Thermal Properties of Graphane: A Greens Function Approach
John Bosco Balaguru Rayappan
Investigations on the hydrogenation of graphane has gained a significant momentum due to its applications in the fields of hydrogen storage, automobiles, etc. The knowledge on lattice vibrations and hence the thermal properties of hydrogenated graphene (graphane) is important in order to use this material for hydrogen storage applications. This study has reported one of the important thermal properties namely Debye-Waller factor and defect modes of graphane by computing phonon frequency modes by considering atomic interactions up to six neighbors in the frame work of Born von Karman formalism and Greens function approach. The estimated defect modes particularly the localized vibrational modes and the values of Debye-Waller factor for a temperature range of 400 to 1300 K have been reported.
June 06, 2012; Accepted: July 09, 2012;
Published: August 09, 2012
Graphane is a two-dimensional covalently bonded hydrocarbon in which hydrogen
(H) atoms are chemically bound to the carbon (C) atoms on alternate sides of
the membrane causing a local buckling of the membrane. Sluiter
and Kawazoe (2003) first predicted Graphane (GA) and it was recently rediscovered
by Sofo et al. (2007). With reference to their
prediction, GA hydrogen atoms are chemically bound to the carbon atoms
on alternating sides of the membrane which causes a local bucking
of the membrane. Flores et al. (2009) has
recently reported such deformations for small membrane size (<1 nm). In the
recent past many experimental (Ao and Peeters, 2010) and
theoretical investigations (Karssemeijer and Fasolino, 2011;
Flores et al., 2009; Tewary and
Yang, 2009) have been carried out on the electronic, thermal properties
of hydrogenated graphene i.e., graphane due to its applications in the fields
of astronomy (Huang et al., 2009), nuclear industries
(Areou et al., 2011) and for designing electronic
components (Huang et al., 2011). The theoretical
insight of the hydrogenation process of graphene helps to predict the electronic
properties of graphane based devices (Bang and Chang, 2010;
Soriano et al., 2010). The lattice thermal properties
of GA such as thermal contraction, roughness and heat capacity are reported
by Neek-Amal and Peeters (2011). Born von Karman formalism
was used to compute the phonon dispersion of graphene by considering first,
second and third neighbor interactions (Falkovsky, 2008),
first and second neighbor interactions with appropriated constraints (Falkovsky,
2007) using the respective force constant values. However, other lattice
vibrations based thermal properties such as defect modes and Debye-Waller factor
are not reported so far for this system. Hence, in this study, the theoretically
calculated results of defect modes and Debye-Waller factor by considering interactions
up to six nearest neighbours are reported.
METHOD OF CALCULATION
In recent experiment, Elias et al. (2009) demonstrated
the fabrication of GA from a graphene (GE) membrane through hydrogenation which
was found to be reversible. Since GA can be derived from GE, as a first step,
the phonon frequencies of GE are calculated using Born von Karman formalism
considering interactions up to sixth neighbors. The following set of force constants
have been assigned for first, second, third, fourth, fifth and sixth neighbor
||Force constants for first:
|A1, B1, A2,
B2, C2 and D2
||Force constants for second:
|A3, B3, C3, D3,
||Force constants for third:
|A5, B5, A6, B6,
C6 and D6
||Force constants for fourth:
|A7, B7, C7, D7,
A8, B8, C8
D8, A9, B9, C9 and D9
||Force constants for fifth:
|A10, B10, C10, D10,
A11 and B11
||Force constants for sixth:
|A12, B12, C12, D12,
A13 and B13
With these parameters the dynamical matrix has been constructed and the elements of dynamical matrix D are as follows:
D (1, 3) = D (1, 6) = D (2, 3) = D (2, 6) = D (3, 1) =
D (3, 2) = D (3, 4) = D (3, 5) = 0
D (4, 3) = D (4, 6) = D (5, 3) = D (5, 6) = D (6, 1) =
D (6, 2) = D (6, 4) = D (6, 5) = 0
D (2, 1) = D (1, 2) = D (4, 5) = D (5, 4)
D (2, 4) = D (1, 5) =-D (4, 2) =-D (5, 1)
D (4, 1) =-D (1, 4)
D (4, 4) = D (1, 1)
D (5, 2) = -D (2, 5)
D (5, 5) = D (2, 2)
D (6, 3) = -D (3, 6)
D (6, 6) = D (3, 3)
Phonon frequencies and eigenvectors are obtained by diagonalising this matrix for 14 wave vector points obtained by uniformly dividing the Brillouin zone. Using the phonon frequencies and eigenvectors, the greens function values are calculated using the formula:
where, ωmax is the maximum frequency among all normal modes of the host crystal.
Using these green function values and the change in dynamical matrix due to the presence of hydrogen, the defect modes are calculated by soling the secular equation:
Debye-Waller factor is calculated using the formula:
The u1 values are calculated using the formula:
RESULTS AND DISCUSSION
By considering the hexagonal close packed coordinates of the system of study
and applying the modified Morse potential (Belytschko et
al., 2002), the force constant values were calculated and are listed
in Table 1.
|| Force constant values in 104 dynes cm-2
|| The defect modes
The computed force constant values have been substituted in the dynamical matrix
and the phonon frequencies and eigenvectors were calculated by diagonalising
the dynamical matrix for 14 representative points obtained by uniformly dividing
the Brillouin zone (Balaguru et al., 2002).
In the presence of defects, the individual frequency levels inside the bands of allowed frequencies are shifted by small amounts and a small number of frequencies which normally lie near the band edges can emerge out of the allowed bands into the gap of the forbidden frequencies. Such normal modes are called Localised Vibrational Modes (LVM) and they have frequencies greater than the maximum frequency of the host crystal. LVMs are observed due to light impurities or impurities which are tightly bound to the surrounding atoms.
A special kind of LVM has been identified with the characteristic that its frequency, instead of lying above the maximum frequency of the host crystal falls in the gap of the two host crystal frequencies. Such modes are called gap modes. In addition to these two modes, resonance type mode occurs, for which, the vibrational amplitude of defect atom is very much higher compared to that of host crystal atoms.
Greens function approach was adapted to estimate the defect mode present in
graphane and calculated resonance and gap modes are given in Table
2. The maximum phonon frequency mode of graphene falls at 1600 cm-1
(Maultzsch et al., 2002; Zimmermann
et al., 2008; Mohr et al., 2007) and
the estimated defect modes after the hydrogenation of graphene falls in the
range of 1600.62 to 6182.82 cm-1.
|| Debye-Waller factor vs. temperature
Since all these modes are greater than the maximum frequency of the host system,
all the listed modes in Table 2 are called as localized vibrational
There are neither other theoretical nor experimental results available to compare
our results with them. The computed Debye-Waller factor values were estimated
at various temperatures and are shown in Fig. 1. It increases
with temperature as expected. As there are no reported works for GA, the results
are compared with that of GE (Tewary and Yang, 2009).
The Debye-Waller factor for GA was found to be much smaller compared to that
of GE (Tewary, 2007). This may be due the presence of
H in GA and this in-turn generated the defect modes. There are three types of
defect modes namely: resonance, gap and localized. Due to the onset of resonance
modes, the H atom vibrates with more amplitude by suppressing the amplitude
of vibration of the surrounding C atoms. More results in this direction for
GA are welcome to check the validity of the present study.
The defect modes namely localized vibrational modes, gap modes and resonance modes of Graphane have been computed employing Greens function technique and scattering matrix formalism. This particular study also supports the fact that lattice defects can perturb the thermal properties of low dimensional materials. The computed values of Debye-Waller factor also support the influence of hydrogen induced defect modes in Graphane.
Ao, Z.M. and F.M. Peeters, 2010. Electric field: A catalyst for hydrogenation of graphene. Applied Phys. Lett., Vol. 96. 10.1063/1.3456384
Areou, E., G. Cartry, J.M. Layet and T. Angot, 2011. Hydrogen-graphite interaction: Experimental evidences of an adsorption barrier. J. Chem. Phys., Vol. 134.
Balaguru, R.J.B., N. Lawrence and S.A.C. Raj, 2002. Lattice Instability of 2H-TaSe2. Int. J. Mod. Phys. B, 16: 4111-4125.
Bang, J. and K.J. Chang, 2010. Localization and one-parameter scaling in hydrogenated graphene. Phys. Rev. B, Vol. 81 10.1103/PhysRevB.81.193412
Belytschko, T., S.P. Xiao, G.C. Schatz and R.S. Ruogg, 2002. Atomistic simulations of nanotube fracture. Phys. Rev. B, Vol. 65. 10.1103/PhysRevB.65.235430
Elias, D.C., R.R. Nair, T.M.G. Mohiuddin, S.V. Morozov and P. Blake et al., 2009. Control of Graphene's properties by reversible hydrogenation: Evidence for Graphane. Science, 323: 610-613.
Falkovsky, L.A., 2007. Phonon dispersion in graphene. J. Exp. Theor. Phys., 105: 397-403.
Falkovsky, L.A., 2008. Symmetry constraints on Phonon dispersion in graphene. Phys. Lett. A, 372: 5189-5192.
Flores, M.Z.S., P.A.S. Autreto, S.B. Legoas and D.S. Galvao, 2009. Graphene to graphane: A theoretical study. Nanotechnology, Vol. 20. 10.1088/0957-4484/20/46/465704
Huang, L.F., M.Y. Ni, G.R. Zhang, W.H. Zhou, Y.G. Li, X.H. Zheng and Z. Zeng, 2011. Modulation of the thermodynamic, kinetic and magnetic properties of the hydrogen monomer on graphene by charge doping. J. Chem. Phys., Vol. 135. 10.1063/1.3624657
Huang, L.F., Y.L. Li, M.Y. Ni, X.L. Wang, G.R. Zhang and Z. Zeng, 2009. Lattice dynamics of hydrogen-substituted graphene systems. Acta Phys. Sin., 58: 306-312.
Direct Link |
Karssemeijer, L.J. and A. Fasolino, 2011. Phonons of graphene and graphitic materials derived from the empirical potential LCBOPII. Surf. Sci., 605: 1611-1615.
Maultzsch, J., S. Reich, C. Thomsen, E. Dobardzic, I. Milosevic and M. Damnjanovic, 2002. Phonon dispersion of carbon nanotubes. Solid State Commun., 121: 471-474.
Mohr, M., J. Maultzsch, E. Dobardzic, S. Reich and I. Milosevic et al., 2007. Phonon dispersion of graphite by inelastic x-ray scattering. Phys. Rev. B, Vol. 76. 10.1103/PhysRevB.76.035439
Neek-Amal, M. and F.M. Peeters, 2011. Lattice thermal properties of graphane: Thermal contraction, roughness and heat capacity. Phys. Rev. B, Vol. 83, No. 23. 10.1103/PhysRevB.83.235437
Sluiter, M.H.F. and Y. Kawazoe, 2003. Cluster expansion method for adsorption: Application to hydrogen chemisorption on graphene. Phys. Rev. B, Vol. 68. 10.1103/PhysRevB.68.085410
Sofo, J.O., A.S. Chaudhari and G.D. Barber, 2007. Graphane: A two-dimensional hydrocarbon. Phys. Rev. B, Vol. 75. 10.1103/PhysRevB.75.153401
Soriano, D., F. Munoz-Rojas, J. Fernandez-Rossier and J.J. Palacios, 2010. Hydrogenated graphenenanoribbons for spintronics. Phys. Rev. B, Vol. 81.
Tewary, V.K. and B. Yang, 2009. Singular behaviour of the Debye-Waller factor of graphene. Phys. Rev. B, Vol. 79. 10.1103/PhysRevB.79.125416
Tewary, V.K., 2007. Theory of nuclear resonant inelastic x-ray scattering from 57Fe in a single-walled carbon nanotube. Phys. Rev. B, Vol. 75.
Zimmermann, J., P. Pavone and G. Cuniberti, 2008. Vibrational modes and low-temperature thermal properties of graphene and carbon nanotubes: A minimal force-constant model Phys. Rev. B, Vol. 78. 10.1103/PhysRevB.78.045410