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Structural stability and electronic properties of β-tetragonal boron: A first-principles study
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Structural stability and electronic properties of β-tetragonal boron: A first-principles study Wataru Hayami
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S0022-4596(14)00454-X http://dx.doi.org/10.1016/j.jssc.2014.10.012 YJSSC18667
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Journal of Solid State Chemistry
Received date: 28 July 2014 Revised date: 6 October 2014 Accepted date: 12 October 2014 Cite this article as: Wataru Hayami, Structural stability and electronic properties of β-tetragonal boron: A first-principles study, Journal of Solid State Chemistry, http://dx.doi.org/10.1016/j.jssc.2014.10.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Structural stability and electronic properties of b-tetragonal boron: A first-principles study Wataru Hayami* National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan
ABSTRACT It is known that elemental boron has five polymorphs: a- and b-rhombohedral, a- and b-tetragonal, and the high-pressure g phase. b-tetragonal (b-t) boron was first discovered in 1960, but there have been only a few studies since then. We have thoroughly investigated, using first-principles calculations, the atomic and electronic structures of b-t boron, the details of which were not known previously. The difficulty of calculation arises from the fact that b-t boron has a large unit cell that contains between 184 and 196 atoms, with 12 partially-occupied interstitial sites. This makes the number of configurations of interstitial atoms too great to calculate them all. By introducing assumptions based on symmetry and preliminary calculations, the number of configurations to calculate can be greatly reduced. It was eventually found that b-t boron has the lowest total energy, with 192 atoms (8 interstitial atoms) in an orthorhombic lattice. The total energy per atom were between those of a- and b-rhombohedral boron. Another tetragonal structure with 192 atoms was found to have a very close energy. The valence bands were fully filled and the gaps were about 1.16 to 1.54 eV, making it comparable to that of b-rhombohedral boron.
Keywords: b-tetragonal boron, density functional theory, Atomic structure, Electronic structure
* Corresponding author. Tel. +81 29 8604319 E-mail address: [email protected]
1. Introduction Boron is the lightest element in Group 13 of the periodic table and usually shows semiconductor characteristics. It is chemically closer to carbon and silicon than to other metallic elements of the same group, such as aluminum, gallium, indium and thallium. At normal pressures, boron is known to have four polymorphs [1 - 4], which are a-rhombohedral (a-rh), b-rhombohedral (b-rh), a-tetragonal (a-t) and b-tetragonal (b-t). A new orthorhombic phase under high pressure called g was discovered several years ago [5,6]. a-rh and b-rh boron have been studied most because a-rh boron has the simplest atomic structure and b-rh boron is the easiest to synthesize. a-t boron has a slightly complicated history. It was first reported by Laubengayer et al. in 1943 [3] but was later challenged by Amberger et al. [7], who claimed that it had contained impurities such as carbon or nitrogen. After many years, pure a-t boron was discovered in nanobelt form by Wang et al. [8], who also measured its physical properties such as electric conductivity [9]. We have proposed a structure model that contains 52 atoms per unit cell, based on first-principle calculations [10]. Recently, a-t boron was successfully synthesized under high pressure by Ekimov et al. [11] and its structure was in a good agreement with our model. In this paper, we mainly study b-t boron, which has been less studied than the other polymorphs. It was first discovered by Tally et al. in 1960 [4], and a structural analysis was conducted by Vlasse et al. [12] which hitherto was the only one on this material. Since these studies were published, there have been a few reports that observed b-t boron [13, 14] but none of its physical properties have yet been investigated. The synthesis of b-t boron itself was originally effected by chemical vapor deposition (CVD) using BBr3 gas [4] and appears not to be particularly difficult. A recent study predicts b-t boron to be stable in a high-pressure region between b-rh and g phases [15]. There have been only a few theoretical studies on b-t boron. We performed first-principles calculations assuming that the unit cell has 196 atoms [16]. Oganov et al. briefly mentioned that it has the lowest energy when comprised of 192 atoms, but did not present any comparison with other configurations [5]. The difficulty of theoretical study arises from the fact that b-t boron has several partially-occupied interstitial sites. Fig. 1 shows the structure model determined by Vlasse et al. [12] It has a tetragonal lattice with space group P41 or P43. The figure is drawn with P43 using the atomic coordinates listed in the paper. The upper figure is viewed from the <001> direction and the lower figure from <100>. The unit cell is indicated by a
green rectangle. The whole structure consists of two different subunits, depicted here in different colors. One is a double icosahedron B21 (orange) which comprises two icosahedra sharing a face. The other is a single icosahedron B12 (grey), which appears ubiquitously in boron and boron-rich solids. The arrays of B12 icosahedra run in the x and y directions. Double and single icosahedra are connected to each other. In addition to these subunits, there are four interstitial sites, namely B(23), B(25), B(25p) and B(26). The notations follow Reference [12]. B(25) (red) and B(25p) (green) are symmetrically placed with respect to B(23) (blue), and B(26) (yellow) is at the center of three icosahedra. Each interstitial site has four equivalent sites due to P43 symmetry. B(23) sites are fully occupied and B(25), B(25p) and B(26) sites are partially occupied. As a result, the number of atoms in the unit cell (N) can be between 184 and 196. N was experimentally estimated to be 192 by Talley et al. [4], and 189.88 by Vlasse et al. [12] Since the unit cell has 12 partially-occupied interstitial sites, it has a large number of configurations of interstitial atoms. When N = 190, for example, 6 interstitial sites of 12 are occupied, generating 12C6 = 924 configurations. The P43 symmetry may reduce the number of configurations to a quarter, but it is still too large a number for all to be calculated from first principles. In this study, we attempted to find the lowest energy configuration of b-t boron by adopting certain assumptions to reduce the number of configurations. We finally calculated 130 configurations from N = 184 196 from first principles and compared their total energies. The electronic structure of the lowest-energy configuration exhibited similar characteristics to those of other boron polymorphs. The methods of calculation and assumptions adopted there will be described in detail in the next section.
2. Computational details The calculations of the total energies were carried out with the lattice constants and geometries optimized. We used CPMD code, version 3.13.2, [17 - 19] which is based on the density functional theory (DFT) with plane waves and pseudopotentials [20, 21]. Norm-conserving Troullier-Martins type pseudopotentials [22] in the Kleinman-Bylander form [23] were used. The generalized gradient approximation (GGA) was included by means of the functional derived by Becke [24] and by Lee, Yang, and Parr [25]. An energy cutoff of 50 Ry was sufficient to provide a convergence of total energies and geometries. The calculations were done using
Monkhorst-Pack sampling [26] of a (2 ´ 2´ 2) mesh for all the calculations. A test calculation with a (3 ´ 3 ´ 3) mesh for a configuration with N = 192 showed that the difference in total energy per atom was about 0.01 meV. While the original structure has a tetragonal symmetry P43, randomly-placed interstitial atoms reduce the symmetry to triclinic. Test calculations, however, showed that interaxial angles hardly deviate from 90˚. This is probably because the interstitial atoms account for a negligible proportion of total atoms, and inter-icosahedral bonds are fairly strong, as shown below in the next section. The lattices were therefore assumed to be orthorhombic and only the lengths of three axes were optimized. As mentioned in the previous section, we adopted three assumptions to reduce the number of configuration to be calculated. They were: i) B(25) and B(25p) sites are energetically equivalent. This is because they are symmetrically placed with relation to the B(23) sites (Fig. 1). ii) N is limited to even numbers. An odd N would make the orbital at the Fermi level open, since boron has three valence electrons. Structures with an open orbital generally have higher energy per atom than those with a closed orbital. iii) N(25), N(25p), N(26) are even, where N(s) means the number of atoms at the interstitial site B(s). When N is even, all of N(s) are even, or one of them is even and the other two are odd. First, suppose N(25) and N(26) are odd, 3 and 1, for example. B(25) and B(26) sites are sufficiently distant from each other and can be regarded as independent. Then, according to the difference in the binding energy, transferring an atom from B(25) to B(26), or from B(26) to B(25), lowers the total energy, and N(25) and N(26) become even. This holds for B(25p) sites as well, since the B(25p) and B(25) sites are energetically equivalent. Second, suppose N(25) and N(25p) are odd: 3 and 1, for example. Our test calculations showed that placing atoms in paired B(25) and B(25p) sites makes the total energy about 8.9 meV higher than when placing them in separate B(25) and B(25p) sites. If the former is the case, transferring an atom from the B(25p) site to the open B(25) site lowers the total energy, and N(25) and N(25p) become even. If the latter is the case, the transfer does not change the total energy, but odd-numbered configurations can be omitted for the purpose of finding the lowest energy configuration.
These assumptions reduce the number of configurations to calculate to 130 as listed in Table 1. We confirmed by hand that the list covers all possible configurations for N from 184 to 196. The numbers in parentheses (1–4) specify the occupied positions in each interstitial site, whose fractional coordinates are provided below. These coordinates are generated from those in Reference [12] by applying P43 symmetry. When there are several parentheses in a line, one of them should be chosen for each site. For example, in the bottom line of N = 188, the B(25) site takes (12) or (13), and the B(26) site takes one out of six parentheses. Thus the configuration is expressed like (12)(-)(12), where the parentheses are for B(25), B(25p) and B(26) in order. The asterisks denote the lowest energy configuration in each N.
3. Results and discussion Calculated total energies per atom for the configurations in Table 1 are plotted in Fig. 2. They are relative to the value of N = 184. The lattices and geometries were optimized. It is observed that the total energy per atom is high at N = 184 and 196, and reaches a minimum at N = 192 with configuration (13)(24)(1234). The lattice parameters are a = 10.16 Å, b = 10.31 Å and c = 14.15 Å (orthorhombic). We compared the total energy per atom of b-t boron with those of a-rh and b-rh boron calculated using the same pseudopotential and cutoff energy. The structure model of b-rh boron, containing 107 atoms in a unit cell, was adopted from a report by Widom et al. [27] We found that b-t boron is positioned between a-rh and b-rh boron: its total energy per atom is 7.1 meV lower than that of a-rh boron, and 10.4 meV higher than that of b-rh boron. This was supported by the CVD experiment in which b-rh boron was mostly deposited with a small amount of b-t boron but no a-rh boron was observed [13]. The configuration for the second lowest energy is (1234)(-)(1234) which is identical to that reported by Oganov et al. [5] The lattice parameters are a = 10.21 Å, b = 10.21 Å and c = 14.18 Å (tetragonal). The energy difference between these configurations (0.12 meV) is very small, but it is significant since the total energies of the lowest and the second lowest energy configuration were calculated using the same number of atoms, cutoff energy, and k-point sampling. Table 2 shows a comparison of the calculated and experimental lattice parameters. They are in good agreement with each other except for the c value of Experiment 3. The sample in Experiment 3 was synthesized under high pressure, while in Experiments 1 and 2, they were synthesized by
CVD: this might be the cause of the difference. Both calculated values are so close that it cannot be determined which one is closer to the experimental values. In fact, b-t boron is synthesized at high temperature, so the free energy instead of the total energy should be compared. The free energy includes contributions from the phonon modes but we do not deal with them in this study. We consider that real samples should have a mixed configuration, since the energy differences between them are so small. The occupancy ratios for B(25), B(25p), and B(26) sites determined by Vlasse et al. [12] were 0.50, 0.50 and 0.47, respectively. This indicates that several configurations are included in addition to the lowest and the second lowest configurations. Fig. 2 reveals that the total energy at N = 190 is statistically lower than that at N = 194. It is therefore likely that the unit cell would have fewer atoms than 192 if several configurations are mixed together. In Reference [12], the unit cell contains 189.88 atoms, which agrees with our estimate. Fig. 3 exhibits the electronic densities of states (DOS) of the lowest configuration for each N. The configurations are marked by an asterisk in Table 1. The Fermi energy is set to zero. When N = 184 - 190, the Fermi energy lies in the valence band and the material is metallic. When N = 192, the Fermi energy rises to the upper edge of the valence band and the material becomes semiconductive. When N = 194 - 196, the Fermi energy is in the conduction band and the material is again metallic. The valence band consists of bonding orbitals and the conduction band consists of antibonding orbitals. The total energy per atom therefore reaches a minimum when the valence band is perfectly filled at N = 192. This mechanism also works in a-t boron [10]. Similarly, it was observed that when metal atoms are doped into b-rh boron, some boron atoms desorb in order to adjust the Fermi energy to the valence-band edge (self-compensation) [28]. When N = 192, the band gap of the lowest energy configuration is 1.16 eV and that of the second lowest is 1.54 eV: this seems to be comparable to that of b-rh boron (1.29–1.50 eV) [29]. It must be noticed, however, the band gap is underestimated in the DFT calculation and the real value should be a little larger. There are as yet no experimental reports on the band gap of b-t boron. The band structure of the lowest energy configuration for N = 192 is shown in Fig. 4. The Fermi energy is set to zero, and the region close to the Fermi energy is enlarged. There is little dispersion due to the large unit cell,
but the top of the valence band and the bottom of the conduction band are observed at the G point, making the gap direct. Fig. 5 shows the electron density distribution of the same structure viewed from the <100> direction. The light blue spheres are boron atoms constituting double and single icosahedra. The left-hand figure illustrates the isosurface at density d = 0.9 electron/a.u.3 (yellow). It is observed that electrons are uniformly distributed within the icosahedra, suggesting that their bonds are three-center bonds. The right-hand figure is the isosurface at d = 1.2 electron/a.u.3 (orange). This indicates that inter-icosahedral bonds have approximately 30 % greater density than intra-icosahedral bonds. These characters are common to a-rh and a-t boron [10, 30] and probably to b-rh boron, judging from the bond lengths [31]. The electron density distribution is reflected in the bond lengths, which are summarized in Table 3. The experimental bond lengths are measured from the atomic coordinates provided by Vlasse et al. [12] that slightly differ from those in Table 2 in the same report. Calculated bond lengths are those of the lowest energy configuration for N = 192, some values of which are listed separately with respect to a and b axes since the lattice is orthorhombic. On the whole, they agree with each other, and in particular, the tendency that inter-icosahedral bonds (B12-B12, B21-B21, B12-B21) are shorter than intra-icosahedral bonds (within B12 and B21) is well reproduced in the calculation. This tendency is expected from the electron distribution in Fig. 5 and is also observed in the other a-rh, b-rh and a-t boron polymorphs. g-boron is different from the others in that the inter-icosahedral bonds are as long as the intra-icosahedral bonds [5]. The only difference between the experiment and calculation is the bond lengths around B(26) sites that are 1.881 Å experimentally but 1.763 Å calculated. B(26) sites are located at the center of three B12 and are represented by yellow spheres in Fig. 1. In the experiment, an atom at B(26) site and the coordinating six atoms form a hexagonal pyramid, whereas they form a planar hexagon in the calculation, which causes the difference in their bond lengths. The reason for the discrepancy is not clear but may be attributable to the difference in the configurations, since we believe the experimental structure has a mixed configuration of interstitial atoms as mentioned before. No structural analysis other than that by Vlasse et al. [12] of b-t boron has been reported so further experimental studies are desired.
4. Conclusions We found, using first-principles calculations, the lowest-energy structure for b-t boron. It has 8 interstitial atoms (N = 192) with the configuration (13)(24)(1234). The lattice parameters are a = 10.16 Å, b = 10.31 Å and c = 14.15 Å (orthorhombic). The total energy per atom is 7.1 meV lower than that of a-rh boron, and 10.4 meV higher than that of b-rh boron. The second-lowest structure also has 8 interstitial atoms with the configuration (1234)(-)(1234), which is identical to that suggested by Oganov et al. [5] The lattice parameters are a = b = 10.21 Å and c = 14.18 Å (tetragonal). The energy difference between these two configurations is so small (0.12 meV) that practically, they will be mixed at finite temperatures. Since the energy at N = 190 is lower than that at N = 194 (Fig. 2), we would expect samples synthesized at high temperatures to have statistically fewer atoms than 192. This is in agreement with the experiments by Vlasse et al. [12] The lowest-energy configuration has the Fermi level just at the top of the valence band (Fig. 3), which explains why it has the lowest energy. The band gap is direct (Fig. 4). The electron density is about 30 % higher around the inter-icosahedral bonds than around the intra-icosahedral bonds (Fig. 5), which influences the inter-icosahedral bonds and makes them shorter than the intra-icosahedral bonds (Table 3). These characteristics relating to electronic structure are common to a-rh, b-rh and a-t boron. There is a discrepancy between the experimental and the calculated atomic structure. In Vlasse et al.’s experiments [12], an atom at the B(26) site and the six coordinating atoms form a hexagonal pyramid, whereas in our calculation, they form a planar hexagon. The reason for this is not clear but might be due to differences in configurations and occupation rates.
Acknowledgments The author wishes to thank Drs. Takaho Tanaka and Shigeki Otani for valuable discussions and their insightful comments.
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[2] R.E. Hughes, C.H.L. Kennard, D.B. Sullenger, H.A. Weakliem, D.E. Sands, J.L. Hoard, J. Amer. Chem. Soc. 85 (1963) 361. [3] A.W. Laubengayer, D.T. Hurd, A. E. Newkirk, J. L.Hoard, J. Am. Chem. Soc. 65 (1943) 1924. [4] C.P. Talley, S. LaPlaca, B. Post, Acta Cryst. 13 (1960) 271. [5] A.R. Oganov, J.H. Chen, C. Gatti Y.Z. Ma, Y.M. Ma, C.W. Glass, Z.X. Liu, T. Yu, O.O. Kurakevych, V.L. Solozhenko, Nature 457 (2009) 863. [6] E.Y. Zarechnaya, L. Dubrovinsky, N. Dubrovinskaia, N. Miyajima, Y. Filinchuk, D. Chernyshov, V. Dmitriev, Sci. Technol. Adv. Mater. 9 (2008) 044209. [7] E. Amberger, K. Ploog, J. Less. Common. Metals 23 (1971) 21. [8] Z. Wang, Y. Shimizu, T. Sasaki, K. Kawaguchi, K. Kimura, N. Koshizaki, Chem. Phys. Lett. 368 (2003) 663. [9] K. Kirihara, Z. Wang, K. Kawaguchi, Y. Shimizu, T. Sasaki, N. Koshizaki, K. Soga, K. Kimura, Appl. Phys. Lett. 86 (2005) 212101. [10] W. Hayami, S. Otani, J. Solid State Chem. 183 (2010) 1521. [11] E.A. Ekimov, I.P. Zibrov, Sci. Technol. Adv. Mater. 12 (2011) 055009. [12] M. Vlasse, R. Naslain, J.S. Kasper, K. Ploog, J. Less-Common Met. 67 (1979) 1. [13] C.S. Park, J.S. Yoo, J.S. Chun, Thin Solid Films, 131 (1985) 205. [14] Y. Ma, C. T. Prewitt, G. Zou,. H-k. Mao, R. J. Hemley, Phys. Rev. B 67 (2003) 174116. [15] J. Qin, T. Irifune, H. Dekura, H. Ohfuji, N. Nishiyama, L. Lei, T. Shinmei, Phys. Rev. B 85 (2012) 014107. [16] W. Hayami, S. Otani, J. Phys. Chem. C 111 (2007) 10394. [17] CPMD, Copyright IBM Corp. 1990-2008, Copyright MPI für Festkörperforschung Stuttgart 1997-2001, (/www.cpmd.org/S). [18] D. Marx, J. Hutter, Modern Methods and Algorithms of Quantum Chemistry, NIC Series Vol. 1, John von Neumann Institute for Computing, Jülich, 2000, pp. 301̽449. [19] W. Andreoni, A. Curioni, Parallel Comput. 26 (2000) 819. [20] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864.
[21] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [22] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [23] L. Kleinman, D.M. Bylander, Phys. Rev. Lett. 48 (1982) 1425. [24] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [25] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [26] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [27] M. Widom, M. Mihalkovič, Phys. Rev. B 77 (2008) 064113. [28] H. Hyodo, S. Araake, S. Hosoi, K. Soga, Y. Sato, M. Terauchi, K. Kimura, Phys. Rev. B 77 (2008) 024515. [29] H. Werheit, M. Laux, U. Kuhlmann, Phys. Stat. Sol. (b) 176 (1993) 415. [30] S. Lee, D.M. Bylander, L. Kleinman, Phys. Rev. B 42 (1990) 1316. [31] G.A. Slack, C.I. Hejna, M.F. Garbauskas, J.S. Kasper, J. Solid State Chem. 76 (1988) 52.
Table 1 Configurations of interstitial atoms for each atom number N. Numbers in parentheses denote occupied sites in each interstitial site whose fractional coordinates are given below. Site
N B(25)
B(25p)
B(26)
184
(-)
(-)
(-)
186
(-)
(-)
(12), (13)*
186
(12), (13)
(-)
(-)
188
(-)
(-)
(1234)
188
(1234)*
(-)
(-)
188
(12)
(12), (23), (34)
(-)
188
(13)
(12), (23), (13), (24)
(-)
188
(12), (13)
(-)
(12), (13), (14), (23), (24), (34)
190
(12), (13)
(-)
(1234)
190
(12), (13)
(1234)
(-)
190
(1234)
(-)
(12)*, (13)
190
(12), (13)
(12), (13), (14), (23), (24), (34)
(12), (13), (14), (23), (24), (34)
192
(1234 )
(1234)
(-)
192
(1234)
(-)
(1234)
192
(12)
(12), (23), (34)
(1234)
192
(13)
(12), (23), (13), (24)*
(1234)
192
(12), (13)
(1234)
(12), (13), (14), (23), (24), (34)
194
(1234 )
(1234)
(12), (13)
194
(12)*, (13)
(1234)
(1234)
196
(1234)
(1234)
(1234)
Fractional coordinates
Site
x
y
z
B(25) (1)
0.145
0.433
0.466
B(25) (2)
0.567
0.145
0.216
B(25) (3)
0.855
0.567
0.966
B(25) (4)
0.433
0.855
0.716
B(25p) (1)
0.061
0.360
0.035
B(25p) (2)
0.640
0.061
0.785
B(25p) (3)
0.939
0.640
0.535
B(25p) (4)
0.360
0.939
0.285
B(26) (1)
0.225
0.244
0.732
B(26) (2)
0.756
0.225
0.482
B(26) (3)
0.775
0.756
0.232
B(26) (4)
0.244
0.775
0.982
Table 2 Comparison of lattice parameters Lattice parameter (Å)
Remarks
a
b
c
Calculation 1
10.16
10.31
14.15
Lowest energy
Calculation 2
10.21
10.21
14.18
Second lowest energy
Experiment 1
10.12
10.12
14.17
CVD, Talley et al. [4]
Experiment 2
10.14
10.14
14.17
CVD, Vlasse et al. [12]
Experiment 3
10.161
10.161
13.71
High pressure, Ma et al. [14]
Table 3 Comparison of bond lengths. a and b denote the direction of the bond. Bond
Average distance (Å) Experiment [12]
Calculation
Within B12 unit
1.796
1.811 (a) 1.811 (b)
B12-B12
1.785
1.742 (a) 1.778 (b)
Within B21 unit
1.810
1.815
B21-B21
1.707
1.710
B12-B21 (horizontal)
1.643
1.674 (a) 1.700 (b)
B12-B21 (vertical)
1.706
1.707
B(23)-B(25)
1.548
1.598
Around B(26)
1.881
1.763
Figure captions
Fig. 1. Structure of b-tetragonal boron. The upper figure is viewed from the <001> direction and the lower is from the <100> direction. Boron atoms are located at the apexes of the icosahedra (grey) and double icosahedra (orange). Colored spheres denote interstitial sites: 23 (blue), 25 (red), 25p (green), and 26 (yellow).
Fig. 2. Total energies per atom for N = 184 to 196. The energy for N = 184 is set to 0.
Fig. 3. Electronic densities of states for N = 184 to 196. The configurations are those having the lowest energy for each N, marked by an asterisk in Table 1. The Fermi energy is set to zero in all the graphs.
Fig. 4. Electronic band structure for the lowest-energy configuration (N = 192). The Fermi energy is set to zero.
Fig. 5. Electronic density distribution for the lowest-energy configuration (N = 192) viewed from the <100> direction. Left: isosurface (yellow) at d = 0.09 electrons/a.u.3 Right: isosurface (orange) at d = 0.12 electrons/a.u.3
TOC legend:
Electronic density distribution for the lowest-energy configuration (N = 192) viewed from the <100> direction. Left: isosurface (yellow) at d = 0.09 electrons/a.u.3 Right: isosurface (orange) at d = 0.12 electrons/a.u.3
Highlights ࣭b-tetragonal boron was thoroughly investigated using first-principles calculations. ࣭The lowest energy structure contains 192 atoms in an orthorhombic lattice. ࣭Another tetragonal structure with 192 atoms has a very close energy. ࣭The total energy per atom is between those of a- and b-rhombohedral boron. ࣭The band gap of the lowest energy structure is about 1.16 to 1.54 eV.
Graphical Abstract (TOC Figure)
Structural stability and electronic properties of β-tetragonal boron: A first-principles study Wataru Hayami
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PII: DOI: Reference:
S0022-4596(14)00454-X http://dx.doi.org/10.1016/j.jssc.2014.10.012 YJSSC18667
To appear in:
Journal of Solid State Chemistry
Received date: 28 July 2014 Revised date: 6 October 2014 Accepted date: 12 October 2014 Cite this article as: Wataru Hayami, Structural stability and electronic properties of β-tetragonal boron: A first-principles study, Journal of Solid State Chemistry, http://dx.doi.org/10.1016/j.jssc.2014.10.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Structural stability and electronic properties of b-tetragonal boron: A first-principles study Wataru Hayami* National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan
ABSTRACT It is known that elemental boron has five polymorphs: a- and b-rhombohedral, a- and b-tetragonal, and the high-pressure g phase. b-tetragonal (b-t) boron was first discovered in 1960, but there have been only a few studies since then. We have thoroughly investigated, using first-principles calculations, the atomic and electronic structures of b-t boron, the details of which were not known previously. The difficulty of calculation arises from the fact that b-t boron has a large unit cell that contains between 184 and 196 atoms, with 12 partially-occupied interstitial sites. This makes the number of configurations of interstitial atoms too great to calculate them all. By introducing assumptions based on symmetry and preliminary calculations, the number of configurations to calculate can be greatly reduced. It was eventually found that b-t boron has the lowest total energy, with 192 atoms (8 interstitial atoms) in an orthorhombic lattice. The total energy per atom were between those of a- and b-rhombohedral boron. Another tetragonal structure with 192 atoms was found to have a very close energy. The valence bands were fully filled and the gaps were about 1.16 to 1.54 eV, making it comparable to that of b-rhombohedral boron.
Keywords: b-tetragonal boron, density functional theory, Atomic structure, Electronic structure
* Corresponding author. Tel. +81 29 8604319 E-mail address: [email protected]
1. Introduction Boron is the lightest element in Group 13 of the periodic table and usually shows semiconductor characteristics. It is chemically closer to carbon and silicon than to other metallic elements of the same group, such as aluminum, gallium, indium and thallium. At normal pressures, boron is known to have four polymorphs [1 - 4], which are a-rhombohedral (a-rh), b-rhombohedral (b-rh), a-tetragonal (a-t) and b-tetragonal (b-t). A new orthorhombic phase under high pressure called g was discovered several years ago [5,6]. a-rh and b-rh boron have been studied most because a-rh boron has the simplest atomic structure and b-rh boron is the easiest to synthesize. a-t boron has a slightly complicated history. It was first reported by Laubengayer et al. in 1943 [3] but was later challenged by Amberger et al. [7], who claimed that it had contained impurities such as carbon or nitrogen. After many years, pure a-t boron was discovered in nanobelt form by Wang et al. [8], who also measured its physical properties such as electric conductivity [9]. We have proposed a structure model that contains 52 atoms per unit cell, based on first-principle calculations [10]. Recently, a-t boron was successfully synthesized under high pressure by Ekimov et al. [11] and its structure was in a good agreement with our model. In this paper, we mainly study b-t boron, which has been less studied than the other polymorphs. It was first discovered by Tally et al. in 1960 [4], and a structural analysis was conducted by Vlasse et al. [12] which hitherto was the only one on this material. Since these studies were published, there have been a few reports that observed b-t boron [13, 14] but none of its physical properties have yet been investigated. The synthesis of b-t boron itself was originally effected by chemical vapor deposition (CVD) using BBr3 gas [4] and appears not to be particularly difficult. A recent study predicts b-t boron to be stable in a high-pressure region between b-rh and g phases [15]. There have been only a few theoretical studies on b-t boron. We performed first-principles calculations assuming that the unit cell has 196 atoms [16]. Oganov et al. briefly mentioned that it has the lowest energy when comprised of 192 atoms, but did not present any comparison with other configurations [5]. The difficulty of theoretical study arises from the fact that b-t boron has several partially-occupied interstitial sites. Fig. 1 shows the structure model determined by Vlasse et al. [12] It has a tetragonal lattice with space group P41 or P43. The figure is drawn with P43 using the atomic coordinates listed in the paper. The upper figure is viewed from the <001> direction and the lower figure from <100>. The unit cell is indicated by a
green rectangle. The whole structure consists of two different subunits, depicted here in different colors. One is a double icosahedron B21 (orange) which comprises two icosahedra sharing a face. The other is a single icosahedron B12 (grey), which appears ubiquitously in boron and boron-rich solids. The arrays of B12 icosahedra run in the x and y directions. Double and single icosahedra are connected to each other. In addition to these subunits, there are four interstitial sites, namely B(23), B(25), B(25p) and B(26). The notations follow Reference [12]. B(25) (red) and B(25p) (green) are symmetrically placed with respect to B(23) (blue), and B(26) (yellow) is at the center of three icosahedra. Each interstitial site has four equivalent sites due to P43 symmetry. B(23) sites are fully occupied and B(25), B(25p) and B(26) sites are partially occupied. As a result, the number of atoms in the unit cell (N) can be between 184 and 196. N was experimentally estimated to be 192 by Talley et al. [4], and 189.88 by Vlasse et al. [12] Since the unit cell has 12 partially-occupied interstitial sites, it has a large number of configurations of interstitial atoms. When N = 190, for example, 6 interstitial sites of 12 are occupied, generating 12C6 = 924 configurations. The P43 symmetry may reduce the number of configurations to a quarter, but it is still too large a number for all to be calculated from first principles. In this study, we attempted to find the lowest energy configuration of b-t boron by adopting certain assumptions to reduce the number of configurations. We finally calculated 130 configurations from N = 184 196 from first principles and compared their total energies. The electronic structure of the lowest-energy configuration exhibited similar characteristics to those of other boron polymorphs. The methods of calculation and assumptions adopted there will be described in detail in the next section.
2. Computational details The calculations of the total energies were carried out with the lattice constants and geometries optimized. We used CPMD code, version 3.13.2, [17 - 19] which is based on the density functional theory (DFT) with plane waves and pseudopotentials [20, 21]. Norm-conserving Troullier-Martins type pseudopotentials [22] in the Kleinman-Bylander form [23] were used. The generalized gradient approximation (GGA) was included by means of the functional derived by Becke [24] and by Lee, Yang, and Parr [25]. An energy cutoff of 50 Ry was sufficient to provide a convergence of total energies and geometries. The calculations were done using
Monkhorst-Pack sampling [26] of a (2 ´ 2´ 2) mesh for all the calculations. A test calculation with a (3 ´ 3 ´ 3) mesh for a configuration with N = 192 showed that the difference in total energy per atom was about 0.01 meV. While the original structure has a tetragonal symmetry P43, randomly-placed interstitial atoms reduce the symmetry to triclinic. Test calculations, however, showed that interaxial angles hardly deviate from 90˚. This is probably because the interstitial atoms account for a negligible proportion of total atoms, and inter-icosahedral bonds are fairly strong, as shown below in the next section. The lattices were therefore assumed to be orthorhombic and only the lengths of three axes were optimized. As mentioned in the previous section, we adopted three assumptions to reduce the number of configuration to be calculated. They were: i) B(25) and B(25p) sites are energetically equivalent. This is because they are symmetrically placed with relation to the B(23) sites (Fig. 1). ii) N is limited to even numbers. An odd N would make the orbital at the Fermi level open, since boron has three valence electrons. Structures with an open orbital generally have higher energy per atom than those with a closed orbital. iii) N(25), N(25p), N(26) are even, where N(s) means the number of atoms at the interstitial site B(s). When N is even, all of N(s) are even, or one of them is even and the other two are odd. First, suppose N(25) and N(26) are odd, 3 and 1, for example. B(25) and B(26) sites are sufficiently distant from each other and can be regarded as independent. Then, according to the difference in the binding energy, transferring an atom from B(25) to B(26), or from B(26) to B(25), lowers the total energy, and N(25) and N(26) become even. This holds for B(25p) sites as well, since the B(25p) and B(25) sites are energetically equivalent. Second, suppose N(25) and N(25p) are odd: 3 and 1, for example. Our test calculations showed that placing atoms in paired B(25) and B(25p) sites makes the total energy about 8.9 meV higher than when placing them in separate B(25) and B(25p) sites. If the former is the case, transferring an atom from the B(25p) site to the open B(25) site lowers the total energy, and N(25) and N(25p) become even. If the latter is the case, the transfer does not change the total energy, but odd-numbered configurations can be omitted for the purpose of finding the lowest energy configuration.
These assumptions reduce the number of configurations to calculate to 130 as listed in Table 1. We confirmed by hand that the list covers all possible configurations for N from 184 to 196. The numbers in parentheses (1–4) specify the occupied positions in each interstitial site, whose fractional coordinates are provided below. These coordinates are generated from those in Reference [12] by applying P43 symmetry. When there are several parentheses in a line, one of them should be chosen for each site. For example, in the bottom line of N = 188, the B(25) site takes (12) or (13), and the B(26) site takes one out of six parentheses. Thus the configuration is expressed like (12)(-)(12), where the parentheses are for B(25), B(25p) and B(26) in order. The asterisks denote the lowest energy configuration in each N.
3. Results and discussion Calculated total energies per atom for the configurations in Table 1 are plotted in Fig. 2. They are relative to the value of N = 184. The lattices and geometries were optimized. It is observed that the total energy per atom is high at N = 184 and 196, and reaches a minimum at N = 192 with configuration (13)(24)(1234). The lattice parameters are a = 10.16 Å, b = 10.31 Å and c = 14.15 Å (orthorhombic). We compared the total energy per atom of b-t boron with those of a-rh and b-rh boron calculated using the same pseudopotential and cutoff energy. The structure model of b-rh boron, containing 107 atoms in a unit cell, was adopted from a report by Widom et al. [27] We found that b-t boron is positioned between a-rh and b-rh boron: its total energy per atom is 7.1 meV lower than that of a-rh boron, and 10.4 meV higher than that of b-rh boron. This was supported by the CVD experiment in which b-rh boron was mostly deposited with a small amount of b-t boron but no a-rh boron was observed [13]. The configuration for the second lowest energy is (1234)(-)(1234) which is identical to that reported by Oganov et al. [5] The lattice parameters are a = 10.21 Å, b = 10.21 Å and c = 14.18 Å (tetragonal). The energy difference between these configurations (0.12 meV) is very small, but it is significant since the total energies of the lowest and the second lowest energy configuration were calculated using the same number of atoms, cutoff energy, and k-point sampling. Table 2 shows a comparison of the calculated and experimental lattice parameters. They are in good agreement with each other except for the c value of Experiment 3. The sample in Experiment 3 was synthesized under high pressure, while in Experiments 1 and 2, they were synthesized by
CVD: this might be the cause of the difference. Both calculated values are so close that it cannot be determined which one is closer to the experimental values. In fact, b-t boron is synthesized at high temperature, so the free energy instead of the total energy should be compared. The free energy includes contributions from the phonon modes but we do not deal with them in this study. We consider that real samples should have a mixed configuration, since the energy differences between them are so small. The occupancy ratios for B(25), B(25p), and B(26) sites determined by Vlasse et al. [12] were 0.50, 0.50 and 0.47, respectively. This indicates that several configurations are included in addition to the lowest and the second lowest configurations. Fig. 2 reveals that the total energy at N = 190 is statistically lower than that at N = 194. It is therefore likely that the unit cell would have fewer atoms than 192 if several configurations are mixed together. In Reference [12], the unit cell contains 189.88 atoms, which agrees with our estimate. Fig. 3 exhibits the electronic densities of states (DOS) of the lowest configuration for each N. The configurations are marked by an asterisk in Table 1. The Fermi energy is set to zero. When N = 184 - 190, the Fermi energy lies in the valence band and the material is metallic. When N = 192, the Fermi energy rises to the upper edge of the valence band and the material becomes semiconductive. When N = 194 - 196, the Fermi energy is in the conduction band and the material is again metallic. The valence band consists of bonding orbitals and the conduction band consists of antibonding orbitals. The total energy per atom therefore reaches a minimum when the valence band is perfectly filled at N = 192. This mechanism also works in a-t boron [10]. Similarly, it was observed that when metal atoms are doped into b-rh boron, some boron atoms desorb in order to adjust the Fermi energy to the valence-band edge (self-compensation) [28]. When N = 192, the band gap of the lowest energy configuration is 1.16 eV and that of the second lowest is 1.54 eV: this seems to be comparable to that of b-rh boron (1.29–1.50 eV) [29]. It must be noticed, however, the band gap is underestimated in the DFT calculation and the real value should be a little larger. There are as yet no experimental reports on the band gap of b-t boron. The band structure of the lowest energy configuration for N = 192 is shown in Fig. 4. The Fermi energy is set to zero, and the region close to the Fermi energy is enlarged. There is little dispersion due to the large unit cell,
but the top of the valence band and the bottom of the conduction band are observed at the G point, making the gap direct. Fig. 5 shows the electron density distribution of the same structure viewed from the <100> direction. The light blue spheres are boron atoms constituting double and single icosahedra. The left-hand figure illustrates the isosurface at density d = 0.9 electron/a.u.3 (yellow). It is observed that electrons are uniformly distributed within the icosahedra, suggesting that their bonds are three-center bonds. The right-hand figure is the isosurface at d = 1.2 electron/a.u.3 (orange). This indicates that inter-icosahedral bonds have approximately 30 % greater density than intra-icosahedral bonds. These characters are common to a-rh and a-t boron [10, 30] and probably to b-rh boron, judging from the bond lengths [31]. The electron density distribution is reflected in the bond lengths, which are summarized in Table 3. The experimental bond lengths are measured from the atomic coordinates provided by Vlasse et al. [12] that slightly differ from those in Table 2 in the same report. Calculated bond lengths are those of the lowest energy configuration for N = 192, some values of which are listed separately with respect to a and b axes since the lattice is orthorhombic. On the whole, they agree with each other, and in particular, the tendency that inter-icosahedral bonds (B12-B12, B21-B21, B12-B21) are shorter than intra-icosahedral bonds (within B12 and B21) is well reproduced in the calculation. This tendency is expected from the electron distribution in Fig. 5 and is also observed in the other a-rh, b-rh and a-t boron polymorphs. g-boron is different from the others in that the inter-icosahedral bonds are as long as the intra-icosahedral bonds [5]. The only difference between the experiment and calculation is the bond lengths around B(26) sites that are 1.881 Å experimentally but 1.763 Å calculated. B(26) sites are located at the center of three B12 and are represented by yellow spheres in Fig. 1. In the experiment, an atom at B(26) site and the coordinating six atoms form a hexagonal pyramid, whereas they form a planar hexagon in the calculation, which causes the difference in their bond lengths. The reason for the discrepancy is not clear but may be attributable to the difference in the configurations, since we believe the experimental structure has a mixed configuration of interstitial atoms as mentioned before. No structural analysis other than that by Vlasse et al. [12] of b-t boron has been reported so further experimental studies are desired.
4. Conclusions We found, using first-principles calculations, the lowest-energy structure for b-t boron. It has 8 interstitial atoms (N = 192) with the configuration (13)(24)(1234). The lattice parameters are a = 10.16 Å, b = 10.31 Å and c = 14.15 Å (orthorhombic). The total energy per atom is 7.1 meV lower than that of a-rh boron, and 10.4 meV higher than that of b-rh boron. The second-lowest structure also has 8 interstitial atoms with the configuration (1234)(-)(1234), which is identical to that suggested by Oganov et al. [5] The lattice parameters are a = b = 10.21 Å and c = 14.18 Å (tetragonal). The energy difference between these two configurations is so small (0.12 meV) that practically, they will be mixed at finite temperatures. Since the energy at N = 190 is lower than that at N = 194 (Fig. 2), we would expect samples synthesized at high temperatures to have statistically fewer atoms than 192. This is in agreement with the experiments by Vlasse et al. [12] The lowest-energy configuration has the Fermi level just at the top of the valence band (Fig. 3), which explains why it has the lowest energy. The band gap is direct (Fig. 4). The electron density is about 30 % higher around the inter-icosahedral bonds than around the intra-icosahedral bonds (Fig. 5), which influences the inter-icosahedral bonds and makes them shorter than the intra-icosahedral bonds (Table 3). These characteristics relating to electronic structure are common to a-rh, b-rh and a-t boron. There is a discrepancy between the experimental and the calculated atomic structure. In Vlasse et al.’s experiments [12], an atom at the B(26) site and the six coordinating atoms form a hexagonal pyramid, whereas in our calculation, they form a planar hexagon. The reason for this is not clear but might be due to differences in configurations and occupation rates.
Acknowledgments The author wishes to thank Drs. Takaho Tanaka and Shigeki Otani for valuable discussions and their insightful comments.
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Table 1 Configurations of interstitial atoms for each atom number N. Numbers in parentheses denote occupied sites in each interstitial site whose fractional coordinates are given below. Site
N B(25)
B(25p)
B(26)
184
(-)
(-)
(-)
186
(-)
(-)
(12), (13)*
186
(12), (13)
(-)
(-)
188
(-)
(-)
(1234)
188
(1234)*
(-)
(-)
188
(12)
(12), (23), (34)
(-)
188
(13)
(12), (23), (13), (24)
(-)
188
(12), (13)
(-)
(12), (13), (14), (23), (24), (34)
190
(12), (13)
(-)
(1234)
190
(12), (13)
(1234)
(-)
190
(1234)
(-)
(12)*, (13)
190
(12), (13)
(12), (13), (14), (23), (24), (34)
(12), (13), (14), (23), (24), (34)
192
(1234 )
(1234)
(-)
192
(1234)
(-)
(1234)
192
(12)
(12), (23), (34)
(1234)
192
(13)
(12), (23), (13), (24)*
(1234)
192
(12), (13)
(1234)
(12), (13), (14), (23), (24), (34)
194
(1234 )
(1234)
(12), (13)
194
(12)*, (13)
(1234)
(1234)
196
(1234)
(1234)
(1234)
Fractional coordinates
Site
x
y
z
B(25) (1)
0.145
0.433
0.466
B(25) (2)
0.567
0.145
0.216
B(25) (3)
0.855
0.567
0.966
B(25) (4)
0.433
0.855
0.716
B(25p) (1)
0.061
0.360
0.035
B(25p) (2)
0.640
0.061
0.785
B(25p) (3)
0.939
0.640
0.535
B(25p) (4)
0.360
0.939
0.285
B(26) (1)
0.225
0.244
0.732
B(26) (2)
0.756
0.225
0.482
B(26) (3)
0.775
0.756
0.232
B(26) (4)
0.244
0.775
0.982
Table 2 Comparison of lattice parameters Lattice parameter (Å)
Remarks
a
b
c
Calculation 1
10.16
10.31
14.15
Lowest energy
Calculation 2
10.21
10.21
14.18
Second lowest energy
Experiment 1
10.12
10.12
14.17
CVD, Talley et al. [4]
Experiment 2
10.14
10.14
14.17
CVD, Vlasse et al. [12]
Experiment 3
10.161
10.161
13.71
High pressure, Ma et al. [14]
Table 3 Comparison of bond lengths. a and b denote the direction of the bond. Bond
Average distance (Å) Experiment [12]
Calculation
Within B12 unit
1.796
1.811 (a) 1.811 (b)
B12-B12
1.785
1.742 (a) 1.778 (b)
Within B21 unit
1.810
1.815
B21-B21
1.707
1.710
B12-B21 (horizontal)
1.643
1.674 (a) 1.700 (b)
B12-B21 (vertical)
1.706
1.707
B(23)-B(25)
1.548
1.598
Around B(26)
1.881
1.763
Figure captions
Fig. 1. Structure of b-tetragonal boron. The upper figure is viewed from the <001> direction and the lower is from the <100> direction. Boron atoms are located at the apexes of the icosahedra (grey) and double icosahedra (orange). Colored spheres denote interstitial sites: 23 (blue), 25 (red), 25p (green), and 26 (yellow).
Fig. 2. Total energies per atom for N = 184 to 196. The energy for N = 184 is set to 0.
Fig. 3. Electronic densities of states for N = 184 to 196. The configurations are those having the lowest energy for each N, marked by an asterisk in Table 1. The Fermi energy is set to zero in all the graphs.
Fig. 4. Electronic band structure for the lowest-energy configuration (N = 192). The Fermi energy is set to zero.
Fig. 5. Electronic density distribution for the lowest-energy configuration (N = 192) viewed from the <100> direction. Left: isosurface (yellow) at d = 0.09 electrons/a.u.3 Right: isosurface (orange) at d = 0.12 electrons/a.u.3
TOC legend:
Electronic density distribution for the lowest-energy configuration (N = 192) viewed from the <100> direction. Left: isosurface (yellow) at d = 0.09 electrons/a.u.3 Right: isosurface (orange) at d = 0.12 electrons/a.u.3
Highlights ࣭b-tetragonal boron was thoroughly investigated using first-principles calculations. ࣭The lowest energy structure contains 192 atoms in an orthorhombic lattice. ࣭Another tetragonal structure with 192 atoms has a very close energy. ࣭The total energy per atom is between those of a- and b-rhombohedral boron. ࣭The band gap of the lowest energy structure is about 1.16 to 1.54 eV.
Graphical Abstract (TOC Figure)