DIFFICULTIES WITH OPEN STRUCTURES

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Submitted on 1 Jan 1972

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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�

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 c o m e un 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 approximate spherical symmetry of the one-electron potential within the core of each atom of the crystal.

For most physical systems in the condensed state it is quite a good approximation to draw about each nucleus an ⁽⁽ atomic sphere >>, inside which the Bloch functions can be represented in terms of eigenfunctions of the angular momentum. Without this simplification, our computations would be almost impossible.

By definition, the atomic spheres on neighbouring sites of a lattice may not overlap. But they often come very close to touching and hence may occupy a large fraction of the total volume of the crystal. What about the remaining volume - the interstitial region. I n a close packed lattice it is a good approximation to assume that we have a strictly mufin-tin potential, in which the interstitial potential is quite flat. This has, of course, the great advantage of allowing us to

expand our Bloch functions in plane waves in this region, with relatively simple rules for matching at the boundaries of the atomic spheres. The whole success of the APW and KKR methods is often sup- posed to depend upon this simplification.

Nevertheless, band structure calculations must not be restricted to close packed structures such as simple metals. As we move towards covalent and molecular crystals, we encounter lattices where the coordination number is much smaller than 8 or 12. In a n open structure such as the diamond lattice, the muffin-tin approximation is certainly not valid. Even if we make our atomic spheres touch one another, they only occupy about 33

%

of the whole volume. The variation of the potential into the centre of an <t interstitial cage ⁾⁾ is then by no means negligible, being compa- rable with, say, the width of the valence band of the 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

capable of channelling electrons along particular directions, with effects as important as the direct interaction between states centred on the atoms them- selves. Any serious study of covalent bonding, in crystals or molecules, must take account of such effects.

With the present interest in amorphous semiconduc- tors and other disordered systems, it is particularly important to have a clear understanding of the role of the interstitial potential in crystalline materials, so as not to lose these effects by over-simplified models. In a series of investigations at Bristol over the past few years, we have tried to assess the magnitude of the contribution of the interstitial potential to the band structure, and to invent new theoretical devices by which it may be taken into account with sufficient accuracy. This lecture is not meant to be a thorough review of the problem, but it is intended merely to provoke discussion. The references, for example, are not complete.

The traditional approach is not to use a muffin-tin potential at all. The one-electron potential in the crystal is represented as a sum of overlapping atomic or ionic potentials, each, of course, spherically symmetrical.

The core region of each atom is then transformed into a weak pseudopotential or model potential, wa(r), so that the total crystal (pseudo) potential may be written as a sum of contributions from the atomic sites R,, i. e.

W (r) = wa(r - Rl)

.

1

This function (or operator) is Fourier transformed

-

and also screened, exchanged, etc. - to give matrix elements

rkk,

^{= S(k - k') wa(k - kt) 7} 0 ) where S(k

-

kf) is a structure factor and w,(k - kf) is an atomic form factor. The OPW method, of course, is equivalent to this procedure, using a special type of analytical pseudopotential.

This is a well defined method, which seems to serve very well for low-coordination lattices. The pseudo- potential transformation leaves the outer part of each atomic potential unchanged, so that the potential in the interstitial region builds up naturally by the overlap of contributions from neighbouring atoms. After suitable screening, etc., the variation of this potential will still be reflected in the matrix elements

rkk*.

^The

correct treatment of interstitial potentials by the OPW method has thus often been claimed as a distinct advantage of this technique.

But this advantage is illusory. It turns out that spatial variations of the interstitial potential can easily be accommodated within the APW and KKRZ schemes [I]. A11 that one needs to do is to calculate the Fourier components of a function, W,,(r), which follows the one-electron potential in the interstitial region, but which is zero across the muffin-tin wells.

These are added to the usual matrix elements repre- senting the effects of the atomic spheres, to make up the full amplitude of the elements

r'.

in the secular determinant.

This is obviously correct in principle. If there were no potential wells within the atomic spheres, then we should simply be carryng out a nearly free electron band structure calculation for the interstitial potential alone. The general proof is quite straightforward : in effect, we match solutions within the atomic spheres to combinations of plane waves in the interstitial region, allowing now for the mixing produced by the Fourier components of W,(r).

This method in fact, is almost exactly equivalent to the use of overlapping atomic potentials, as in the OPW method. Suppose that V,,(r) had actually been constructed by superposing the parts of the neigh- bouring atomic potentials outside the corresponding atomic spheres. The matrix elements of this function can be written in the form of a structure factor multi- plied by the Fourier transform of this outer potential for a single atom. At the same time, the APW or KKRZ matrix elements can be represented as Fourier components of a model potential within the atomic sphere. Combining these two terms, atom by atom we find that we have simply calculated the Fourier trans- form of a complete atomic pseudopotential, with a model potential for the interior of the atom and the outer part intact. This is almost identical with the result obtained by the OPW method, where only the inner part of the atomic potential is really affected by the pseudopotential transformation. The only diffe- rence is that the overlap from neighbouring atoms may be sufficient to be noticeable inside the next atomic sphere thus modifying the depth or sphericity of the muffin-tin well. This small effect could be dealt with as a perturbation, or, more elaborately, by the use of warped muffin-tin wells [2].

In practice, of course, exponents of the OPW and APWIKKRZ techniques use different schemes of screening, etc., which may make the detailed results of computations on a particular material rather different.

That is a different type of problem, which I do not wish to discuss here. The main point is that there is no essential advantage of either method, when applied to a particular material, except the mathematical convergence of the pseudopotential matrix ; the OPW method is not, in fact, especially favourable for open structures.

The question remains : can this general method be applied to any type of open structure ? How far can we go before we reach the practical computational limits of allowing for the interstitial potential by its Fourier components ?

In any attempt to delineate the obstacles to such a scheme, we began about 3 years ago an attack on an extreme case - the band structure of cc crystalline polyethylene

)>.

The thesis work of Mr. L. Griffiths 131 on this subject, although not successful in a conventio-

nal sense, has proved a most instructive cc negative experiment

>>.

Let me remind you that the cr infinite >> chains of -CH,- pack together into a relatively open lattice, containing 12 atoms (4 carbon

+

8 hydrogen) per unit cell. Each chain is, of course, a zig-zag skeleton of carbon atoms, with pairs of hydrogens at each joint.

In principle, therefore, with 12 atomic spheres and a scheme for the variation of the interstitial potential, the calculation can be attempted using the KKRZ method. The aim was merely to find a band structure that would look something like the results of an atomic orbital or molecular orbital computation, which would of course be the normal way of dealing with this sort of compound.

The first difficulty was to deal with the hydrogen atoms, which lie too closely within the atomic spheres of the carbons to be treated as separate muffin-tins.

This is a general obstacle to any approach to organic materials and requires special treatment in its own right. Liberman [4], in his discussion of molecular hydrogen, has suggested a device that might be useful also for the -CH,- complex - but I must confess that we don't yet know what to do about it, and will not discuss this point further. The methods used by Johnson (see his review in this conference) for mole- cular systems are obviously applicable in this case.

The major difficulty was, however, that the band structure computation conspicuously failed to converge on well-defined energy eigenvalues, even when as many as 101 plane waves were included in the basis functions. For example, the bands were still about 4 e.v wide in the direction normal to the chain axes, where tight-binding theory would predict that they should be very nearly flat. In other words, there was still far too much apparent penetration of the potential barriers, which are, in fact, quite high enough to isolate each t< chain ⁾⁾ from its neighbours. Since the cc interstitial region >> now occupies 90

%

of the volume of the crystal, this has now become the dominant fea- ture of the potential landscape, and must have a most serious influence on the wave functions. In fact, it has become so large that the slow convergence of the plane wave expansion in the interstitial region now hampers the computation.

To see this, we can carry out a band structure calcu- lation for a one-dimensional lattice by the same method. Consider, for example, a squarewell lattice, for which the eigenvalues are easily determined exactly by the Kronig-Penney method. Now compute these same eigenvalues by a Fourier expansion of the potential and of the Bloch functions : it turns out that one needs a great many plane waves to represent the decay of the functions into the barriers. Griffiths found, for example, with barriers as high as in his polyethylene model, that he had to go to a 9 x 9 matrix to get qualitative agreement, and as far as 16 x 16 for 3 significant figures in the eigenvalue. In three dimensions, where the number of significant reciprocal

points would be cubed, this means that several thou- sand waves would be required for reasonable accuracy.

Despite the marvellous capabilities promised to us for the new computers, I can only say of such a ven- ture : ⁽⁽ C'est magnz$que, mais ce n'est pas la phy- sique ! >>.

Nor can the situation be improved significantly by an arbitrary variation of the tt muffin-tin zero >>

-

a device that has sometimes been used to adjust the results obtained for some diamond structures with a muffin-tin potential. If we set the zero too low - e. g.

at the lip of each atomic well - then we get a free- electron-like band structure in all directions, which cannot be corrected adequately by large components of interstitial potential. If we set this zero very high, so as to force an expansion in negative energy plane wave functions in the interstitial region, we introduce high thin barriers between neighbouring atoms along the chain, where in fact there is only a low ((pass >>

to be crossed. In simple physical terms, the difficulty is to reconcile the ^(< freedom ⁾⁾ of the electrons along the chain with their cr boundedness )> away from the main skeleton of carbon and hydrogen atoms.

In the face of these obstacles, we return to the arche- type of all covalent crystals-diamond. Here, in fact, we know that a direct OPW computation seems to converge very slowly, But a KKRZ calculation [5]

with interstitial potentials gives quite reliable results (within the uncertainties of the one-electron potential in the lattice), so that the convergence of the plane wave expansion in the interstitial region is adequate.

Nevertheless, this calculation can be significantly improved and simplified [6]. It will be noted that the interstitial region in a diamond lattice approximates to a spherical t( cage P, within which, for example, a sphere as large as an atomic sphere could be enclosed.

Within this sphere, indeed, the interstitial potential itself is approximately symmetrical. The trick is simply to treat this region as another type of spherical t( atom >>

in the lattice, with its own characteristic phase shifts produced by a spherically symmetrical c< repulsive ⁾⁾ potential therein. The two types of tr atom )) now form a face-centred cubic lattice occupying 68

%

of the crystal volume, so that the residual interstitial potential does not now vary significantly from place to place. For this system, in fact, we now have a muffin- tin potential, so that we can return to the KKR repre- sentation, which is often more rapidly convergent than the corresponding plane wave expansions. This device also has the estimable advantage that it can be used in studies of glassy tetrahedral structures, such as amorphous silicon and germanium. The cluster method, of Klima and McGill [7], was originally applied to clusters of simple muffin-tin wells in a flat potential, but there is no difficulty in augmenting these with spherical ⁽⁽ interstitions ^)> to represent the varia- tions of potential in the interstitial regions.

The next case to consider is that of graphite. To a first approximation, this is a two-dimensional system,

C3-212 J. M. ZIMAN since the potential barrier between successive hexagonal

layers is about 2

A

thick, and about Zryd high. The usual band structure calculation is therefore an LCAO scheme, with adjustable parameters for overlap inte- grals etc., in a two-dimensional zone. But we know that the cr n-electrons >> are relatively free to move in the plane of the atoms, so that a nearly-free-electron model is not inappropriate along these two dimensions.

To calculate the parameters of such a model from first principles, we must make a full three-dimensional computation of the band structure, with proper model potentials in the atomic spheres.

It is obvious that the interstitial potential will play a most important part in the band structure. As we have already seen in the case of polyethylene, mere empirical adjustment of the muffin-tin zero is not adequate to represent these effects. If this is set too low, for example [8], the bands of the <c two-dimen- sional zone n are split by several volts, because the interaction between adjacent planes is too strong ; if the muffin-tin zero is raised by about 1 ryd to match the average potential in the interstitial regions, the bands become too narrow in the basal plane, showing that electron transfer between neighbouring atoms in a layer is now being inhibited.

A more elaborate calculation, with Fourier compo- nents of the interstitial potential, would probably converge somewhat better in this case than in the case of polyethylene. But Keller and Spiridonov have been investigating the use of the cc interstition ^)> device in this more complex lattice. Around every local maxi- mum of the interstitial potential, a sphere is drawn, as before, and transformed into an equivalent ⁽⁽ repul- 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 plane between two successive atomic layers. Suitable spherically symmetrical potential distributions can be constructed in each sphere to mimic the local behaviour of the interstitial potential and then treated as peculiar atoms in a band structure calculation with 14 << atoms >>

per unit cell. This is, of course, a fairly complicated computing program, but the results obtained so far are quite consistent with what is known about the band structure of graphite. The point is that we can,

by this device, represent fairly accurately, and without difficulties of convergence, the effects of very strong interstitial potentials of the kind normalIy found in molecular crystals.

An alternative approach, which has not yet been worked out in detail, is to return to the cellular method, in which the volume of the crystal is divided into polyhedral cells, and the wave function made to satisfy the usual conditions of continuity at each boundary. If each cell contains just one atomic sphere, then the potential may be treated approximately as a function only of the distance from the nucleus of the atom, right out into the corners of the cell. I n this way, the high potentials in the interstitial regions far from any atom can be reproduced

-

yet we may still use the angular momentum eigenfunctions as basis functions for the Bloch function within each cell. The arbitrariness of the choice of matching points on irre- gular polyhedral surfaces is an objection to this method on purist mathematical terms, but we know that it converges satisfactorily in practice for the diamond lattice. Could this method be applied to more complex systems, such as graphite or polyethy- lene ? The question is worth a considered answer.

The results discussed in this lecture do not, as yet, add up to a solution of the general problem of finding an alternative to the tight binding method for open structures. But perhaps they show that this project is not altogether hopeless. In the long run, it is possible that the theoretical chemists may even come to bless us for robbing them of their c< two-centre integrals ))

and << three-centre integrals >> and the other conven-

tional paraphenalia of their trade.

In the discussion after this lecture, the following significant points were made : A. R. Williams reported that his compu- tations indicated that warping of the muffin-tin wells was more important than the cc interstitions ^>> in diamond lattices.

B. Segall and 0. K. Anderson pointed out that a variational approach using muffin-tin orbitals deduced from the KKR formalism could yield better convergence in the interstitial

region than a plane wave expansion.

Several speakers also stated that the cellular method did not in fact work satisfactorily even in the diamond lattice, because very high components of angular momentum are needed in 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.

[4] LIBERMAN (D.), private communication.

to be published.

KELLER (J.), J. Phys. C : Solid State Phys., 1971, 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.