J. Phys. Chem. B 2008, 112, 10217–10220
10217
Thermodynamic Stability of Novel Boron Sheet Configurations Kah Chun Lau*,† and Ravindra Pandey*,‡ Department of Chemistry, George Washington UniVersity, Washington, District of Columbia 20052, and Department of Physics, Michigan Technological UniVersity, Houghton, Michigan 49931 ReceiVed: April 16, 2008
Thermodynamic stability together with the vibrational spectrum of several structural motifs of the pristine 2D boron sheet are investigated. The results suggest that the nature of the chemical bonding, rather than thermal effects, appears to be the prime factor in determining the stability of atomic monolayers of boron. The most stable configuration is predicted to be composed of a hybrid of triangular and hexagonal configuration. A boron sheet composed of either perfect triangular or perfect hexagonal motifs is unlikely to be synthesized. The distinctive features predicted in vibrational and thermodynamic properties for boron sheet configurations are expected to exhibit their signatures in the corresponding pristine nanotubes and crystalline bundles. I. Introduction The stability of two-dimensional (2D) structures is known to be the long-standing challenging research problem in the field of materials science. It is well-known that the long-wavelength thermal fluctuations can destroy the long-range order of a 2D lattice.1 It therefore remains questionable, in principle, whether a free-standing 2D atomic layer could exist.1–6 In recent years, graphene, a single layer of carbon atoms densely packed in a honeycomb crystal lattice, has sparked much scientific interest.6–9 Owing to its superior form in the quasi one-dimensional (1D) structure when rolled up to form a nanotube,10 fundamental understanding of novel 2D sheet configurations becomes an important supplement to predict the properties of the corresponding tubular configurations. Several theoretical studies have focused on the 2D boron sheet11–16 as a precursory step to form boron nanostructures. These studies have, however, reported only the structural and electronic properties without looking into the question of their stability at ambient conditions. This is what we propose to do in this article: perform a detailed analysis of the thermodynamic stability of boron sheets. Specifically, we will focus on first principles calculations of the vibrational frequency, free energy, and heat capacity of several sheet configurations, and we will compare the results with those obtained for the well-studied graphene to gain insight in the stability of the 2D boron sheet configurations.
successfully the structural parameters of the boron crystalline solid, R-B12. For example, the calculated lattice constant is 4.99 Å as compared to the GGA value of 5.05 Å15 and experimental value of 5.06 Å.18,19 To simulate a 2D boron sheet, a supercell was constructed by placing a basic unit in the xy-plane inside a rectangular grid with a surface-to-surface separation of ∼15 Å in the z-direction. The sheet configuration was built by repeating the basic unit consisting of six to eight boron atoms depending on a given configuration. Calculations were deemed converged when changes in total energy were less than 10-5 eV and those in the interatomic forces were less than 0.01 eV/Å. The optimized sheet configurations were then used to calculate the frequencies with the convergence criterion of 10-6 eV and 10-4 eV/Å for total energy and interatomic forces, respectively. The vibrational free energy is calculated within the harmonic approximation using the direct method.17 This method employs the forces on the atoms induced by the displacements of other atoms in the supercell. The calculated forces are used to obtain elements of the dynamic matrix related to the interatomic force constants. The vibrational frequencies are then obtained via the diagonalization of the dynamic matrix in which the atomic displacements may correspond to a particular phonon mode. If frequency of a particular mode is imaginary, the configuration is said to be not stable, indicating it to be in a transition state, instead of representing a local minimum on the energy surface. III. Results and Discussion
II. Methodology Electronic structure calculations were performed within the local density approximation (LDA) to density functional theory (DFT). A plane wave basis set was used with a cutoff of 430 eV, and the valence-core interaction was described by the projector augmented wave potential as implemented in the Vienna ab initio simulation package (VASP).17 The 23 × 23 × 1 Monkhorst-Pack grid was used, and the k-space integrations were carried out using the Gaussian smearing width of 0.1 eV. We note that the model parameters employed reproduced * Corresponding authors. K.C.L. e-mail:
[email protected]. R.P. e-mail:
[email protected]. † George Washington University. ‡ Michigan Technological University.
Figure 1 displays the configurations of R-sheet, β-sheet, flat and buckled triangular sheets, and flat and distorted hexagonal sheets considered in this work. The relative energy differences (Table 1) among the sheet configurations are consistent with the reported values in the previous studies.11–16 The cohesive energy (or binding energy) of the sheet configurations is predicted to be less than that of the boron rhombohedral solid. The binding energy of the most stable R-sheet is ∼93% of the R-B12 solid. A. Stability. First, we performed calculations on the hexagonal graphene sheet to set the benchmark for comparison with the results obtained for the boron sheet configurations. For graphene, the frequencies spanning from 383 to 1598 cm-1 are found to be consistent with the previously reported values.20
10.1021/jp8052357 CCC: $40.75 2008 American Chemical Society Published on Web 07/29/2008
10218 J. Phys. Chem. B, Vol. 112, No. 33, 2008
Lau and Pandey
Figure 2. Vibrational frequency spectrum (in cm-1) of the boron sheet configurations.
Figure 1. Two-dimensional sheet configurations of boron: (top) R-sheet and β-sheet, (center) buckled triangular and distorted hexagonal, and (bottom) perfect triangular and perfect hexagonal configurations.
TABLE 1: Binding Energy (Eb) and Structural Parameters (Bond Length (d) and Buckling Height (∆z)) of Sheet Configurations structural parameters sheet configuration
Eb
d (Å)
∆z (Å)
R-sheet β-sheet buckled triangular sheet perfect triangular sheet distorted hexagonal sheet perfect hexagonal sheet
6.74 6.71 6.64 6.46 6.44 5.72
1.66-1.69 1.63-1.74 1.61, 1.83 1.69 1.62, 1.65 1.67
0.84 -
The highest vibration mode, ν ∼ 1598 cm-1 is associated with the symmetric stretching mode of the strong σ-bond. The outof-plane transverse polarization in the optical mode (i.e., ZO at Γ) emerges at ∼896 cm-1 consisting of in-phase and antiphase out-of-plane atomic oscillations. In the low-frequency regime (i.e., ∼383-685 cm-1), the modes are related to either transverse out-of-plane or bending modes of atomic oscillations in the 2D lattice. For the hexagonal boron sheet, the calculated results find several imaginary frequencies, thus confirming its instability. The largest imaginary frequency of about νc ∼ 531 cm-1 shows the instability of its in-plane symmetric stretching mode of the σ-bond. Similarly, the out-of-plane ZA bending modes are also associated with the imaginary frequencies. The results therefore conclusively rule out the existence of the pristine hexagonal boron sheet at ambient conditions.
In the previous study of the hexagonal boron sheet,15 we predicted that the induced distortions, breaking the D6h symmetry of the hexagonal lattice to a lower D2h symmetry of the orthorhombic lattice, make the sheet configuration relatively stable. The present calculations further confirm the stability of the distorted hexagonal sheet where no imaginary frequencies were found. At Γ, the eigenvalue distribution is relatively more dispersive than the perfect hexagonal sheet as shown in Figure 2. The highest vibrational mode at 1401 cm-1 is associated with the symmetric stretching of the σ-bond with the bond distance of 1.62 Å. It is slightly smaller than that in graphene (∼1598 cm-1), thereby exhibiting similar strength of σ bonds present in the distorted hexagonal boron sheet and graphene. We now consider the configurations based on the triangular networks which resemble the icosahedral B12 networks of the elemental boron solids (Figure 1). In the perfect triangular boron sheet, both the high atomic coordination and the electronicdeficient character of boron yield a nearly homogeneous electron density distribution in the 2D lattice.15 Its Fermi level is high enough to force some of the electrons to occupy antibonding molecular orbitals which, in turn, induce a destabilizing effect in the 2D lattice making it to be highly chemically reactive.16 The calculated frequency spectrum with several imaginary modes also show the flat triangular sheet to be highly unstable. In particular, all of its transverse out-of-plane and ZA bending modes are not stable with imaginary νc ranging from 261 to 193 cm-1. It therefore appears that that the three-center bonds present in the perfect triangular lattice are not strong enough to bind the boron atoms in a 2D lattice. The results find that the buckling (∆z ) 0.84 Å, Table 1) removes the instability of a perfect triangular boron sheet.12–15 The symmetry-reducing distortions lead to the formation of the σ-bonds with an increase in the stability by 0.18 eV/atom in the buckled sheet. The presence of σ-bonds with Rσ ) 1.61 Å is well-represented by the atomic oscillations having stretching modes in the range of 1284-1348 cm-1 (Figure 2). For frequencies in the range of 541-741 cm-1, oscillations are dominated by the symmetric and asymmetric of the diagonal bonds with Rdiag ) 1.83 Å (Figure 1, Table 1). The longwavelength modes (214-439 cm-1) are associated with the outof-plane atomic oscillations. In general, the results confirm that distortions induced in a perfect 2D lattice (e.g., hexagonal or triangular) yield stable configurations. The vibrational spectrum of the energetically stable R- and β-sheet configurations, composed of triangular and hexagonal motifs, shows features of the isolated triangular and hexagonal
Thermodynamic Stability of Boron Sheet
J. Phys. Chem. B, Vol. 112, No. 33, 2008 10219
Figure 3. Thermodynamic stability of the boron sheets (BS) in terms of the calculated Helmholtz free energy. The dotted lines are the computed free energy including zero-point vibration energy as stated in eq 1.
sheet configurations (Figure 2). In the spectrum ranging from 128 to 1146 cm-1, the highest mode is the symmetric stretching mode associated with three-center bonds in the 2D lattice. This is a signature of the triangular motif (Figure 2). The next highest mode exhibits an asymmetric stretching along the edges of hexagonal hole (Figure 1), resembling the one predicted for the hexagonal sheet at about ∼910 cm-1. The long-wavelength oscillations in the range of 128-283 cm-1 are associated with the out-of-plane oscillation and bending modes. In general, the ZA bending modes are predicted to be lower in energy relative to those in graphene. The long-range ordering in the boron sheet is, therefore, likely to be more vulnerable to the long-wavelength thermal fluctuations as compared to graphene, though the sheet configurations of boron and graphene are thermodynamically feasible. B. Thermodynamic Properties. From the computed phonon spectrum of the sheet configurations, the temperature-dependent vibrational Helmholtz free energy and entropy can be calculated in the harmonic approximation.21 F(V, T), the free energy of a crystal, can be written as a sum of E(V), the energy of the static lattice at the equilibrium configuration, and Fvib(V, T), the vibrational free energy associated with the modes, ω(q).
F ) E + Fvib ) E +
- ST ∑ pω(q) 2
(1)
q
Here, S is entropy, ∑q pω(q)/2 is the summation of contributions of the zero-point energy from each phonon mode, and T is the temperature. Considering the first-order corrections to be small, S can be calculated20 as follows:
S ) -kB
∑ ln(1 - e-pω(q)/k T) B
(2)
q
Figure 3 shows the variation of the total free energy with temperature for the configurations considered. The order of the stability remains nearly the same as predicted from energies calculated at zero temperature; R-sheet being the most stable. Thus, the results suggest that the nature of the chemical bonding rather than the thermal effects determines the relative stability of the 2D boron lattice. We note that contributions from the vibrational term do affect the ordering of the stability of the distorted hexagonal and flat triangular boron sheet. The distorted hexagonal sheet becomes relatively more stable at about 350 K relative to the flat triangular sheet.
Figure 4. Calculated constant volume heat capacity (in J K-1 mol-1) vs temperature plots of boron sheets and graphene. The black solid lines are as follows: (cross) graphene, (diamond) R, (star) buckled triangular, (triangle) distorted hexagonal, (square) perfect triangular, and (circle) hexagonal boron sheets. The red dotted horizontal asymptote is the Cv ) 3R (where R is the gas constant) value, governed by the Dulong-Petit law.
The lattice heat capacity per unit cell at constant volume can be defined as follows:
CV(T) )
∑ cv(q) ) kB∑ q
q
( ) ( ) pω(q) 2kBT
2
1 pω(q) sinh 2kBT
(3)
2
In order to compare the results of boron sheets with carbon graphene, the normalized heat capacity CV (i.e., J K-1 mol-1) is computed. In the high-temperature regime, as shown in Figure 4, CV is bound by the asymptote stated by the Dulong-Petit law, CV ) 3R. In the low-temperature regime, CV of the boron sheets is different from that of graphene. Note that CV of the thermodynamic stable boron sheets (i.e., R, buckled triangular and distorted hexagonal) is generally larger than that of graphene. The heat capacity values of R, buckled triangular, and distorted hexagonal sheets are nearly identical at high temperatures. It is a manifestation of the law of corresponding states for different materials with essentially very similar Debye temperatures.20 IV. Conclusions In summary, first-principles calculations based on density functional theory are performed to calculate the thermodynamic properties of the several configurations of the pristine 2D boron sheet. The results suggest that a boron sheet composed of either perfect triangular or perfect hexagonal motifs is unlikely to be synthesized. The induced distortions in either a perfect hexagonal or triangular sheet configuration are found to provide the stability of the 2D lattice. The thermodynamic stable configurations are predicted to be composed of a hybrid of triangular and hexagonal configurations and the buckled triangular and the distorted hexagonal sheets. It is expected that distinctive features predicted in vibrational and thermodynamic properties for sheet configurations will exhibit their signatures in the corresponding pristine nanotubes and bundles of boron. Finally, the results find that the nature of chemical bonding, not the thermal effects, is the prime factor in determining the stability of boron sheet configurations. Acknowledgment. The work at Michigan Technological University was supported by DARPA (contract number ARL-
10220 J. Phys. Chem. B, Vol. 112, No. 33, 2008 DAAD17-03-C-0115). We thank Prof. M. A. Blanco, S. Gowtham, Haiying He, and Prof. R. Orlando for helpful comments during this work. References and Notes (1) Mermin, N. D. Phys. ReV. B 1968, 176, 250. (2) Pierls, R. E. HelV. Phys. Acta 1934, 7, 81. (3) Pierls, R. E. Ann. Inst. Henri Poincare 1935, 5, 177. (4) Landau, L. D. Phys. Z. Sowjetunion 1937, 11, 26. (5) Landau, L. D.; Lifshitz, E. M. Statistical Physics Part I; Pergamon: Oxford, 1980. (6) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. Proc. Natl Acad. Sci. USA 2005, 102, 10451. (7) Novoselov, K. S.; et al. Nature 2005, 438, 197. (8) Zhang, Y.; Tan, J. W.; Stormer, H. L.; Kim, P. Nature 2005, 438, 201. (9) Meyer, J. C.; Geim, A. K.; Katsnelson, M. I.; Novoselov, K. S.; Booth, T. J.; Roth, S. Nature 2007, 446, 60.
Lau and Pandey (10) Iijima, S. Nature 1991, 354, 56. (11) Boustani, I.; Quandt, A.; Hernandez, E.; Rubio, A. J. Chem. Phys. 1999, 110, 3176. (12) Evans, H. M.; Joannopoulos, D. J.; Pantelides, S. T. Phys. ReV. B 2005, 72, 045434. (13) Cabria, I.; Lopez, J. M.; Alonso, J. A. Nanotechnology 2006, 17, 778. (14) Kunstmann, J.; Quandt, A. Phys. ReV. B 2006, 74, 035413. (15) Lau, K. C.; Pandey, R. J. Phys. Chem. C 2007, 111, 2906. (16) Tang, H.; Ismail-Beigi, S. Phys. ReV. Lett. 2007, 99, 115501. (17) Kresse, G.;; Hafner, J. VASP (Vienna ab initio simulation package) software. Kresse, G.; Hafner, J. Phys. ReV. B 1993, 47, 558. Kresse, G.; Furthmuller, J. Phys. ReV. B 1996, 54, 11169. (18) Kittel, C. Introduction to Solid State Physics, 7th ed.; Wiley: New York, 1996. (19) Donohue, J. The Structure of The Elements; Wiley: New York, 1974. (20) Mounet, N.; Marzari, N. Phys. ReV. B 2005, 71, 205214. (21) Rickman, J. M.; LeSar, R. Annu. ReV. Mater. Res. 2002, 32, 195.
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