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Electronic structure of disclinated polytopes
S. Nicolis, R. Mosseri, J.-F. Sadoc
To cite this version:
S. Nicolis, R. Mosseri, J.-F. Sadoc. Electronic structure of disclinated polytopes. Journal de Physique, 1988, 49 (4), pp.599-604. �10.1051/jphys:01988004904059900�. �jpa-00210734�
Electronic structure of disclinated polytopes
S. Nicolis, R. Mosseri (1) and J.-F. Sadoc
Laboratoire de Physique des Solides, Université Paris-Sud, Orsay Cedex 91405, France
(1) Laboratoire de Physique des Solides, CNRS, 1 place A. Briand, Meudon Principal Cedex 92190, France
(Requ le 29 juillet 1987, accepte le 22 dgcembre 1987)
Résumé. 2014 Des structures discrètes, définies sur un espace courbe, sont des modèles utiles pour des
configurations complexes d’empilement compact, comme des verres métalliques. L’idée est de définir une structure idéale sur un espace courbe, où un empilement local peut être continué globalement, et, ensuite, d’introduire de défauts qui « décourbent » l’espace, pour obtenir des structures réalistes dans l’espace physique (plat). Dans ce travail on étudie l’influence de certaines configurations de défauts, qui « décourbent »
partiellement, sur la densité d’états électroniques dans l’approximation d’un modèle des liaisons fortes.
Abstract. 2014 Discrete curved-space models are a useful paradigm for complex dense-packed structures such as
metallic glasses. The idea is to define an ideal structure in a curved space where a given local packing arrangement may propagate and then introduce decurving defects to map it onto a realistic flat-space structure.
In the following we study the effect of certain configurations of (partially) decurving defects on the electronic
density of states within the tight-binding approximation.
Classification
Physics Abstracts
61.40 - 71.20
There has been much activity, in the last few years, on the subject of geometrical models of non- crystalline systems. One such model is the polytope
model [1].
This model is based on the idea that the structure of any system results from a compromise between
some short-range order (i.e. a local packing arrange-
ment) and some boundary-condition constraints (i.e.
that we obtain a space-filling structure) ; if the short- range order is compatible with crystallographic con- straints, then a crystal is a natural solution. The greatest interest, lies, of course, in the cases where the short-range order is incompatible with the crys-
tallography in the flat space. The way out proposed
in these cases is the following : one tries to find a
space where this short-range order may become
long-range. One then must devise a scheme for introducing defects in this ideal structure so as to recover, in some appropriate limit, a realistic model for the flat-space structure one started with. The defects introduced will then describe the topological
defects that result from the incompatibility between
the local packing arrangement and the space-filling requirement.
This line of reasoning was followed (as hindsight shows !) in the case of metallic glasses : it was
JOURNAL DE PHYSIQUE. - T. 49, N° 4, AVRIL 1988
observed [2], that many undercooled liquids contain
icosahedral clusters ; this was observed experimen- tally as well as by computer simulations in the case of metallic glasses [3]. This may be understood by the
fact that an icosahedral packing has lower cohesive
energy than cubic packings if one assumes that the atoms interact via simple pair potentials (e.g. Len- nard-Jones). The next step was to identify a space where icosahedral short-range order could propa- gate. It was found that there exist a « polyhedron »
in a three-dimensional curved space (i.e. inscribed
on the surface of S3) with the property that each and every vertex had 12 nearest-neighbours forming a perfect icosahedron. This « polyhedron » (the stan-
dard term is « polytope ») has 120 vertices. It corres-
ponds to a perfect tiling of S3 by regular tetrahedra.
Its mathematical properties are well known [4] and
its physical properties have been extensively studied
the last seven years [5]. It is called polytope {3, 3, 5} .
The next problem to be tackled, then, is the introduction of topological defects in this structure
so as to return to flat space. It splits into two (related) questions : (a) what are the appropriate
defects ? and (b) what is the procedure to follow in
order to introduce them ? Finally, of course, one
39
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004904059900
600
would like to relate the results of this procedure to
« universal » properties of icosahedral metallic glas-
ses (for an example, see [6]) since the specific properties of a particular alloy have nowhere been used.
As regards the first question, it has been suggested [7] that disclinations (lines in 3-D, points in 2-D) are
the relevant defects to introduce. The reason is that disclinations carry curvature : for example, in 2-D, introducing a disclination in a honeycomb lattice changes a hexagon to pentagon or a heptagon (depending on whether matter was removed or added) ; in the first case, a conical point appears
(concentration of positive curvature), in the second,
a saddle point (negative curvature). Since the initial structure lives in a space of positive curvature, and the final structures, (those one wants to describe !)
live in flat space, it is only natural to adopt the idea
of the introduction of negative curvature defects to
cancel the curvature one started with.
We now come to the second question : how may
one construct an algorithm for introducing defect
lines in polytope {3, 3, 5} ? No general answer has
been given to this question.
There have been two, partial, answers : in [8] was proposed a method of constructing hierarchically organized, interpenetrating, defect networks ; it gen- erates, very rapidly, structures with a large number
of sites. It has recently been studied as first step towards understanding many physical properties of glasses [6].
In [9], we proposed another method with which
one may introduce individual defect lines in
polytope {3, 3, 5} ; it also has the advantage that
one may study local effects more readily. We
constructed polytopes with 2, 3 and 4 such lines, that
are non-intersecting great circles of S3. It should be
kept in mind that the two methods are not contradic- tory but, rather, complementary ; one may, for
example, take a configuration, obtained from the second method, as seed for the first (however, the
converse will not necessarily be possible).
A polytope that contains a single disclination line has also been constructed, but by a different method [10].
In this note we examine the influence of these defects on the electronic density of states. In this, first attempt we study a very simple model: an « s-
band » Hamiltonian in the tight-binding approxi-
mation (The quotes mean that on S3 the terms « s-
band », « d-band » etc., are used in a somewhat
figurative sense). The method we shall use was
introduced in [11] to calculate the electronic band structure of a polytope model for a : Si-H. This
Hamiltonian has also been studied, by different methods, in [13], where the s-band density of states
of the structure here called D14 was also calculated.
The structures we shall study are the three polytopes
introduced in [9], a new one with six disclination
lines, constructed by the same principle (called
« D18 » [8]) and the first iterate of the procedure
introduced in [8] with seed the {3, 3, 5}. This
structure has 2160 vertices and will be designated polytope Pl.
We shall now construct the matrix elements of the
polytope Hamiltonian (generic case). We will exploit
the existence of symmetry operations, that form an
Abelian subgroup of the full polytope group, to split
the NV polytope vertices in to N closed families (of
M members each), in order to reduce the NV x
NV Hamiltonian matrix to block-diagonal form (MN x N blocks). Each block may be diagonalized seperately, at far less cost and effort (since the form
of each block is the same ; a block is parametrized by
a block index, similar to the wavenumber index of Bloch functions). This reduction is accomplished as
follows : define a first basis {II>, 1 = 1, ..., NV } , corresponding to a located s-band atomic orbital on
each vertex. However, the afore-mentioned sym- metry operation implies that every vertex 1 may be
designated by a pair (m, n) of integers, where
n denotes the family and m the position within the family ; it also implies that one may find N distinct vertices (0, n ) such that any vertex (m, n ), in family
n say, may be obtained from (0, n ) by application of
an operator T, representing the operation. Thus :
The closure condition implies : Tm 10, n) = 10, n).
In this basis our Hamiltonian may be written in the
following form :
where the sum runs over nearest-neighbours and
t will be normalized to -1 in what follows. Let us now perform a linear transformation to the following
basis :
It may be shown that these kets are eigenkets of
T :
the closure property of each family implies that
By construction JC commutes with T ; therefore, JC
is diagonal in the indices p, p’ (in the basis
): q
therefore,
(The factor M will be absorbed in the normalization of the L basis). The sum over R gives a non-zero
contribution for values of R such that vertex
(R, q’ ) is nearest-neighbour to (0, q ). For q = q’,
R = 1, M - 1 ; for q :A q’ the values of R were
determined numerically. For the polytopes with 2, 3,
4 great circle defect lines M = 12 while for the D18 M = 15 ; they were deduced by geometrical argu- ments related to the composition rules of disclination lines. For the {3, 3, 5} and the PI M = 10 from the
polytope symmetry group.
We may now compute the energy spectrum and the eigenvectors (in the basis of the families, for any
block). This completes the solution of the one-
electron problem. Below we shall fill in these levels to calculate the electron density as a function of the band filling. As regards the densities of states, there
are several points worth mentioning :
1) They show how new energy levels are created
as the polytope size is increased. We may see quite clearly, therefore, the effect of the added defects
(especially in the dilute cases).
2) We observe that all show an absolute maximum at E = + 2 (for hopping parameter - 1). We may understand why all polytopes studied here have this
eigenvalue by constructing the following vector in
the polytope vertex basis : choose coefficients of
equal absolute value on all the families but of
alternating sign along each family (for simplicity, we
may normalize this global absolute value to 1). It
may be shown that this vector is an eigenvector of
the polytope Hamiltonian with eigenvalue + 2. That
the energy corresponding to this vector is + 2 may be
seen from the following argument: due to the fact that every vertex in a given family has two nearest-
neighbours on its own family and two on each nearest-neighbour family, this configuration cancels
the contribution from the nearest-neighbour families
and leaves only that from the two nearest-
neighbours on its own family to the energy. The
polytopes D18 and PI require some care : in the case
of the D18, the number of vertices per family is odd, which means that it is not possible to realize a perfect alternation of signs along each family (a typical frustrated situation) ; this will lead to shift of
a number of levels that contribute to the peak at
E = + 2 to lower values, and suppression of a pronounced maximum to more modest size (relative
to the other peaks). The effects of frustration (due to
the parity of the number of families) should become less pronounced with increasing size. On the other
hand, the existence of an absolute maximum at E = + 2 seems to depend on an organization of the polytope vertices that may not subsist in general
cases ; what should subsist, however, are local arrangements of atoms in antiprismatic configur-
ations that induce a concentration of states around E = + 2.
This leads us to the case of the P1; the reasoning
breaks down in this case since a given vertex does
not have two nearest-neighbours on its own family.
However, the exist many antiprismatic configur-
ations in this polytope, so the argument may still be considered as a useful guide. To summarize, the previous argument does not explain the high degen-
eracy of the E = + 2 level, but offers an understand-
ing of why certain structures should possess it.
Symmetry considerations that allow freedom of choice for the phases of the coefficients in the state- vector expansion may provide hints to this degenera-
cy.
3) The behaviour of the upper band limit merits
some comment. Let us first recall certain established features of this limit [12] (reminder : we take the hopping parameter normalized to -1 in what fol-
lows).
If the lattice under study has only even-numbered circuits, the upper band limit is non-degenerate, has
energy E = + z, where z is the coordination number,
and corresponds to a state with vector that may be expanded in the basis of an atomic orbital on each vertex with coefficients + 1 and -1. It realizes the
antibonding state i.e. the state with the highest
energy that admits such an expansion. If the lattice
has odd circuits, the antibonding state is not realized
due to frustration (not every circuit admits an
alternating covering with + 1 and - 1). The upper band limit has energy less than E = + z and may be
degenerate. Even circuits relieve the frustration.
With this in mind, we recall that the {3, 3, 5} has
no hexagonal circuits (save twice-circumnavigated
602
triangles), the D14 has two great circle lines thread-
ing genuine hexagonal circuits, the D15 has three,
the D16 four and the D18 six. The PI has entire networks of lines threading genuine hexagonal cir-
cuits. It should also be kept in mind that all these
polytopes have many triangular circuits as well as pentagonal ones. We find the following values for the upper band limit :
Insight to this behaviour may be gained by noting
that the decurving procedure (leading from the {3, 3, 5} to polytopes with ever more negative-
curvature disclination lines) leads to an increase in the number of triangular circuits as well as to a (more modest) increase in the number of genuine hexagonal ones. Both are due to the fact that
introducing negative-curvature defects, that corre- spond to lines threading hexagonal circuits, raises
the (mean) coordination. The interplay between the two effects (of the triangles and the hexagons) may be thought of as important in the determination of the upper band limit.
Finally, it should perhaps be noted that the upper band limit of the D16, D18, PI is beyond 4, the limit for the FCC. This means that these structures are
less frustrated than the FCC.
Fig. 1. - s-band densities of states of disclinated polytopes : (a) (3, 3, 5} (N = 12, M = 10), (b) D14 (N = 14, M = 12), (c) D15 (N = 15, M = 12), (d) D16 (N = 16, M = 12), (e) D18 (N = 18, M = 15), (f) Pl (N = 216,
M = 10) ; the (isolated) peaks are binned into channels of width 0.5 (in units of the hopping parameter).
4) We would now like to comment on the Pl. It is the largest structure studied within the polytope
model to date. It has an intricate defect organization
and may be considered as a best approximation to flat-space structures due to the fact that it contains many more defects and thus is much less curved.
This leads to hopes that its density of states, for instance, may prove most relevant for the under-
standing of related physical properties of complex closepacked structures. It is noteworthy that if we
bin the density of states in channels of width 0.5 (in
units of the hopping parameter) there appears a strong concentration of states at E = + 3 over-
shadowing the peak at E = + 2.
It has recently been suggested [13] that negative-
curvature defects may act as attractive centres for electrons. What is presumably meant by such a
statement is that the electron density is higher on the
defect than on the non-defect sites. We shall test this
hypothesis on the structures studied here. To do so we shall compute the (electron) density on the defect
and on the non-defect sites as a function of the Fermi energy (i.e. the energy of the highest filled level in
the band). The levels are filled as follows : to each energy level corresponds one (if it is non-degener- ate) or more (if it is degenerate) states. Two (non- interacting) electrons for every state of a given
energy are introduced until all states at this energy
are filled before those of the next level. We recall
that, since all members of the same family are equivalent, we may perform our computation in the
basis of the families. We sum, therefore, the squared
modulus of the relevant component of the eigenvec-
tors corresponding to energies lying between the
bonding edge and the Fermi level. It is obvious that the occupancy of the families will remain constant between two successive energy levels (there are no
states to fill). In figure 2 we display, as an example
of such a calculation, our results for the D14 (that
contains two negative-curvature defect lines (or families)). We observe that there exist values of the Fermi level (that correspond to special values of the
number of electrons in the band), for which the
density on the defect families is less than on the non-
defect ones. However, for Fermi energies below
zero, these values remain the exception, while beyond they become the rule. This result may be
expressed in the following terms : in a nearly empty band electrons are attracted to disclination lines with sites of high coordination number, because their wavefunctions can spread out and reduce their kinetic energy. In a nearly full band, holes are
attracted to disclination lines for the same reason.
This picture raises the question of the effect of
charge transfer between defect and non-defect sites.
This perturbation should be calculated in a self- consistent manner ; in most cases (for an example
see [14]) the effect is very small ; it is not obvious
whether this is valid in the cases considered here. To
conclude, the existence of these exceptional values
shows that the hypothesis (and the picture presented above), while helpful in outline, is not the whole story as far as the effect of the defects is concerned ;
other features of each structure must also be taken
Fig. 2. - Electron density of defect (- ) and non-defect (+ ) families.
604
into account (it should be added that similar calcu-
lations performed on the other polytopes support these conclusions). Naturally, all our results are subject to the restrictions of our model, most conspicuous among them being the assumption of a
uniform hopping parameter. A next step would take this into account. In figure 2 we have displayed one
defect family vs one non-defect family since, due to symmetry, all defect families give the same results as
do all the non-defect ones. In table I we display
occupancy values when the Fermi level is very close to the bonding edge. A remarkable feature in
figure 2 is the appearance of plateaus in the diagram
for the density on the defect family. This may be inferred from the fact that there exist regions where
the density remains constant on the defect while increasing always on the non-defect sites. The exist-
ence of these plateaus signals that of energy levels
whose corresponding eigenvectors have zero compo- nent on the defects.
In conclusion we have calculated the electronic
density of states of structures containing decurving
defects that are relevant for the description of dense- packed structures (such as metallic glasses) on the
basis of a simple model that, nonetheless, shows a
fair amount of intriguing features. It is an open
question what happens in a more sophisticated
model (example : distance-dependent hopping inte- gral).
The numerical calculations were performed on the
Centre Inter-regional de Calcul Electronique (C.I.R.C.E.) computers. The Laboratoire de Physi-
que des Solides at Orsay is a Laboratoire associe au
C.N.R.S. (LA 2) and the Laboratoire de Physique
des Solides at Meudon is a Laboratoire propre du C.N.R.S.
References
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