INTRODUCTION TO THE PHYSICS OF

QUASICRYSTALS

Clement Sire1 and Denis Gratias2

1 Laboratoire de Physique Quantique Universite Paul Sabatier 31062 Toulouse Cedex, France

2CECM-CNRS 15, rue George Urbain 94407 Vitry jSeine Cedex, France

INTRODUCTION

In condensed matter physics the notion of geometrical order is often associated to the much more restricted notion of periodicity. Amorphous metals are seen as "disordered" materials whereas crystalline phases are the best representation of this periodic order. As an illustration, group theory was introduced, now a long time ago, to classify the different structures compatible with this order. In crystallography, the experimentally observed diffraction patterns are analyzed and understood with the help of group theory, one of the best achieved mathematical theory. One of the elementary results of this theory is that the diffraction pattern of a crystal in 2D or 3D can only display 2, 3, 4 and 6-fold axes.

In 1984, the discovery of a metastable icosahedral quasicrystalline phase in the binary alloy Als6Mn14 by Schechtman et al. 1 has slightly altered this perfect picture as can be seen of fig. 1. In addition to 2 and 3-fold axes, one observes a 5-fold axis which is forbidden by classical crystallography. Moreover, the fact that one obtains Bragg peaks clearly show that some long range order exists in these materials. To explain this apparent contradiction, Pauling suggested2 that this AlMn phase was nothing but twinned crystals with a large unit cell. This hypothesis was ruled out by the discovery shortly afterwards of many other metastable and stable quasi crystalline phases of increasing structural quality. For instance, the length characterizing the long range order in the stable AlCuFe3 phase is at least of the order 30000 A, and the resolution of the observed Bragg peaks is of the same order as the one of the used devices. Finally, we shall see that the description of the diffraction patterns of quasicrystals in terms of structure factors of quasiperiodic lattices works so convincingly well that this approach is now commonly accepted.

Among the icosahedral phases which are known nowadays, one should mention AlMn(Si) (a few percent of Si stabilizes the structure4), AlMgZn5, AlCu(Fe3, Li6, ... )

Statics and Dynamics of Alloy Phase TransjormaJions, Edited by P.E.A. Turchi and A. Oonis, Plenum Press, New York, 1994 127

Fig. 1. 2, 3, and 5-fold axes of the icosahedral Als6Mn14 phase (LEPES).

which are the most studied. Apart from icosahedral quasicrystals, other symmetries have now been observed. Some of the preceding alloys and others, like ALCuCo or AlNiCo, display a stable T-phase for which the structure is described by stacked 2D decagonal quasicrystals7. Moreover, 8-fold9 and even 12-foldlO quasicrystals have also been obtained, but their structural quality is not as good as in the previ­ously mentioned cases. It is interesting to note that the two last phases coexist in V 15NilOSi. When studying the geometry of quasicrystals, we shall see that a perfect quasicrystal can be arbitrary well approximated by crystalline approximants. The higher the order of the approximant, the larger the unit cell . It is noteworthy that some of these icosahedral approximants have been observed for AICuFe and very recently high order approximants were obtained in TiZrNi alloys22.

This short lecture does not claim to cover the whole field of quasicrystals. We have tried to select common features which are shared by most of them. More references and interesting matter could be found in 0. When dealing with particular examples, we often only comment on the case of the icosahedral phases, but some theoretical examples concerns 2D (and sometimes even ID) quasicrystals which are much more easy to visualize on our poor stupid 2D paper sheet.

The plan is arranged as follows. First, we shall be interested in describing the new geometrical tools in terms of high-dimensional spaces, which were introduced in order to understand the structure of quasicrystals and their diffraction patterns. This approach will be generalized to provide a unified picture for crystals, quasicrystals and modulated phases. Secondly, this formalism is applied to the example of the structure determination of A163Cu25Fe12. After this section, we hope that the reader will be convinced that the new concepts introduced previously correspond to the physical reality. The next section will be devoted to the question of stability of these

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phases. The importance of the geometrical properties of quasicrystals will be quite clear when we describe the different mechanisms for their stability (energetic and/or en tropic stabilization). The last section will describe some physical properties, and especially transport and mechanical ones. We will try to present some properties which seem to be shared by a wide class of quasicrystals. For this reason, we will not develop magnetic properties.

GEOMETRY OF QUASICRYSTALS

High-dimensional Spaces in Physics

In order not to frighten the reader, we would like to recall that in many areas of physics, the use of high-dimensional or curved spaces was a very efficient way to describe physical reality. We forget about general relativity which states that we are living in a space locally curved by the density of matter. We do not develop the fact that billiards on hyperbolic surfaces are among the most chaotic systems, and are nice paradigms for the study of classical and quantum chaos, as well as an interesting field towards the proof of the Riemann-Hilbert conjecturell . In fact, we shall only describe examples in condensed matter physics for which the use of high-dimensional or curved spaces helps to understand the apparent complexity of various systems.

First consider the case of amorphous metals: it is known that a dense packing of spheres reproduces reasonably well the data of amorphous metals in real or re­ciprocal space12 (e.g. correlation functions). In this model, the atoms tend to form tetrahedra locally, since four spheres occupy the smallest volume in this configura­tion. Unfortunately, the tetrahedral dense packing problem is frustrated in R3, as the problem of covering the plane with pentagons. Consider the sphere S3 in the 4-dimensional flat space13 . The surface of the sphere has dimension 3 and can be covered by tetrahedra as can the usual sphere S2 by pentagons (forming a dodec­ahedron). In our case, the obtained structure is called a polytope and consists of 120 sites. One can cut this polytope by R3 planes with different distances from the center of the sphere (see fig. 2-a).

R'

Fig. 2: a) schematic representation of the cut of a polytope on S3 by a 3D plane

leading to different polyhedra; b) one can generate amorphous clusters by projecting

a finite portion of a polytope on the flat space.

For some of these distances, the flat 3D space intersects some sites of the poly­topes. The first layer is an icosahedron, the second a dodecahedron, the third another icosahedron, and the equatorial cut is a icosidodecahedron of 30 vertices. The other

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cuts are symmetric with respect to the equator. This shows that the local order is icosahedral for all vertices. Now, to build an amorphous structure in the real flat space, one has to decurve the sphere. A simple way is simply to project a region of any given polytope on the flat space14 as shown on fig. 2-b. With different kinds of polytopes, this will be well adapted to the construction of amorphous clusters (e.g. the Mackay cluster), but if one wants to form a bulk macroscopic sample one is limited by the finite number of sites on S3. In this case, one can introduce an iterative decoration process15 which will increase at each step the number of atoms on S3, and will thus decurve the space. One can show that a consequence of this process is to generate a network of disclinations.

This brief section was intended to show that amorphous systems can be seen as a realization of perfect local tetrahedral and icosahedral order which is altered by the decurving process. We shall see that, in a sense, the quasicrystalline icosahe­dral (decagonal) phases are another realization of this order, but by keeping perfect (distorted) icosahedra in real space. Their geometrical construction will involve flat high-dimensional spaces. Note finally that quite large local parts of differ­ent polytopes are involved in some structural models for decagonal or icosahedral quasicrystals16.

We end this section by some other examples that we do not develop very far. The first concerns cholesteric blue phases. In these phases, a pair of neighboring chiral molecules will have a minimum energy when the molecules sit at a slight angle with respect to each other. If one modelizes this system by a continuous approach, where the local direction of the molecules is described by a headless vector, one can show that it is impossible to propagate this local order in a flat 3D space. Once again, on S3 with a properly chosen radius (in relation to the tilt angle) the local order is no longer frustrated 17. The real structure will be obtained from the curved space structure by introducing disclinations.

In order to describe modulated phases of crystals, Janner and Jansen27 have introduced a fictitious representation of these phases involving spaces of higher di­mensions. We do not describe this method since we shall see that it has something to do with the modern theory of atomic surfaces that we will shortly describe hereafter.

Finally, high-dimensional space approaches have also been efficient in dealing with membranes and minimal (soap) surfaces 18. In the next section, we briefly introduce the basic method to build quasicrystals.

The Cut and Project Method

We call Ell the physical space of dimension d in which the quasicrystal we wish to build is embedded. The subscript will be clear in a moment. We first treat the case d = 1 for which the minimal dimension needed for the high-dimensional space is n = 2. The principle of the cut and project method19 (CP) is that quasiperiodic tilings can be seen as a small region selected from a crystal of high dimension, and then projected on the physical space Ell' In our example, consider a 2D square lattice (fig. 3), and draw a line which represents Ell' For convenience, the line passes through a site of the square lattice. The space orthogonal to Ell is naturally called El..' The second step consists in translating the unit cell of the 2D lattice along Ell' This procedure defines a strip which encloses a unique path along the vertical and horizontal bonds of the square lattice. Then, the sites inside this strip are projected on the physical space Ell' Depending whether it comes from the projection of a vertical or horizontal bond, the distance between two neighboring atoms in Ell is short (S) or long (L). If the slope of Ell is irrational, the obtained structure cannot

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be a crystal, since this slope is also equal to the ratio of the concentrations of S and L bonds: this is a quasicrystal. If the slope is a rational number p/q, one can show that this procedure leads to a crystal for which the unit cell contains p + q atoms. Since all irrational numbers are arbitrary well approximated by a sequence of rational ones, we conclude that a sequence of approximants of increasing unit cell exists, which tends to the perfect infinite quasicrystal. Note finally that many 1D studies on quasicrystals are performed by means of the Fibonacci chain which is associated to a strip of slope equal to the golden mean. The corresponding sequence of Land S is called the Fibonacci sequence and can be built by the following inflation rules.

L --t LS,

One obtains successively, L, LS, LSL, LSLLS, LSLLSLSL ...

LV

L V ~ Y E1. V

V ~ Y Ell

V V 1\ [;v V

/'"

\ V I~ y V

./"

~ 'f I~ V

~ ~

~ \ Fig. 3. description of the CP method with d = 1 and n = 2.

Now, we shall shortly describe the generalization of the CP method for 2D or 3D quasicrystals. Depending on the symmetry we want for our quasicrystal, we need to use a periodic lattice of dimension n. The relation between the minimal n needed and the desired order of symmetry s (for instance 5, 8, 10, 12 but why not 7, 11 ... ) is found from group theory arguments26. The minimal n is exactly the number of integers prime with s, lower than s. For instance, for s = 8 and s = 10, one finds n = 4 (check it !). So, in order to build the 2D Penrose tiling20 or the 2D octagonal tiling21, one must use at least a crystal of dimension 4. The periodic lattice needed for the octagonal lattice is a simple hypercubic lattice but not for the Penrose lattice. However, the later can be built from a simple hypercubic lattice in dimension 5. The CP method is now exactly the same as in 1D. One specifies the direction of Eli by means of d vectors of dimension n. E 1. is again defined as the orthogonal space to Eli' The unit cell of the periodic lattice is translated along Eli and the vertices selected inside the strip are projected on the physical space. Generally, the vectors on which Eli is spanned involve irrational numbers, which are

for instance T = ( V5 + 1)/2 (golden mean) for the 2D or 3D Penrose tilings, and J2 in the case of the 8-fold octagonal lattice. Rational approximants of these structures are obtained by approximating these numbers by rational ones. For instance, for

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the golden mean the natural sequence of approximants is defined as the ratio of two successive Fibonacci numbers defined by

Po = 0, PI = 1 (1)

The first approximants of T = 1.618 ... are 0/1, 1/1, 2/1, 3/2, 5/3, 8/5... In the case of the icosahedral phase of TiZrNi22 alloys, approximants up to 5/3 have been observed, whereas for AICuFe23 the 3/2 one is known to exist. This is quite remark­able since the unit cells of these large approximants contain several thousands atoms ! For instance, a fictitious 5/3 cubic approximant of the AIZnMg icosahedral phase has a unit cell of 60 A.

On fig. 4, we show some finite regions of the 2D Penrose and octagonal lattice. These tilings also display inflation rules20,21. Most of the structural models for the decagonal phases essentially consist of atomic decorations of a IO-fold 2D lattice Penrose24 (or its generalizations) and the octagonal tiling can be fairly well compared to experimental lattice images of VNiSi as shown in 25. Note finally that quasicrystals can also be generated by the grid methodlOI in real space. Unfortunately, it does not allow to compute the diffraction patterns and is less convenient to classify the local environments.

Fig. 4. (a) the 2D Penrose tiling and (b) the octagonal lattice.

Before studying the diffraction patterns of quasicrystals, we describe an impor­tant feature of quasicrystal: the existence of a particular kind of defect. These defects called phasons are associated to shifts of the band which can be local or global. We shall describe the local phason strain in the section devoted to the random tiling model and only describe the global case now. Consider, say in the 2D-ID example of fig. 3, a small shift of the band in the perpendicular direction (a parallel shift does not affect the structure). A vertex which was very close to the band edge can disappear in this procedure whereas another atom enters the band by the other edge. The effective change in the projected structure can be seen as the exchange of a long and short bond or a discrete hopping of an atom. This is shown in fig. 5, in the ID case and for the 2D lattices of fig. 4. In this last case, an elementary phason in real space consists in the hopping of an atom inside a hexagon of the structure.

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") r~ .--- /~" // ", ,Ii ,/ \ /

-:. D ~-1 ~ e( ·'1 ;.

--- -0-------4 '.L \ / ' .... ,

.. ~ .. £!. \} J 1 e •

e-- ---------0---------

~<) \~ /-

~~. ~ ?--'t ~ ';v )-" './ '0 ./ a b c

Fig. 5. an elementary phason in a) a 1D quasicrystal and in hexagons of b) the

octagonal lattice and c) the Penrose lattice.

As we shall see later, these phasons could help to stabilize quasicrystals by pro­viding a source of entropy, and could be at the origin of the curious mechanical properties of those materials. Very recently, evidence for phason hopping in the icosahedral AICuFe quasicrystal 56 has been obtained, even if the interpretation of the experiments is still controverted.

Diffraction Patterns

One of the great successes of the cut and project method19 was to provide a simple and efficient way of understanding the observed diffraction patterns of quasicrystals50. The derivation of the structure factor from the CP method has now become a classical exercise and will not be described here l9 . We will only state the result which is illustrated on fig. 6.

Fig. 6. the structure factor of a quasi crystal is obtained from the cut of the Fourier

transform of the acceptance domain.

The CP method selects all the points in the strip for which the projection on E.1 falls into the domain which is the projection of the unit hypercube on E.1' This bounded domain is called the acceptance domain and will turn out to playa crucial role in the modern generalization of the cut and project method. For a perfect quasicrystal, the mass density, when seen in the strip, turns out to be the product of a hypercubic lattice of Dirac peaks by the characteristic function of the acceptance

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domain, which only depends on the perpendicular coordinate. As a consequence, one can prove that the Fourier transform of this mass density is obtained by the following procedure. To Ell and E.1. one can associate their Fourier reciprocal space. The physical reciprocal space corresponds to k.1. = O. Now to all sites of the reciprocal space of zn (this is another hypercubic space zn), attach the Fourier transform of the characteristic function of the acceptance zone as shown on fig. 6. The structure factor of the considered quasicrystal is now given by the restriction (the cut) of this set of functions to the physical reciprocal space. This situation which is illustrated on fig. 6, in the case of lD quasicrystals, can be written in the general case

S(k) = L 6(k - kll)X( -k.1.) (kll,k.dEZn

(2)

where the sum is over all the vertices of the periodic reciprocal lattice, and the coordinates are expressed in the perpendicular and parallel subspaces. X is the Fourier transform of the acceptance domain and thus, only depends on k.1.'

This result explains the basic properties of the observed properties of the diffrac­tion patterns of quasicrystals. (2) confirms the idea that the quasicrystalline struc­tures are leading to Dirac peaks. For the icosahedral phase, we note that each of these peaks can be indexed by 6 integers in the reciprocal lattice. Moreover, the symmetry of the diffraction pattern will essentially depend on the symmetries of the acceptance domain. For instance, as we shall see, the acceptance zone is an octagon for the octagonal lattice or a triacontahedron for the 3D Penrose lattice, explaining the observed symmetries of their diffraction patterns. Another consequence of (2) concerns the support of the structure factor, that is the vectors kll for which one has a Bragg peak. Since the projection of a hypercubic lattice on a generic subspace is dense on it, we conclude that the support consists in a dense set of vectors in the physical reciprocal space. However, since the acceptance domain is bounded, the intensity of the Bragg spots IX( -k.1.)12 vanishes when I k.1. I is large. This explains why experimentally, one can only observe the brightest peaks leading to an effective cut-off in the perpendicular reciprocal space.

Atomic Surfaces

In this section, we briefly describe the acceptance domain of some quasicrystals. This leads us to introduce a more general method to build them, which transpires to be appropriate in the structural study of real quasicrystals.

The first interest of working with acceptance domain is that it provides a simple classification of the local environments in quasicrystals102. For instance, in the lD example, one can see that depending on the region where x.1. falls in the acceptance domain w, the local environment of the point XII can be LS, SL or LL. To be more precise, consider e~, e~ (resp. e[, ef) the projections of el and e2, the unit vectors of Z2, on the physical space Ell (resp. the perpendicular space E .1.). A point x = (xII' x.1.) has a physical projection surrounded by, sayan L and an S, if and only if the points x - el and x + e2 also belong to the strip. An equivalent statement is that X.1. - e[ and X.1. + ef are inside the acceptance domain. The subset of the acceptance domain w for which this property is fulfilled is the acceptance domain of the environment LS. It can be built by taking the intersection of w, w + e[ and w - ef. The acceptance domains for the three kinds of environments are shown on fig. 7-a. Of course, one could draw the acceptance domains corresponding to largest environments. On fig. 7-b, the acceptance domain of the 2D octagonal lattice is

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shown with all the possible local environments and their associated acceptance zones. In this example, one can note that the symmetry of the tiling is very reminiscent of the symmetry of the acceptance zone.

b

c? LS

Fig. 7. a) acceptance domain of a ID chain, and b) environments of the octagonal

lattice and (c) their associated sub domains (from 103).

Now, we can introduce the atomic surface method which is a generalization of the CP method which makes use of the acceptance zones. Indeed, consider the CP method in its simplest expression. It is completely equivalent to the following procedure. Attach to each point of the initial hypercubic lattice a surface similar to the acceptance domain. This will be called an atomic surface. A point belongs to the physical quasicrystal if its associated atomic surface intersects the physical space (see fig. 8). By this procedure, the selection and projection are realized in one direct step. This method can lead to simple generalizations. One is not limited in the shapes of the atomic surfaces which can display a parallel component. Moreover, to each vertex in the high-dimensional space, one can associate different types of atomic surface.

We would like to describe two short examples to illustrate the power of this method. First, it provides a unified picture for crystals, quasicrystals and the crys­talline modulated phases. Crystals (and approximants for quasicrystals) can be built by giving rational directions to the atomic surfaces. Quasicrystals can be built by giving irrational directions to it, and if necessary, by associating different atomic surfaces to different sites. Finally, modulated phases can be built by associating smooth atomic surfaces displaying a parallel component27 ,28.

The second example concerns structural models for quasicrystals. Let us take the example of AICuFe. The phase diagram at 680°C consists of a small icosahedral phase region, surrounded by crystalline phases and regions where these phases are mixed. The -', f3 and w phases of fig. 9 are respectively, monoclinic, simple cubic and tetragonal crystalline phases (see e.g. 57).

In the case of the icosahedral phase of A163Cu25Fe12, one starts from a 6D face-centered lattice. We differentiate the vertices in 6D for which the sum of the coordinates are even (n 1 sites) and those for which this sum is odd (n2 sites). In the same way, the body centers are labelled as bel or bc2 according to their neighbors.

13S

Fig. 8. the atomic surface method for ID quasicrystals.

-- Cu(o/%)

Fig. 9. phase diagram of AlCuFe around the icosahedral region (from 57).

The atomic surfaces that can be associated to each kind of site are shown on fig. 1029.

The atomic surface of the n2 sites is slightly smaller than that of the nl sites in order to prevent too small unphysical distances between neighboring atoms. The physical relevancy of these surfaces is discussed in 30. The authors have measured the correlation functions between AI, Cu and Fe by isotopic substitutions. These correlation functions are shown along the five-fold axis in the perpendicular space on fig. 11. The first remark is that the obtained surfaces are compact and well defined which illustrates that the high-dimensional approach is physically relevant. Note the difference in occupancy of the be vertices, when the correlation functions involve copper and when they do not. A simple conclusion is that the be sites are only occupied by Cu atoms. We will not comment this atomic model any further, and refer the reader to 29,30 for more details.

This first part was intended to introduce the reader to some new geometrical ideas which have been developed to understand the structure of quasicrystals in real and reciprocal space. These phases can now be definitively considered as a new state

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Fig. 10. the atomic surfaces which are attached to the n2, n1 and be vertices (from

left to right).

Fig. 11. correlation functions between AI, eu and Fe in EJ.. Note the presence of

intensity at the be vertices in the cases involving copper.

of matter and deserve their own geometrical tools. In the following, we are interested in the stability of the quasicrystalline phases and present the two main approaches which have been proposed to explain it.

STABILITY OF QUASICRYSTALS

Molecular Dynamics, Total Energy Calculations

In this section devoted to the stability of quasicrystals, we will try to focus on the different mechanisms which could account for it. Thus, we will be very brief concerning the molecular dynamics approach, or total energy calculations which give an a posteriori justification for the stability of quasicrystals.

The first molecular dynamics calculation in this field was performed on a over­simplified model for the 2D decagonal phase. A binary alloy consisting of two kinds of disks interacting through nearest-neighbor Lennard-Jones potentials is considered in 31. The radii of the disks (or the equilibrium distance of the potentials) are suit-

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Fig. 12. (a) the decoration of the tiles of a Penrose lattice and a (b) MD calculation

leading to a (disordered) Penrose lattice.

ably chosen so that one can decorate the tiles of the Penrose lattice as show on fig. 12.

Now, more realistic calculations are performed by extracting effective pair po­tentials from experimental pair correlation function32,33. Note that MD calculation could help to discriminate between the various structural models present in the literature33. We refer the reader to the lectures of J. Haffner but also A. Zunger and K. Binder for more details on these methods.

Concerning total energy calculations, we only state that the energy per atom has been compared in observed quasicrystalline phases and (sometimes fictitious) other crystalline states34,35. The basic conclusion is that the quasicrystalline state may be a ground state in certain narrow domains of concentrations. Moreover, the estimate for a quantity such as the cell parameter are correctly given up to a few percent by this kind of approach.

Matching Rules

Now, we will shortly describe the matching rules approach. Crystals can be built by periodically repeating a unit cell. It is an interesting physical (and even mathematical) question to know whether some growth rules or weaker rules are compatible with the long range quasiperiodic order. A natural condition required for such a rule is locality. One does not need to know the structure far from the surface of growth to know how the atoms must be placed to build a perfect quasicrystal. Only the inspection of finite distance environments should be necessary. Local growth rules seem to be too strong or restrictive to exist in quasi crystals. However, to a physical structure one could demand the existence of a weaker type of rule: a posteriori local rules. This kind of rule known as matching rules are locally verified in the perfect structure, but are not sufficient to grow it. For instance, in the case of the 2D penrose lattice such rules exist, which are expressed on fig. 1320. Decorate the edge of the two rhombi of the Penrose lattice by single or double arrows. The matching rules consist in forcing two tiles sharing an edge to have the same decoration on it.

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Fig. 13. matching rules for a piece of the perfect Penrose lattice.

In the perfect Penrose lattice, one can show that this rule is fulfilled everywhere. Unfortunately, it is not strong enough to grow a perfect lattice. However, one can expect that atomic diffusion can relax a non perfectly grown structure to a perfect one. Matching rules and more general rules are available for many quasicrystalline tilings98-100 and their existence can be a physical constraint on structural models of quasicrystals.

The relation to the stability is, then, the following. If one associates an energy cost to any mismatch, it results that the quasicrystalline state is the ground state of a local classical Hamiltonian. In this approach, generating phasons also creates mis­matches and a quasicrystal with phason strains have a higher energy. This situation should be compared to the random tiling model approach which is described in the next section. Finally, we would like to point out that the energy costs associated with mismatches have not received any microscopic justification. It would be interesting to compute the cohesive energy (by quantum or classical potentials methods) of two different configurations: one with no defect, the other with one mismatch of each type. This could give an estimate of the energy involved, which could be compared to the entropy per atom in the random tiling model.

The Random Tiling Model

Theoretical grounds. The approach developed in the framework of the random tiling model (RTM) is somewhat the opposite to the one described in the previous section. Quasicrystals are not seen as the ground state of any Hamiltonian. Their stability is due to the fact that the entropy is maximum in this state. Therefore, it should be stabilized at high temperature. Contrary, to the energetic approaches, the model starts from the assumption that introducing phasons in the structure does not cost any energy, or at least that this contribution is negligible at high temperature. We recall that in the high-dimensional space, one can generate phasons by allowing the strip to fluctuate around its mean position. In the space, these phasons can be seen as the exchange of long and short bonds in ID, as the hopping of a site in a hexagon in 2D (exchanging a set of tiles), or in a polyhedron in 3D. Before going to the precise formulation of the RTM, one can give a simple argument to explain why the quasicrystalline state has the largest entropy in reserve. For the sake of simplicity, consider the 2D-ID picture. It can be shown that the number of LS or SL environments is maximum, and these environments are the most equally distributed, when the strip is flat (perfect quasicrystal), so that one can easily introduce a defect

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by locally exchanging an S and an L. On the contrary, if the band between the vertices (0,0) and (p, q) consists in a vertical part between (0,0) and (p, 0), followed by an horizontal part between (p,O) and (p, q), the obtained structure would be

~~ p S's q L's

Even if the band associated with this structure is far from being the flat one of slope p/q, the entropy is certainly very small, since one can only introduce a defect between both crystalline sequences.

Now, we would like to give a short introduction on the mathematical formulation of the RTM. Consider a realization of a random tiling built by using a non-flat strip. In the high-dimensional space, one can coarse grain the positions of atoms over lengths much larger than the rhombus edge length. Then, one introduces h.L(xlI)' which in the naive previous picture, describes the perpendicular deviation of the strip (or of the coarse grained density of atoms) from the perfect quasicrystal. So, h.L( xII) is essentially constant for a perfect quasicrystal. Thus, a random tiling differs from a perfect quasi crystal through a non-zero phason strain defined as

ah.f-Eij = _to (3)

aXIl As stated previously, fluctuations of the surface create configurational entropy which could be at the origin of the stability of quasicrystals. More precisely, one can define u(E), the entropy density per unit surface (for convenience we use the term "surface" for the (n - d)-dimensional space defined by X.L = h.L(xlI))' Elser36 and Henley37

have conjectured that this entropy density is maximum for a zero phason strain, and that it is quadratic in E. This, added to group theory considerations can lead to the most general form of u(E) consistent with the considered quasicrystalline symmetry (see 38 for the decagonal 2D case). Generally, the total entropy is obtained by integrating u(E) over the parallel coordinate. In the Fourier space, one obtains typically an expression of the form

KJ 2.L 2 S = So - 2: Iqlll Ih (qll)1 dqll (4)

where K is a phason elastic constant. (4) is clearly equivalent to the assumption that the entropy is maximum in the (zero phason strain) perfect quasicrystal. On (4), we see that the dimensionality of the tiling plays a crucial role in the dynamics of the surface. If one is interested in the fluctuations of the strip (deviation from the perfect quasicrystal), it is easy to obtain the following qualitative properties by applying a type of equipartition theorem. In ID the perpendicular fluctuation of the strip is of the same order as the size of the quasi crystal in Ell: the quasicrystalline state is certainly completely unstable even if one can show that the random tiling hypothesis for u(E) is exact. This phenomenon is reminiscent of the absence of shaped ID surfaces in 2D.

In 2D, the surface is always rough, but the fluctuations are only logarithmically growing with the size of the sample:

.L .L 2 1 < Ih (XII) - h (0)1 >'" 27rK In IXIII (5)

The first consequence is that the diffraction pattern of a 2D random tiling has power­law peaks instead of Bragg peaks. The density-density correlation function falls off with exponent

(6)

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All these theoretical predictions have been confirmed numerically on various models including binary alloys interacting via a Lennard-Jones potential, and on random tilings generated by the elementary flips shown on fig. 5-c 39,40. In the first case, it has also been shown that, the entropy per area without any phason strain, 0'0, is maximum when the concentrations of big and small atoms are those of the perfect quasicrystal39 (see fig. 11). Finally, one can extract K from the comparison of (5)(6) with numerical simulations. It is intriguing that this value is found to be close to 0.6 for any kind of models for the decagonal phase.

In the case of the 3D icosahedral phase, the conclusion of the random tiling model is that below a finite temperature, deviations from the perfect strip are bounded and the random tiling model leads to a structure possessing long range order. The diffraction pattern presents Bragg peaks plus an intrinsic continuous diffuse part.

Before commenting the relation of the RTM with experiments, it should be mentioned that this kind of model is related to very famous problems in statistical mechanics. For instance, the phase diagram of the octagonal lattice can be shown to share similar features with the one of the ID Hubbard model, and the computation of the entropy in an incommensurate phase of its phase diagram is a problem which reduces to the exactly solvable six-vertex model41 . Moreover, the exact computation of the entropy of various tilings can be related to the number of lattice coverings by dimers42 or polygons43.

Experimental predictions. Now, we shall briefly review some of the predic­tions of the random tiling model. Unfortunately, one must admit that most of them are very difficult to observe, especially in three dimensions for which the distinction between a perfect quasicrystal (possibly including some defects), and an entropic random tiling is even harder to make. One may hope that the decagonal T-phases could provide the best materials to test this model.

The first prediction that can be experimentally tested concerns the temperature dependance of high resolution X-ray diffraction peaks. The Debye-Waller effect associated to the peak at qll is given by

I = Ioexp [_! (~l + q,l)] 2 /\11 /\1..

(7)

where KII and K 1.. are combinations of elastic constants. The RTM coupled to the elastic theory predicts that the phason elastic constant increases linearly with the temperature. This provides a mechanism for a possible instability of the quasicrystal state at low temperature, which should enter a crystalline state through a first order transition44 ,45. A consequence is that the intensity of peaks with large q1.. should increase with temperature and saturate at high-temperature since K 1.. becomes pro­portional to the temperature. Even if this effect was qualitatively observed in 46, several other groups were not able to reproduce it. It seems that the dynamics play a crucial role which was perhaps underestimated in 46. In the random tiling model, the diffuse scattering is also explained in terms of phason strain47 . The prediction is that in addition to Bragg peaks, one should observe lorentzian shoulders. Un­fortunately, one should be able to separate the different contributions in the diffuse scattering to conclude experimentally. It is interesting to note that the complex diffuse scattering observed in AILiCu48 is well understood in the framework of the CP method invoking a sharp cut-off relevant to the strip model 49.

Lattice images give direct structural information and thus could be helpful to test the RTM. There is no indication of the presence of strip fluctuations. Instead, the obtained lattice images reproduce beautiful and apparently perfect patterns, with

141

no evidence of any phason strain50,51. This observation could be an indication of very small phason fluctuations if any. However, this is not enough to rule out the random tiling model for the following reasons52. First of all, not all the scattered electrons are used in forming the image, excluding some diffuse scattering. The multiple scattering certainly obscures the relation between the real structure and the observed lattice images. Finally, the finite thickness of the samples could lead to an averaging of the structure, even if great improvements in this direction have not revealed any difference in the conclusions.

Concluding remarks. Before ending this section devoted to the random tiling model, we would like to put forward some interesting (as far as we can be interesting) questions and comments. First of all, it is clear that one cannot imagine that defects, including phasons, are not present in a "perfect" (=not random tiling) quasicrystal. The RTM states that these defects are crucial to understand the physical properties and the stability of quasicrystals. For the RTM to be physically relevant, one should exclude the occurrence of very small fluctuations of the strip as it could be tantalizing and dangerous to conclude from the observation of lattice images. This would lead to a too small entropy per atom which would rule out the RTM mechanism for the stabilization.

Concerning this point, it would be necessary to have a better estimate of this entropy per atom in real materials (say AICuFe). In fact, the entropy per atom of a random tiling obtained by MC simulations with short range (cut) Lennard-Jones dominates the long range contribution of the potential above O.25TM39 (melting temperature). However, in order to describe real materials, one must consider real­istic quite long range interatomic potentials including Friedel oscillations, or atomic decorations of the tiles of a random tiling. Both procedures have the same effect of rigidly binding some atoms. Therefore, the entropy per atom in a realistic ran­dom tiling model should be considerably lower than the previous estimate. Finally, similarly to matching rules theory, the RTM would deserve a microscopic treatment.

As it is now clear, quasicrystalline binary alloys interacting through cut Lennard­Jones potentials in a thermal MC bath are well described by the RTM. It is known that adding some matching rules constraints does not change drastically the RTM picture at high temperature. For instance, the 2D Penrose lattice has still a rough strip at any temperature, even if one associates an energy cost to a mismatch53,54. However, interaction between these mismatches can drive the roughening tempera­ture to a non-zero value and increases this temperature in 3D54. Once again, it seems necessary to look for the consequence of a mismatch on a real structure in order to have physically correct estimates for the involved energy and entropy per atom. It would also be interesting to understand the mechanism which could generate this interaction between mismatches.

Finally, it should be pointed out that all these approaches assume that the ef­fective classical interactions (matching rules, pair potentials) are short range. In fact, the quantum treatment generates effective quite long range interactions which are more or less correctly described by the many Friedel oscillations of the classical potentials. Moreover, the classical potentials used in usual MC or MD simulations are radial ones which could be considered as a questionable assumption at the mi­croscopic scale. It would certainly be interesting to study the RTM coupled to a quantum Hamiltonian. Of course, the computation time would be certainly much bigger if the problem were treated numerically. However, it can be shown exactly that even a ID "quantum" quasicrystal can have a flat strip at sufficiently low tem­perature, whereas it always has a very fluctuating strip with a short range classical Hamiltonian55. This is believed to be a fortiori also true for 2D or 3D quasicrystals, and the underlying reason will be seen in the next section.

142

Electronic Stability of Quasicrystals

In this section, we develop another approach in which the stability is seen as a consequence of the electronic structure of quasicrystals. This interpretation does not exclude the preceding one, even if phason disorder would certainly weaken the electronic effect. The basic idea that we shall develop in this section is that for all materials displaying, say, an icosahedral phase, the density of states at the Fermi level is quite small, due to a systematic matching between an effective Fermi vector and the most intense peaks in the reciprocal space58,59,60. This is a mark of a Hume-Rothery61 type of stabilization that we describe in the following.

Qualitative features. In studying the electronic properties of quasicrystals it apparently seems that the tools developed for crystals, like the Brillouin zone or the Bloch theorem, cannot be of any help. Some specific models have been introduced in order to find the qualitative features of the electronic spectra of quasicrystals. We shall shortly describe these models before showing why the proposed mechanism is relevant for real quasicrystals.

The first quasiperiodic Hamiltonians where studied in ID. They came from the study of 2D electrons in a magnetic field62 or dynamical systems63 and were intro­duced before the discovery of quasicrystals. The simplest model is a tight-binding binary model for which the sequence of on-site potentials (or/and hopping integrals) follows exactly the sequence of long and short bonds of the Fibonacci chain63. An exact renormalization group can be found for a large class of models including this one. The energy spectrum is a Cantor set of zero measure with an infinite number of gaps, and the wave functions are critical, that is between localized and extended (power localized). One can show that if one associates different valences to the two kinds of atoms, the Fermi energy always falls into a gap. More precisely, one can show that for a finite range of the electronic filling factor, the Fibonacci chain is the ground state of a very simple Hamiltonian64• This is due to the occurrence of a large gap at the Fermi level, which lowers the energy of the Fermi sea.

In 2D, numerical simulations have been performed for simple tight-binding Ha­miltonians on the Penrose lattice65 and the octagonal lattice66 . A basic feature is that as a function of the increasing intensity of the potential, the spectrum consists of one band with large pseudo-gaps, or a finite number of gaps, and for very strong potentials, the ID fractal behaviour is recovered. The transition between the different regimes were also obtained in a solvable model, on a sublattice of the octagonal tiling67• This lattice can be the ground state of a very simple Hamiltonian in a small region of the electronic filling64• Finally, the strong potential limit was studied by renormalization group technics, for a general Hamiltonian on the octagonallattice103. The conclusions in 2D or 3D are that for physically relevant parameters, the Fermi energy is likely to be in a gap and the wave-functions are badly extended.

Pseudo-Brillouin zone. Now, we will show that the qualitative features de­scribed above are relevant in real quasicrystals. In crystalline materials, the opening of a (pseudo-)gap is associated to the matching of the Brillouin zone and the Fermi surface for certain reciprocal vectors. This condition reads for a spherical Fermi surface

(8)

which is equivalent to saying that a free electron at the Fermi surface, scattered by

143

the harmonic q of the potential will have the same energy after the scattering. Thus, the effective Hamiltonian for an electron of momentum k (with Ikl = k F) is

( e(k) Vq ) V~ e(q - k)

(9)

The degenerate states of the free electron split into two states separated by 2Vq . After summing on k (and q), one thus obtains a pseudo-gap or a gap at the Fermi level.

This argument can be strictly repeated for a quasicrystal after defining a pseudo­Brillouin zone (PBZ). It can be constructed from the perpendicular bisecting planes of reciprocal vectors related to prominent X-ray-diffraction peaks. For the icosahe­dral phases, the obtained PBZ are almost spherical, since due to the high symmetry order, many equivalent diffraction peaks are involved in their construction. The effect described previously (see (8)(9)) is certainly enhanced by summing up the numerous equivalent or nearly equivalent q contributions.

Hume-Rothery rule and pseudo-gap. We shall shortly describe some ex­perimental confirmations of the relevance of the Hume-Rothery rule for quasicry­stals58,59,60. We shall also comment on other experimental and numerical indications for the existence of a pseudo-gap at the Fermi level.

Wagner et al. have measured the Hall coefficient and the electronic coefficient of the specific heat for icosahedral single-phased AllOO-x-yCuxMgy (x = 4 - 19, Y = 35 - 39 and GalOO-x-yZnxMgy (x = 37 - 46, Y = 33 - 37)68. Scanning different compositions allows to vary the average valence Z. The result is shown on fig. 14 for the Hall coefficient, and on fig. 15 for the specific heat coefficient.

0.0 0.0

-o.~

0 0 -0.5

0 en-1.0 00 •••• (!J

u 0 00 0

~ -1.5 ..

+ N T: -1.0 , ~ 0

...-t -2,0

~

I-2.~ -1.5 IT

-3.0 • (iI! 0 (10)

-3.5 -2.0 2.2 2.3 2.2 2.3 2.4 2.5

Z (e/at. )

Fig. 14. room temperature Hall coefficient of (a) i-GaZnMg and (b) i-AICuMg. (from 68).

For both quantities, and also for the thermopower vs Z, one observes abrupt changes in their behaviors when Z ~ 2.25 (both materials) and Z ~ 2.4 (AICuMg). By comparing experimental values of reciprocal lattice vectors with the associated 2kF, the Fermi sphere is found to make contact with the PBZ for q with 6D label equal to (222100) and (311111)/(222110) for Z ~ 2.2 and Z ~ 2.4 respectively (index notation of 69). The associated PBZ are shown on fig. 15.

Gratias et al.57 have noticed that the region corresponding to the perfect icosa­hedral structure in the phase diagram is a narrow strip elongated along the con­centration line AI62Cu25.5Fe12.5 - AI62.6CU22.4Fe13' This corresponds to a constant ratio of e/ a ~ 1.862, with electronic valency for AI, Cu and Fe respectively equal to

144

1.0 -.-----r--.--.---, 1.4

0.9

~ ~ 1.3

...; 0.8

~ 1.2

'" I 0>

@J '--, 0.7 1.1 .§

>-0.6 1.0

a 0.5 0.9

2.2 2.3 2.2 2.3 2.4 2.5

Z Ie/at. )

Fig. 15. electronic coefficient of specific heat of (a) i-GaZnMg and (b) i-AICuMg.

The PBZ 1 and 2, as discussed in the text are shown in the upper and lower side

of (b) (from 68).

3, 1 and -2. Once again, this can be associated with the nesting of the PBZ and the free electron Fermi surface. The restricted composition range for the thermody­namically stable phase, related to the electronic structure, is a strong argument in favor of a Hume-Rothery type of stabilization.

Other direct observations of the pseudo-gap have been realized on AlCuFe( Cr)10 and AlMn(Si)71 alloys of different compositions, by soft-X-ray emission and pho­toabsorption spectroscopies. In all these cases, the DOS at the Fermi energy E F is found to be very low. Moreover, the existence of a wide pseudo-gap is evidenced and is found to be larger in the quasicrystalline state than in the related crystalline counterparts, even if it already exists in these phases. Finally, the low DOS at EF was also observed by specific heat and susceptibility measurements72. The estimated DOS at E F is of the order of one third of the free electrons value, or less. We shall see in the following section, the consequence of this wide pseudo-gap due to the nesting of the pseudo-Brillouin zone and the Fermi surface.

Numerical simulations are now able to account for this pseudo-gap. We have already mentioned that simple tight-binding models for quasicrystalline lattices al­ready provide a simple interpretation of it in the framework of a Hume-Rothery stabilization process64. Of course, it is also very interesting to confirm these trends by more realistic theoretical approaches. LMTO calculations where performed for small periodic approximants of AIMn, AICuLi73 and AIZnMg74. Very recently, tight­binding LMTO technics were applied to approximants of AIZnMg with up to 12380 atoms for the 5/3 one75 . All these works confirm the presence of a pseudo-gap at EF and show that the energy per electron decreases monotonically with the order of the approximant 75.

This is related to the proof that for simple models the perfect quasicrystal is the ground-state of the electronic Hamiltonian. Natural periodic approximants have very close cohesive energies and are the most stable periodic structures for a given size of the unit ce1l64 .

Other works have been carried out to understand the influence of d-states (from transition metals) in stable quasicrystals76. The authors in 76 have considered a Friedel-Anderson model including the contribution of Bragg peaks and of the hy­bridization. The main conclusion is that the strong variation of the DOS near EF and the hybridization tend to populate the partial d-DOS just below the Fermi level (see fig. 17), strengthening the stabilization effect. They were also able to account for the effective negative valency of transition metals, which suggests that it is strongly affected by sp-d hybridization. Their calculations are in good agreement with LMTO simulations.

145

1.0

A! ZnMg

0.5

E 1/1 r 0 ~

-< 5C

-U C f11

> TB

" c.

" n. TB

(fl O.

3/2 S

0

0 TB

o. 5/3 5

TB

o. ~ 1 5 ~ 1 0 ~ 5

E-Er in leV)

Fig. 16. electronic DOS from k and real (TB) space LMTO, for the 1/1, 2/1, 3/2,

5/3 approximants of AIZnMg, containing respectively, 162, 688, 2920 and 12380 atoms in the unit cell. The dashed lines are the results for approximants differing only by a shift of the acceptance domain (from 75).

60

50 ( a )

40

E 30 c ;:; 20 c

:~ 10

c: 0 '" l:: » 60 -e:, - 50 ,..

" 4J -;:; 3C r/i

20

'" -0.2 0.2 O • .;

Energy (Ry)

Fig. 17. (a) local d-DOS of a model considering a PBZ with 42 Bragg planes

and d orbitals coupled to the bonding sp state (full line), antibonding (dashed) and neglecting the contribution of Bragg planes (dotted) obtained from a Friedel­

Anderson model; (b) comparison with a LMTO calculation for the partial d-DOS

of Fe in AI7Cu2Fe (from 76).

Discussion. In this last section devoted to stability, we have presented the Hume-Rothery approach, for which the notion of pseudo-Brillouin zone has been introduced. The Fermi vector is locked at half the value of the radius of a shell of intense peaks in the reciprocal lattice. There are now strong experimental, theoret­ical and numerical evidences for the existence of the expected pseudo-gap in most of quasicrystalline phases (not to say all). Moreover, it has been shown in simple models to be sufficient to stabilize the structure.

146

It is interesting to discuss the connection of this interpretation with the random tiling model. Of course, the stabilization could be due to the joint effects of phason entropy and electronic contributions. However, both effects are also antagonist. Indeed, as already stated, atoms locally bound by strong energetic constraints would certainly have a lower entropy per atom. On the other hand, the occurrence of a strong phason disorder could more or less fill the pseudo-gap at the Fermi level and increase the energy. As already quoted, even in a ID system one can show that the electronic contribution leads to a transition to a disordered chain only above a non-zero temperature55 . In higher dimensions, it would be interesting to evaluate this temperature for a more realistic Hamiltonian. It could be substantially higher than the one derived from oversimplified short-range Lennard-Jones binary alloys.

PHYSICAL PROPERTIES

In this section, we present some typical physical properties of quasicrystals. The first years in the field of quasicrystals were specially devoted to the search for new phases, structural determination, geometry, and elementary models for the electronic or phononic spectra o. In fact, a few experimental data concerned the measurements of physical properties. Now, some of them are available which show that quasicrys­tals display quite peculiar properties. The transport properties are certainly the most interesting and are quite universal for quasicrystalline materials. We shall also describe the strange mechanical properties of AICuFe which could be due to phasons.

Transport Properties

From the occurrence of a pseudo-gap at the Fermi level, one expects a low con­ductivity at low temperature. Indeed, the lowest reported conductivities, measured in the samples of best structural quality, are of the order of a few hundreds (ncm )-1 77 -82, and are at least 10 times lower than in the crystalline and amorphous coun­terparts. As shown on fig. 18, the conductivity decreases with increasing annealing time at higher temperature. This may be due to the fact that the pseudo-gap is filled by the effects of defects, including phasons which are frozen at low temperature and can relax in the slowly annealed samples, from high temperature.

Moreover, the conductivity increases with respect to the temperature by a factor up to 10 between lOoK and 10000 K 82. The origin of the very low conductivity and of its temperature dependence is an interesting problem which has motivated many works recently. The low DOS cannot account by itself for the weak conductivities observed, since we recall that the DOS at E F is estimated to be close to one third of the free electrons value in most quasicrystals. Moreover, increasing the temperature could fill the pseudo-gap by phasonic effect, but not in a ratio sufficient to explain the fast temperature dependence. Note finally that the conductivity at low temperature (below 200 K) presents some effects of electron-electron interaction 79.

In order to illustrate this phenomenon, we show on fig. 19, a now famous picture83 ,80 showing the value of a 40 J( versus a3000 J( (40 K and 3000 K are arbi­trary values). The data for many samples come along a line that may be extended up to the zero a 40 J( axis. The minimum of the conductivity is close to the Mott minimum conductivity for metallic samples. This suggests that good quasicrystals can be considered as being close to a metal/insulator transition (towards a quasicrys­talline semiconductor ?). The dynamical conductivity also reveals similar trends84.

147

600 r--[ .....,..,..~,-.-.-,-.-,-.-,-~~~

1 -............. 6OO'C-2h 400 800'C -3h

~

5001 i ___ AI63Cu24.5FeI2.5:

8 300~>~1t12~'~~:~125: g ~~~4----= 8OO'C-3h

o 200~i~162~~~::12.5: 100~ 1 OL...f~ ............ ~t............~...4.......Jj o 50 100 150 200 250 300

T (K)

Fig. 18. the conductivity of pure i-AlCuFe, with (600DC) and without (800DC) structural defects. Note that all the curves are parallel and that only the residual conductivity seems to be a function of the disorder and the composition (from 82)_

- AICuFe 0 AIPdMn . AICuRu I 250

-Metal-insulator transition -200 . -~

~ 150 oe-e -"

,-S 100

, rue

o~ 50

1 0 0 100 200 300 400 500

cr300K «nCm),l)

Fig. 19. conductivity U40K as a function of U3000K for different samples of icosa­hedral quasicrystals (adapted from 80).

Indeed, a( w) almost vanishes when w -+ 0 and the Drude peak is not even visible at the scale of the plot. These data are quite similar to the result for a zero-gap semiconductor.

148

f: > 1= u ~ Cl Z o u

6000

FREQUENCY (eV)

Fig_ 20. conductivity u(w) of AI62.5Cu25Fe12.5 (from 84), compared to a theoretical fit in 85.

In addition to the pseudo-gap effects, other different theoretical arguments have been proposed to explain the behaviour of the conductivity. It has been proposed that the scattering by d-states86 or the induced relative localization by the pseudo­gap, of the states near the Fermi level87 could favor low conductivity. Recently, it was proposed that the anomalous diffusive properties of wave functions in a perfect quasicrystal87 could qualitatively explain the conductivity behaviour above 100° ]{82.

Another approach due to Burkov et al.85 treats the problem in the nearly free ap­proximation. They have shown by standard perturbation theory that the effect on the conductivity of the quasiperiodic potential is certainly stronger than on the DOS at the Fermi level. The conductivity should be much more reduced than the DOS at EF' This is due again to quasisphericity of the pseudo-Brillouin zone. Their result is certainly qualitatively correct, but one must note that their calculation has a reduced range of validity, especially in the pseudo-gap where the DOS is strongly affected by the potential. Moreover, they state that the AC conductivity is strongly affected by intra-band effects. Their result fits the experimental data quite well (but 3 free parameters), even if the obtained scattering time is rather short.

The low conductivity and its behaviour with respect to the temperature and the structural quality of the samples can also be deduced from the following argument. 88. The velocity at the Fermi level, VF, plays a very important role for transport properties. Consider a periodic structure which has a large unit cell of linear size L. One can associate a Brillouin zone of typical size (271"1 L)d to this crystal. If the structure inside the cell is itself periodic with a much smaller unit cell of typical linear size 1. Then, a typical band has a dispersion width bE of order tilL so that the velocity VF is of order tl (we recall that Vk = V'kE(k) and that n = 1 !). On the contrary, if the unit cell of size L is highly disordered, we are in the localized Anderson regime, and the bands shrink fast:

bE ".,f exp( - L I 0, bk '" 271" L

and VF '" tLexp(-L/~F) (10)

where ~F is a typical localization length. VF goes exponentially to 0 when L --t +00, and the completely disordered system is an insulator. In a quasicrystal, the width of the dispersion bE is indeed found to be small in simple models63,66 and even in realistic LMTO calculations 75. The reason is that the bands are slowly shrinking with the size of the approximant: 8E '" t(ll LY\ where 0' is a typical exponent greater than 1, depending on the intensity of the potential 89, and 1 is a typical length of a rhombus edge. This explains why in ID the measure of the spectrum is always zero since it is given by f-l '" t(LIl)l-a. In higher dimensions, this measure can be finite or zero depending on the value of 0' which is related to the intensity of the potential67. One can see that VF slowly goes to 0 when L --t +00. In a real sample, L must be taken as a typical scattering length for the electrons, which increases by lowering the temperature or improving the structural quality of the samples. We find finally that

a(T) '" L(T)-211- apl (11 )

which reproduces qualitatively the observed features. This treatment is of course already valid for approximants for which the bands are already shrinking. Note that a similar expression has been found by the authors of 82 by considering the contribution of sub diffusing electrons. This treatment presented in this lecture seems more general, since it does not make this hypothesis of sub diffusion, which is realized in simple models only for very strong potentials87 . Moreover, it should be pointed out

149

that the approach developed in 85 in terms of almost spherical PBZ, is a perturbative approach of the presented non-perturbative phenomenon.

We end the section devoted to transport properties by a few comments concern­ing the magnetotransport77- 79 ,89. We have already discussed the band structure effects on the transport properties at quite high temperature. At low temperature the interelectronic interactions are very important in quasicrystals79,89. Usually, these contributions are neglected in disordered alloys of much higher conductivity. They become more and more important as the conductivity decreases, leading to a linear relationship between ~u and -Vii. The slope is positive and temperature in­dependent. Localization theories also predict such a linear behaviour but of opposite sign, which is observed in AIMgZn90. In AICuFe, the agreement is shown to be good, with a strong negative D.u confirming the role of the interelectronic interactions79. Their influence can also be seen on the conductivity at very low temperature, for which the electron-electron interactions generate a D.u '" .jT at low temperature. Both results are illustrated for AICuFe on fig. 21.

b

~ 0 r--------------I S g .g -3

-6 '---'--"~~~_L_~~~__'__'__~~~ 0.5 1 1.5

{T (K1I2)

Fig. 21. (a) magnetoconductivity as a function of,fij for different temperatures

(from 90) and (b) low temperature conductivity in different AlCuFe samples.

2

Mechanical Properties

It has been very early recognized that quasicrystals are extremely brittle, which was at that time considered as a direct consequence of the absence of periodicity91. Despite of this lack of periodicity, it is possible to define dislocations92 and more generally disvections93, by means of the high-dimensional periodic structure asso­ciated to these materials. Theses dislocations has been observed and characterized by electron microscopy analyses94 ,95 , and it would be interesting to study their role in the plastic deformation process of quasicrystals. Plastic deformation of AICuFe has been studied96 by compression tests. At room temperature, no ductility is ob­served, whereas the samples can be homogeneously deformed up to 130% at high temperatures, with no final hardening stage.

The conclusion of 96 is that the plastic deformation in AICuFe is not due to nucle­ation and migration of linear defects moving in well defined slip planes. The authors argue that high atomic diffusion could be responsible for this phenomenon. They are guided in their conclusions by the relaxation curves, and direct SEM observations.

These results could be analyzed in terms of "easy phasons". Atomic sites in a quasicrystal can be classified by their perpendicular coordinates. We have already seen that sites near the surface of the strip can easily disappear after a small (local or global) shift of the band, and be replaced by an other atom coming from an other

150

Fig. 22: true stress-strain curves of AICuFe ranging from room temperature to 750°C (from96).

Trut SI .... (MP.)

!

!-;;;oc

! I, I~-----r-----+

part of the strip. In real space, it corresponds to a discrete jump of an atom. It is natural to associate a large degree of stability to atoms with a small perpendicular component (unlikely to hop). The quasicrystal can be described as an intricate set of quasiperiodic set of "hard" and "soft" zones, the relative sizes of which depend on temperature. At low temperature the soft zones consists of small bubbles in the hard background. At sufficiently high temperature (characteristic of a phason flip), these soft zones percolate, which could induce atomic transport to long distances through these zones, leading to a macroscopic deformation of the sample. This and the result of 56 provide examples of physical possible consequences of the existence of phasons. Similar results, stressing the possible important role of phasons compared to classical dislocations was also obtained in 97.

CONCLUSION

This short lecture on quasicrystals was intended to stress a few new ideas which have penetrated this field. We hope that the importance of the new geometrical tools introduced (CP method, atomic surfaces, matching rules, random tiling, PBZ ... ) to understand the structure, diffraction patterns, stability, physical properties, is now clear to the reader. We have shown how icosahedral (and even decagonal) quasicrystals preserve a local icosahedral order, which is only partially realized in amorphous systems.

The stability of these phases, whatever the exact mechanism for it, is intrinsically related to their geometry. The stabilization of quasicrystals was shown to be possibly due to the fact that they are large entropy states, when introducing the phasons. The random tiling model accounts for this mode of stabilization. On the other hand, energetic stabilization mechanisms are provided by the matching rules theory, and perhaps more physically, by an electronic Hume-Rothery process. The Fermi vector is locked at half the value of the module of an intense Bragg peak. This implies the creation of a pseudo-gap at the Fermi level which is clearly seen experimentally. The quasisphericity of the pseudo-Brillouin zone enhances this effect.

This phenomenon induces interesting transport properties. For instance, the conductivity strongly increases with the temperature and is lowered by improving the structural quality of the samples. This can be understood by invoking the influence of d-states, bad diffusion properties, high symmetry of the PBZ, or small Fermi velocity. Some other physical properties (magnetotransport, plastic deformation, ... ) display specific features which make quasicrystals a new interesting state of matter.

151

ACKNOWLEDGMENTS

C.S. is grateful to the organizers for their invitation, and acknowledges the par­ticipants (especially F. Ducastelle, J. Haffner, V.V. Kamyshenko and A. Zunger) for, interesting discussions. We would like to thank C. Berger, E. Belin, and L. Billard for providing their original figures, and for useful comments. We also thank M. Duneau and C. Oguey who have supplied most of the figures concerning the geometry of quasicrystals, and B. Worth alias "Bridgette" for her diligent proof-reading!

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