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Electronic properties of dehydrogenated nanodiamonds: A first-principles study
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Diamond & Related Materials 17 (2008) 204 – 208 www.elsevier.com/locate/diamond
Electronic properties of dehydrogenated nanodiamonds: A first-principles study C. Wang, B. Zheng, W.T. Zheng ⁎, Q. Jiang Department of Materials Science and Key Laboratory of Automobile Materials, MOE, Jilin University, Changchun 130012, People's Republic of China Received 26 July 2007; received in revised form 29 November 2007; accepted 6 December 2007 Available online 15 December 2007
Abstract First-principles calculations using quantum-mechanical density functional theory (DFT) are carried out to study the geometric structure and electronic properties of dehydrogenated nanodiamonds with diameters varying from 0.8 nm to 1.6 nm. The results show that the electronic properties of dehydrogenated nanodiamond are quite different from those of bulk diamond or hydrogenated nanodiamond. Surface atoms play an important role in the electronic structure, especially the states near the Fermi level, for dehydrogenated nanodiamond. In addition, it has been revealed that the size-dependent feature in the electronic properties for dehydrogenated diamonds is also contributed by the surface effect, in addition to the quantum confinement effect. © 2007 Elsevier B.V. All rights reserved. Keywords: Nanodiamond; DFT; Electronic property
1. Introduction Nanometer-sized diamond can be found in meteorites [1], interstellar dusts [2], solid detonation products[3], and diamondlike films [4,5]. The sizes of diamond nanoparticles vary from 2 to 5 nm, with a distribution peak around 2.6 nm [1]. Since its first discovery, nanodiamond has attracted much interest from both experimental and theoretical sides. Using ab-initio calculations, Raty et al. [6] have shown that dehydrogenated nanodiamond with a diameter of about 1.4 nm has a fullerene-like surface and a diamond core. This structure is consistent with many experimental findings. They have called these carbon nanoparticles bucky-diamonds. Another family of nanometer-sized diamond is hydrogenated nanodiamond, whose surface is fully hydrogenated. Studies on the stability of carbon nanoparticles [7] show that dehydrogenated nanodiamonds become more stable than hydrogenated nanodiamonds when the size of the particle is larger than about 2.5 nm. Furthermore, the result of molecular dynamic simulation shows that at very high temperature dehydrogenated nanodiamonds can transform to tube-shaped fullerenes [8]. ⁎ Corresponding author. Tel./fax: +86 431 85168246. E-mail address: [email protected] (W.T. Zheng). 0925-9635/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2007.12.024
Density functional theory (DFT) becomes one of the most useful tools to investigate the nanoparticles because of its ability in explaining and predicting the experimental results. Many DFT calculations have been performed to study the properties of both hydrogenated [9–11] and dehydrogenated nanodiamonds [12–14]. However, most of the studies on dehydrogenated nanodiamonds are focused on their stability, and many properties of this family of carbon clusters are mostly unexplored. Thus, a further study of dehydrogenated nanodiamond, as an important form of nanometer-sized diamonds, is necessary. In this paper, we present our DFT calculations on the geometric structure and electronic properties of dehydrogenated nanodiamonds. The diameters of the nanoparticles we have studied vary from 0.8 nm to 1.6 nm. Our results show that surface atoms play an important role in the electronic properties for dehydrogenated nanodiamonds. The size-dependent feature in the electronic properties for nanodiamonds is also discussed. 2. Theoretical details In our work, four dehydrogenated nanodiamonds with 66, 147, 275, and 476 atoms were selected. Initially, they are ideally terminated diamond particles with truncated-octahedral morphology
C. Wang et al. / Diamond & Related Materials 17 (2008) 204–208
cut from bulk diamond (Fig. 1). This morphology has been identified as a low energy shape of nanodiamond [15]. The diameters of these nanoparticles are 0.8 nm, 1.2 nm, 1.4 nm, and 1.6 nm, respectively. Each of them has eight (111) facets and six (100) facets. Geometric optimization was performed to make all the structures fully relaxed before the electronic property calculation. Both geometric optimization and electronic property calculation were performed by DMol3 program [16,17]. DMol3 uses DFT to obtain a high accuracy calculation while keeping the computational cost fairly low. In our DFT calculations, the all-electron Kohn–Sham wavefunctions were expanded in the local atomic orbital basis set represented by the double numerical polarized (DNP) basis set with orbital cutoff of 3.7 Å. The generalized gradient approximation (GGA) with Perdew and Wang parameterization [18] was used to describe the exchange and correlation energy in all calculations. Self-consistent field procedure was done with a convergence criterion of 1e-6 Hartree on the energy. Optimization of the structures was done until the change in energy was less than 1e-5 Hartree. During the geometric optimization for the largest cluster, a less precise quality was used to reduce the computational cost. These conditions allowed us to find the ground state geometric and electronic structure accurately. The electronic properties of dehydrogenated nanodiamonds were studied by computing the electronic density of states (DOS) as well as the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) states. Mulliken charge populations of dehydrogenated nanodiamonds were also computed. In addition, we compared the electronic structures between the clusters with different sizes to study the size-dependent feature in the properties for nanodiamonds.
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3. Results and discussion 3.1. Structures of dehydrogenated nanodiamonds During the geometric optimization of all these nanoparticles, we find that the surface reconstruction occurs spontaneously. It means that the structure after surface reconstruction is more stable than the initial one. We can observe the graphitization of the (111) facets and the formation of dimers on the (100) facets during the surface reconstruction. In contrast, the inner atoms have only a few changes in their positions and keep the diamondlike structure through the geometric optimization. The clusters transform to “bucky-diamond” after the surface reconstruction (Fig. 1), which is in good agreement with the previous results obtained by DFT calculations using pseudopotential plane wave approach [6,14]. We calculate the average length of C–C bonds for these fully reconstructed nanodiamonds after the geometric optimization. The calculated average bond lengths are all between 1.47 Å and 1.52 Å for these clusters, which are shorter than that of sp3 bonds in bulk diamond, but longer than the sp2-like bonds in fullerene. To have a further understanding of the bonds in dehydrogenated nanodiamond, we focus on the structure of the second largest nanoparticle C275. After the geometric optimization, the eight (111) facets of C275 transform to eight fullerene-like caps, four of which are larger than the others. Fig. 2 shows the structure around one of the smaller fullerene-like caps in C275. In this figure, we find that the lengths of the bonds at the fullerenelike surface reduce to about 1.44 Å after the reconstruction [19], which is close to that of sp2 bonds, while the bonds in the diamond core are much sp3-like. The bonds link the surface and the core are quite particular, and their average length is about
Fig. 1. The structures of C66, C147, C275, and C476 nanoparticles before (top) and after (bottom) the geometric optimization. The (111) facets transform to fullerene-like caps after the surface reconstruction, and the inner atoms keep the diamond-like structure. This figure is viewed from the [100] direction.
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Fig. 2. The lengths of C–C bonds near one of the smaller fullerene-like caps in C275. The gray lines represent the C–C bonds in the surface of C275 and the black ones denote the inner bonds.
1.60 Å (Fig. 2), which is even longer than that of sp3 C–C bonds in bulk diamond. These bonds are most likely to be elongated by the effective tensile stress between surface and the core of fully reconstructed nanodiamonds [6]. The core of fully reconstructed nanodiamond almost keeps the diamond-like structure. However, from Fig. 2 one can see that the atoms near the fullerene-like surface distribute much disorderly. The length of C–C bonds changes rapidly from about 1.44 Å in the fullerene-like cap to about 1.60 Å in the interface. 3.2. Electronic properties of dehydrogenated nanodiamonds The calculated DOS for dehydrogenated nanodiamonds is shown in Fig. 3, and the calculated DOS for bulk diamond is also given for comparison. The Fermi level is fixed at the zero point of energy. The DOS for dehydrogenated nanodiamonds is very different from that for bulk diamond or hydrogenated nanodiamonds [10,11]. There is no visible gap in the DOS for dehydrogenated nanodiamonds. The vanishing of gap can also be seen from the fact that the energy differences between the
Fig. 3. Density of states (DOS) for dehydrogenated nanodiamond (a) C66, (b) C147, (c) C275, and (d) C476. The DOS of bulk diamond is also presented in (e) for comparison. The Fermi level is fixed at the zero point of energy.
LUMO and HOMO for the dehydrogenated nanodiamonds are all smaller than 0.2 eV. This value is much smaller than the band gap of bulk diamond, which is about 5 eV, and even much smaller than the calculated gap of 1.7 eV for C60. In order to find out the origin of the states around the Fermi level which eliminate the gap for nanodiamond, we have calculated the LUMO and HOMO states for C275. LUMO and HOMO are two well-known orbitals near the Fermi level, and hence we can have an understanding of the states around the Fermi level from the characteristic of these two orbitals. Fig. 4 shows the locations of LUMO and HOMO states in C275. One can see that the LUMO and HOMO states are mostly distributed around the bigger and smaller fullerene-like caps of C275, respectively. Moreover, both LUMO and HOMO states are located at the surface and the interface between surface and the core of dehydrogenated nanodiamond. Any contribution of these orbitals can hardly be found in the diamond-like core. We have also calculated some other orbitals near the Fermi level (ten levels above and below the Fermi level are calculated) and find that all of them show this characteristic. This means that most of the states around the Fermi level of dehydrogenated nanodiamonds come from the atoms at the surface as well as the interface, and have little to do with the atoms in the diamond core. Calculation on the orbitals near the Fermi level for C476 supports this conclusion. In a further analysis, we find that the states near the Fermi level are localized at the defects in the surface as well as the interface, and can be considered as the defect states. There are several kinds of defects in the surface, such as the edges of fullerene-like caps and the dimers on the surface. Most of the surface states around the Fermi level locate at these sites. The interfacial defects could be found below the reconstructed (111) facets. Some interfacial atoms are 3-fold coordinated (Fig. 5), but all of these 3-fold coordinated atoms are linked to three 4fold coordinated atoms, namely, they are isolated 3-fold coordinated atoms. This kind of carbon atoms have been discussed previously in the amorphous carbon [20,21]. The pz electronic orbitals of these atoms do not have a nearest-neighbor pz orbital to form π and π⁎ states, and finally become dangling bonds. The unpaired electrons of these atoms come into defect states near the Fermi level. These surface and interfacial states trap electrons or holes and affect the electrical and optical properties of the material [22].
Fig. 4. HOMO (a) and LUMO (b) states for C275 dehydrogenated nanodiamond, in which most of the states are located at the surface and interface.
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Fig. 5. Topological structure of atoms near one of the bigger fullerene-like caps in C275. The black atoms below the fullerene-like cap are 3-fold coordinated, and their neighboring atoms are all 4-fold coordinated.
We have calculated the Mulliken charge populations of dehydrogenated nanodiamonds to study the charge distribution in the surface of these clusters. Since it is widely accepted that the absolute value of the obtained atomic charges has little physical meaning as they are extreme sensitive to the atomic basis set [23], only the relative values are considered during our discussion. Our result shows that the total charges of the fullerene-like caps for these clusters are positive. Furthermore, the cores of the reconstructed nanodiamonds are also positively charged. Fig. 6 shows the Mulliken charge populations for the atoms near one of the larger cap in C275. We find that the 3-fold coordinated atoms in the surface as well as the interface and the surface dimers are all positively charged. On the other hand, the 4-fold coordinated atoms which are linked to the 3-fold coordinated atoms are all negatively charged. This result implies that in such structure, the electrons transfer from both the sp2 region and sp3 region to the atoms that link these two regions. These negatively charged atoms form C–C bonds with their neighboring positively charged atoms, having partially polar character. Such bonds are not as stable as perfectly covalent C–C bonds and prefer to break more easily than the other bonds in nanodiamond. 3.3. Size-dependent features Many DFT calculations have been performed to study the quantum confinement effects in hydrogenated nanodiamonds [10,11], which show that the energy gap of hydrogenated nano-
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diamond reduces when the size of the nanoparticle increases. We do not observe this phenomenon in our DFT study of dehydrogenated nanodiamonds, because the surface states near the Fermi level make the gap of dehydrogenated nanodiamond vanish. However, the properties of dehydrogenated nanodiamond show some other size-dependent features. The size-dependent features in the electronic properties can be seen in Fig. 3, which can be understood from the DOS of dehydrogenated nanodiamonds. Firstly, when the size of the cluster increases from 0.8 nm (C66) to 1.6 nm (C476), the DOS becomes more continuous. This is a result of the quantum confinement effect. As the radius of the nanoparticles becomes very small, the energy levels for electrons change from continuous band to a ladder of discrete levels [24]. Simultaneously, the DOS of smaller cluster becomes more discrete. Secondly, the relative intensity of the peaks near the Fermi level becomes weaker when the size of dehydrogenated nanodiamond increases (Fig. 3). We believe that this phenomenon is induced by the surface effect. According to the conclusions mentioned above, the peaks around the Fermi level can be considered as surface defect states, which come from the defects at the surface and are only related to the surface atoms and interfacial atoms. When the size of the clusters increases, the percentage of surface atoms gets smaller, and the relative intensity of these states will become weaker. As the size of dehydrogenated nanodiamond gets large enough, the relative intensity of peaks near the Fermi level will become negligible, and the gap will finally appear again. Bulk diamond is just the limit case in which the surface atoms can completely be negligible. In order to make a comparison of the relative stabilities between dehydrogenated nanodiamonds, we calculate the cohesive energies of nanodiamonds per carbon atom. We defined the cohesive energy per carbon atom by ΔE = [Etot(Cn)−nEC]/n, where Etot(Cn) is the total energy of nanodiamond, EC is the energy of a single carbon atom, n is the number of carbon atoms in the nanodiamond. The calculated cohesive energies for dehydrogenated nanodiamonds per atom are listed in Table 1. From our result, it is observed that as the size of nanoparticle increases from 0.8 nm to 1.6 nm, the absolute value of the cohesive energy for the related dehydrogenated nanodiamonds per atom increases gradually. In contrast, our DFTcalculation and other investigations [10,15] on nanodiamonds with fully hydrogenated surface show that the cohesive energy of hydrogenated nanodiamond decreases as the size increases. For examples, according to our DFT study, the cohesive energies per carbon atom for hydrogenated nanodiamonds with 35, 66, and 147 Table 1 Cohesive energies of dehydrogenated nanodiamonds per atom calculated using DFT
Fig. 6. Mulliken charge populations for atoms near one of the bigger fullerenelike caps in C275. The white atoms are 3-fold coordinated, while the gray ones denote the 4-fold coordinated atoms.
Nanodiamond
Energy (eV)
C66 C147 C275 C476 Diamond
−8.18 −8.52 −8.58 −8.83 −9.16
The cohesive energy for bulk diamond is also presented.
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nanodiamonds, most of which (including the LUMO and HOMO) come from the defects of the surface and interface. These states can be considered as localized defect states appeared in the gap. In addition to the quantum confinement effects, the size-dependent feature in the electronic properties of dehydrogenated nanodiamonds is contributed by the surface effect. This is different from hydrogenated nanodiamond, where size-dependent feature completely comes from the quantum confinement effects. Our results demonstrate that the surface atoms play an important role in the properties of dehydrogenated nanodiamonds. Acknowledgements
Fig. 7. Cohesive energies per atom calculated using semi-empirical tight-binding potential versus the number of carbon atoms in dehydrogenated nanodiamond. The broken line is just to guide the eyes.
carbon atoms are −10.5 eV, −10.2 eV, and −9.6 eV, respectively. This difference implies that surface energy plays an important role in the size-dependent stability of nanodiamonds. Because DFT method has a difficulty in dealing with very large system, we employ the environment-dependent tightbinding (TB) potential [25] to calculate the cohesive energies for nanodiamonds with larger sizes. This TB potential has been proven to be transferable enough to describe various structures of carbon, including nanodiamond [6,9]. As the TB potential is just a semi-empirical method, we use this potential just to have a qualitative study of the stability for nanodiamond. The four structures mentioned above, and another two dehydrogenated nanodiamonds with 741 and 1033 carbon atoms are relaxed using the steepest descent method with this TB potential. The cohesive energies per atom versus the number of carbon atoms are shown in Fig. 7. The sizes of these nanodiamonds vary from 0.8 nm to 2.2 nm. As the size of dehydrogenated nanodiamond increases, the absolute value of the obtained cohesive energy per atom increases in the whole size range. This means the larger the size for the dehydrogenated nanodiamonds, the more stable for the structure. 4. Conclusions The geometric structure and electronic properties of dehydrogenated nanodiamonds can be studied by quantum-mechanical density functional theory. The calculated results show that the core of dehydrogenated nanodiamond keeps diamond-like structure, and is quite ordered. However, the structure of the surface atoms is much disordered. Many defects appear in the surface as well as the interface. Calculations on the electronic properties exhibit that there are many states near the Fermi level and no visible gap is found in the DOS of dehydrogenated
The authors gratefully appreciate the financial support by the National Natural Science Foundation of China under Grant No. 50525204, the National Key Basic Research and Development Program (grant No 2004CB619301), Project 985—Automotive Engineering of Jilin University, and the Teaching and Research Award Program for the Outstanding Young Teachers in High Education Institutions (No. 2002359). References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
R.S. Lewis, M. Tang, J.F. Wecker, E. Anders, E. Stell, Nature 326 (1987) 160. R.S. Lewis, E. Anders, B.T. Draine, Nature 339 (1989) 117. N.R. Greiner, D.S. Phillips, J.D. Johnson, A.F. Volk, Nature 333 (1989) 440. Y.K. Chang, H.H. Hsieh, W.F. Pong, M.H. Tsai, F.Z. Chien, P.K. Tseng, L.C. Chen, T.Y. Wang, K.H. Chen, D.M. Bhusari, J.R. Yang, S.T. Lin, Phys. Rev. Lett. 82 (1999) 5377. K.E. Spear, J. Am. Ceram. Soc. 72 (1998) 171. J.Y. Raty, G. Galli, C. Bostedt, T. Van Buuren, L. Terminello, Phys. Rev. Lett. 90 (2003) 037401. J.Y. Raty, G. Galli, Comput. Phys. Commun. 169 (2005) 14. G.D. Lee, C.Z. Wang, J. Yu, E. Yoon, K.M. Ho, Phys. Rev. Lett. 91 (2003) 265701. G.C. McIntosh, M. Yoon, S. Berber, D. Tománek, Phys. Rev. B 70 (2004) 045401. A.J. Lu, B.C. Pan, J.G. Han, Phys. Rev. B 72 (2005) 035447. N.D. Drummond, A.J. Williamson, R.J. Needs, G. Galli, Phys. Rev. Lett. 95 (2005) 096801. A.S. Barnard, S.P. Russo, I.K. Snook, Phys. Rev. B 68 (2003) 073406. A.S. Barnard, S.P. Russo, I.K. Snook, Diamond Relat. Mater. 12 (2003) 1867. N. Park, S. Park, N.M. Hwang, J. Ihm, S. Tejima, H. Nakamura, Phys. Rev. B 69 (2004) 195411. A.S. Barnard, P. Zapol, J. Chem. Phys. 121 (2004) 4276. B. Delley, J. Chem. Phys. 92 (1990) 508. B. Delley, J. Chem. Phys. 113 (2000) 7756. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. C.Q. Sun, H.L. Bai, B.K. Tay, S. Li, E.Y. Jiang, J. Phys. Chem. B 107 (2003) 7544. C.H. Lee, W.R.L. Lambrecht, B. Segall, P.C. Kelires, Th. Frauenheim, U. Stephan, Phys. Rev. B 49 (1994) 11448. B. Zheng, W.T. Zheng, K. Zhang, Q.B. Wen, J.Q. Zhu, S.H. Meng, X.D. He, J.C. Han, Carbon 44 (2006) 962. A.P. Alivisatos, Science 271 (1996) 933. M.D. Segall, R. Shah, C.J. Pickard, M.C. Payne, Phys. Rev. B 54 (1996) 16317. A.D. Yoffe, Adv. Phys. 42 (1993) 173. M.S. Tang, C.Z. Wang, C.T. Chan, K.M. Ho, Phys. Rev. B 53 (1996) 979.
Diamond & Related Materials 17 (2008) 204 – 208 www.elsevier.com/locate/diamond
Electronic properties of dehydrogenated nanodiamonds: A first-principles study C. Wang, B. Zheng, W.T. Zheng ⁎, Q. Jiang Department of Materials Science and Key Laboratory of Automobile Materials, MOE, Jilin University, Changchun 130012, People's Republic of China Received 26 July 2007; received in revised form 29 November 2007; accepted 6 December 2007 Available online 15 December 2007
Abstract First-principles calculations using quantum-mechanical density functional theory (DFT) are carried out to study the geometric structure and electronic properties of dehydrogenated nanodiamonds with diameters varying from 0.8 nm to 1.6 nm. The results show that the electronic properties of dehydrogenated nanodiamond are quite different from those of bulk diamond or hydrogenated nanodiamond. Surface atoms play an important role in the electronic structure, especially the states near the Fermi level, for dehydrogenated nanodiamond. In addition, it has been revealed that the size-dependent feature in the electronic properties for dehydrogenated diamonds is also contributed by the surface effect, in addition to the quantum confinement effect. © 2007 Elsevier B.V. All rights reserved. Keywords: Nanodiamond; DFT; Electronic property
1. Introduction Nanometer-sized diamond can be found in meteorites [1], interstellar dusts [2], solid detonation products[3], and diamondlike films [4,5]. The sizes of diamond nanoparticles vary from 2 to 5 nm, with a distribution peak around 2.6 nm [1]. Since its first discovery, nanodiamond has attracted much interest from both experimental and theoretical sides. Using ab-initio calculations, Raty et al. [6] have shown that dehydrogenated nanodiamond with a diameter of about 1.4 nm has a fullerene-like surface and a diamond core. This structure is consistent with many experimental findings. They have called these carbon nanoparticles bucky-diamonds. Another family of nanometer-sized diamond is hydrogenated nanodiamond, whose surface is fully hydrogenated. Studies on the stability of carbon nanoparticles [7] show that dehydrogenated nanodiamonds become more stable than hydrogenated nanodiamonds when the size of the particle is larger than about 2.5 nm. Furthermore, the result of molecular dynamic simulation shows that at very high temperature dehydrogenated nanodiamonds can transform to tube-shaped fullerenes [8]. ⁎ Corresponding author. Tel./fax: +86 431 85168246. E-mail address: [email protected] (W.T. Zheng). 0925-9635/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2007.12.024
Density functional theory (DFT) becomes one of the most useful tools to investigate the nanoparticles because of its ability in explaining and predicting the experimental results. Many DFT calculations have been performed to study the properties of both hydrogenated [9–11] and dehydrogenated nanodiamonds [12–14]. However, most of the studies on dehydrogenated nanodiamonds are focused on their stability, and many properties of this family of carbon clusters are mostly unexplored. Thus, a further study of dehydrogenated nanodiamond, as an important form of nanometer-sized diamonds, is necessary. In this paper, we present our DFT calculations on the geometric structure and electronic properties of dehydrogenated nanodiamonds. The diameters of the nanoparticles we have studied vary from 0.8 nm to 1.6 nm. Our results show that surface atoms play an important role in the electronic properties for dehydrogenated nanodiamonds. The size-dependent feature in the electronic properties for nanodiamonds is also discussed. 2. Theoretical details In our work, four dehydrogenated nanodiamonds with 66, 147, 275, and 476 atoms were selected. Initially, they are ideally terminated diamond particles with truncated-octahedral morphology
C. Wang et al. / Diamond & Related Materials 17 (2008) 204–208
cut from bulk diamond (Fig. 1). This morphology has been identified as a low energy shape of nanodiamond [15]. The diameters of these nanoparticles are 0.8 nm, 1.2 nm, 1.4 nm, and 1.6 nm, respectively. Each of them has eight (111) facets and six (100) facets. Geometric optimization was performed to make all the structures fully relaxed before the electronic property calculation. Both geometric optimization and electronic property calculation were performed by DMol3 program [16,17]. DMol3 uses DFT to obtain a high accuracy calculation while keeping the computational cost fairly low. In our DFT calculations, the all-electron Kohn–Sham wavefunctions were expanded in the local atomic orbital basis set represented by the double numerical polarized (DNP) basis set with orbital cutoff of 3.7 Å. The generalized gradient approximation (GGA) with Perdew and Wang parameterization [18] was used to describe the exchange and correlation energy in all calculations. Self-consistent field procedure was done with a convergence criterion of 1e-6 Hartree on the energy. Optimization of the structures was done until the change in energy was less than 1e-5 Hartree. During the geometric optimization for the largest cluster, a less precise quality was used to reduce the computational cost. These conditions allowed us to find the ground state geometric and electronic structure accurately. The electronic properties of dehydrogenated nanodiamonds were studied by computing the electronic density of states (DOS) as well as the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) states. Mulliken charge populations of dehydrogenated nanodiamonds were also computed. In addition, we compared the electronic structures between the clusters with different sizes to study the size-dependent feature in the properties for nanodiamonds.
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3. Results and discussion 3.1. Structures of dehydrogenated nanodiamonds During the geometric optimization of all these nanoparticles, we find that the surface reconstruction occurs spontaneously. It means that the structure after surface reconstruction is more stable than the initial one. We can observe the graphitization of the (111) facets and the formation of dimers on the (100) facets during the surface reconstruction. In contrast, the inner atoms have only a few changes in their positions and keep the diamondlike structure through the geometric optimization. The clusters transform to “bucky-diamond” after the surface reconstruction (Fig. 1), which is in good agreement with the previous results obtained by DFT calculations using pseudopotential plane wave approach [6,14]. We calculate the average length of C–C bonds for these fully reconstructed nanodiamonds after the geometric optimization. The calculated average bond lengths are all between 1.47 Å and 1.52 Å for these clusters, which are shorter than that of sp3 bonds in bulk diamond, but longer than the sp2-like bonds in fullerene. To have a further understanding of the bonds in dehydrogenated nanodiamond, we focus on the structure of the second largest nanoparticle C275. After the geometric optimization, the eight (111) facets of C275 transform to eight fullerene-like caps, four of which are larger than the others. Fig. 2 shows the structure around one of the smaller fullerene-like caps in C275. In this figure, we find that the lengths of the bonds at the fullerenelike surface reduce to about 1.44 Å after the reconstruction [19], which is close to that of sp2 bonds, while the bonds in the diamond core are much sp3-like. The bonds link the surface and the core are quite particular, and their average length is about
Fig. 1. The structures of C66, C147, C275, and C476 nanoparticles before (top) and after (bottom) the geometric optimization. The (111) facets transform to fullerene-like caps after the surface reconstruction, and the inner atoms keep the diamond-like structure. This figure is viewed from the [100] direction.
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Fig. 2. The lengths of C–C bonds near one of the smaller fullerene-like caps in C275. The gray lines represent the C–C bonds in the surface of C275 and the black ones denote the inner bonds.
1.60 Å (Fig. 2), which is even longer than that of sp3 C–C bonds in bulk diamond. These bonds are most likely to be elongated by the effective tensile stress between surface and the core of fully reconstructed nanodiamonds [6]. The core of fully reconstructed nanodiamond almost keeps the diamond-like structure. However, from Fig. 2 one can see that the atoms near the fullerene-like surface distribute much disorderly. The length of C–C bonds changes rapidly from about 1.44 Å in the fullerene-like cap to about 1.60 Å in the interface. 3.2. Electronic properties of dehydrogenated nanodiamonds The calculated DOS for dehydrogenated nanodiamonds is shown in Fig. 3, and the calculated DOS for bulk diamond is also given for comparison. The Fermi level is fixed at the zero point of energy. The DOS for dehydrogenated nanodiamonds is very different from that for bulk diamond or hydrogenated nanodiamonds [10,11]. There is no visible gap in the DOS for dehydrogenated nanodiamonds. The vanishing of gap can also be seen from the fact that the energy differences between the
Fig. 3. Density of states (DOS) for dehydrogenated nanodiamond (a) C66, (b) C147, (c) C275, and (d) C476. The DOS of bulk diamond is also presented in (e) for comparison. The Fermi level is fixed at the zero point of energy.
LUMO and HOMO for the dehydrogenated nanodiamonds are all smaller than 0.2 eV. This value is much smaller than the band gap of bulk diamond, which is about 5 eV, and even much smaller than the calculated gap of 1.7 eV for C60. In order to find out the origin of the states around the Fermi level which eliminate the gap for nanodiamond, we have calculated the LUMO and HOMO states for C275. LUMO and HOMO are two well-known orbitals near the Fermi level, and hence we can have an understanding of the states around the Fermi level from the characteristic of these two orbitals. Fig. 4 shows the locations of LUMO and HOMO states in C275. One can see that the LUMO and HOMO states are mostly distributed around the bigger and smaller fullerene-like caps of C275, respectively. Moreover, both LUMO and HOMO states are located at the surface and the interface between surface and the core of dehydrogenated nanodiamond. Any contribution of these orbitals can hardly be found in the diamond-like core. We have also calculated some other orbitals near the Fermi level (ten levels above and below the Fermi level are calculated) and find that all of them show this characteristic. This means that most of the states around the Fermi level of dehydrogenated nanodiamonds come from the atoms at the surface as well as the interface, and have little to do with the atoms in the diamond core. Calculation on the orbitals near the Fermi level for C476 supports this conclusion. In a further analysis, we find that the states near the Fermi level are localized at the defects in the surface as well as the interface, and can be considered as the defect states. There are several kinds of defects in the surface, such as the edges of fullerene-like caps and the dimers on the surface. Most of the surface states around the Fermi level locate at these sites. The interfacial defects could be found below the reconstructed (111) facets. Some interfacial atoms are 3-fold coordinated (Fig. 5), but all of these 3-fold coordinated atoms are linked to three 4fold coordinated atoms, namely, they are isolated 3-fold coordinated atoms. This kind of carbon atoms have been discussed previously in the amorphous carbon [20,21]. The pz electronic orbitals of these atoms do not have a nearest-neighbor pz orbital to form π and π⁎ states, and finally become dangling bonds. The unpaired electrons of these atoms come into defect states near the Fermi level. These surface and interfacial states trap electrons or holes and affect the electrical and optical properties of the material [22].
Fig. 4. HOMO (a) and LUMO (b) states for C275 dehydrogenated nanodiamond, in which most of the states are located at the surface and interface.
C. Wang et al. / Diamond & Related Materials 17 (2008) 204–208
Fig. 5. Topological structure of atoms near one of the bigger fullerene-like caps in C275. The black atoms below the fullerene-like cap are 3-fold coordinated, and their neighboring atoms are all 4-fold coordinated.
We have calculated the Mulliken charge populations of dehydrogenated nanodiamonds to study the charge distribution in the surface of these clusters. Since it is widely accepted that the absolute value of the obtained atomic charges has little physical meaning as they are extreme sensitive to the atomic basis set [23], only the relative values are considered during our discussion. Our result shows that the total charges of the fullerene-like caps for these clusters are positive. Furthermore, the cores of the reconstructed nanodiamonds are also positively charged. Fig. 6 shows the Mulliken charge populations for the atoms near one of the larger cap in C275. We find that the 3-fold coordinated atoms in the surface as well as the interface and the surface dimers are all positively charged. On the other hand, the 4-fold coordinated atoms which are linked to the 3-fold coordinated atoms are all negatively charged. This result implies that in such structure, the electrons transfer from both the sp2 region and sp3 region to the atoms that link these two regions. These negatively charged atoms form C–C bonds with their neighboring positively charged atoms, having partially polar character. Such bonds are not as stable as perfectly covalent C–C bonds and prefer to break more easily than the other bonds in nanodiamond. 3.3. Size-dependent features Many DFT calculations have been performed to study the quantum confinement effects in hydrogenated nanodiamonds [10,11], which show that the energy gap of hydrogenated nano-
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diamond reduces when the size of the nanoparticle increases. We do not observe this phenomenon in our DFT study of dehydrogenated nanodiamonds, because the surface states near the Fermi level make the gap of dehydrogenated nanodiamond vanish. However, the properties of dehydrogenated nanodiamond show some other size-dependent features. The size-dependent features in the electronic properties can be seen in Fig. 3, which can be understood from the DOS of dehydrogenated nanodiamonds. Firstly, when the size of the cluster increases from 0.8 nm (C66) to 1.6 nm (C476), the DOS becomes more continuous. This is a result of the quantum confinement effect. As the radius of the nanoparticles becomes very small, the energy levels for electrons change from continuous band to a ladder of discrete levels [24]. Simultaneously, the DOS of smaller cluster becomes more discrete. Secondly, the relative intensity of the peaks near the Fermi level becomes weaker when the size of dehydrogenated nanodiamond increases (Fig. 3). We believe that this phenomenon is induced by the surface effect. According to the conclusions mentioned above, the peaks around the Fermi level can be considered as surface defect states, which come from the defects at the surface and are only related to the surface atoms and interfacial atoms. When the size of the clusters increases, the percentage of surface atoms gets smaller, and the relative intensity of these states will become weaker. As the size of dehydrogenated nanodiamond gets large enough, the relative intensity of peaks near the Fermi level will become negligible, and the gap will finally appear again. Bulk diamond is just the limit case in which the surface atoms can completely be negligible. In order to make a comparison of the relative stabilities between dehydrogenated nanodiamonds, we calculate the cohesive energies of nanodiamonds per carbon atom. We defined the cohesive energy per carbon atom by ΔE = [Etot(Cn)−nEC]/n, where Etot(Cn) is the total energy of nanodiamond, EC is the energy of a single carbon atom, n is the number of carbon atoms in the nanodiamond. The calculated cohesive energies for dehydrogenated nanodiamonds per atom are listed in Table 1. From our result, it is observed that as the size of nanoparticle increases from 0.8 nm to 1.6 nm, the absolute value of the cohesive energy for the related dehydrogenated nanodiamonds per atom increases gradually. In contrast, our DFTcalculation and other investigations [10,15] on nanodiamonds with fully hydrogenated surface show that the cohesive energy of hydrogenated nanodiamond decreases as the size increases. For examples, according to our DFT study, the cohesive energies per carbon atom for hydrogenated nanodiamonds with 35, 66, and 147 Table 1 Cohesive energies of dehydrogenated nanodiamonds per atom calculated using DFT
Fig. 6. Mulliken charge populations for atoms near one of the bigger fullerenelike caps in C275. The white atoms are 3-fold coordinated, while the gray ones denote the 4-fold coordinated atoms.
Nanodiamond
Energy (eV)
C66 C147 C275 C476 Diamond
−8.18 −8.52 −8.58 −8.83 −9.16
The cohesive energy for bulk diamond is also presented.
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nanodiamonds, most of which (including the LUMO and HOMO) come from the defects of the surface and interface. These states can be considered as localized defect states appeared in the gap. In addition to the quantum confinement effects, the size-dependent feature in the electronic properties of dehydrogenated nanodiamonds is contributed by the surface effect. This is different from hydrogenated nanodiamond, where size-dependent feature completely comes from the quantum confinement effects. Our results demonstrate that the surface atoms play an important role in the properties of dehydrogenated nanodiamonds. Acknowledgements
Fig. 7. Cohesive energies per atom calculated using semi-empirical tight-binding potential versus the number of carbon atoms in dehydrogenated nanodiamond. The broken line is just to guide the eyes.
carbon atoms are −10.5 eV, −10.2 eV, and −9.6 eV, respectively. This difference implies that surface energy plays an important role in the size-dependent stability of nanodiamonds. Because DFT method has a difficulty in dealing with very large system, we employ the environment-dependent tightbinding (TB) potential [25] to calculate the cohesive energies for nanodiamonds with larger sizes. This TB potential has been proven to be transferable enough to describe various structures of carbon, including nanodiamond [6,9]. As the TB potential is just a semi-empirical method, we use this potential just to have a qualitative study of the stability for nanodiamond. The four structures mentioned above, and another two dehydrogenated nanodiamonds with 741 and 1033 carbon atoms are relaxed using the steepest descent method with this TB potential. The cohesive energies per atom versus the number of carbon atoms are shown in Fig. 7. The sizes of these nanodiamonds vary from 0.8 nm to 2.2 nm. As the size of dehydrogenated nanodiamond increases, the absolute value of the obtained cohesive energy per atom increases in the whole size range. This means the larger the size for the dehydrogenated nanodiamonds, the more stable for the structure. 4. Conclusions The geometric structure and electronic properties of dehydrogenated nanodiamonds can be studied by quantum-mechanical density functional theory. The calculated results show that the core of dehydrogenated nanodiamond keeps diamond-like structure, and is quite ordered. However, the structure of the surface atoms is much disordered. Many defects appear in the surface as well as the interface. Calculations on the electronic properties exhibit that there are many states near the Fermi level and no visible gap is found in the DOS of dehydrogenated
The authors gratefully appreciate the financial support by the National Natural Science Foundation of China under Grant No. 50525204, the National Key Basic Research and Development Program (grant No 2004CB619301), Project 985—Automotive Engineering of Jilin University, and the Teaching and Research Award Program for the Outstanding Young Teachers in High Education Institutions (No. 2002359). References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
R.S. Lewis, M. Tang, J.F. Wecker, E. Anders, E. Stell, Nature 326 (1987) 160. R.S. Lewis, E. Anders, B.T. Draine, Nature 339 (1989) 117. N.R. Greiner, D.S. Phillips, J.D. Johnson, A.F. Volk, Nature 333 (1989) 440. Y.K. Chang, H.H. Hsieh, W.F. Pong, M.H. Tsai, F.Z. Chien, P.K. Tseng, L.C. Chen, T.Y. Wang, K.H. Chen, D.M. Bhusari, J.R. Yang, S.T. Lin, Phys. Rev. Lett. 82 (1999) 5377. K.E. Spear, J. Am. Ceram. Soc. 72 (1998) 171. J.Y. Raty, G. Galli, C. Bostedt, T. Van Buuren, L. Terminello, Phys. Rev. Lett. 90 (2003) 037401. J.Y. Raty, G. Galli, Comput. Phys. Commun. 169 (2005) 14. G.D. Lee, C.Z. Wang, J. Yu, E. Yoon, K.M. Ho, Phys. Rev. Lett. 91 (2003) 265701. G.C. McIntosh, M. Yoon, S. Berber, D. Tománek, Phys. Rev. B 70 (2004) 045401. A.J. Lu, B.C. Pan, J.G. Han, Phys. Rev. B 72 (2005) 035447. N.D. Drummond, A.J. Williamson, R.J. Needs, G. Galli, Phys. Rev. Lett. 95 (2005) 096801. A.S. Barnard, S.P. Russo, I.K. Snook, Phys. Rev. B 68 (2003) 073406. A.S. Barnard, S.P. Russo, I.K. Snook, Diamond Relat. Mater. 12 (2003) 1867. N. Park, S. Park, N.M. Hwang, J. Ihm, S. Tejima, H. Nakamura, Phys. Rev. B 69 (2004) 195411. A.S. Barnard, P. Zapol, J. Chem. Phys. 121 (2004) 4276. B. Delley, J. Chem. Phys. 92 (1990) 508. B. Delley, J. Chem. Phys. 113 (2000) 7756. J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. C.Q. Sun, H.L. Bai, B.K. Tay, S. Li, E.Y. Jiang, J. Phys. Chem. B 107 (2003) 7544. C.H. Lee, W.R.L. Lambrecht, B. Segall, P.C. Kelires, Th. Frauenheim, U. Stephan, Phys. Rev. B 49 (1994) 11448. B. Zheng, W.T. Zheng, K. Zhang, Q.B. Wen, J.Q. Zhu, S.H. Meng, X.D. He, J.C. Han, Carbon 44 (2006) 962. A.P. Alivisatos, Science 271 (1996) 933. M.D. Segall, R. Shah, C.J. Pickard, M.C. Payne, Phys. Rev. B 54 (1996) 16317. A.D. Yoffe, Adv. Phys. 42 (1993) 173. M.S. Tang, C.Z. Wang, C.T. Chan, K.M. Ho, Phys. Rev. B 53 (1996) 979.