J. Phys. Chem. C 2008, 112, 2711-2715
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Effect of Codoping in r-Rhombohedral Boron Wataru Hayami* and Shigeki Otani AdVanced Nanomaterials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan ReceiVed: October 23, 2007; In Final Form: December 5, 2007
R-Rhombohedral (R-rh) boron is the most stable polymorph of elemental boron at low temperatures and exhibits p-type semiconductive properties. Although R-rh boron has some voids of sufficient size in its structure, it does not accept alkali metals as donor dopants. An n-type or metallic R-rh boronlike material so far has not been found. We considered substituting Li for either of P or As in the stable R-rh boronlike compounds B12P2 and B12As2 and calculated the electronic structures from first principles to investigate the possibility that R-rh boron becomes an n-type semiconductor or a metal. The idea is similar to the case of diamond, where the codoping of a donor and acceptor creates an n-type semiconductor. Our results showed that B12PLi and B12AsLi became intrinsic narrow-gap semiconductors with band gaps of 0.478 and 0.536 eV, respectively. In both materials, the curvature of the conduction band was higher than that of the valence band, suggesting that the electron mobility may be higher than the hole mobility.
1. Introduction Elemental boron is known to have four main polymorphs.1 These are the R-rhombohedral (R-rh), β-rhombohedral (β-rh), R-tetragonal, and β-tetragonal polymorphs. The two rhombohedral polymorphs are more stable than the tetragonal versions and have been studied more. R-rh Boron is more stable than β-rh boron below 1400 K.2 Both R- and β-rh boron are p-type semiconductors with a band gap of about 2 eV3,4 and about 1.5 eV,5 respectively. Some attempts have been made to dope other elements into R- and β-rh boron. The doping of alkali metals has been successful in β-rh boron but not in R-rh boron.6 This fact is related to the atomic and electronic structure of R-rh boron. The structure of R-rh boron comprises B12 icosahedral units that connect to each other to form a three-dimensional framework.7 There are some voids between icosahedra, and dopant atoms usually occupy these interstitial sites. As an example, the structures of B12P2 and B12As2 are given in Figure 1. The rhombohedral lattice has a B12 icosahedron on its vertices, and two P or As atoms are located on the diagonal line of the 〈111〉 direction. The structure of R-rh boron is that without P or As. Experimentally, C, P, As, and O can be inserted stably into R-rh boron. These atoms have higher electronegativity than that of boron. The resulting materials are B4C, B12P2, B12As2, and B12O2 (B4C has a slightly different structure from the others), which have a wider band gap than R-rh boron according to the electronic structure calculations.8-12 Their electrical conduction is also p-type.13-16 A discussion of why these materials become p-type was presented by Emin based on the molecular orbital scheme.17 The hole doping to B4C, B12P2, and B12As2 to achieve superconductivity was proposed by Calandra et al.18 On the contrary to the above case, it is difficult to dope alkali metal atoms,6 which have lower electronegativity than that of boron. A theoretical work showed that an alkali metal atom (Li) is to donate its electron to the boron framework without changing the band structure very much (the rigid band model).19 * Corresponding author. E-mail:
[email protected].
Figure 1. Structure of B12P2 and B12As2. Boron atoms are at the vertices of the icosahedra and two P or As atoms (spheres) are aligned on the 〈111〉 diagonal. View from the 〈11h0〉 direction (left), and from the 〈111〉 direction (right).
The calculated density of states of R-rh boron shows that the valence bands are fully occupied and the conduction bands are empty, so that on Li doping, an electron enters the conduction band. From the view of molecular orbital theory, the conduction bands consist mainly of antibonding orbitals, so that an electron in the conduction band lowers the binding energy between B atoms, resulting in a low binding energy between Li and B.19-21 This is what prevents alkali metal atoms from being doped into R-rh boron. It would be interesting if we could create a metallic or n-type R-rh-boronlike material by alkali metal doping. In recent years, the superconductivity of boron and borides has attracted much attention.22,23 If R-rh boron becomes metallic by doping, it may become a superconductor. From the point of view of device applications, the n-type semiconductive R-rh boron would be useful to create a p-n junction. Because, as mentioned above, it is difficult to dope an alkali metal into R-rh boron directly, we have tried doping simultaneously Li and P or As. The idea of codoping was borrowed from the case of diamond, where an n-type semiconductor was successfully produced by codoping a donor and an acceptor.24 In diamond, it is difficult to dope solely a donor, but the
10.1021/jp710239f CCC: $40.75 © 2008 American Chemical Society Published on Web 01/26/2008
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Hayami and Otani Concerning band calculations, the notation of the symmetrical points of the Brillouin zone of the rhombohedral lattice varies from one reference to another.10,11,19 We followed the notation used by Morrison et al.10 as shown in Figure 2. Using the rhombohedral reciprocal lattice vectors, point A ) (1/2)〈100〉 (and equivalent points), B ) (1/2)〈110〉 and Z ) (1/2)〈111〉. 3. Results and Discussion
Figure 2. Brillouin zone of the rhombohedral lattice. The notation of the points follows the reference.10
Figure 3. Structure of B12PLi. The upper sphere is P and the lower is Li. Compared with Figure 1, the B12 icosahedra are distorted.
codoping of donor and acceptor lowers the formation energy due to the Coulomb interaction between them. Although there are voids surrounded by four icosahedra, the Li atoms at this site are unstable.19,20 We therefore considered replacing one P or As with Li and made B12PLi and B12AsLi for testing. The atomic structures were first optimized from first principles and then the electronic structures were selfconsistently calculated. The binding energies, the band structures, the densities of states, and the contour maps of the density of electrons are presented below.
One P or As atom in a unit cell was replaced by a Li atom, after which both the lattice parameters and the atomic positions were fully optimized. The optimized structure of B12PLi is shown in Figure 3. P (upper sphere) and Li (lower sphere) are aligned on the diagonal 〈111〉 line. Unlike the B12P2 structure (Figure 1), the icosahedra are slightly distorted. The distance between Li and B is larger than that between P and B. The structure of B12AsLi is very similar to that of B12PLi. The details of the structures are summarized in Table 1. The values in the table are the results of calculations, and the values in the parentheses are from experiments.34 The calculated lattice parameters of B12P2 and B12As2 agree well with the experimental values, which guarantees the accuracy of the calculation. The lattice parameter of B12PLi is 3.4% larger than that of B12P2, and that of B12AsLi is 1.9% larger than that of B12As2. The interaxial angle R decreases slightly by 1.5° for B12PLi and by 0.7° for B12As2. The change in lattice parameters may be small but not negligible. For B12PLi, the Li-B distance is larger than the P-B distance by 19%, which causes the deformation of the B12 icosahedra. In B12As2, the Li-B distance is larger than the As-B distance by 11%: less than in B12PLi. The covalent radii for B, P, As, and Li are 0.81, 1.10, 1.21, and 1.23 Å, respectively. The P-B and As-B distances (1.91 and 2.03 Å) are very close to the sum of their covalent radii, whereas the Li-B distances (2.28 and 2.26 Å) are about 10% larger than the sum of their covalent radii, suggesting that Li-B bonding is weak in both materials. For B12PLi, the distance between P-Li is slightly larger than P-P in B12P2, and the same applies to B12AsLi. As a whole, both materials exhibit similar tendencies. When the total energy of atom or molecule X is written as E(X), the binding energy of the system is expressed as
E(B12) + E(P) + E(Li) f E(B12PLi) + 3.32 eV
2. Computational Details The calculation of electronic structures and geometry optimization were performed using the CPMD code, version 3.9.1.25,26 This code is based on the density functional theory with plane waves and pseudopotentials.27,28 Norm-conserving Troullier-Martins-type pseudopotentials29 in the KleinmanBylander form30 were used. The generalized gradient approximation was included by means of the functional derived by Becke31 and by Lee, Yang, and Parr.32 An energy cutoff of 50 Ry was sufficient to provide a convergence for total energies and geometries. Geometry optimization and total energy calculations were done using Monkhorst-Pack sampling33 of a (4 × 4 × 4) mesh. The test calculation was compared with that for a (5 × 5 × 5) mesh. The difference in total energy per atom was about 4 × 10-4 eV. Calculations were performed on a parallel computer (Hitachi SR11000) using a message-passing interface.
E(B12P) + E(Li) f E(B12PLi) + 1.89 eV
(2)
it is possible to dope Li into the P defects in the crystal. This energy is much higher than in the case where a single Li is doped into R-rh boron.19-21 Experimentally, no R-rh B12P single phase has been reported. On the other hand, the negative energy of the reaction,
E(B12P2) + E(Li) f E(B12PLi) + E(P) - 7.13 eV (3) indicates that it is impossible to simply replace P with Li under equilibrium conditions. The reaction,
2E(B12PLi) f E(B12P2) + E(B12) + E(Li2) + 4.66 eV
TABLE 1: Calculated Lattice Parameters and Some Bond Lengths of B12P2, B12PLi, B12As2, and B12AsLi, r Is the Interaxial Angle, the Bond Lengths Are in Å, Values in Parentheses Are from Experiments34 B12P2 B12PLi B12As2 B12AsLi
a (Å)
R (°)
5.27(5.256) 5.45 5.34(5.333) 5.44
69.5(69.619) 68.0 70.4(70.5) 69.7
(1)
The positive energy means that it is theoretically possible to synthesize B12PLi from B12, P, and Li. Considering another reaction,
P(As)-P(As)
P(As)-Li
2.27(2.240) 2.29 2.41(2.390) 2.51
P(As)-B 1.92(1.911) 1.91 2.01(2.000) 2.03
Li-B 2.28 2.26
(4)
Effect of Codoping in R-Rhombohedral Boron
J. Phys. Chem. C, Vol. 112, No. 7, 2008 2713
Figure 4. Band structures of B12P2 (top) and B12PLi (bottom). The Fermi energy is set to zero.
Figure 5. Band structures of B12As2 (top) and B12AsLi (bottom). The Fermi energy is set to zero.
suggests that B12PLi can decompose into B12P2, B12, and Li2. Judging from these reaction energies, the synthesis of B12PLi is theoretically possible but may not be experimentally easy because B12PLi exists only in a metastable state. Similar tendencies were observed in B12AsLi. The reaction energies corresponding to eqs 1-4 are 2.22, 1.77, -5.80, and 4.43 eV, respectively. B12PLi thus appears to be energetically more stable with Li doping than B12AsLi. Figure 4 shows the electronic band structure of B12P2 and B12PLi. The Brillouin zone is illustrated in Figure 2. The Fermi energy is set to 0 eV. The result of B12P2 is similar to the work by Li et al.11 The top of the valence band is at the Z point and the bottom of the conduction band is at the A point. This indirect band gap, 2.79 eV, is a little better than the result by Li et al. (2.63 eV) but is still underestimated compared to the experimental value of 3.35 eV.35 The band structure of B12PLi exhibits a quite different feature, especially around the Fermi level. It is still semiconductive, and both the top of the valence band and the bottom of the conduction band come to the A point having a direct band gap of 0.478 eV, which is much narrower than before doping. The dispersion curve from the A point to the Γ point appears to
show that the curvature of the lowest conduction band is larger than that of the highest valence band. This suggests that when an electron is excited from the valence to the conduction band, the mobility of electrons may be higher than that of holes. Figure 5 shows the electronic band structures of B12As2 and B12AsLi. The features are very similar to those of B12P2 and B12PLi. The bands of B12As2 are almost identical to the result obtained by Morrison et al.10 and similar to that by Li et al.11 In the case of B12As2, as in B12P2, the top of the valence band is at the Z point and the bottom of the conduction band is at the A point. The indirect band gap, 2.947 eV, is wider than that of B12P2 and again is still lower than the experimental value 3.47 eV,35 though it is a little better than the results obtained by Morrison et al. (2.609 eV) and Li et al. (2.78 eV). When Li is doped, B12AsLi exhibits similar features to that of B12PLi, but the energy differences between the bands around the Fermi level are slightly different. Both the top of the valence band and the bottom of the conduction band come to the A point. This direct band gap is 0.536 eV, much narrower than before doping and somewhat wider than that of B12PLi. At the A point, as in B12PLi, the curvature of the lowest conduction band
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Hayami and Otani
Figure 7. Density of electrons of B12P2 (top left), B12PLi (top right), B12As2 (bottom left), and B12AsLi (bottom right). The unit is electron/ (au)3
Figure 7 exhibits contour maps of the density of electrons on the cross section, the (11h0) plane that includes the 〈111〉 axis and the P-B (As-B) bond on it. The B12 icosahedra are seen as ringlike patterns. In both B12P2 and B12As2, the intensity of the intericosahedral B-B bonds is about 50% higher than the intraicosahedral B-B bonds, which agrees with the result by Morrison et al.10 In B12P2, covalent bonds are observed between P and B that seem as strong as the intericosahedral B-B bonds, while in B12As2, the As-B bonds are not as strong as the intericosahedral B-B bonds. Covalent bonds are observed between P-P and between As-As, and the P-P bonds seem to be relatively stronger than the As-As bonds. When P or As is replaced by Li, there are almost no electrons around Li, so that Li atoms are almost fully ionized and make no covalent bonds with neighboring atoms. In B12PLi, intraicosahedral B-B bonds appear to be slightly weakened by Li doping, while it is not clear in B12AsLi. Li atoms donate electrons to the B12P and the B12As system but not according to the rigid band model, as would be expected from the band structures (Figures 4 and 5). Although the system preserves its rhombohedral lattice after Li doping, the B12 icosahedra are distorted (Figure 3), which removes the degeneracy of the electronic states within B12. 4. Conclusions Figure 6. Densities of states of B12P2, B12As2, B12PLi, and B12AsLi. The Fermi energies are set to zero.
appears larger than that of the highest valence band, suggesting again that the electron mobility may be higher than the hole mobility. The densities of states (DOS) are exhibited in Figure 6. The Fermi level is set to 0 eV. The DOS of B12P2 and B12As2 are similar to each other, as expected from the band structures. When Li is doped, the states around the Fermi energy are a little different between B12PLi and B12AsLi. In the valence band just below the Fermi level (-2 to ∼0 eV), B12AsLi has a pseudo gap at about -1 eV, while B12PLi has a continuous valence band. In the conduction band just above the Fermi level (0 to ∼ 2 eV), two peaks are observed in B12PLi, while only one is observed in B12AsLi. Both have a gap at about 2 eV.
To investigate the possibility of donor doping to R-rh boron, we substituted a Li atom for either P in B12P2 or As in B12As2, expecting that codoping would stabilize the dopants, and calculated the electronic structures from first principles. Total energy calculations showed that B12PLi and B12AsLi can theoretically be synthesized from their components, B12, P, or As and Li, but it was only in a metastable state. Although B12P2 and B12As2 are originally semiconductors with a wide band gap, 3.35 and 3.47 eV, respectively, after Li doping, B12PLi and B12AsLi became intrinsic narrow-gap (0.478 and 0.536 eV) semiconductors. The band gaps were direct at the A point in the Brillouin zone. At the A point, the conduction band had a larger curvature than the valence band, suggesting the mobility of electrons may be higher than that of the holes. The band structures of B12PLi and B12AsLi were similar to each other but different from the original B12P2 and B12As2. This was
Effect of Codoping in R-Rhombohedral Boron probably caused by the distortion of the B12 icosahedra that removed the degeneracy of the electronic states within the B12 icosahedra. The densities of states of B12PLi and B12AsLi were slightly different around the Fermi energy. The contour maps of electron density showed that a Li atom had almost no electrons around it, so that it seemed to fully donate its electron to the boron framework. References and Notes (1) Adams, R. M. Boron, Metallo-Boron Compounds and Boranes; Interscience Publishers, John Wiley & Sons: New York, 1964. (2) Runow, P. J. Mater. Sci. 1972, 7, 499. (3) Golikova, O. A.; Solovev, N. E.; Ugai, Y. A.; Feigelman, V. A. J. Less-Common Met. 1981, 82, 362. (4) Terauchi, M.; Kawamata, Y.; Tanaka, M.; Takeda, M.; Kimura, K. J. Solid State Chem. 1997, 133, 156. (5) Werheit, H.; Laux, M.; Kuhlmann, U. Phys. Status Solidi B 1993, 176, 415. (6) Soga, K.; Oguri, A.; Araake, S.; Terauchi, M.; Fujiwara, A.; Kimura, K. J. Solid State Chem. 2004, 177, 498. (7) Emin, D. Phys. Today 1987, 40, 55. (8) Bylander, D. M.; Kleinman, L.; Lee, S. Phys. ReV. B 1990, 42, 1394. (9) Lee, S.; Kim, S. W.; Bylander, D. M.; Kleinman, L. Phys. ReV. B 1991, 44, 3550. (10) Morrison, I.; Bylander, D. M.; Kleinman, L. Phys. ReV. B 1992, 45, 1533. (11) Li, D.; Ching, W. Y. Phys. ReV. B 1995, 52, 17073. (12) Li, D.; Ching, W. Y. Phys. ReV. B 1996, 54, 1451. (13) Wood, C.; Emin, D. Phys. ReV. B 1984, 29, 4582. (14) Kumashiro, Y.; Yokoyama, T.; Sato, K.; Ando, Y.; Nagatani, S.; Kajiyama, K. J. Solid State Chem. 2000, 154, 33.
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