H ppσ (R) H spσ (R) H ssσ (R) R (Bohr)

Transcription

1 TIGHT-BINDING HAMILTONIANS FOR CARBON AND SILICON 1 D. A. PAPACONSTANTOPOULOS, M. J. MEHL, S. C. ERWIN, AND M. R. PEDERSON Complex Systems Theory Branch, Naval Research Laboratory, Washington DC ABSTRACT We demonstrate that our tight-binding method { which is based on tting the energy bands and the total energy of rst-principles calculations as a function of volume { can be easily extended to accurately describe carbon and silicon. We present equations of state that give the correct energy ordering between structures. We also show that quantities that were not tted, such as elastic constants and the band structure of C 60, can be reliably obtained from our scheme. INTRODUCTION Over the past decade, many authors have developed tight-binding parametrizations for carbon [1, 2] and silicon [3, 4, 5, 6, 7, 8]. Because these parametrizations lead to small size Hamiltonians, they are particularly useful for the study of large unit cell systems such as the fullerenes [1, 9]. We have developed another tight-binding total energy method [10, 11] wherein the parameters are chosen to reproduce both the rst-principles total energy and electronic band structure of face-centered and body-centered cubic crystals at several volumes. This method has been shown to work quite well in transition metals, where it correctly predicts the correct ground state and lower lying crystal states for all of the non-ferromagnetic transition metals. This includes the hcp metals and the phase of manganese [12], even though these structures were not included in the t. We also predict elastic constants, phonon frequencies, surface energies, and vacancy formation energies in agreement with experiment and rst-principles results. The method has recently been extended to cover some of the sp bonded metals in column IIIA of the periodic table, aluminum, gallium, and indium [13]. For these elements the rst-principles database had to be extended to include other than fcc and bcc crystals, but it did not include the body-centered tetragonal indium ground state or the complex Ga structure. Nevertheless, the method correctly predicts the ground state properties of all of these materials. The method also generates an electronic density of states for Ga which is very close to the results of rst-principles calculations. [14] Can this method be extended even further to the right of the periodic table, specically to carbon and silicon? In this paper we develop a slightly modied version of the parametrization scheme developed in Ref. [11], applied to carbon and silicon. In the sections below we introduce the method, and discuss the properties predicted by our parameters. METHOD 1 To appear in Tight-Binding Approach to Computational Materials Science, P.E.A. Turchi, A. Gonis, and L. Colombo, eds., MRS Proceedings 491, (Materials Research Society, Warrendale, PA, 1998).

2 Table I: The Slater-Koster tight-binding parameters for carbon, generated from the database described in the text. On-Site Parameters (Eqs. (1) and (2)) Orbital (Ry) (Ry) (Ry) (Ry) s p Hopping Parameters (Eqs. (3)) Orbital a (Ry) b (Ry/Bohr) c (Ry/Bohr 2 ) d (Bohr?1=2 ) H ss H sp H pp H pp Overlap Parameters (Eqs. (4)) Orbital p (Bohr?1 ) q (Bohr?2 ) r (Bohr?3 ) s (Bohr?1=2 ) S ss S sp S pp S pp The tight-binding parametrization of Refs. [10, 11] consists of a prescription for specifying the behavior of the on-site parameters as a function of the local environment, and a parametrization of the hopping and overlap matrix elements. We begin by discussing the behavior of the on-site terms, which are allowed to vary depending upon the local environment of each atom. This environment is determined by dening a pseudo-atomic density for each atom, X = e jr?2 j?r ij f(jr j? R i j) ; (1) i j where R k is the position of the k th atom, the sum is over all neighbors of atom i, and where f(r) is a cuto function as dened in Ref. [11]. In this paper we choose the cuto so that f(r) vanishes when R > 10:5 atomic units for carbon, and when R > 12:5 atomic units for silicon. The on-site terms on each atom are given by a Birch-like equation h i` = ` + ` 2=3 i + ` 4=3 i + ` 2 i : (2) In both carbon and silicon we naturally consider only ` = s; p, so, including the in equation (1), there are a total of nine parameters which determine the on-site terms on each atom. The two-center Slater-Koster hopping terms for the Hamiltonian are simply polynomials times an exponential cuto, H``0(R) = (a``0 + b``0r + c``0r 2 ) exp(?d 2``0R)f(R) ; (3) where f(r) is the same cuto as above. The overlap parameters have been modied from our previous work[10, 11] to exhibit the proper behavior as the atoms get close together.

3 Table II: The Slater-Koster tight-binding parameters for silicon, generated from the database described in the text. On-Site Parameters (Eqs. (1) and (2)) Orbital (Ry) (Ry) (Ry) (Ry) s p Hopping Parameters (Eqs. (3)) Orbital a (Ry) b (Ry/Bohr) c (Ry/Bohr 2 ) d (Bohr?1=2 ) H ss H sp H pp H pp Overlap Parameters (Eqs. (4)) Orbital p (Bohr?1 ) q (Bohr?2 ) r (Bohr?3 ) s (Bohr?1=2 ) S ss S sp S pp S pp Thus we write S``0(R) = (``0 + p``0r + q``0r 2 + r``0r 3 ) exp(?s 2``0R)f(R) ; (4) where ``0 is the Kronecker delta function. In this paper we use only s and p orbitals, so we need only consider the Slater-Koster parameters (``0) = (ss); (sp); (pp) and (pp). The tting procedure is described in [11]. Since we only consider s and p orbitals, we have the parameter from eq. (1); eight parameters `; `; ` and ` from eq. (2); sixteen parameters a``0; b``0; c``0 and d``0 from eq. (3); and sixteen parameters p``0; q``0; r``0 and s``0 from eq. (4). This gives us a total of forty-one parameters. While this is a much larger number of parameters than used by some other methods [5, 7], this method is designed to map the electronic structure as well as the total energy information from a series of rst-principles calculations onto the Slater-Koster parametrization, requiring a larger number of parameters for an accurate t. For each element, a database of rst-principles density functional calculations is constructed. For carbon, this database contains the the one-electron eigenvalues as a function of k-point and volume, along with the total energy as a function of volume, for the diamond, graphite, and simple cubic lattice. In addition we include the eigenvalues and total energy of the C 2 dimer as a function of atomic separation. For silicon we t only to the diamond, simple cubic, fcc, and bcc structures. In each case we shift the eigenvalues by a structure-dependent and volume dependent constant V 0, so that the total energy of a given structure S at a volume V is given by E(S; V ) = X [" i + V 0 (S; V )] = X " 0 i ; (5) where the sums are over the occupied states. The tight-binding parameters are then chosen so as to reproduce both the total energies and the shifted eigenvalues " 0 as closely i

4 as possible. A justication of this shift was recently proposed by McMahan and Klepeis.[8] The nal tight-binding parameters of our scheme for carbon and silicon are listed in Tables I and II, respectively H spσ (R) H ppσ (R) H(R) (Ry) H ppπ (R) H ssσ (R) R (Bohr) Figure 1: Two center Slater-Koster hopping matrix elements for carbon, based on the tting procedure described in the text. RESULTS Carbon We obtained our tight-binding parameters for carbon by tting to a set of rst-principles band structures and total energies, as outlined in Ref. [11]. Results for the bulk diamond, graphite, and the simple cubic structures were obtained using the full-potential Linearized Augmented Plane Wave (LAPW) method,[15, 16] and the Wigner form of the Local Density Approximation. We also found it desirable to include results for the C 2 molecule in our database. These calculations used the all-electron density functional based NRLMOL codes [17, 18, 19]. In the dimer there is a crossing between the u state and the 3 g state near the experimental equilibrium distance.[19] We dealt with this problem by tting only in the regions where the ordering of the states is unambiguous. Thus we t to the fully -bonded molecule at short bondlengths, and to the partially ( u,3 g )-bonded state at large bondlengths. The nal tight-binding parameters for carbon are shown in Table I. In Fig. 1 we show the two-center Slater-Koster hopping matrix elements ss, sp, pp and pp derived from our parameters in Table I as a function of the interatomic distance R. It should be noted that these parameters decrease smoothly to zero for R 10:5 Bohr and also follow the correct sign convention, that is, ss and pp are negative, while sp and pp are positive.[20] In Fig. 2 we show the distance dependence of the corresponding overlap matrix elements. In the interaction region (R > 2 Bohr) each matrix element has the opposite sign from its respective hopping matrix element, as it should.[20] However, note that the overlap matrix element pp must approach 1 as the distance approaches zero, as required by (4) and on physical grounds. Finally, note that the overlap matrix elements also approach zero at large distances. In Fig. 3 we show our TB results for the total energy of carbon as a function of the

5 0.4 S(R) S ssσ (R) S ppπ (R) S ppσ (R) S spσ (R) R (Bohr) Figure 2: Two center Slater-Koster overlap matrix elements for carbon, based on the tting procedure described in the text. nearest-neighbor distance in several structures. It is clear that our tight-binding Hamiltonian works very well; it shows the cubic diamond and graphitic plane structures with the lowest energy, and the hexagonal diamond structure nearly degenerate with cubic diamond. It does not, however, predict the correct ground state of graphite. When we vary the volume and c/a ratio of the graphite structure to nd the minimum energy conguration, we nd that the structure stabilizes into a set of innitely separated graphite planes with an in-plane lattice constant a = 2:472A. This occurs because our current t does not adequately account for the interaction between the graphite planes. For reference, the experimental equilibrium lattice constant for graphite is a = 2:464A.[21] From various rst-principles calculations we estimate that the energy of the C 60 molecule is about 0.4eV/atom above the energy of the diamond lattice. Using our TB parameters we nd this energy to by 0.38eV/atom. It should be emphasized that we did not t to the C 60 molecule and that this result is an output of our TB method. We further investigated the behavior of C 60 by calculating the energy bands of an fcc crystal of C 60 molecules, with an eye to studying the superconducting phase transition in K 3 C 60.[22, 23] In Fig. 4 we show the band structure of an fcc crystal of C 60 molecules calculated with our tightbinding parameters, and compare it to a rst-principles calculation for K 3 C 60. Our TB valence and conduction bands compare quite favorably with the LDA results, although the TB bands are slightly narrower. We note that the rst band above the Fermi level is signicantly narrower than the LDA, and the next band is somewhat lower in energy than the LDA. However, in superconductivity the important band is the threefold band crossing the Fermi level. These bands are well reproduced in TB which will allow us to carry out further studies of the electronic density of states in this system for dierent orientational phases of the C 60 molecules. We determined the elastic constants of diamond at the experimental lattice constant by imposing a nite strain on the crystal, calculating the total energy, and obtaining the corresponding elastic constant from the curvature of the energy-strain relation at zero strain.[25, 26] In calculating the bulk modulus (C 11 +2C 12 )=3 and tetragonal shear C 11?C 12 we can impose strains so that the diamond basis remains xed by symmetry. In calculating the shear constant C 44, however, we must allow the carbon atoms to move as we strain the

6 Energy/atom (Ry) sc bcc C 2 C graphite plane 0.2 C 60 diamond Nearest Neighbor Distance (Bohr) fcc C 4 Figure 3: Energy of carbon predicted by our tight-binding parameters. Several dierent structures are shown, with the relative energy per atom shown as a function of the nearest neighbor separation for that structure. All other internal and external parameters were allowed to relax to minimize the total energy. The structures marked by solid lines were used in the t, while the structures marked by dotted lines are predictions of the method. The unlabeled curve for the hexagonal diamond structure is nearly degenerate with the cubic diamond curve. Figure 4: The left graph shows TB band structure of fcc C 60, compared to a the rstprinciples band structure of K 3 C 60 determined in Ref. [22]. The Fermi level in both pictures is set to correspond to a system with three electrons in the conduction band, as in K 3 C 60, and is then set to zero. Note: Unfortunately, this figure won't display properly using dvips or xdvi. A hard or electronic copy of this figure can be obtained by contacting mehl@dave.nrl.navy.mil. lattice. For comparison, we also calculated the elastic moduli using the LAPW method with the Wigner parametrization of the local density approximation. In this case we did not allow the carbon atoms to move, so we can only calculate an upper bound on C 44. When we calculated this upper bound from the TB parameters, however, we found that there is very little relaxation, as shown in Table III. Our results are shown in Table III. The TB parameters underestimate the bulk modulus by about 15% compared to experiment (10% compared to the LAPW result) and overestimate C 11? C 12 by 4% (7%). These combine to produce the very small result for C 12, which is 60% lower than experiment (57% compared to LAPW). The shear modulus C 44 is 4% above the experimental value (8% above LAPW). Silicon We performed similar rst-principles calculations for silicon using the LAPW method and the Hedin-Lundqvist prescription for exchange and correlation. We tted our TB Hamiltonian parameters to four structures (diamond, simple cubic, fcc, and bcc) using a non-orthogonal sp basis. In Fig. 5 we show a comparison of TB and LAPW results for theband structure. It is evident that the valence bands t very well. However, the conduction bands are poorly t, and in TB the smallest gap is at the symmetry point L

7 Table III: The elastic constants of diamond structure carbon and silicon at the indicated room temperature lattice constants, using the TB parameters of Tables I and II. For comparison, we also show the results of the full-potential LAPW calculations described in the text, and the experimental values.[24] Since the LAPW calculations only give an upper bound for C 44, we also give the upper bound found using our TB parameters. Carbon Silicon a = 3.567A a = 5.430A TB LAPW Exp. TB LAPW Exp. B C 11? C C C C 44 Upper Bound C 44 Relaxed instead of on the line near X. We have done further work using an spd representation which dramatically improves the conduction band. These results will be reported elsewhere. The present t is still useful because by restricting the TB to s and p orbitals we have a more convenient model for use in molecular dynamics simulations. Results of similar quality were recently reported by Cohen et al.[6] using the same approach but with some dierences in the form of the on-site terms. In Fig. 6 we plot the equation of state for various phases of silicon. In this plot the hexagonal diamond, Sn, and hcp phases are predictions of the method. This plot should be compared to the rst-principles results of Yin and Cohen.[27] We see that we have the correct ordering of all structures except the Sn state, which is too high in energy. In Table III we compare the calculated TB and rst-principles elastic constants of silicon to experiment. Again, the rst-principles value of C 44 is only an upper bound. We see that in this case the elastic constants of silicon are nearly as accurate as the rstprinciples results. Note that there is a considerable change in C 44 when we relax the internal parameters. SUMMARY We have proposed an extension of our parameterized tight-binding method { which was originally formulated for highly coordinated metals { to the more open structures of carbon and silicon. We are able to accurately describe the relative energies of several dierent structural phases, the elastic constants of the ground state phase, and the occupied electronic band structure relative to rst-principles methods. Future work will focus on the electronic structure and total energies for geometries far outside the tting database, including silicon surface reconstructions and the C 60 orientational phase diagram. ACKNOWLEDGMENTS This work was supported by the U.S. Oce of Naval Research and the Common High Performance Computing Software Support Initiative (CHSSI) of the United States De-

8 ε - ε(γ 25 ) (Ry) Γ X W L Γ K Figure 5: The electronic band structure of silicon at the experimental lattice constant, 5.430A. The solid lines represent the results of the rst-principles LAPW method. The dashed lines show the results obtained from the TB parameters in Table II. partment of Defense High Performance Computing Modernization Program (HPCMP). Partial computer support was provided by the Aeronautical Systems Center (ASC) at Wright-Patterson Air Force Base, Dayton, Ohio. REFERENCES 1. M. Menon and K. R. Subbaswamy, Phys. Rev. Lett. 67, 3487 (1991). 2. M. S. Tang, C. Z. Wang, C. T. Chan, and K. M. Ho, Phys. Rev. B 53, 979 (1996). 3. Th. Frauenheim, F. Weich, Th. Kohler, and S. Uhlmann, Phys. Rev. B 52, (1995). 4. T. J. Lenosky, J. D. Kress, I. Kwon, A. F. Voter, B. Edwards, D. F. Richards, S. Yang, and J. B. Adams, Phys. Rev. B 55, 1528 (1997). 5. M. Menon and K. R. Subbaswamy, Phys. Rev. B 55, 9231 (1997). 6. R. E. Cohen, L. Stixrude, and E. Wasserman, Phys. Rev. B 56, 8575 (1997). 7. N. Bernstein and E. Kaxiras, Phys. Rev. B 56, (1997). 8. A. K. McMahan and J. E. Klepeis, Phys. Rev. B 56, (1997). 9. D. Porezag, G. Jungnickel, T. Frauenheim, G. Seifert, A. Ayuela A, M. R. Pederson, Applied Physics A { Materials Science and Processing 64, 321 (1997). 10. R. E. Cohen, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 50, (1994). 11. M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54, 4519 (1996). 12. M. J. Mehl and D. A. Papaconstantopoulos, Europhysics Letters 31, 537 (1996).

9 0.13 Energy/Atom (Ry) fcc hcp bcc simple cubic βsn hexagonal diamond diamond Volume/Atom (Bohr 3 ) Figure 6: The TB equation of state for several structures silicon, using the parameters given in Table II. The structures marked by solid lines were used in the t, while the structures marked by dotted lines are predictions of the method. 13. S. Yang, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B, to appear (1997). 14. All of the tight-binding parameters based on Refs. [11] and [13], as well as the parameters used in this paper, are available on the World Wide Web at O. K. Andersen, Phys. Rev. B 12, 3060 (1975). 16. S. H. Wei and H. Krakauer, Phys. Rev. Lett. 55, 1200 (1985). 17. M. R. Pederson and K. A. Jackson, Phys. Rev. B 41, 7453 (1990) 18. M. R. Pederson and K. A. Jackson, Phys. Rev. B 42, 3276 (1990). 19. M. R. Pederson and K. A. Jackson, Phys. Rev. B 43, 7312 (1991). 20. W. A. Harrison, Electronic Structure and the Properties of Solids, (Dover, New York, 1989). 21. P. Villars and L.D. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases, 2 nd Edition (ASM International, Materials Park, Ohio, 1991), p S. C. Erwin and W. E. Pickett, Science 254, 842 (1991). 23. S. Satpathy, V. P. Antropov, O.K. Andersen, O. Jepsen, O. Gunnarsson, and A. I. Liechtenstein, Phys. Rev. B 46, 1773 (1992). 24. H. L. Anderson, Ed., A Physicist's Desk Reference, The Second Edition of Physics Vade Mecum, (American Institute of Physics, New York, 1989). 25. M. J. Mehl, Phys. Rev. B 47, 2493 (1993).

10 26. M. J. Mehl, B. A. Klein, and D. A. Papaconstantopoulos, in Intermetallic Compounds: Principles and Applications, J. H. Westbrook and R. L. Fleischer, eds. (Wiley, London, 1994). Vol. I, Chapter M. T. Yin and M. L. Cohen, Phys. Rev. B 26, 5668 (1982).