac c. Chem. Res, 33, 695, (2000).
16. M. Prato, J. Mater. Chem., 7, 1097, (1997).
17. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim,
Rev. Mod. Phys., 81, 109, (2009).
18. M. I. Katsnelson, Mater. Today, 10, 20, (2007).
19. C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li,
J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, W. A. de Heer,
Science, 312, 1191, (2006).
20. M. Y. Han, B. Ozyilmaz, Y. Zhang, P. Kim, Phys. Rev. Lett., 98, 206805,
(2007).
21. F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katnelson,
K. S. Novoselov, Nat. Mater., 6, 652, (2007).
22. B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, D. Goldhaber-
Gordon, Phys. Rev. Lett., 98, 236803, (2007).
23. E. H. Leib,
Phys. Rev. Lett., 62, 1201, (1989).
24. M. Fujita et al., J. Phys. Soc. Jpn., 65, 8271, (1999).
25. R. A. de Groot et al., Phys. Rev. Lett., 50, 2024, (1983).
26. J. -H. Park et al., Nature (London), 392, 794, (1998).
27. S. M. Mallajosyula, S. K. Pati, J. Phys. Chem. B, 111, 13877, (2007); 112,
16982, (2008).
28. Y. W. Son et al., Nature (London), 444, 347, (2006).
29. S. Dutta, S. K. Pati, J. Phys. Chem. B, 112, 1333, (2008).
30. S. Dutta, S. Lakshmi, S. K. Pati, Phys. Rev. B, 77, 073412, (2008).
31. W. Y. Kim, Y. C. Choi, K. S. Kim, J. Mater. Chem., 18, 4510, (2008).
32. W. Y. Kim, K. S. Kim, J. Comput. Chem., 29, 1073, (2008).
33. W. Y. Kim, K. S. Kim, Nat. Nanotechnol., 3, 408, (2008).
34. S. M. Sairam, P. Parida, S. K. Pati, J. Mater. Chem., 19, 1761, (2009).
35. S.Dutta,A.K.Manna,S.K.Pati,Phys. Rev. Lett., 102, 096601, (2009).
36. A. Das, S. Pisana, B. chakraborty, S. Piscanec, S. K. Saha, U. V. Waghmare,
K.S.Novoselov,H.R.Krishnamurthy,A.K.Geim,A.C.Ferrari,A.K.Sood,
Nat. Nanotechnol., 3, 210, (2008).
37. M. Kalbac, L. Kavan, L. Dunsch, M. S. Dresselhaus, Nano. Lett., 8, 1257,
(2008).
January 20, 2011 13:57 World Scientific Review Volume - 9in x 6in Chap4
Doping of Graphene: A Computational Study 75
38. M. A. Pimenta, G. Dresselhaus, M. S. Dresselhaus, L. G. Cancado, A. Jorio,
R. Saito, Phys. Chem. Chem. Phys., 9, 1276, (2007).
39. J. Kotakoski, A. V. Krasheninnikov, Y. Ma, A. S. Foster, K. Nordlund,
R. M. Nieminen, Phys. Rev. B, 71, 205408, (2005).
40. L. Ci, L. Song, C. Jin, D. Jariwala, D. Wu, Y. Li, A. Srivastava, Z. F. Wang,
K. Storr, L. Balicas, F. Liu, P. M. Ajayan, Nat. Mater., 9, 430, (2010).
41. J. Jhou, Q. Wang, Q. Sun, X. S. Chen, Y. Kawazoe, P. Jena, Nano Lett., 9,
3867, (2009).
42. E.-J.Kan,Z.Li,J.Yang,J.G.Hou,J. Am. Chem. Soc., 130, 4224, (2008).
43. B. Das, R. Voggu, C. S. Rout, C. N. R. Rao, Chem. Commun., 5155, (2008);
C. N. R. Rao, A. K. Sood, K. S. Subrahmanyam, A. Govindaraj, Angew.
Chem., Int. Ed., 48, 7752, (2009).
44. R. Voggu, B. Das, C. S. Rout, C. N. R. Rao, J. Phys. Condens. Mater., 20,
472204, (2008).
45. a. K. S. Subrahmanyam, A. K. Manna, S. K. Pati, C. N. R. Rao, Chem.
Phys. Lett., 497, 70, (2010).
b. A. K. Manna, S. K. Pati, Chem. Asian J, 4, 855, (2009).
46. J. M. Soler, E. Artacho, J. D. Gale, A. Garcia, J. Junquera, P. Ordejon,
D. Sanchez-Portal, J. Phys. : Cond. Matt., 14, 2745, (2002); P. Ordejon,
E. Artacho, J. M. Soler, Phys. R ev. B, 53, R10441, (1996).
47. K. Burke, J. P. Perdew, M. Ernzerhof, Int. J. Quantum Chem., 61, 287,
(1997).
48. N. Troullier, J. L. Martins, Phys. Rev. B, 43, 1993, (1991).
49. L.Kleinman,D.M.Bylander,Phys. Rev. Lett., 48, 1425, (1982).
50. J. P. K. Doye, D. J. Wales, New. J. Chem., 22, 733, (1998).
51. R. Voggu, S. Paul, S. K. Pati, C. N. R. Rao, J. Phys. : Cond. Matt., 20,
215211, (2008).
52. A. K. Manna, S. K. Pati, Nanoscale, 2, 1190 (2010).
53. O. Dubay, G. Kresse, Phys. Rev. B, 67, 035401, (2003).
54. K. V. Christ, H. R. Sadeghpour, Phys. Rev. B, 75, 195418 (2007).
55. D. Sanchez-Portal, E. Artacho, J. M. Soler, A. Rubio, P. Ordejon, Phys. Rev.
B, 59, 12678 (1999).
December 8, 2010 15:42 World Scientific Review Volume - 9in x 6in Chap5
Chapter 5
Vibrations and Buckling of Uni-Axially Strained Graphene
and BN-Monolayer: A First-Principles Study
K. P. S. S. Hembram and Umesh V. Waghmare
Theoretical Sciences Unit,
Jawaharlal Nehru Centr e for Advanced Scientific Research,
Jakkur Campus, Bangalore 560064, India
We present a comparative first-principles analysis of phonons and buck-
ling in graphene and BN-monolayer arising from compressive uni-axial
strains along zigzag (Z) direction. We find that the changes in vibra-
tional frequencies with compressive strain depend on (a) in-plane versus
out-of-plane polarization of modes, (b) optic versus acoustic nature of
the modes. While the responses of graphene and BN-monolayer are
similar qualitatively, graphene exhibits a remarkable anisotropy in the
changes in inter-atomic forces due to uni-axial strain compared to those
of BN-monolayer. For modes with in-plane atomic displacements, optic
modes at all the wave-vectors harden, while acoustic modes soften with
compressive uni-axial strain. This hardening of optic modes is stronger
at all wave vectors for graphene than for BN-monolayer, except at the
Γ point. On the other hand, softening of acoustic modes are stronger
in BN-monolayer than in graphene, revealing stronger hyperelastic na-
ture of BN than graphene. For the modes with out-of-plane polarization,
most modes of BN-monolayer and graphene soften with compressive uni-
axial strain exhibiting a strong dependence on the direction of wave vec-
tor. Flexural acoustic modes with out-of-plane polarization, relevant
to buckling, exhibit instabilities for compressive uni-axial strain greater
than a nonzero value, demonstrating an intrinsic buckling strength of
graphene and BN-monolayer. Analogus to strong electron-phonon cou-
pling in graphene, we find that the electronic band-gap of BN-monolayer
depends sensitively on strain and even vanishes for large enough com-
pressive uni-axial strain of 0.2.
1. Introduction
Graphene composed of a layer of sp
2
bonded carbon (C) atoms arranged on
a honeycomb lattice, possesses topologically different edge profiles such as
77
February 1, 2011 10:39 World Scientific Review Volume - 9in x 6in Chap5
78 K. P. S. S. Hembram and U. V. Waghmar e
zigzag (Z) and armchair (A), which stay orthogonal to each other. Its fasci-
nating electronic and vibrational properties have a great potential for vari-
ety of applications.These include effectively relativistic dynamics of charge
carriers,
1
anomalous quantum Hall effect.
2
These phenomena discovered
earlier, with different materials possess different structure and composition,
but only carbon based material such as graphene showing the similar phe-
nomena is quite interesting. Its use as a semiconductor in nano-electronic
circuits is however limited due to its vanishing bandgap. In contrast to
graphene which possess as zero bandgap, BN with a similar structure is
an insulator. B and N atoms are one electron deficient and one electron
rich compared to C, leading to polar bonding in BN which is analogous to
graphene. Aming at the improved properties of BN over graphene various
researchers have synthesized BN for various applications.
3–5
Mermin and Wagner’s argument for non-existence of truly long range
2D ordering crystal,
6
is refreshed by the discovery of graphene
7
with little
buckling or corrugation.
8,9
The underlying mechanisms and quantitative
analysis of buckling are necessary to answer fundamental features probed
through sophisticated experiments, and also for technological applications.
The buckling strength can be estimated by applying compressive strain
and measuring the critical stress. Critical strain is the amount of strain
that graphene can withstand without buckling. Quantitatively thermal
fluctutation is a measure for buckling at finite temperature. But at 0K, the
linear along with the quadratic terms in the dispersion of flexural modes are
responsible for buckling.
10
Various researchers have investigated graphene
subjected to the tensile uni-axial strain to understand the underlying mi-
croscopic picture, not only theoretically,
11–16
but also with the support
from properties, observed experimentally.
15–17
Theoretically, large strain
shows clear features of opening of band gap, but for small strain it becomes
controversial.
18
In the reality the maximum tensile strain realised exper-
imentally has been a few percent. In this article, we study the flexural
modes of uni-axially compressive strained BN and compare its results with
graphene.
2. Methods
We use plane wave self consistent field (PWSCF)
19
implementation of den-
sity functional theory (DFT), with a local density approximation (LDA)
to the exchange correlation energy of electrons, and ultrasoft pseudopoten-
tials,
20,21
to represent interaction between ionic cores and valence electrons.
February 1, 2011 10:39 World Scientific Review Volume - 9in x 6in Chap5
Vibrations and Buc kling of Uni-Axially Strained Gr aphene and BN-Monolayer 79
Kohn-Sham wave functions were represented with a plane wave basis with
an energy cutoff of 40 Ry and charge density cutoff of 480 Ry. To facilitate
a direct comparison, graphene and BN were simulated using the same peri-
odic supercell with a vacuum of 16
˚
A in the direction perpendicular to the
plane. Integration over the Brillouin zone was sampled with a 20 X 10 X 1
Monkhorst-Pack mesh
22
grid points and occupation numbers were semeard
using Methfessel-Paxton scheme
23
with a broadening of 0.01 Ry. Conver-
gence was checked with finer meshes of k-points in sampling the Brillouin
zone.
3. Results
3.1. Structure
We use a unit cell containing 4 atoms as shown in Fig. 1(a) for graphene
and in Fig. 1(b) for BN-monolayer. To ensure that no two B or N atoms
are the nearest neighbors, we consider the homogeneous distribution of B
and N in the ab-plane which is expected to be favored from configurational
entropy minimization in the experimental synthesis procedure. Properties
like conductivity differ from Z-toA-direction. Similarly the properties
may be drastically different for the uni-axial strain applied along different
directions.
When we apply a series of compressive strain along Z-andA-direction
and simultaneously relax the structure, the system minimizes the energy
Fig. 1. (Color online) Two dimensional honeycomb structure with 4 atoms in the unit
cell of (a) Graphene, (b) BN-monolayer.
December 8, 2010 15:42 World Scientific Review Volume - 9in x 6in Chap5
80 K. P. S. S. Hembram and U. V. Waghmar e
using forces on each atom retaining the minimal residual stress. The stress
on the graphene and BN-monolayer is shown in the Fig. 2(a). It is clear
that the graphene possess slightly higher stress than that of BN at all values
of applied strain.
Fig. 2. (Color online) (a) Stress on graphene and BN-monolayer as a function of uniaxial
compressive strain. (b) Phonon dispersion of graphene and BN-monolayer.
3.2. Phonons
Phonon dispersion of graphene and BN-monolayer over the symmetry point
on the Brillouin zone is shown in the Fig. 2(b). It is evident from the
phonon dispersion that flexural acoustic modes are significantly softer in BN
(310 cm
−1
at K point) than those in graphene (527 cm
−1
at K point). The
Brillouin zone along with symmetry points of graphene and BN-monolayer
with 4 atoms in the unit cell, with strain along Z-direction is shown in the
Fig. 3. Comparison of flexural modes of graphene and BN-monolayer when
the compressive strain is applied along Z-direction is shown in Fig. 4. It is
clearly seen that the negative frequencies shown by both BN-monolayer and
graphene indicate the structures are unstable with the applied compressive
strain. The instability of BN-monolayer is stronger than that of graphene.
The dependence of frequency on wave vector gives an indication of the
elastic nature of the graphene or BN-monolayer. The best fit to omega (ω)
vs q needs the following terms at the lowest order.
ω
2
flex
= α ∗ q
2
+ β ∗ q
4
The value of α does not change much for small strains, but drops to
negative value for large enough compressive strain (see Fig. 5(a)). In the
February 1, 2011 10:39 World Scientific Review Volume - 9in x 6in Chap5
Vibrations and Buc kling of Uni-Axially Strained Gr aphene and BN-Monolayer 81
unstrained case the value of α for graphene is 74 while that for BN-
monolayer is 65 (Z-direction) with critical strains of -1.93 % and -1.77 %
respectively.
Long-range structural order of an isolated 2D graphene sheet can be
understood from the existence of the flexural modes. The number of flexural
modes (if one assumes purely quadratic dispersion) per unit area at a certain
temperature is given by
N
flex
=
1
2π
∞
0
dk
k
e
β
√
κ/σk
2
− 1
with each flexural mode oscillating like quantum harmonic oscillator
24
with
the frequency given by
ω
flex
=
κ
σ
1
2
k
2
The parameter σ is graphene’s 2D mass density, κ is the bending stiff-
ness, β =1/k
β
T .ForT = 0 the above integral is logarithmically divergent
in the long wavelength region indicating a divergent number of phonons in
the thermodynamic limit. For a system with finite size L,withL>>L
T
,
(L
T
is the thermal wavelength of the flexural modes) one finds the num-
ber of flexural modes diverges logarithmically with the size of the system.
Hence
N
flex
=
2π
L
2
T
ln(L/L
T
)
where
L
T
=
2π
k
β
T
(κ/σ)
1
4
indicating that the system cannot be structurally ordered at any finite tem-
perature. This is nothing but the buckling instability.
24
For L<<L
T
,one
finds that N goes to zero exponentially with the size of the system, indicat-
ing that systems with finite size can be flat at sufficiently low temperature.
Since our calculations here are based on 0K, we expect the similar char-
acteristics. With applied compressive strain of a finite nonzero value, the
scenario becomes different and hence the nature of the flexural modes.
When compressive strain is applied uni-axially on the graphene, the atoms
in the graphene adjust collectively remaining in the plane till the critical
strain, beyond which they move out-of plane, causing the corrugation or
buckling.
December 8, 2010 15:42 World Scientific Review Volume - 9in x 6in Chap5
82 K. P. S. S. Hembram and U. V. Waghmar e
Fig. 3. Schematic diagram of compressive strained Brillouin Zone along the Z-direction.
Fig. 4. (Color online) Branch of flexural modes with compressive strain along
Z-direction.
The values of acoustic and optical phonon frequency at the symmetry
point on the Brillouin zone (listed in the Tables 1 and 2 for uni-axially
strained along Z-andA-direction) suggest that the optical phonon modes
at all the symmetry point harden and the acoustic phonon modes soften.
Interestingly, optical phonon modes harden more for graphene than that
of BN-monolayer in all the cases except at the Γ point, where the optical
December 8, 2010 15:42 World Scientific Review Volume - 9in x 6in Chap5
Vibrations and Buc kling of Uni-Axially Strained Gr aphene and BN-Monolayer 83
phonon modes harden more for BN-monolayer than for graphene. On the
other hand the acoustic phonon modes soften more for BN-monolayer than
those of graphene.
Table 1. Calculated phonon frequencies of graphene and BN-monolayer (at different
symmetry points on the Brillouin zone) with different uni-axial compressive strain along
Z-direction.
System Γ (optic) X (acoustic) X (optic) M (acoustic) M (optic)
with strain cm
−1
cm
−1
cm
−1
cm
−1
cm
−1
Gr 0.00 % 1618 383 1435 474 1435
Gr-Z -0.75 % 1635 377 1465 470 1458
Gr-Z -1.50 % 1655 366 1487 462 1492
Gr-Z -2.25 % 1657 353 1514 454 1516
Gr-Z -3.00 % 1657 341 1543 446 1532
BN 0.00 % 1523 274 1337 313 1323
BN-Z -0.75 % 1543 263 1358 303 1337
BN-Z -1.50 % 1566 248 1380 298 1362
BN-Z -2.25 % 1587 234 1399 277 1381
BN-Z -3.00 % 1610 215 1422 269 1407
Table 2. Calculated phonon frequencies of graphene and BN-monolayer (at different
symmetry points on the Brillouin zone) with different uni-axial compressive strain along
A-direction.
System Γ (optic) X (acoustic) X (optic) M (acoustic) M (optic)
with strain cm
−1
cm
−1
cm
−1
cm
−1
cm
−1
Gr 0.00 % 1618 383 1435 474 1435
Gr-A -0.75 % 1651 382 1450 474 1457
Gr-A -1.50 % 1666 380 1463 469 1494
Gr-A -2.25 % 1674 374 1493 470 1518
Gr-A -3.00 % 1726 369 1523 466 1548
BN 0.00 % 1523 274 1337 313 1323
BN-A -0.75 % 1535 269 1354 307 1337
BN-A -1.50 % 1551 267 1372 303 1352
BN-A -2.25 % 1568 256 1394 299 1358
BN-A -3.00 % 1587 248 1419 291 1401
Figure 5(b) shows atomic displacements in flexural modes of graphene
and BN-monolayer at different compressive strain. At M point, the flexural
modes for compressive strain along Z-direction are larger than that of flex-
ural modes for compressive strain along A-direction for both graphene and