ISSN 0021-3640, JETP Letters, 2017, Vol. 106, No. 2, pp. 110–115. © Pleiades Publishing, Inc., 2017. Original Russian Text © A.I. Podlivaev, L.A. Openov, 2017, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2017, Vol. 106, No. 2, pp. 98–103.
CONDENSED MATTER
Elementary Defects in Graphane A. I. Podlivaev and L. A. Openov* National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, 115409 Russia *e-mail:
[email protected] Received May 25, 2017; in final form, June 14, 2017
The main zero-dimensional defects in graphane, a completely hydrogenated single-layer graphene, having the chair-type conformation have been numerically simulated. The hydrogen and carbon−hydrogen vacancies, Stone–Wales defect, and “transmutation defect” resulting from the simultaneous hoppings of two hydrogen atoms between the neighboring carbon atoms have been considered. The energies of formations of these defects have been calculated and their effect on the electronic structure, phonon spectra, and Young modulus has been studied. DOI: 10.1134/S0021364017140090
INTRODUCTION In graphene, which is a hexagonal single layer formed by carbon atoms [1], three valence electrons of each atom give rise to strong valence bonds with the nearest neighbors, whereas the p orbitals perpendicular to the layer form π bands of itinerant states with the conical dispersion law and zero effective mass of charge carriers. As a result, graphene combines high mechanical strength [2] with high mobility of electrons and holes [3]. This makes it a promising material for the usage in various electronic and electromechanical devices. In the applications for electronics, the noticeable defect of graphene is the absence of band gap. There exist several ways for transforming graphene to the insulating state. The simplest and the most efficient one corresponds to the adsorption of a singly charged cation (e.g., hydrogen) onto each carbon atom. This leads to the replacement of the sp 2 hybridization of carbon orbitals by the sp 3 hybridization and to the saturation of dangling bonds. Then, instead of the conducting π bands, we have a band gap corresponding to the insulating state. The completely hydrogenated graphene is referred to as graphane. This quasi-twodimensional hydrocarbon material has been theoretically predicted in [4] and then it has been actually produced [5]. In particular, it has been shown that the hydrogenation of graphene is a reversible process: after annealing in an inert atmosphere, all characteristics of the initial sample, including even the quantum Hall effect, are restored. By alternating the hydrogenated graphene segments with the nonhydrogenated ones [6], it is possible to obtain different combinations of quantum wells and barriers (including superlattices) useful for applications in electronics. The calculations
demonstrate that, in such nanostructures, the graphene/graphane interfaces are stable with respect to the thermally induced disordering [7]. In the theoretical studies, various modifications of graphane with different arrangements of hydrogen atoms were considered [4, 8, 9]. Among them, the chair-type conformation is accepted to be the most stable one [4]. In this conformation, hydrogen atoms adsorbed on neighboring carbon atoms are located at different sides of the layer plane (that is why every two nearest carbon atoms are shifted perpendicularly to the layer in opposite directions). As a result, the carbon backbone becomes buckled (see Fig. 1). The experimental studies of graphane are complicated by low quality of the samples [5, 10–12]. Therefore, currently it is not possible to provide unambiguous conclusions concerning its structure. In interpretation of experimental data, one should have in mind that the results can be significantly affected by the presence of different structural defects. This is clearly seen in the case of graphene, which is a
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Fig. 1. Graphane С160Н160 supercell. Large and small circles denote carbon and hydrogen atoms, respectively.
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close relative of graphane [13]. The present work is aimed at the computer simulations of the main point defects existing in the chair-type conformation of graphene, including the analysis of their effect on the electronic structure, phonon spectra, and mechanical stiffness. We modeled the chair-type conformation of graphane using the 320-atom supercell С160 Н160 (see Fig. 1) with periodic boundary conditions within the ( x, y) plane and free ones in the transverse (z ) direction. The periods of the initial supercell and of those containing defects were determined by finding their minimum energies after the relaxation with respect to the positions of all atoms. The configurations with defects considered in this paper are metastable; i.e., they correspond to local minima of the potential energy surface. An indication of this fact is the absence of imaginary frequencies in the spectra of eigenmodes, which were calculated by the numerical diagonalization of the corresponding Hessian. The interatomic interactions were described in the framework of the nonorthogonal tight-binding model [14], which explicitly takes into account the contributions of all valence electrons of carbon and hydrogen to the total energy and involves many sites; i.e., it is not reduced to either pair or three-body interactions. Being inferior in accuracy with respect to the ab initio approaches, this model does not require a large computational burden and allows for a detailed analysis of the potential energy surface in the systems consisting of 300–500 atoms. Earlier, a similar method was implemented for the numerical simulations in the cases of graphane [15], hydrocarbon hypercubane [16], graphone [17], diamond nanofilaments [18], and other carbon-based nanothreads. HYDROGEN VACANCY To perform the simulations of the hydrogen vacancy, we removed one hydrogen atom from the С160Н160 supercell. This corresponds to the desorption of about 0.6% hydrogen from graphane. The formation energy for such vacancy was assumed to be equal to the energy needed to remove a hydrogen atom from the supercell
E f = E (С160Н159 ) + E (H) − E (С160Н160 ), where E (С160Н159 ) and E (С160Н160 ) are the energies of the С160Н159 and С160Н160 supercells, respectively, and E(H) is the energy of an isolated (moved to infinity) hydrogen atom. As a result, we obtain E f = 3 . 65 eV. Such energy can come either from external sources (e.g., from electron irradiation) or (at high temperature) from thermal fluctuations. JETP LETTERS
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Fig. 2. Carbon−hydrogen vacancy in the С160Н160 supercell.
CARBON−HYDROGEN VACANCY Such vacancy is formed by removing a carbon atom with a hydrogen atom adsorbed on it (i.e., of a CH molecule) from their equilibrium position in the crystal lattice (Fig. 2) but not from the sample. To calculate its formation energy, we use the expression [19]
E f = E (С159Н159 ) + μ − E (С160Н160 ), where E(С159Н159) and E(С160Н160) are the supercell energies with the vacancy and without it, respectively, and μ is the chemical potential of the molecule, which is equal to the energy per CH molecule in the supercell without defects.
Eventually, we have E f = 4 . 61 eV. This value is much lower than the vacancy formation energy in graphene, E f ≈ 7 . 5 eV [13], which is due to the weakening of the С–С bonds accompanying the hydrogen adsorption onto graphene. The vacancy shown in Fig. 2 is symmetric. Note that a symmetric vacancy in graphene undergoes the Jahn−Teller reconstruction. As a result, the vacancy symmetry becomes lower and one of the atoms shifts in the transverse direction [20]. For the carbon−hydrogen vacancy in graphane, we did not find such reconstruction. This can be related either to the insufficient accuracy of the model or to the specific features of graphane. The carbon−hydrogen vacancies can appear either under the irradiation of graphane by high-energy particles (ions or electrons) or at the stage of synthesis. CARBON VACANCY As a hypothesis, we can imagine that, at the removal of a carbon atom from the lattice site, the hydrogen atom adsorbed on it remains near the formed vacancy and saturates one of three remaining carbon bonds. However, such situation seems to be quite improbable and we do not consider it in the present paper.
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Fig. 3. SW defect in the С160Н160 supercell. The С–С bond rotated in the course of SW transformation is highlighted in black.
STONE–WALES DEFECT In graphane, as well as in graphene, we can observe the topological Stone–Wales (SW) defects, which can be formed because of the SW transformation, namely, by the rotation of a С–С bond at an angle of ≈90 ° [21]. The specific feature of such defects in graphane is the simultaneous rotation of carbon atoms and hydrogen atoms adsorbed on them. As a result, there appear two pairs on neighboring carbon atoms, at which hydrogen is adsorbed not from the different sides of the carbon layer, but at the same side (Fig. 3), as in the boat-type conformation [8]. The formation energy of such defect (the energy difference between the supercells with and without the defect) is equal to E f = 3 . 18 eV (in graphene, we have E f ≈ 5 eV [13]). TRANSMUTATION DEFECT In graphane, all valence bonds of each carbon atom are completely saturated. Therefore, hydrogen atoms cannot be individually moved from one to another carbon atom. However, it is possible to have a situation where the neighboring carbon atoms exchange their hydrogen atoms (Fig. 4a). As a result, there appears a defect, which can be called a transmutation defect. Such defect is characterized by the existence of two groups consisting of three carbon atoms in each one, where hydrogen is adsorbed only from one side of the carbon layer plane (Fig. 4b). This resembles a fragment of the stirrup-type conformation [8]. The calculation of the formation energy for the transmutation defect yields E f = 0 . 50 eV. This value turns out to be lower than that for the other defects. ELECTRONIC STRUCTURE Graphane without defects is an insulator (see Fig. 5). For the band gap width defined as the energy difference between the lower unoccupied (LUMO) and higher occupied (HOMO) molecular orbitals, we
Fig. 4. (a) Schematic illustration for the formation of the transmutation defect at the simultaneous hoppings of two hydrogen atoms between neighboring carbon atoms. (b) Transmutation defect in the С160Н160 supercell.
found that E g = 5 . 34 eV, which is in agreement with the value calculated by other authors, E g = 5 . 4 eV [22]. The formation of a hydrogen vacancy gives rise to a local energy level within the band gap (see Fig. 5). The main contribution to the wavefunction corresponding to this level comes from the 2Pz orbital of the carbon atom, on which the removed hydrogen atom was adsorbed. With the growth in the number of hydrogen vacancies, the number of such levels also grows and they form an impurity band, within which the Fermi level is located. Note that the formation of the mid-gap states with the nonzero density of states at the Fermi level accompanying the partial dehydration of graphane was considered theoretically in [23], whereas the specific form and the details of the density of states are sensitive both to the density of hydrogen vacancies and to their arrangement in the sample. At the same time, the recent experimental work [24] revealed the dependence of the band gap width of the hydrated graphene on the hydrogen content rather than the mid-gap states. To clarify this problem, we first need further experiments preferably performed using single-phase samples and second the comparison of experimental results with the calculations made for different graphane conformations. The formation of the carbon−hydrogen vacancy is accompanied by the simultaneous arising of three JETP LETTERS
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Fig. 5. Electron density of states g(ε) per spin and per unit cell in the С160Н160 supercell (solid line). The energy is measured from the Fermi level. The thin vertical line depicts the local level in the С160Н159 supercell with the hydrogen vacancy.
local levels (Fig. 6). The wavefunctions corresponding to these levels are different linear combinations of orbitals belonging to three carbon atoms nearest to the vacancy and hydrogen atoms adsorbed on them. The dominant contribution comes from the 2Px , 2Py , and 2Pz orbitals. We have not revealed any appreciable effect of SW and transmutation defects on the electron density of states.
Fig. 6. Same as in Fig. 5. The short and long vertical lines correspond to the separate level and to two close (Δε ~ 10−2 eV) levels in the С159Н159 supercell with the carbon−hydrogen vacancy.
the desorption of 50% hydrogen atoms. At the formation of defects, the position of PDOS peaks changes only slightly, within 3 cm−1.
MECHANICAL CHARACTERISTICS The Young modulus for quasi-two-dimensional materials is given by the expression [25] Y = S −1(∂ 2U / ∂ε 2 ) ε→0,
PHONON SPECTRUM In Fig. 7, we show the calculated phonon density of states (PDOS), where the main specific feature is a high-frequency peak at ω ~ 3100 cm −1 stemming from atomic vibrations at the С–Н bonds. Two other peaks at ω ~ 1050 and 1250 cm −1 are also related to different hydrocarbon modes. Defects lead to smearing of these features in the PDOS and to the decrease in the height of peaks (see Table 1). The SW defects produce the most clearly pronounced effect on the phonon spectrum. The very strong (almost by a factor of 2) decrease in the PDOS peak at ω ~ 1250 cm −1 manifests itself because the main contribution to this peak comes from vibrations in the carbon subsystem. The SW transformation significantly affects the latter subsystem owing to the relatively small supercell size. The growth in the number of defects results in the further decrease in the height of the PDOS peaks. In particular, for the С160Н144 supercell (10% of hydrogen vacancies), the height decreases by about 35–45%. The high-frequency peak almost completely disappears at JETP LETTERS
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Fig. 7. Phonon density of states in the С160Н160 supercell.
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Table 1. Data demonstrating the relative decrease (in percent) in the heights of the peaks in the phonon density of states due to the presence of one defect in the С160Н160 supercell Decrease in the height of peaks (in %) due to different defects −1
Frequency (cm ) ~1050 ~1250 ~3100
Hydrogen vacancy 3.2 4.7 2.8
Carbon−hydrogen vacancy 4.9 10.0 9.3
where S is the sample area, U is its elastic energy, and ε is the strain under the tensile force. In anisotropic materials, one can distinguish Y x and Y y moduli corresponding to the strains along the x and y axes, respectively (in an isotropic material, we have Y x = Y y ≡ Y ). For the supercell without defects, we obtained Y = 249 N/m. This value is far below than Y = (340 ± 50 ) N/m characteristic of graphene [2]. Our result is in agreement with the value Y ≈ 245 N/m calculated by the other authors [8]. For the supercell with one hydrogen vacancy, we have Y = 245 N/m. With the growth in the number of vacancies, the values of Y x and Y y become sensitive to the arrangement of vacancies within the supercell. At a vacancy content of 10%, these moduli range within 176–233 N/m. The Y x and Y y values averaged over the configurations of vacancies achieve their minimum of about 130 N/m at the vacancy content near 50%. Then, they again begin to grow to their maximum in graphene (100% of vacancies). The other defects also lead to the lowering of the mechanical stiffness of graphane. For the supercells with the carbon−hydrogen vacancy and SW and transmutation defects, we determined (Y x , Y y ) = (208, 94) , (236, 216), and (240, 248) N/m, respectively. The anisotropy of the Young modulus is related to the anisotropic character of the supercell distortions produced by such defects. In a macroscopic sample with the random distribution of defects, the anisotropy should become smaller. CONCLUSIONS Let us briefly summarize the results. In the chairtype conformation of graphane, the lowest formation energy (0.5 eV) corresponds to the transmutation defects arising because of the exchange of hydrogen atoms between two neighboring carbon atoms. For vacancies and SW defects, this energy exceeds 3 eV. The electronic structure is strongly affected by the hydrogen and carbon−hydrogen vacancies. They lead to the mid-gap energy levels and to the formation of the impurity conduction band. All these defects result in the decrease in the height of the characteristic peaks
SW defect
Transmutation defect
8.2 40.8 9.9
7.4 7.2 3.3
in the phonon density of states and in the lowering of the mechanical stiffness. At the same time, the concentration dependence of the Young modulus is nonmonotonic. Finally, note that the SW and transmutation defects in the chair-type conformation can be treated as the nucleation centers of the boat- and stirrup-type conformations, respectively. This work was supported by the Russian Foundation for Basic Research, project no. 15-02-02764. REFERENCES 1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 2. C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321, 385 (2008). 3. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). 4. J. O. Sofo, A. S. Chaudhari, and G. D. Barber, Phys. Rev. B 75, 153401 (2007). 5. D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. Morozov, P. Blake, M. P. Halsall, A. C. Ferrari, D. W. Boukhvalov, M. I. Katsnelson, A. K. Geim, and K. S. Novoselov, Science 323, 610 (2009). 6. L. A. Chernozatonskii, P. B. Sorokin, E. E. Belova, J. Bruning, and A. S. Fedorov, JETP Lett. 85, 77 (2007). 7. L. A. Openov and A. I. Podlivaev, JETP Lett. 90, 459 (2009). 8. H. Sahin, O. Leenaerts, S. K. Singh, and F. M. Peeters, WIREs Comput. Mol. Sci. 5, 255 (2015). 9. T. E. Belenkova, V. M. Chernov, and E. A. Belenkov, RENSIT 8, 49 (2016). 10. H. L. Poh, F. Sanek, Z. Sofer, and M. Pumera, Nanoscale 4, 7006 (2012). 11. C. Zhou, S. Chen, J. Lou, J. Wang, Q. Yang, C. Liu, D. Huang, and T. Zhu, Nanoscale Res. Lett. 9, 26 (2014). 12. M. V. Kondrin, N. A. Nikolaev, K. N. Boldyrev, Y. M. Shulga, I. P. Zibrov, and V. V. Brazhkin, arXiv:1608.07221v1 (2016). 13. F. Banhart, J. Kotakoski, and A. V. Krasheninnikov, ACS Nano 5, 26 (2011). JETP LETTERS
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Translated by K. Kugel
