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Difficulties with Open Structures J DIFFICULTIES WITH OPEN STRUCTURES J. Ziman To cite this version: J. Ziman. DIFFICULTIES WITH OPEN STRUCTURES. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-209-C3-212. 10.1051/jphyscol:1972331. jpa-00215065 HAL Id: jpa-00215065 https://hal.archives-ouvertes.fr/jpa-00215065 Submitted on 1 Jan 1972 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C3, suppl6ment au no 5-6, Tome 33, Mai-Juin 1972, page C3-209 DIFFICULTIES WITH OPEN STRUCTURES J. M. ZIMAN H. H. Wills Physics Laboratory, Royal Fort, Tyndall Avenue, Bristol BS8 ITL R6sum6. - L'approximation << muffin-tin u est suffisamment bonne pour les structures relati- vement compactes typiques des mktaux. Cependant, dans les structures a liaisons directives ou chaque atome peut ne pas avoir plus de quatre premiers voisins, les <( vallks >) et (( collines D du potentiel interstitiel jouent un rBle fondamental dans la structure electronique. Une partie de notre travail rkcent a Bristol concerne cette difFicultk. Dans la plupart des <( methodes >> de structure de bandes, les deviations par rapport au potentiel <( muffin-tin )) peuvent &re reprksentkes par les composantes de Fourier du potentiel interstitiel ajoutees aux contributions dues a la non-sphericit6 du potentiel sur la sphkre limite. Mais l'8tude du cas extrgme qu'est le polyethylbne montre que ce proc6d6 ne convient pas quand la rkgion interstitielle contient des barribres que les klectrons de valence ne pknktrent presque pas. Dans le rkseau du diamant, on pourra tenir compte de ces effets par la methode cellulaire en utilisant une cellule tktraedrique. Un moyen plus pratique est de remplir la plus grande part du vide interstitiel par une << anti-barribre >> spherique qui se comporte comeun autre type d'atome dans un calcul KKR ou APW. Mais dans les structures en couches ou en chdnes, telles que le graphite ou le poly8thylbne, il semble essentiel d'introduire de nouvelles fonctions de base avec des conditions aux limites approprikes sur les barribres de potentiel planes ou cylindriques. La signification de ces id& pour les theories de la liaison chimique est kvident. Abstract. - The (< muffi-tin >> approximation works well enough for the relatively close-packed structures typical of metals. In ((bonded >) structures, however, where each atom may have no more than four neighbours, the << valleys )) and << hills )) of the interstitial potential play a fundamental part in the electronic structure. Some of our recent work at Bristol has been concerned with this difficulty. In most of the band structure <( methods >>, deviations from the simple muffin-tin potential can be represented by the Fourier components of the interstitial potential, together with contributions from non-sphericity of the muffin-tin wells. But a study of the extreme case of polyethylene shows that this procedure fails when the interstitial region contains barriers through which the valence electrons can scarcely penetrate. In the diamond lattice, these effects could be taken care of by the cellular method, using a tetra- hedral cell. A more practical procedure is to fill the major part of the interstitial void with a sphe- rical << anti-well )>, which behaves like another type of atom in a KKR or APW calculation. But in layer or chain structures, such a graphite or polyethylene, it seems essential to introduce new basis functions with appropriate boundary conditions on plane or cylindrical potential barriers. The significance of these ideas for theories of chemical bonding is obvious. Band structure calculations rely heavily upon the expand our Bloch functions in plane waves in this approximate spherical symmetry of the one-electron region, with relatively simple rules for matching at potential within the core of each atom of the crystal. the boundaries of the atomic spheres. The whole For most physical systems in the condensed state it is success of the APW and KKR methods is often sup- quite a good approximation to draw about each posed to depend upon this simplification. nucleus an (( atomic sphere >>, inside which the Bloch Nevertheless, band structure calculations must not functions can be represented in terms of eigenfunctions be restricted to close packed structures such as simple of the angular momentum. Without this simplification, metals. As we move towards covalent and molecular our computations would be almost impossible. crystals, we encounter lattices where the coordination By definition, the atomic spheres on neighbouring number is much smaller than 8 or 12. In an open sites of a lattice may not overlap. But they often come structure such as the diamond lattice, the muffin-tin very close to touching and hence may occupy a large approximation is certainly not valid. Even if we make fraction of the total volume of the crystal. What about our atomic spheres touch one another, they only the remaining volume - the interstitial region. In a occupy about 33 % of the whole volume. The variation close packed lattice it is a good approximation to of the potential into the centre of an <t interstitial assume that we have a strictly mufin-tin potential, in cage )) is then by no means negligible, being compa- which the interstitial potential is quite flat. This has, rable with, say, the width of the valence band of the of course, the great advantage of allowing us to material. In other words, the interstitial potential Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972331 C3-210 J. M. ZIMAN becomes a significant feature of the crystal potential These are added to the usual matrix elements repre- capable of channelling electrons along particular senting the effects of the atomic spheres, to make up directions, with effects as important as the direct the full amplitude of the elements r'. in the secular interaction between states centred on the atoms them- determinant. selves. Any serious study of covalent bonding, in This is obviously correct in principle. If there were crystals or molecules, must take account of such effects. no potential wells within the atomic spheres, then we With the present interest in amorphous semiconduc- should simply be carryng out a nearly free electron tors and other disordered systems, it is particularly band structure calculation for the interstitial potential important to have a clear understanding of the role of alone. The general proof is quite straightforward : in the interstitial potential in crystalline materials, so as effect, we match solutions within the atomic spheres not to lose these effects by over-simplified models. In to combinations of plane waves in the interstitial a series of investigations at Bristol over the past few region, allowing now for the mixing produced by the years, we have tried to assess the magnitude of the Fourier components of W,(r). contribution of the interstitial potential to the band This method in fact, is almost exactly equivalent to structure, and to invent new theoretical devices by the use of overlapping atomic potentials, as in the which it may be taken into account with sufficient OPW method. Suppose that V,,(r) had actually been accuracy. This lecture is not meant to be a thorough constructed by superposing the parts of the neigh- review of the problem, but it is intended merely to bouring atomic potentials outside the corresponding provoke discussion. The references, for example, are atomic spheres. The matrix elements of this function not complete. can be written in the form of a structure factor multi- The traditional approach is not to use a muffin-tin plied by the Fourier transform of this outer potential potential at all. The one-electron potential in the crystal for a single atom. At the same time, the APW or is represented as a sum of overlapping atomic or ionic KKRZ matrix elements can be represented as Fourier potentials, each, of course, spherically symmetrical. components of a model potential within the atomic The core region of each atom is then transformed sphere. Combining these two terms, atom by atom we into a weak pseudopotential or model potential, wa(r), find that we have simply calculated the Fourier trans- so that the total crystal (pseudo) potential may be form of a complete atomic pseudopotential, with a written as a sum of contributions from the atomic model potential for the interior of the atom and the sites R,, i. e. outer part intact. This is almost identical with the result obtained by the OPW method, where only the W (r) = wa(r - Rl) . 1 inner part of the atomic potential is really affected by the pseudopotential transformation. The only diffe- This function (or operator) is Fourier transformed - rence is that the overlap from neighbouring atoms may and also screened, exchanged, etc. - to give matrix be sufficient to be noticeable inside the next atomic elements sphere thus modifying the depth or sphericity of the muffin-tin well. This small effect could be dealt with = S(k - k') wa(k - kt) 7 rkk, 0) as a perturbation, or, more elaborately, by the use of where S(k - kf) is a structure factor and w,(k - kf) warped muffin-tin wells [2]. is an atomic form factor. The OPW method, of course, In practice, of course, exponents of the OPW and is equivalent to this procedure, using a special type APWIKKRZ techniques use different schemes of of analytical pseudopotential.
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