- Email: [email protected]
Quasicrystals as alloys with short-range order
Physica B 405 (2010) 3885–3889
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Quasicrystals as alloys with short-range order K.W. Sulston a,, B.L. Burrows b a b
Department of Mathematics & Statistics, University of Prince Edward Island, Charlottetown, PE, Canada C1A 4P3 Mathematics Section, Faculty of Computing, Engineering and Technology, Staffordshire University, Beaconside, Stafford ST18 0DG, UK
a r t i c l e in fo
abstract
Article history: Received 20 April 2010 Accepted 9 June 2010
The electronic structure of quasiperiodic lattices is studied. An alloy theory, including short-range order effects, is used to approximate Fibonacci and Thue–Morse lattices. Short-range order is treated by embedding small clusters in an alloy that itself incorporates a two-site approximation, and the probabilities of these clusters are used to construct an efficient procedure for the calculation of electronic properties. This approach allows easy identification of the contributions of particular clusters to the electronic density of states. As the short-range order is increased via the number of clusters, the density of states can be clearly seen to transition from that of an alloy to that of a quasicrystal. It is shown that the techniques may be applied to other lattices defined by substitution rules. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quasicrystal Alloy Short-range order Fibonacci Thue–Morse
1. Introduction The discovery of quasicrystals (QC) in 1984 [1] produced a good amount of interest in solid structures with properties intermediate between those of a crystal, with its perfect long-range translational and orientational order, and a substitutional alloy, with its inherent disorder. The icosohedral structure of the QC may be conceptualized more simply by a 2-dimensional tiling, such as the Penrose tiling, or by a 1-dimensional quasiperiodic (QP) lattice, such as that generated by the Fibonacci sequence. Ab initio work on quasicrystals [2,3] has indicated that the density of states of these systems are ‘‘spiky’’, which is characteristic of localized states, perhaps localized over clusters of atoms rather than at one particular atom. In order to gain insight into the electronic structure and to model the different possible behaviours, it is necessary to determine how the features of the density of states are influenced by particular subsequences in the lattice. In this paper, we construct a simple and fairly general model to facilitate this analysis. To illustrate this model, we consider a binary lattice, whose components are designated by A and B, generated by the Fibonacci (or golden mean) substitution rule A-AB
B-A,
ð1Þ
which produces the chain ABAABABAABAABABAABABA . . . : Of particular interest here is the fact that, although the Fibonacci lattice does not possess the long-range positional order of a crystal, it does possess short-range order (SRO) in the sense that the identity of the neighbours of a specific component are not Corresponding author. Tel.: + 1 902 566 0391; fax: +1 902 566 0466.
E-mail address: [email protected] (K.W. Sulston). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.06.021
completely random, as would be the case in an pure alloy, but are determined probabilistically by way of the governing substitution rule. In this paper, we examine the electronic structure of the Fibonacci lattice, by approximating it as an alloy with SRO. Work along these lines has been done by Karmakar et al. [4], who applied the cluster-Bethe-lattice method [5] to calculate the electronic density of states (DOS) of the Fibonacci lattice. They did so by embedding a Fibonacci chain in an effective medium modelled by the coherent-potential approximation, but modified to incorporate SRO exactly within the single-site approximation. They showed that using a suitably long chain could approximate the DOS with arbitrary accuracy. In this paper, we take a somewhat different tactic, wherein we apply a variation of the Falicov–Yndurain (FY) theory of binary alloys with SRO [6] to short chains, using the probabilities of these chains to obtain an efficient procedure for the calculation of the DOS. This procedure allows us to identify the contributions of specific small clusters to the DOS. In this way, we can examine how various combinations of components contribute to the well-known hierarchy of self-similar sub-bands [7], that comprises the DOS. Unlike Karmakar et al, who used a single-site approximation to model their effective medium, we construct a simplified version of Butler’s [8] self-consistent boundary-site approximation (SCBSA) to provide an effective medium, into which are embedded clusters, of relatively short length, by means of the FY theory. Thus SRO is incorporated into the model, through both the embedding medium and the clusters themselves; mathematical details of this are given in the appendix.
2. Theory of alloys with short-range order Here we utilize a variation of the SRO theory of FY [6], based on the cluster-Bethe-lattice method [5]. It is applied to a linear chain
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(i.e. a Bethe lattice with coordination number z¼ 2) composed of A and B atoms, as shown in Fig. 1(a). The basic idea is to preserve a central cluster (C) of N atoms and to replace the rest of the chain by effective atoms attached to the end atoms of C, as shown in Fig. 1(b). In this way, the system can be mathematically reduced to a finite (and presumably small) number of equations, for which the solution can more easily be calculated than for the infinite original. The details lie in the choice of clusters C and in modelling the effective atoms. We assume that we have a substitutional binary solid of constituents A and B having concentrations xA and xB, with xA +xB ¼1. These represent the probabilities P(A) and P(B) of a site being occupied by an A or B atom, respectively. In a purely random alloy, the occupancies of adjacent sites are independent of each other, so that the probability of a site and its neighbour being occupied by A and B atoms, respectively, is P(AB)¼P(A)P(B). However, if the alloy possesses SRO, then the probable occupancy of a site is affected by the occupancies of its neighbours, so that PðABÞ a PðAÞPðBÞ. In considering the Fibonacci lattice as an alloy with SRO, despite its deterministic nature, the substitution rule can be used to calculate the relevant probabilities P(AB), etc, as well as those for longer chains, such as P(ABA) [9]. We take the Hamiltonian to be of the tight-binding form X X H1 ¼ Ui jiS/ij þ Vij jiS/jj, ð2Þ i
ij
where jiS represents the atomic orbital on site i. The i-summation runs over all atoms in the chain, while the j-summation includes only nearest neighbours. Ui takes the value UA or UB according to whether site i is occupied by an A or B atom, while Vij takes the value VAA, VAB or VBB to correspond to the occupancy of adjoining sites. We now seek to calculate the average DOS at a particular site, say site i ¼0 to be specific. Referring again to Fig. 1, we isolate a cluster C of N atoms, centred at site 0, and attach effective atoms (representing the rest of the lattice) to the ends of this cluster. The Green function (GF) G for this system can be found using the Dyson equation [10] in the form G1 0 G ¼ I þ VG, where G0 is the GF for the Hamiltonian X Ui jiS/ij H0 ¼
ð3Þ
ð4Þ
and X Vij jiS/jj:
ð8Þ
As the sites i¼ 2 and 2 are to be occupied by effective atoms, elements involving them are removed from these equations by making the replacements V 7 1, 7 2 -V 7 1, 7 2 ,G 7 20 ¼ T 7 2, 7 1 G 7 10 :
ð9Þ
In Eq. (9), V 7 1, 7 2 are, in general, effective potentials V A or V B , which represent the connection of an A or B atom to the effective lattice, but which are here taken to be constant V, and equal to the other Vij, as we ignore off-diagonal disorder. Meanwhile, T 7 2, 7 1 are transfer matrices between atoms in the effective medium, which for a one-dimensional periodic lattice take the form [10] 2
TðEÞ T 7 2, 7 1 ¼ x þ sðx 1Þ1=2 ,
ð10Þ
where
x ¼ ðsEÞ=2V
ð11Þ
and s ¼ 71 so that jTjo 1. The identity of the quantity s depends on the nature of the medium. For example, in the simplest case of an ordered monatomic lattice, s would simply equal the atomic potential VA. However, for the case of an alloy under consideration here, we take s to be the coherent potential (CP) calculated using a two-site version of the SCBSA [8]. This approximation is an extension of the widely used single-site coherent-potential approximation (CPA) [11], and for the one-dimensional tightbinding model used here, it is equivalent to the molecular CPA of Tsukada [12]. The basic idea of the SCBSA is to set up a central cluster of atoms in an effective lattice of CP’s s, in such a way as to require consistency between the diagonal element of the GF for a boundary site of the cluster and that for the surrounding effective medium. The approximation uses a boundary site (rather than the central site) because it is in close contact with the surrounding medium, and thus the DOS there is strongly dependent on that medium, whereas the DOS at a central site is almost independent of the medium for sufficiently large clusters. For the two-site SCBSA adopted here, the self-consistency condition for the energy-dependent parameter a takes the form 1 2 1 X V2 V PðC2 Þ EE1 a ¼ a , ð12Þ EE2 a a C 2
i
V¼
ðEU1 ÞG10 ¼ V10 G00 þ V1,2 G20 :
ð5Þ
ij
Calculating the GF matrix elements, for the case N ¼3 to be specific, leads to the trio of equations ðEU0 ÞG00 ¼ 1 þ V01 G10 þ V0,1 G10 ,
ð6Þ
ðEU1 ÞG10 ¼ V10 G00 þ V12 G20 ,
ð7Þ
where the summation is over valid clusters C2 of length 2, P(C2) is the probability of a cluster’s occurrence, and E1 and E2 are the site energies ( ¼UA or UB) for the first and second atoms in the cluster. Hence the CP s is given in terms of a by
s ¼ Ea
V2
a
:
ð13Þ
The theoretical details are given in the appendix, where we present an adaptation of Butler’s [8] SCBSA method. Upon substituting Eqs. (9)–(13) into Eqs. (6)–(8), these equations can be solved numerically for G00, for any energy E. The DOS at the central site 0 for the chosen cluster C is found directly from G00(E,C) [10], and hence the average DOS at site 0 is found by taking an average over all contributing clusters, weighted by the probability P(C) of each cluster’s occurrence, producing X r0 ðEÞ ¼ p1 PðCÞ Im G00 ðE þ i0,CÞ: ð14Þ C
Fig. 1. Schematic showing the model: (a) segment of Fibonacci chain with central cluster of three atoms (labelled C) isolated and (b) central cluster preserved and rest of chain replaced by effective atoms.
G00(E+ i 0,C) can be calculated for a particular cluster C by solving the set of linear equations exemplified by Eqs. (6)–(8), incorporating the energy-dependent value of s, which is evaluated by numerical solution of nonlinear Eq. (12) for a followed by substitution into Eq. (13). In the calculations reported here, the value of N is always taken to be odd, so that the central site of the cluster is well-defined. The problem is now reduced to selection of
K.W. Sulston, B.L. Burrows / Physica B 405 (2010) 3885–3889
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the relevant clusters C and the calculation of their weights P(C). Note that averaging could be avoided in Eq. (14) by choosing one cluster with very large N; however, we emphasize that we have used an efficient procedure by taking small N and exact values of P(C) to obtain our estimates of the DOS.
3. Application to quasiperiodic chains Although the theory of Section 2 was originally developed to treat disordered alloys, we apply it here to a pair of quasiperiodic lattices, Fibonacci and Thue–Morse, which are deterministic structures. The goal is to be able to see the connection between the ordered lattice and the alloys which approximate it, and in particular, to see how clusters of various lengths contribute to the DOS. The Fibonacci lattice is constructed according to the substitution rule (1), which acts to determine the values of the various probabilities. The Fibonacci chain is governed by the golden mean pffiffiffi t ¼ ð 5 þ1Þ=2 1:618, from which it can be shown [9] that PðAÞ ¼ t1 0:618 and PðBÞ ¼ t2 0:382. It is also straightforward [9] to determine the clusters of a particular length which can appear in a Fibonacci chain, and to calculate their probabilities of occurrence. For clusters of length 2, which underlie the effective medium in Eq. (12), we have that PðABÞ ¼ PðBAÞ ¼ t2 and PðAAÞ ¼ t3 0:236, while P(BB)¼0 as the pair BB never appears in a Fibonacci chain. The embedded clusters appearing in Eq. (14) are of length N. For N ¼3 for example, the only clusters (and their probabilities) which can appear are ABA with PðABAÞ ¼ t2 , AAB and BAA both with PðAABÞ ¼ PðBAAÞ ¼ t3 and BAB with PðBABÞ ¼ t4 . The Thue–Morse lattice is determined by the substitution rule A-AB
B-BA,
Fig. 2. Average DOS rðEÞ for the Fibonacci lattice, calculated exactly using rescaling.
ð15Þ
which generates the chain ABBABAABBAABABBA . . . : From the form of the substitution rule, it is obvious that P(A)¼P(B)¼0.5, and hence the probabilities of longer clusters can be calculated [9]. In particular, all possible clusters of length 2 can occur, with probabilities P(AB) ¼P(BA) ¼ 16 and P(AA)¼P(BB) ¼ 13. Furthermore, all clusters of length 3 (except AAA and BBB) occur, all with equal probability 16. We have performed numerical calculations on both chains, using as parameter values UA ¼ UB ¼1 and VAA ¼ VAB ¼ 0.5 (to allow for diagonal disorder only).
4. Results and discussion For comparison purposes, the average DOS for the Fibonacci lattice, calculated exactly using rescaling [13], is shown in Fig. 2. The graph shows the division of the DOS into four sub-bands, which divide further into sub-sub-bands, and so forth, producing the well-known self-similar structure characteristic of quasiperiodic lattices. Fig. 3 shows the bulk DOS for the effective medium, a periodic lattice of CP’s (13). The DOS is split into upper and lower bands, which closer analysis shows are associated with the A and B sites, respectively, originating from the fact that UA ¼1 and UB ¼ 1. The upper (A) band shows partial division into 3 sub-bands, the middle one of which is primarily associated with AB clusters. The outer sub-bands originate mainly from AA clusters, in accordance with the standard [14] quantum chemical interpretation of them as being bonding (lower) and anti-bonding (upper).
Fig. 3. Bulk DOS rðEÞ for the effective medium corresponding to the Fibonacci lattice.
Fig. 4 shows the main results for the Fibonacci lattice, those of the full calculations for the model via (14), for several values of the cluster size N. (The case N ¼2 is simply that shown in Fig. 3.) For N ¼3 the basic 4-band structure of Fig. 2 can already be seen to be emerging, especially with respect to the upper sub-bands which are now much more well-defined than was the case in Fig. 3. The cluster types associated with each band, as indicated above, are reiterated by noticing that the contributions to the lower band are mainly from ABA clusters (recalling that the DOS is at the centre site of the cluster); the middle peak of the upper band arises mainly from BAB clusters, while its 2 outer peaks come primarily from AAB and BAA clusters. For N ¼5, the 4 main peaks are more accentuated, but splitting into further sub-bands is now becoming apparent. Detailed examination of the cluster contributions to each peak confirms the previous interpretations. The lower band, being associated with B sites, receives contributions primarily from clusters with a B at the centre (namely, AABAA, BABAA and AABAB). The middle peak of the upper band, with its connection to single A sites, arises mainly from ABABA clusters, the only type with a single A at the centre. The outer peaks of the upper band are associated with AA clusters, and receive their structure almost exclusively from those clusters with AA at their centre, namely BAABA and ABAAB. For N¼ 7 and N ¼ 9,
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Thue–Morse lattice. When the results for the two lattices are considered together, they show the strength of the theory to reproduce diverse QC features: the lattices have different SRO parameters, one disallows clusters containing BB, they exhibit symmetric versus non-symmetric DOS, etc. Although these calculations are essentially qualitative, the free parameters may be chosen to match particular atomic or molecular structures leading to more quantitative results. As shown in the appendix, the theory may also be applied to different substitution rules so that the range of applications is wider than the particular sequences examined here. The probabilistic properties of such rules have been discussed in Ref. [16].
Appendix A
Fig. 4. Model calculations of DOS rðEÞ of the Fibonacci lattice for N ¼3, 5, 7, 9 (front to back).
The general structure of a tight-binding model is described by the tridiagonal matrix 0 1 a b 0 0 0 : Bb a b 0 0 :C B C B C C An ¼ B ðA:1Þ B 0 b a b 0 : C, B C @0 0 b a b :A :
:
:
:
:
:
with n rows and columns and An 1 and An 2 denote similar systems with n 1 and n 2 rows and columns, respectively. Now the (11) element of An 1 is given by ðA1 n Þ11 ¼
detðAn1 Þ detðAn1 Þ ¼ : detðAn Þ a detðAn1 Þb2 detðAn2 Þ
ðA:2Þ
Consequently we have the relationship 1 2 ðA1 n Þ11 ¼ ab
detðAn2 Þ : detðAn1 Þ
ðA:3Þ
If we now consider a semi-infinite tight-binding system, obtained by taking n-1 so that An -A then this relationship becomes 1 ¼ ab2 A1 ðA1 11 Þ 11 :
ðA:4Þ
Applying this to A¼(E H) for the effective surrounding medium so that a ¼ Es, b ¼V and defining a ¼ V 2 A1 11 we have Fig. 5. Model calculations of DOS rðEÞ of the Thue–Morse lattice for N ¼ 3, 5, 7, 9 (front to back). Only region for E Z 0 is shown.
the trend towards further subdivision of bands is seen to continue, and the exact DOS of Fig. 2 is clearly emerging. Fig. 5 shows the results for the Thue–Morse lattice, analogous to those of Fig. 4. Because the DOS is symmetric about E¼0, results are only shown for EZ 0. For the case N ¼3, the tri-peak structure, that is a familiar trait of the Thue–Morse DOS [15], is already visible. A detailed look at the cluster contributions shows that, as would be expected, the central peak is associated with clusters centred by a single A, while the end peaks are associated with AA. A similar interpretation holds for the lower bands (not shown) and B and BB clusters. As N increases, the peaks are clearly seen to heighten and narrow, while splitting into increasingly numerous subpeaks. In summary, we have utilized a theory of alloys, including SRO effects, to approximate the DOS of a Fibonacci quasicrystal. The degree of SRO can be controlled by setting the number of sites in the cluster around the central site. With no SRO, the DOS shows two bands with limited fine structure, but still vaguely reminiscent of the basic appearance of that of the Fibonacci lattice. As SRO is increased, the splitting of bands into sub-bands is observed, eventually resulting in the self-similar appearance of the Fibonacci band structure. Similar comments also apply to the
V2
a
¼ Esa ) s ¼ Ea
V2
ðA:5Þ
a
and we recover Eq. (13) which effectively parameterizes the coherent potential s in terms of a. We now seek to express the diagonal element of the GF both for the cluster and the effective medium in terms of the parameter a. To achieve this we may use a general result for partitioned matrices and, assuming that all the inverses exist, perform the following analysis: 0 10 1 0 1 A1 B1 0 D W1 C1 I 0 0 T B BT A C B C B B2 A@ C1 W2 C2 A ¼ @ 0 I 0 C 2 ðA:6Þ @ 1 A: 0
BT2
A3
DT
C2T
W3
0
0
I
Here we may denote Aj as a mj mj matrix so that the sizes of Bj are determined and the inverse matrix may be calculated as a matrix with the same structure. Three particular equations are BT1 C1 þ A2 W2 þB2 C2T ¼ I,
ðA:7Þ
T BT2 W2 þA3 C2T ¼ 0 ) C2T ¼ A1 3 B2 W2 ,
ðA:8Þ
A1 C1 þ B1 W2 ¼ 0 ) C1 ¼ A1 1 B1 W2 :
ðA:9Þ
Substituting into the first of these equations gives 1 T ðBT1 A1 1 B1 B2 A3 B2 þ A2 ÞW2 ¼ I:
ðA:10Þ
K.W. Sulston, B.L. Burrows / Physica B 405 (2010) 3885–3889
From this general result we can now specialize to a tight-binding situation where the Bus only have non-zero elements in their last row and first column. Thus for example for a 3 3 matrix B has the form 0 1 0 0 0 B C B¼@ 0 0 0A ðA:11Þ V 0 0 and in general 2 1 ðBT1 A1 1 B1 Þpq ¼ V ðA1 Þm1 m1 dp1 dq1 ¼ adp1 dq1 ,
BT1
ðA:12Þ
A1 1B1
simply adds one additional term a so that the term to the (11) element of A2 in Eq. (A.10). There is a similar analysis for the term B2 A3 1BT2 and this simply adds an additional term a1 to the (m2 m2) element of A2 in Eq. (A.10). Now applying this analysis to the effective medium A2 ¼ Es, with A1 and A3 both semi-infinite tight-binding matrices (so that a1 ¼ a), we have the form 0 1 A B 0 B T C Es B A ðA:13Þ @B A 0 BT and ð2a þEsÞW2 ¼ 1:
ðA:14Þ
The term W2 is the required diagonal GF element for the effective medium and is given by W2 Geff ¼ ðEs2aÞ1 ¼
V2
a
1 a
ðA:15Þ
(see Eq. (13) in the text). For the cluster we may use the same analysis except that A2 is not the single term Es but the matrix (EI h) where h is the tight-binding matrix for the cluster. Thus for a two-site cluster, we have ! EE1 a V ðA:16Þ W2 ¼ I V EE2 a and we require the (11) GF element of W2 for the cluster which is given by ðW2 Þ11 Gcl ¼ EE1 a
1 V2 : EE2 a
ðA:17Þ
3889
In the theory the values used for E1 and E2 differ for the set of 2clusters and the criterion for finding a is that the expected value of Gcl is consistent with Geff so that we have 2 1 X 1 V V2 a ¼ PðC2 Þ EE1 a , ðA:18Þ a EE2 a C 2
where P(C2) are the probabilities of the possible 2-clusters C2 (see Eq. (12) in the main text). Once a, and thus s, is calculated, the transfer matrix T(E) (10) for the effective medium is then available to be (see Eqs. (6)–(9) in the main text). Of course, alternatively we may use the expected value of the (22) element of W2 but this gives rise to the same Eq. (A.18) since P(AB)¼P(BA). These probabilities are determined for the infinite Fibonacci sequence which is invariant under the left-to-right substitution rule A-AB,B-A and can be generated recursively from this rule starting from the symbol A ðA-AB-ABA-ABAAB . . .Þ. The probability, P(C), of any string C is uniquely determined by the invariance. The infinite sequence and the probability of any string can also be generated by the equivalent right-to-left substitution rule BA’A,A’B ð. . . BAABA’ABA’BA’AÞ and so by the symmetrical process PðCuÞ ¼ PðCÞ where Cu is the reverse string of C. Thus in particular P(AB) ¼P(BA). This symmetry is not confined to Fibonacci sequences and may be applied to any infinite sequence produced by a substitution rule. Other substitution rules have been considered in Ref. [16] and in all cases the 2-cluster probabilities are such that P(XY)¼P(YX). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. T. Fujiwara, Phys. Rev. B 40 (1989) 942. T. Fujiwara, T. Yokokawa, Phys. Rev. Lett. 66 (1991) 333. S.N. Karmakar, A. Chakrabarti, R.K. Moitra, J. Phys.: Condens. Matter 1 (1989) 1423. J.D. Joannopoulos, F. Yndurain, Phys. Rev. B 10 (1974) 5164. L.M. Falicov, F. Yndurain, Phys. Rev. B 12 (1975) 5664. Y. Liu, R. Riklund, Phys. Rev. B 35 (1987) 6034. W.H. Butler, Phys. Rev. B 8 (1973) 4499. B.L. Burrows, K.W. Sulston, J. Phys. A: Math. Gen. 24 (1991) 3979. E.N. Economou, Green’s Functions in Quantum Physics, second ed., SpringerVerlag, Berlin, 1983. P. Soven, Phys. Rev. 156 (1967) 809. M. Tsukada, J. Phys. Soc. Japan 26 (1969) 684. J.A. Ashraff, R.B. Stinchcombe, Phys. Rev. B 37 (1988) 5723. Q. Niu, F. Nori, Phys. Rev. Lett. 57 (1986) 2057. C.S. Ryu, G.Y. Oh, M.H. Lee, Phys. Rev. B 46 (1992) 5162. K.W. Sulston, B.L. Burrows, A.N. Chishti, Physica A 217 (1995) 146.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Quasicrystals as alloys with short-range order K.W. Sulston a,, B.L. Burrows b a b
Department of Mathematics & Statistics, University of Prince Edward Island, Charlottetown, PE, Canada C1A 4P3 Mathematics Section, Faculty of Computing, Engineering and Technology, Staffordshire University, Beaconside, Stafford ST18 0DG, UK
a r t i c l e in fo
abstract
Article history: Received 20 April 2010 Accepted 9 June 2010
The electronic structure of quasiperiodic lattices is studied. An alloy theory, including short-range order effects, is used to approximate Fibonacci and Thue–Morse lattices. Short-range order is treated by embedding small clusters in an alloy that itself incorporates a two-site approximation, and the probabilities of these clusters are used to construct an efficient procedure for the calculation of electronic properties. This approach allows easy identification of the contributions of particular clusters to the electronic density of states. As the short-range order is increased via the number of clusters, the density of states can be clearly seen to transition from that of an alloy to that of a quasicrystal. It is shown that the techniques may be applied to other lattices defined by substitution rules. & 2010 Elsevier B.V. All rights reserved.
Keywords: Quasicrystal Alloy Short-range order Fibonacci Thue–Morse
1. Introduction The discovery of quasicrystals (QC) in 1984 [1] produced a good amount of interest in solid structures with properties intermediate between those of a crystal, with its perfect long-range translational and orientational order, and a substitutional alloy, with its inherent disorder. The icosohedral structure of the QC may be conceptualized more simply by a 2-dimensional tiling, such as the Penrose tiling, or by a 1-dimensional quasiperiodic (QP) lattice, such as that generated by the Fibonacci sequence. Ab initio work on quasicrystals [2,3] has indicated that the density of states of these systems are ‘‘spiky’’, which is characteristic of localized states, perhaps localized over clusters of atoms rather than at one particular atom. In order to gain insight into the electronic structure and to model the different possible behaviours, it is necessary to determine how the features of the density of states are influenced by particular subsequences in the lattice. In this paper, we construct a simple and fairly general model to facilitate this analysis. To illustrate this model, we consider a binary lattice, whose components are designated by A and B, generated by the Fibonacci (or golden mean) substitution rule A-AB
B-A,
ð1Þ
which produces the chain ABAABABAABAABABAABABA . . . : Of particular interest here is the fact that, although the Fibonacci lattice does not possess the long-range positional order of a crystal, it does possess short-range order (SRO) in the sense that the identity of the neighbours of a specific component are not Corresponding author. Tel.: + 1 902 566 0391; fax: +1 902 566 0466.
E-mail address: [email protected] (K.W. Sulston). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.06.021
completely random, as would be the case in an pure alloy, but are determined probabilistically by way of the governing substitution rule. In this paper, we examine the electronic structure of the Fibonacci lattice, by approximating it as an alloy with SRO. Work along these lines has been done by Karmakar et al. [4], who applied the cluster-Bethe-lattice method [5] to calculate the electronic density of states (DOS) of the Fibonacci lattice. They did so by embedding a Fibonacci chain in an effective medium modelled by the coherent-potential approximation, but modified to incorporate SRO exactly within the single-site approximation. They showed that using a suitably long chain could approximate the DOS with arbitrary accuracy. In this paper, we take a somewhat different tactic, wherein we apply a variation of the Falicov–Yndurain (FY) theory of binary alloys with SRO [6] to short chains, using the probabilities of these chains to obtain an efficient procedure for the calculation of the DOS. This procedure allows us to identify the contributions of specific small clusters to the DOS. In this way, we can examine how various combinations of components contribute to the well-known hierarchy of self-similar sub-bands [7], that comprises the DOS. Unlike Karmakar et al, who used a single-site approximation to model their effective medium, we construct a simplified version of Butler’s [8] self-consistent boundary-site approximation (SCBSA) to provide an effective medium, into which are embedded clusters, of relatively short length, by means of the FY theory. Thus SRO is incorporated into the model, through both the embedding medium and the clusters themselves; mathematical details of this are given in the appendix.
2. Theory of alloys with short-range order Here we utilize a variation of the SRO theory of FY [6], based on the cluster-Bethe-lattice method [5]. It is applied to a linear chain
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(i.e. a Bethe lattice with coordination number z¼ 2) composed of A and B atoms, as shown in Fig. 1(a). The basic idea is to preserve a central cluster (C) of N atoms and to replace the rest of the chain by effective atoms attached to the end atoms of C, as shown in Fig. 1(b). In this way, the system can be mathematically reduced to a finite (and presumably small) number of equations, for which the solution can more easily be calculated than for the infinite original. The details lie in the choice of clusters C and in modelling the effective atoms. We assume that we have a substitutional binary solid of constituents A and B having concentrations xA and xB, with xA +xB ¼1. These represent the probabilities P(A) and P(B) of a site being occupied by an A or B atom, respectively. In a purely random alloy, the occupancies of adjacent sites are independent of each other, so that the probability of a site and its neighbour being occupied by A and B atoms, respectively, is P(AB)¼P(A)P(B). However, if the alloy possesses SRO, then the probable occupancy of a site is affected by the occupancies of its neighbours, so that PðABÞ a PðAÞPðBÞ. In considering the Fibonacci lattice as an alloy with SRO, despite its deterministic nature, the substitution rule can be used to calculate the relevant probabilities P(AB), etc, as well as those for longer chains, such as P(ABA) [9]. We take the Hamiltonian to be of the tight-binding form X X H1 ¼ Ui jiS/ij þ Vij jiS/jj, ð2Þ i
ij
where jiS represents the atomic orbital on site i. The i-summation runs over all atoms in the chain, while the j-summation includes only nearest neighbours. Ui takes the value UA or UB according to whether site i is occupied by an A or B atom, while Vij takes the value VAA, VAB or VBB to correspond to the occupancy of adjoining sites. We now seek to calculate the average DOS at a particular site, say site i ¼0 to be specific. Referring again to Fig. 1, we isolate a cluster C of N atoms, centred at site 0, and attach effective atoms (representing the rest of the lattice) to the ends of this cluster. The Green function (GF) G for this system can be found using the Dyson equation [10] in the form G1 0 G ¼ I þ VG, where G0 is the GF for the Hamiltonian X Ui jiS/ij H0 ¼
ð3Þ
ð4Þ
and X Vij jiS/jj:
ð8Þ
As the sites i¼ 2 and 2 are to be occupied by effective atoms, elements involving them are removed from these equations by making the replacements V 7 1, 7 2 -V 7 1, 7 2 ,G 7 20 ¼ T 7 2, 7 1 G 7 10 :
ð9Þ
In Eq. (9), V 7 1, 7 2 are, in general, effective potentials V A or V B , which represent the connection of an A or B atom to the effective lattice, but which are here taken to be constant V, and equal to the other Vij, as we ignore off-diagonal disorder. Meanwhile, T 7 2, 7 1 are transfer matrices between atoms in the effective medium, which for a one-dimensional periodic lattice take the form [10] 2
TðEÞ T 7 2, 7 1 ¼ x þ sðx 1Þ1=2 ,
ð10Þ
where
x ¼ ðsEÞ=2V
ð11Þ
and s ¼ 71 so that jTjo 1. The identity of the quantity s depends on the nature of the medium. For example, in the simplest case of an ordered monatomic lattice, s would simply equal the atomic potential VA. However, for the case of an alloy under consideration here, we take s to be the coherent potential (CP) calculated using a two-site version of the SCBSA [8]. This approximation is an extension of the widely used single-site coherent-potential approximation (CPA) [11], and for the one-dimensional tightbinding model used here, it is equivalent to the molecular CPA of Tsukada [12]. The basic idea of the SCBSA is to set up a central cluster of atoms in an effective lattice of CP’s s, in such a way as to require consistency between the diagonal element of the GF for a boundary site of the cluster and that for the surrounding effective medium. The approximation uses a boundary site (rather than the central site) because it is in close contact with the surrounding medium, and thus the DOS there is strongly dependent on that medium, whereas the DOS at a central site is almost independent of the medium for sufficiently large clusters. For the two-site SCBSA adopted here, the self-consistency condition for the energy-dependent parameter a takes the form 1 2 1 X V2 V PðC2 Þ EE1 a ¼ a , ð12Þ EE2 a a C 2
i
V¼
ðEU1 ÞG10 ¼ V10 G00 þ V1,2 G20 :
ð5Þ
ij
Calculating the GF matrix elements, for the case N ¼3 to be specific, leads to the trio of equations ðEU0 ÞG00 ¼ 1 þ V01 G10 þ V0,1 G10 ,
ð6Þ
ðEU1 ÞG10 ¼ V10 G00 þ V12 G20 ,
ð7Þ
where the summation is over valid clusters C2 of length 2, P(C2) is the probability of a cluster’s occurrence, and E1 and E2 are the site energies ( ¼UA or UB) for the first and second atoms in the cluster. Hence the CP s is given in terms of a by
s ¼ Ea
V2
a
:
ð13Þ
The theoretical details are given in the appendix, where we present an adaptation of Butler’s [8] SCBSA method. Upon substituting Eqs. (9)–(13) into Eqs. (6)–(8), these equations can be solved numerically for G00, for any energy E. The DOS at the central site 0 for the chosen cluster C is found directly from G00(E,C) [10], and hence the average DOS at site 0 is found by taking an average over all contributing clusters, weighted by the probability P(C) of each cluster’s occurrence, producing X r0 ðEÞ ¼ p1 PðCÞ Im G00 ðE þ i0,CÞ: ð14Þ C
Fig. 1. Schematic showing the model: (a) segment of Fibonacci chain with central cluster of three atoms (labelled C) isolated and (b) central cluster preserved and rest of chain replaced by effective atoms.
G00(E+ i 0,C) can be calculated for a particular cluster C by solving the set of linear equations exemplified by Eqs. (6)–(8), incorporating the energy-dependent value of s, which is evaluated by numerical solution of nonlinear Eq. (12) for a followed by substitution into Eq. (13). In the calculations reported here, the value of N is always taken to be odd, so that the central site of the cluster is well-defined. The problem is now reduced to selection of
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the relevant clusters C and the calculation of their weights P(C). Note that averaging could be avoided in Eq. (14) by choosing one cluster with very large N; however, we emphasize that we have used an efficient procedure by taking small N and exact values of P(C) to obtain our estimates of the DOS.
3. Application to quasiperiodic chains Although the theory of Section 2 was originally developed to treat disordered alloys, we apply it here to a pair of quasiperiodic lattices, Fibonacci and Thue–Morse, which are deterministic structures. The goal is to be able to see the connection between the ordered lattice and the alloys which approximate it, and in particular, to see how clusters of various lengths contribute to the DOS. The Fibonacci lattice is constructed according to the substitution rule (1), which acts to determine the values of the various probabilities. The Fibonacci chain is governed by the golden mean pffiffiffi t ¼ ð 5 þ1Þ=2 1:618, from which it can be shown [9] that PðAÞ ¼ t1 0:618 and PðBÞ ¼ t2 0:382. It is also straightforward [9] to determine the clusters of a particular length which can appear in a Fibonacci chain, and to calculate their probabilities of occurrence. For clusters of length 2, which underlie the effective medium in Eq. (12), we have that PðABÞ ¼ PðBAÞ ¼ t2 and PðAAÞ ¼ t3 0:236, while P(BB)¼0 as the pair BB never appears in a Fibonacci chain. The embedded clusters appearing in Eq. (14) are of length N. For N ¼3 for example, the only clusters (and their probabilities) which can appear are ABA with PðABAÞ ¼ t2 , AAB and BAA both with PðAABÞ ¼ PðBAAÞ ¼ t3 and BAB with PðBABÞ ¼ t4 . The Thue–Morse lattice is determined by the substitution rule A-AB
B-BA,
Fig. 2. Average DOS rðEÞ for the Fibonacci lattice, calculated exactly using rescaling.
ð15Þ
which generates the chain ABBABAABBAABABBA . . . : From the form of the substitution rule, it is obvious that P(A)¼P(B)¼0.5, and hence the probabilities of longer clusters can be calculated [9]. In particular, all possible clusters of length 2 can occur, with probabilities P(AB) ¼P(BA) ¼ 16 and P(AA)¼P(BB) ¼ 13. Furthermore, all clusters of length 3 (except AAA and BBB) occur, all with equal probability 16. We have performed numerical calculations on both chains, using as parameter values UA ¼ UB ¼1 and VAA ¼ VAB ¼ 0.5 (to allow for diagonal disorder only).
4. Results and discussion For comparison purposes, the average DOS for the Fibonacci lattice, calculated exactly using rescaling [13], is shown in Fig. 2. The graph shows the division of the DOS into four sub-bands, which divide further into sub-sub-bands, and so forth, producing the well-known self-similar structure characteristic of quasiperiodic lattices. Fig. 3 shows the bulk DOS for the effective medium, a periodic lattice of CP’s (13). The DOS is split into upper and lower bands, which closer analysis shows are associated with the A and B sites, respectively, originating from the fact that UA ¼1 and UB ¼ 1. The upper (A) band shows partial division into 3 sub-bands, the middle one of which is primarily associated with AB clusters. The outer sub-bands originate mainly from AA clusters, in accordance with the standard [14] quantum chemical interpretation of them as being bonding (lower) and anti-bonding (upper).
Fig. 3. Bulk DOS rðEÞ for the effective medium corresponding to the Fibonacci lattice.
Fig. 4 shows the main results for the Fibonacci lattice, those of the full calculations for the model via (14), for several values of the cluster size N. (The case N ¼2 is simply that shown in Fig. 3.) For N ¼3 the basic 4-band structure of Fig. 2 can already be seen to be emerging, especially with respect to the upper sub-bands which are now much more well-defined than was the case in Fig. 3. The cluster types associated with each band, as indicated above, are reiterated by noticing that the contributions to the lower band are mainly from ABA clusters (recalling that the DOS is at the centre site of the cluster); the middle peak of the upper band arises mainly from BAB clusters, while its 2 outer peaks come primarily from AAB and BAA clusters. For N ¼5, the 4 main peaks are more accentuated, but splitting into further sub-bands is now becoming apparent. Detailed examination of the cluster contributions to each peak confirms the previous interpretations. The lower band, being associated with B sites, receives contributions primarily from clusters with a B at the centre (namely, AABAA, BABAA and AABAB). The middle peak of the upper band, with its connection to single A sites, arises mainly from ABABA clusters, the only type with a single A at the centre. The outer peaks of the upper band are associated with AA clusters, and receive their structure almost exclusively from those clusters with AA at their centre, namely BAABA and ABAAB. For N¼ 7 and N ¼ 9,
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Thue–Morse lattice. When the results for the two lattices are considered together, they show the strength of the theory to reproduce diverse QC features: the lattices have different SRO parameters, one disallows clusters containing BB, they exhibit symmetric versus non-symmetric DOS, etc. Although these calculations are essentially qualitative, the free parameters may be chosen to match particular atomic or molecular structures leading to more quantitative results. As shown in the appendix, the theory may also be applied to different substitution rules so that the range of applications is wider than the particular sequences examined here. The probabilistic properties of such rules have been discussed in Ref. [16].
Appendix A
Fig. 4. Model calculations of DOS rðEÞ of the Fibonacci lattice for N ¼3, 5, 7, 9 (front to back).
The general structure of a tight-binding model is described by the tridiagonal matrix 0 1 a b 0 0 0 : Bb a b 0 0 :C B C B C C An ¼ B ðA:1Þ B 0 b a b 0 : C, B C @0 0 b a b :A :
:
:
:
:
:
with n rows and columns and An 1 and An 2 denote similar systems with n 1 and n 2 rows and columns, respectively. Now the (11) element of An 1 is given by ðA1 n Þ11 ¼
detðAn1 Þ detðAn1 Þ ¼ : detðAn Þ a detðAn1 Þb2 detðAn2 Þ
ðA:2Þ
Consequently we have the relationship 1 2 ðA1 n Þ11 ¼ ab
detðAn2 Þ : detðAn1 Þ
ðA:3Þ
If we now consider a semi-infinite tight-binding system, obtained by taking n-1 so that An -A then this relationship becomes 1 ¼ ab2 A1 ðA1 11 Þ 11 :
ðA:4Þ
Applying this to A¼(E H) for the effective surrounding medium so that a ¼ Es, b ¼V and defining a ¼ V 2 A1 11 we have Fig. 5. Model calculations of DOS rðEÞ of the Thue–Morse lattice for N ¼ 3, 5, 7, 9 (front to back). Only region for E Z 0 is shown.
the trend towards further subdivision of bands is seen to continue, and the exact DOS of Fig. 2 is clearly emerging. Fig. 5 shows the results for the Thue–Morse lattice, analogous to those of Fig. 4. Because the DOS is symmetric about E¼0, results are only shown for EZ 0. For the case N ¼3, the tri-peak structure, that is a familiar trait of the Thue–Morse DOS [15], is already visible. A detailed look at the cluster contributions shows that, as would be expected, the central peak is associated with clusters centred by a single A, while the end peaks are associated with AA. A similar interpretation holds for the lower bands (not shown) and B and BB clusters. As N increases, the peaks are clearly seen to heighten and narrow, while splitting into increasingly numerous subpeaks. In summary, we have utilized a theory of alloys, including SRO effects, to approximate the DOS of a Fibonacci quasicrystal. The degree of SRO can be controlled by setting the number of sites in the cluster around the central site. With no SRO, the DOS shows two bands with limited fine structure, but still vaguely reminiscent of the basic appearance of that of the Fibonacci lattice. As SRO is increased, the splitting of bands into sub-bands is observed, eventually resulting in the self-similar appearance of the Fibonacci band structure. Similar comments also apply to the
V2
a
¼ Esa ) s ¼ Ea
V2
ðA:5Þ
a
and we recover Eq. (13) which effectively parameterizes the coherent potential s in terms of a. We now seek to express the diagonal element of the GF both for the cluster and the effective medium in terms of the parameter a. To achieve this we may use a general result for partitioned matrices and, assuming that all the inverses exist, perform the following analysis: 0 10 1 0 1 A1 B1 0 D W1 C1 I 0 0 T B BT A C B C B B2 A@ C1 W2 C2 A ¼ @ 0 I 0 C 2 ðA:6Þ @ 1 A: 0
BT2
A3
DT
C2T
W3
0
0
I
Here we may denote Aj as a mj mj matrix so that the sizes of Bj are determined and the inverse matrix may be calculated as a matrix with the same structure. Three particular equations are BT1 C1 þ A2 W2 þB2 C2T ¼ I,
ðA:7Þ
T BT2 W2 þA3 C2T ¼ 0 ) C2T ¼ A1 3 B2 W2 ,
ðA:8Þ
A1 C1 þ B1 W2 ¼ 0 ) C1 ¼ A1 1 B1 W2 :
ðA:9Þ
Substituting into the first of these equations gives 1 T ðBT1 A1 1 B1 B2 A3 B2 þ A2 ÞW2 ¼ I:
ðA:10Þ
K.W. Sulston, B.L. Burrows / Physica B 405 (2010) 3885–3889
From this general result we can now specialize to a tight-binding situation where the Bus only have non-zero elements in their last row and first column. Thus for example for a 3 3 matrix B has the form 0 1 0 0 0 B C B¼@ 0 0 0A ðA:11Þ V 0 0 and in general 2 1 ðBT1 A1 1 B1 Þpq ¼ V ðA1 Þm1 m1 dp1 dq1 ¼ adp1 dq1 ,
BT1
ðA:12Þ
A1 1B1
simply adds one additional term a so that the term to the (11) element of A2 in Eq. (A.10). There is a similar analysis for the term B2 A3 1BT2 and this simply adds an additional term a1 to the (m2 m2) element of A2 in Eq. (A.10). Now applying this analysis to the effective medium A2 ¼ Es, with A1 and A3 both semi-infinite tight-binding matrices (so that a1 ¼ a), we have the form 0 1 A B 0 B T C Es B A ðA:13Þ @B A 0 BT and ð2a þEsÞW2 ¼ 1:
ðA:14Þ
The term W2 is the required diagonal GF element for the effective medium and is given by W2 Geff ¼ ðEs2aÞ1 ¼
V2
a
1 a
ðA:15Þ
(see Eq. (13) in the text). For the cluster we may use the same analysis except that A2 is not the single term Es but the matrix (EI h) where h is the tight-binding matrix for the cluster. Thus for a two-site cluster, we have ! EE1 a V ðA:16Þ W2 ¼ I V EE2 a and we require the (11) GF element of W2 for the cluster which is given by ðW2 Þ11 Gcl ¼ EE1 a
1 V2 : EE2 a
ðA:17Þ
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In the theory the values used for E1 and E2 differ for the set of 2clusters and the criterion for finding a is that the expected value of Gcl is consistent with Geff so that we have 2 1 X 1 V V2 a ¼ PðC2 Þ EE1 a , ðA:18Þ a EE2 a C 2
where P(C2) are the probabilities of the possible 2-clusters C2 (see Eq. (12) in the main text). Once a, and thus s, is calculated, the transfer matrix T(E) (10) for the effective medium is then available to be (see Eqs. (6)–(9) in the main text). Of course, alternatively we may use the expected value of the (22) element of W2 but this gives rise to the same Eq. (A.18) since P(AB)¼P(BA). These probabilities are determined for the infinite Fibonacci sequence which is invariant under the left-to-right substitution rule A-AB,B-A and can be generated recursively from this rule starting from the symbol A ðA-AB-ABA-ABAAB . . .Þ. The probability, P(C), of any string C is uniquely determined by the invariance. The infinite sequence and the probability of any string can also be generated by the equivalent right-to-left substitution rule BA’A,A’B ð. . . BAABA’ABA’BA’AÞ and so by the symmetrical process PðCuÞ ¼ PðCÞ where Cu is the reverse string of C. Thus in particular P(AB) ¼P(BA). This symmetry is not confined to Fibonacci sequences and may be applied to any infinite sequence produced by a substitution rule. Other substitution rules have been considered in Ref. [16] and in all cases the 2-cluster probabilities are such that P(XY)¼P(YX). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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