Modeling the Properties of Quartz with Clusters. Abstract

Transcription

1 Modeling the Properties of Quartz with Clusters James R. Chelikowsky Department of Chemical Engineering and Materials Science Minnesota Supercomputer Institute University of Minnesota, Minneapolis, MN Nadia Binggeli Institut de Physique Appliquée, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland Abstract We illustrate a cluster approach to calculate the electronic properties of α- quartz. Cluster approaches to modeling defects and other localized states within crystalline matter have a number of computational advantages when compared to traditional energy band methods. However, the implementation of cluster models has been inhibited by the poor convergence of bulk properties with cluster size. Here we illustrate recent developments in algorithm construction which allow one to consider larger clusters than previously accessible. In particular, we use clusters such as Si 18 O 23 H 26 and Si 25 O 34 H 32 to model quartz. We compare these cluster calculations to plane-wave-pseudopotential calculations for crystalline quartz and find very good agreement. 1

2 I. INTRODUCTION Historically, the electronic structure problem for crystalline matter has been solved by capitalizing on the translational periodicity of the system, e.g., utilizing energy-band theory based on Bloch s theorem. It would appear to be more than counter intuitive to discard this well established and very successful procedure. However, there are some good reasons for considering alternatives. In particular, localized excitations or defects can invalidate periodic symmetry and, as such, are difficult to model within a crystalline environment. Although one can retain a band-structure approach to examine defects, e.g., by employing a supercell configuration 1, this approach can present one with a number of drawbacks. Supercells, unless very large, frequently result in unwanted cell-cell interactions. These interactions are strong because they involve interactions of states with similar energy and wave-function character. For example, if one considers a vacancy in a supercell, then unless the supercell is very large, vacancy-vacancy interactions will result in a large dispersion of the defect levels. This makes it difficult to assign a precise position of the defect state within the band gap. If the vacancy is charged, the situation is even more complicated, as a spurious neutralizing medium has to be inserted to remove the divergent Coulombic repulsion energy. Another difficulty centers on constructing formalisms for response functions and optical excitations in crystalline matter. Calculating the response of crystalline matter to an external electric field is highly nontrivial when starting from a Bloch-function formulation 2. Clusters avoid many of these problems 3. Because clusters are finite, charged states and localized excitations are easy to quantify. Unlike supercells, there are no cell-cell interactions to introduce an artificial dispersion into discrete energy levels. Of course, one might argue that clusters possess a surface which can introduce surface states or other unwanted interactions into the system in a manner similar to unwanted cell-cell interactions in a supercell configuration. However, the free surface of a cluster can be passivated by adding hydrogens or other artificial potentials. One has great flexibility in designing passivating mechanisms or mediums for the surface cluster. One needs only to find a prescription which 2

3 gently terminates the surface of the cluster. This is not the case for supercells where such flexibility is restricted by the requirement of periodic symmetry. A more serious arises because the electronic and structural properties of clusters converge slowly to bulk values as a function of cluster size. To obtain accurate bulk properties, a very large cluster is required. This problem can be especially acute in obtaining accurate band gaps and band edges from cluster geometries. Here we illustrate how new algorithms for the electronic structure problem can overcome this obstacle. 4 We illustrate that one can obtain accurate electronic properties for quartz by considering large clusters of silica. Since one expects strong local reconstruction around defects in quartz, cluster models of defects would appear to be a very efficacious approach. A first step in such a procedure is to establish that bulk quartz properties can be accurately replicated by large silica clusters. Owing to the technological and fundamental importance of silica, this is a useful test case. 5 II. BUILDING A CLUSTER MODEL The construction of a cluster model for a given crystal is a non-unique operation. One would like the cluster to be large enough to retain bulk like properties, but small enough to be tractable in terms of numerical calculations. We have proceeded as follows. We consider a fragment of quartz which is encompassed within a sphere whose size is taken to be approximately 15 a.u. (1 a.u. = Å). Such a sphere size should be sufficiently large to capture the electronic properties of crystalline silica. We constructed two different clusters of this approximate size. One cluster is centered on an oxygen atom; the other on a silicon atom. Of course, the surface of the fragments contain silicon and oxygen species which are under-coordinated. It is possible to simply cap off any broken bond at the surface with a hydrogen atom, but we wanted to avoid having bond configurations such as silicon atoms with only one oxygen neighbor (or oxygens with only one silicon neighbor.) These bonding configurations correspond to short fragments of Si O chains on the cluster surface. Therefore, we removed any atom from the cluster with 3

4 fewer than two neighbors. In doing so, we retain fully coordinated oxygen atoms and silicon atoms which contain at most two Si H bonds. Our procedure for constructing clusters introduces only one new type of hydrogen bond, the Si H bond, and does not introduce any O H bonds. We maintain the oxygen bonding environment at the expense of the silicon bonding environment. This choice is appropriate as the Si O bond contains a large ionic contribution with most of the valence charge on the oxygen atom. The stoichiometry of the resulting clusters is Si 25 O 34 H 32 for the origin at a silicon atom and Si 18 O 23 H 26 for the origin at an oxygen atom. Ball and stick models of these clusters are shown in Figure 1. We believe these two clusters will provide a reasonable first approach at determining the electronic properties of quartz from cluster models. III. COMPUTATIONAL METHODS Our theoretical approach centers on ab initio pseudopotentials 6 which have been constructed within the local density approximation using the procedure of Troullier and Martins. 7 The exchange-correlation potential was from the work of Ceperley and Alder 8 as parameterized by Perdew and Zunger. 9 The atomic configuration used for the construction of the Si potential was 3s 2 3p 2 with the core-size parameters fixed to r s = r p = 2.50 a.u. For the O potential, the configuration was 2s 2 p 4 with the core size taken to be r s = 1.30 a.u. and r p = 1.65 a.u.. The local part of the pseudopotential was taken to be l = 0; only l = 1 nonlocal terms were included in the pseudopotential. The H pseudopotential was taken to be a simple local potential as we need not replicate the Si H bond with great accuracy. All we require is that the dangling bond state be localized and electrically inactive. We followed our previous work on quantum dots and used H-like pseudopotentials. 10 Additional computational details on the pseudopotentials can be found elsewhere. 6,10 12 The resulting one-electron Schrödinger equation was solved in real space on a uniform grid via a higher-order finite difference method. 4 The grid spacing, h, was fixed to be a.u.. 4

5 We can roughly estimate the plane-wave cut-off as 1 2 (π/h)2 or about 90 Ry. This can be compared to a plane wave cut-off of 64 Ry commonly used for crystalline silica. 12 Our higher-order finite difference expansion of the kinetic energy operator includes terms up to 12th order. The resulting eigenvalue problem was solved using a Generalized Davidson procedure. 13 This method takes advantage of the sparsity and well defined structure of the Hamiltonian matrix. A block diagonalization procedure was used to find the eigenvalue/eigenvector pair. Preconditioning consisted of averaging over neighboring grid points. This simple preconditioning accelerates the convergence by approximately 30% in terms of the computing time. The boundary conditions for the eigenvalue problem demand the wave function vanish outside a sphere which contains the cluster. The size of the sphere was set to be at least 4-5 a.u. removed from any atom within the cluster. A multi-pole expansion was performed to determine the Hartree potential outside of this domain. This expansion was used to fix the boundary condition in solving Poisson s equation with a conjugate gradient method. 4,13 Up to l =9 components were kept in the multi-pole expansion. IV. ELECTRONIC DENSITY OF STATES In this work, we focus on the electronic properties of the clusters as measured by the energy levels of the cluster, i.e., the eigenvalue spectrum. For both clusters, we calculated all the occupied states and the empty states within a few ev of the lowest empty state. We wish to examine whether the energy spectrum of the cluster resembles our previous work for the density of states of α-quartz. 12 The eigenvalue spectra for the two clusters were broadened by a convolution with a Gaussian (half width of 0.15 ev at half maximum). This broadening will facilitate comparisons with band-structure results from crystalline quartz. In Figure 2, we illustrate the broadened eigenvalue spectra for both of our model clusters. The highest occupied state is taken to be the energy zero in both cases. The eigenvalue spectra for the two clusters are not expected 5

6 to be identical as the cluster stoichiometry and size differ. Nonetheless, the essential features are very similar. One noticeable difference concerns the empty states. In Si 25 O 34 H 32 the gap is slightly smaller than for Si 18 O 23 H 26. This is to be expected from confinement effects, i.e., the optical gap for quantum dots increases as the dot size shrinks. In previous work, we examined the density of states for α-quartz using a plane-wave basis and pseudopotentials. 12 We compare our density of states from this older work to our clustereigenvalue spectra (Figure 2). Again, the eigenvalue spectra and density of states are not expected to be quantitatively similar owing to a number of differences in the calculations. For example, the two calculations used slightly different pseudopotentials and a different basis, i.e., the crystalline case employed a plane-wave basis and the cluster used a grid and higher-order finite differencing. Also, the clusters contain H-like atoms used to passivate the surface. Nonetheless, the essential features are well replicated by the clusters. From our work on quartz, the spectral features of the density of states in quartz can be decomposed as follows: The contributions between -20 to -17 ev corresponds to quasicore-like states associated with the O 2s levels. The contribution between -10 to -4 ev correspond to Si-O bonding orbitals. The region near the top of the valence band (-4 to 0 ev) corresponds to oxygen lone-pair states. The calculated band gap for quartz is about 6 ev. The measured band gap of quartz is roughly 9 ev. The difference between the calculated and measured gap in quartz is consistent with what one would expect from the local density functional approximation. One can perform a local density of states (LDOS) analysis for the cluster-eigenvalue spectra to assess whether the spectral features in the cluster are similar to the crystalline case. We constructed a LDOS for the clusters by weighting the eigenvalue spectra with the degree of overlap of the corresponding eigenfunction with a Gaussian centered on the atomic site of interest. If we wished to examine the Si p-state contribution to the eigenvalue spectrum, we placed a localized Gaussian with p-state symmetry on a Si atom site. For example, a p x state might be represented by the function: xexp( αr 2 ) which is centered on a silicon site. α is chosen to localize the Gaussian within a bond length or so of the 6

7 silicon; typically α = 0.2 a.u. for all of the atomic species. The relative contributions to the LDOS in this method are somewhat sensitive to α and the functional form used. However, independent of these issues, the resulting LDOS provides a useful picture of the energetic and spatial decomposition of the eigenvalue spectrum. In Figure 3, the LDOS is illustrated for the Si 18 O 23 H 26 cluster. The relative decomposition of the silicon and oxygen states is consistent with the crystalline case. One feature which merits special mention is the distribution of hydrogen-like states. These states are removed from the gap, although the conduction band states exhibit a rather large hydrogen composition. It is important to remove H-like states from the energy gap if one wants to examine defect levels, e.g., such states might pin the Fermi level instead of the defect state of interest. A more quantitative comparison to the crystalline density of states can be made by constructing a different LDOS: namely, one which is guaranteed to converge to the crystalline case for an infinite cluster. If one weights the eigenvalue spectra by the square of the corresponding eigenfunction integrated over a quartz unit cell constructed within the cluster, then as the cluster grows this LDOS will converge to the quartz density of state. In Figure 4, LDOS curves for the Si 18 O 23 H 26 and Si 25 O 34 H 32 clusters are illustrated and compared to the density of states for crystalline quartz. Many of the density of state features which are commonly attributed to critical points in the energy bands are accurately replicated in the cluster LDOS. In particular, the density of states peaks in the energy interval -10 to -5 ev for the crystal are well reproduced in the LDOS for the cluster. This supports the view that the fundamental electronic features of quartz are well localized within the size of this cluster. The strongest differences between the crystalline and cluster densities of states (and between the clusters) occurs for the empty sates. This is not surprising. In the cluster or quartz crystal, these states tend to be more delocalized than the occupied states. As such, they are more sensitive to the boundary conditions and the hydrogen-like states on the cluster boundary. The H-like states do contribute significantly to the LDOS of the cluster near the lowest empty states (Figure 3). 7

8 V. CONCLUSIONS In this paper, we have illustrated the utility of a cluster approach to modeling crystalline matter. We focused on quartz owing to the technological interest on this material and the localized nature of the electronic states. For clusters of atoms which corresponded to fragments of the quartz crystal, we found that the electronic spectrum of the cluster agreed very well with the density of states from energy-band calculations of quartz. Our work strongly suggests the feasibility of employing cluster models for analyzing defects and localized excitations in quartz and related materials. ACKNOWLEDGMENTS We would like to acknowledge support for this work by the U.S. Department of Energy under Grant No. DE-FG02-89ER45391, and the Minnesota Supercomputer Institute. We would like to thank S. Öğüt for helpful discussions. 8

9 REFERENCES 1 J.R. Chelikowsky, and S.G. Louie, editors: Quantum Theory of Real Materials, (Kluwer Press, 1996). 2 S. Baroni, P. Giannozi, and A. Testa, Phys. Rev. Lett. 58, 1861 (1987); A. Dal Corso, F. Mauri, and A. Rubio, Phys. Rev. B 53, (1996). 3 K. Jackson, M.R. Pederson, D. Proczay, Z. Hajnal and T. Frauenheim, Phys. Rev. B 55, 2549 (1997); K. Raghavachari and C.M. Rohlfing, J. Chem. Phys. 84, 5672 (1991); U. Röthlisberger, W. Andreoni and M. Parrinello Phys. Rev. Lett. 72, 665 (1994); J.R. Chelikowsky, S. Öğüt, I. Vasiliev, A. Stathopoulos, and Y.Saad, Predicting the Properties of Semiconductor Clusters, Theory of Atomic and Molecular Clusters, J. Jellinek, editor, [Springer, to be published]; E. Kaxiras and K. Jackson, Phys. Rev. Lett. 71, 727 (1993), and references therein. 4 J.R. Chelikowsky, N. Troullier, and Y. Saad, Phys. Rev. Lett. 72, 1240 (1994); J.R. Chelikowsky N. Troullier, Y. Saad, Phys. Rev. B 50, (1994); J.R. Chelikowsky, N. Troullier, K. Wu, and Y. Saad, Algorithms for Predicting Properties of Real Materials on High Performance Computers, Proceedings of the Toward Teraflop Computing Conference and New Grand Challenge Applications, Baton Rouge, LA 1994, editors: R.K. Kalia and P. Vashista, (Nova, New York), 1995, p. 13; J.R. Chelikowsky, N. Troullier, X. Jing, D. Dean, N. Binggeli, K. Wu and Y. Saad, Computer Physics Communications 85, 325 (1995); J.R. Chelikowsky and Y. Saad, Chemical Design Automation News 11, 1 (1996); J.R. Chelikowsky, X. Jing, K. Wu and Y. Saad, Phys. Rev. B 53, (1996). 5 G.V. Gibbs, F.C. Hill, and M.B. Boisen Jr., The SiO Bond and Electron Density Distributions, Modeling of Minerals and Silicated Materials, ed. B. Silvi and P. D Arco, Kluwer Academic Publishers, 1997, p. 157 and G.V. Gibbs, J.W. Downs, M.B. Boisen, Jr., Rev. Mineral. 29, 331 (1994). 6 J.R. Chelikowsky and M.L. Cohen, Ab Initio Pseudopotentials for Semiconductors, 9

10 Handbook on Semiconductors, ed. P. Landsberg, Elsevier, Amsterdam, 1992, Volume 1, p N. Troullier and J.L. Martins, Phys. Rev. B 43, 8861 (1991). 8 D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1986). 9 J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 10 S. Öğüt, J.R. Chelikowsky, and S.G. Louie, Phys. Rev. Lett. 79, 1770 (1997). 11 J.R. Chelikowsky, N. Troullier, J.L. Martins, and H.E. King, Jr., Phys. Rev. B44, 489 (1991). 12 N. Binggeli, N. Troullier, J.L. Martins, and J.R. Chelikowsky, Phys. Rev. B44, 4771 (1991). 13 Y. Saad, A. Stathopoulos, J.R. Chelikowsky, K. Wu, and S. Öğüt, BIT 36, 563 (1996). 10

11 FIGURES FIG. 1. Cluster models for α-quartz. Si 25 O 34 H 32 is illustrated in (a) looking down the c-axis and (b) with the c-axis in the plane of the figure. Si 18 O 23 H 26 is illustrated in (c) looking down the c-axis and (d) with the c-axis in the plane of the figure. The Si atoms correspond to large gray balls, O to smaller open balls and H to small black balls. FIG. 2. The density of states for crystalline quartz (top panel) and the corresponding broadened eigenvalue spectra for quartz-like clusters. FIG. 3. The projected local density of states for the Si 18 O 23 H 26 cluster. FIG. 4. The quartz-cell local density of states (see text) for the Si 18 O 23 H 26 and Si 25 O 34 H 32 clusters. Also shown is the density of states from an energy-band calculation of α-quartz. 11

12

13 α-quartz Si 25 O 34 H 32 Si 18 O 23 H Energy (ev)

14 50 40 Total s-like 30 p-like 20 Oxygen LDOS (arb. units) s-like Silicon 30 p-like s-like p-like Hydrogen Energy (ev)

15 Si 18 O 23 H 26 Si 25 O 34 H 32 α-quartz Energy (ev)