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

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. screening, etc., which may make the detailed results of This is a well defined method, which seems to serve computations on a particular material rather different. very well for low-coordination lattices. The pseudo- That is a different type of problem, which I do not potential transformation leaves the outer part of each wish to discuss here. The main point is that there is atomic potential unchanged, so that the potential in no essential advantage of either method, when applied the interstitial region builds up naturally by the overlap to a particular material, except the mathematical of contributions from neighbouring atoms. After convergence of the pseudopotential matrix ; the suitable screening, etc., the variation of this potential OPW method is not, in fact, especially favourable for will still be reflected in the matrix elements rkk*.The open structures. correct treatment of interstitial potentials by the The question remains : can this general method be OPW method has thus often been claimed as a distinct applied to any type of open structure ? How far can advantage of this technique. we go before we reach the practical computational But this advantage is illusory. It turns out that limits of allowing for the interstitial potential by its spatial variations of the interstitial potential can Fourier components ? easily be accommodated within the APW and KKRZ In any attempt to delineate the obstacles to such a schemes [I]. A11 that one needs to do is to calculate scheme, we began about 3 years ago an attack on an the Fourier components of a function, W,,(r), which extreme case - the band structure of cc crystalline follows the one-electron potential in the interstitial polyethylene )>.The thesis work of Mr. L. Griffiths 131 region, but which is zero across the muffin-tin wells. on this subject, although not successful in a conventio- DIFFICULTIES WITH OPEN STRUCTURES C3-211 nal sense, has proved a most instructive cc negative points would be cubed, this means that several thou- experiment >>. sand waves would be required for reasonable accuracy. Let me remind you that the cr infinite >> chains of Despite the marvellous capabilities promised to us -CH,- pack together into a relatively open lattice, for the new computers, I can only say of such a ven- containing 12 atoms (4 carbon + 8 hydrogen) per unit ture : (( C'est magnz$que, mais ce n'est pas la phy- cell. Each chain is, of course, a zig-zag skeleton of sique ! >>. carbon atoms, with pairs of hydrogens at each joint. Nor can the situation be improved significantly by In principle, therefore, with 12 atomic spheres and a an arbitrary variation of the tt muffin-tin zero >> - a scheme for the variation of the interstitial potential, device that has sometimes been used to adjust the the calculation can be attempted using the KKRZ results obtained for some diamond structures with a method. The aim was merely to find a band structure muffin-tin potential. If we set the zero too low - e. g. that would look something like the results of an atomic at the lip of each atomic well - then we get a free- orbital or molecular orbital computation, which electron-like band structure in all directions, which would of course be the normal way of dealing with cannot be corrected adequately by large components this sort of compound. of interstitial potential. If we set this zero very high, The first difficulty was to deal with the hydrogen so as to force an expansion in negative energy plane atoms, which lie too closely within the atomic spheres wave functions in the interstitial region, we introduce of the carbons to be treated as separate muffin-tins. high thin barriers between neighbouring atoms along This is a general obstacle to any approach to organic the chain, where in fact there is only a low ((pass >> materials and requires special treatment in its own to be crossed. In simple physical terms, the difficulty right. Liberman [4], in his discussion of molecular is to reconcile the (< freedom )) of the electrons along hydrogen, has suggested a device that might be useful the chain with their cr boundedness )> away from the also for the -CH,- complex - but I must confess main skeleton of carbon and hydrogen atoms. that we don't yet know what to do about it, and will In the face of these obstacles, we return to the arche- not discuss this point further. The methods used by type of all covalent crystals-diamond. Here, in fact, Johnson (see his review in this conference) for mole- we know that a direct OPW computation seems to cular systems are obviously applicable in this case. converge very slowly, But a KKRZ calculation [5] The major difficulty was, however, that the band with interstitial potentials gives quite reliable results structure computation conspicuously failed to converge (within the uncertainties of the one-electron potential on well-defined energy eigenvalues, even when as in the lattice), so that the convergence of the plane many as 101 plane waves were included in the basis wave expansion in the interstitial region is adequate. functions. For example, the bands were still about Nevertheless, this calculation can be significantly 4 e.v wide in the direction normal to the chain axes, improved and simplified [6]. It will be noted that the where tight-binding theory would predict that they interstitial region in a diamond lattice approximates should be very nearly flat. In other words, there was to a spherical t( cage P, within which, for example, a still far too much apparent penetration of the potential sphere as large as an atomic sphere could be enclosed. barriers, which are, in fact, quite high enough to Within this sphere, indeed, the interstitial potential isolate each t> now occupies 90 % of the volume to treat this region as another type of spherical t( atom >> of the crystal, this has now become the dominant fea- in the lattice, with its own characteristic phase shifts ture of the potential landscape, and must have a most produced by a spherically symmetrical c< repulsive )) serious influence on the wave functions. In fact, it has potential therein. The two types of tr atom )) now become so large that the slow convergence of the plane form a face-centred cubic lattice occupying 68 % of wave expansion in the interstitial region now hampers the crystal volume, so that the residual interstitial the computation. potential does not now vary significantly from place To see this, we can carry out a band structure calcu- to place. For this system, in fact, we now have a muffin- lation for a one-dimensional lattice by the same tin potential, so that we can return to the KKR repre- method. Consider, for example, a squarewell lattice, sentation, which is often more rapidly convergent for which the eigenvalues are easily determined exactly than the corresponding plane wave expansions. This by the Kronig-Penney method. Now compute these device also has the estimable advantage that it can be same eigenvalues by a Fourier expansion of the used in studies of glassy tetrahedral structures, such potential and of the Bloch functions : it turns out that as amorphous silicon and germanium. The cluster one needs a great many plane waves to represent the method, of Klima and McGill [7], was originally decay of the functions into the barriers. Griffiths applied to clusters of simple muffin-tin wells in a flat found, for example, with barriers as high as in his potential, but there is no difficulty in augmenting these polyethylene model, that he had to go to a 9 x 9 matrix with spherical (( interstitions )> to represent the varia- to get qualitative agreement, and as far as 16 x 16 tions of potential in the interstitial regions. for 3 significant figures in the eigenvalue. In three The next case to consider is that of graphite. To a dimensions, where the number of significant reciprocal first approximation, this is a two-dimensional system, C3-212 J. M. ZIMAN since the potential barrier between successive hexagonal by this device, represent fairly accurately, and without layers is about 2 A thick, and about Zryd high. The difficulties of convergence, the effects of very strong usual band structure calculation is therefore an LCAO interstitial potentials of the kind normalIy found in scheme, with adjustable parameters for overlap inte- molecular crystals. grals etc., in a two-dimensional zone. But we know An alternative approach, which has not yet been that the cr n-electrons >> are relatively free to move in worked out in detail, is to return to the cellular the plane of the atoms, so that a nearly-free-electron method, in which the volume of the crystal is divided model is not inappropriate along these two dimensions. into polyhedral cells, and the wave function made to To calculate the parameters of such a model from first satisfy the usual conditions of continuity at each principles, we must make a full three-dimensional boundary. If each cell contains just one atomic sphere, computation of the band structure, with proper then the potential may be treated approximately as a model potentials in the atomic spheres. function only of the distance from the nucleus of the It is obvious that the interstitial potential will play atom, right out into the corners of the cell. In this a most important part in the band structure. As we way, the high potentials in the interstitial regions far have already seen in the case of polyethylene, mere from any atom can be reproduced - yet we may still empirical adjustment of the muffin-tin zero is not use the angular momentum eigenfunctions as basis adequate to represent these effects. If this is set too functions for the Bloch function within each cell. The low, for example [8], the bands of the device in that the theoretical chemists may even come to bless this more complex lattice. Around every local maxi- us for robbing them of their c< two-centre integrals )) mum of the interstitial potential, a sphere is drawn, and << three-centre integrals >> and the other conven- as before, and transformed into an equivalent (( repul- tional paraphenalia of their trade. sive atom D. It is obvious, for example, that the centre of each hexagonal ring should acquire its sphere and that various further spheres must be packed into the In the discussion after this lecture, the following significant points were made : A. R. Williams reported that his compu- plane between two successive atomic layers. Suitable tations indicated that warping of the muffin-tin wells was spherically symmetrical potential distributions can be more important than the cc interstitions >> in diamond lattices. constructed in each sphere to mimic the local behaviour B. Segall and 0. K. Anderson pointed out that a variational of the interstitial potential and then treated as peculiar approach using muffin-tin orbitals deduced from the KKR atoms in a band structure calculation with 14 << atoms >> formalism could yield better convergence in the interstitial region than a plane wave expansion. per unit cell. This is, of course, a fairly complicated Several speakers also stated that the cellular method did not computing program, but the results obtained so far in fact work satisfactorily even in the diamond lattice, because are quite consistent with what is known about the very high components of angular momentum are needed in band structure of graphite. The point is that we can, the corners of the cell.

References

[I] SCHLOSSER(H. C.) and MARCUS(P. M.), Phys. Rev., [6] KELLER(J.), J. Phys. C : Solid State Phys., 1971, 1963, 131, 2529. 4, L 85. BELEZNAY(F.) and LAWRENCE(M. J.), J. Phys. This method was also in fact proposed indepen- C : Solid State Phys., 1968, 1, 1288. dently by the MIT group working on molecules. [7] MCGILL(T. C.) and KLIMA(J.), J. Phys. C : Solid [2] EVANS(R.) and KELLER(J.), J. Phys. C : to be State Phys., 1970, 3, L 163-4. published. MCGILL(T. C.) and KLIMA(J.), Phys. Rev., 1971, [3] GRIFFITHS(L.), to be submitted for publication. to be published. KELLER(J.), J. Phys. C :Solid State Phys., 1971, [4] LIBERMAN(D.), private communication. to be vublished. [5] BELEZNAY(F.), J. Phys. C : Solid State Phys., 1970, 181 SPIRIDONOV?T.) and KELLER(J.), to be submitted for 3, 1884. publication.