Electronic structure in three-dimensional quasicrystals

844

Journal of Non-Crystalline Solids 117/118 (1990) 844-847 North-Holland

ELECTRONIC STRUCTURE IN THREE-DIMENSIONAL QUASICRYSTALS

Takeo FUJIWARA Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Electronic structure in three-dimensional quasicrystals is discussed both for simple tight-binding models of Penrose lattice and rationally approximated A1-Mn alloys. In three-dimensional Penrose lattice, there are several infinitelly degenerated states though they are not generic in quasi-periodicity. The electronic structure in the latter system is calculated by the LMTO-ASA method with the local density functional theory. The density of states in AI-Mn alloys consists of a set of spiky peaks, and the stability and the role of the vacant center of the Mackay icosahedron are discussed. 1. INTRODUCTION The Fibonacci lattice is the one-dimensional cor-

view on the electronic structure in a three-dimensional

respondence of the quasicrystals.

The energy spec-

alloy and to show a little additional data on the role

trum in the Fibonacci lattice is purely singular contin-

of the glue atoms and the center of so-called 54-atom

uous with zero Lebesgue measure and the wavefunc-

Mackay icosahedron in A1-Mn quasicrystals.

Penrose lattice and in a realistic quasicrystalline AlMa

tions are critical, i.e. neither extended nor localized. 1 The idea of multifractal is quite useful to analyze these

2. TIGHT-BINDING MODELS IN THREE-

properties. 2 The two-dimensional Penrose lattices have

DIMENSIONAL PENROSE LATTICE

been analyzed extensively by the present author and coworkers. 3 They analyzed the energy spectrum nu-

2.1. Periodic Penrose lattice The three-dimensional Penrose lattice can be gen-

merically in the two-dimensional 'periodic' Penrose

erated by the generalized grid method.

lattices with varying the system size and showed tlle

space, the icosahedral basis vectors gt = (0, 1,r)~.p.

anomalous distribution of the energy spacings between

for g = 1,2,3 and gt = (0, - 1 , r)cp. for g = 4,5,6 were

adjacent eigenstates. The energy spectrum shows ill-

chosen, where c.p. stands for cyclic permutation and

smoothed density of states and they put the conjecture

r is the golden mean ( v / 5 +

that the energy spectrum in two-dimensional Penrose

are defined as

lattice can be singular continuous. The infinitely degenerate confined states and string states have been

G6 -- {x

e

E31x-~, - ~

gt/lgtl.

1)/2.

-- k t l g t l ; e

In the grid

Then the hexagrids

= 1,2,

..,6;kt E Z},

analyzed inclusively and their wavefunctions and the

where I~t =

frequencies of existence are given rigorously, 4 The con-

of the origin. Each non-singular intersection point x0

fined states were also found in the three-dimensional

defines a rhombohedron in the real space. The icosahe-

Penrose latticesJ The present author presented the

dral packings of the Penrose local isopmorphism class

The parameter 7t defines the shift

electronic structure in a crystalline analogue of a re-

(PLI) cannot be generated by dual to the hexagrids

alistic quasicrystalline AI-Mn alloy calculated by the

but by the introduction of the unit-cell shape for each

Linear Muffin-Tin Orbital Method with Atomic-Sphere

intersection type. s

Approximation (LMTO-ASA) 6 and discussed the co-

If the golden mean r is replaced by its minimal

hesive properties and stability of the so-called 54-atom

rational approximation r,~ = F,,+I/F,~, we can get the

Mackay icosahedron as a constituent unit. 7

'periodic' Penrose lattice, where n is an integer and

The purpose of this paper is to present a brief re0022-3093/90/$03.50 @ Elsevier Science Publishers B.V. (North-Holland)

F,~ is the Fibonacci number (F0 = Ft =1, F,~+t = F,~

845

T. Fujiwara / Electronic structure in three-dimensional quasicrystals

+ F,~-t ). The parameters "/t can be chosen here to

the already mentioned large holes. Around this hole (a

get non-singular hexagrids.

This 'periodic' lattice is

MI center), 12 Al(a), 12 Mn , 24 Al(/31) and 6 A1(/32)

of bcc symmetry for n = 1,4, 7, .. and otherwise of sc

form the 54-atom Mackay icosahedron ( i I ) . An Al(/31)

symmetry. The period of the simple cubic symmetry

is on a face of P R and then in a interconnected chain

is (2 + 2/,¢f5)l/2r ~ in units of the length of an edge of

of MI's. An A1(/32) is on other edges of MI and one

a rhombohedron. 5

of its nearest neighbor sites are a &site. The 7 and

2.2. Thigt-binding model and confined states

&sites are for glue A1 and Si atoms or vacancies. In

We can define two simple tight-binding models in

a(A1MnSi), all 7-sites and half of 6-sites are occupied

the three-dimensional Penrose lattice, considering one

by A1 or Si atoms but the other half of &sites (six sites)

s-orbital per one atom.

are unoccupied. Therefore, the a(A1MnSi) is not the

Atoms are either placed on

the centers of rhombohedra with constant hopping in-

bcc but the simple cubic crystal.

teractions ( - 1 ) between adjacent rhombohedra (center

sublattice with the vacant atoms on 6-sites is denoted

model) or on vertices with constant hopping interac-

by "2" in the parenthesis and the other sc-sublattice

tions ( - 1 ) along edges (vertex model). We can see the

with A1 atoms on b-sites by "1", if it is necessary.

Hereafter, the sc-

degeneracy at the energy E = 2 in the center model and that at E = 0 in the vertex m o d e l } The wavefunctions

8402

3,10 3

for these states are strictly localized in a sense that their amplitude becomes abruptly zero outsides some finite region, and have a constant amplitude on some specific sites with alternating signs. We thus call these

O3

states confined states. The sites with non-zero amplitudes form some high symmetric shape and, owing to

Et3

the Conway's theorem, we can find the same configurations uniformly distributing in a system. Therefore, the confined states are infinitely degenerated in the infinite Penrose lattice and the degeneracy is proportional to the system size. 3. A1-Mn I C O S A H E D R A L QUASICRYSTALS

¢o

0--1"0Energy

(Ryd)

0

0.50~

3.1. Crystalline analogue of AI-Mn alloy Elser and Henley 9 pointed out that the crystalline a ( h l M n S i ) is the three-dimensional (1/1)-periodic Penrose lattice with a uniquely decorated unit, i.e. rhombic dodecahedron. In ce(AIMnSi) structure, there are six classes of atomic sites: one Mn atmnic site and five A1 or Si atomic sites ( a, /31,/32, 7 and 6 ). The large holes are at (0,0,0) and (1, ½, ½), forming a perfect bcc lattice, which is two interpenetrating simple cubic (sc) sublattice. The unit cell is a packing of six rhombic dodecahedra (RD) and eight prolate rhombohedra (PR), where the atomic decoration is only for a RD. The faces of a PR are shared with RD's. Two vertices of one RD connected by the longest body diagonal are positions of

FIGURE 1 DOS and integrated DOS of a(AIMn) in a unit cell. 3.2. Electronic structure of A1-Mn alloy We calculated the electronic structure of the ElserHenley model of c~(A1Mn) selfconsistently by the LMTOASA method with local density functional theory. 7 For spacially vacant region, i.e.

on centers of MI's and

PR's, vacant atoms are placed artificially and totally we use 154 atoms, 114 A1, 24 Mn and 16 vacant atoms, in a simple cubic unit cell, but Si atoms are not distinguished from A1 atoms because the favorable positions of Si atoms have not been experimentally determined.

11:5

T. Fujiwara/ Electronic structure in three-dimensional quasicrystals

846

every band is highly dispersionless. The Fermi energy EF deviates from the high DOS peak due to the pseu-

dogap at E ~ 0.14Ryd. Thus, we may say, the pseudogap are formed so that the MI is relatively stable. The

50

resonance peak at E "~ 0.1 ~ 0.13Ryd can be seen in

<

Mn, A l a and Alfl2(2) but not in A1/32(1).

o

<

~o

projected DOSs and IDOSs are shown in Figs.2a and

5

p

The local

2b for Mn(1) and Ala(1).

Those for other A1 atoms

can be seen in Ref.7 and those in the sublattice (2) are

~5

quite similar to corresponding ones in the sublattice

~0

(1) except A1/32(2) and the vacancy 5(2).

~o

$

The local DOS's at a MI center, a P R center and a vacant 5 site are quite different from each other. The local DOS's at the P R center and the vacant 5 site do

~ 2

not have any characteristic structure and are broad, flat and low bands, which means that the neighboring

~o

0

--]iO

Energy

(Ryd)

0

wavefunctions extend scarecely inside the P R center

OB5

and the vacant 5 site. On the other hand, that at a MI center shows specific s- and p-type character in lower

F I G U R E 2a Projected DOS and integrated DOS of Mn(1) site.

energy region but near the Fermi energy no DOS is seen. The center of a MI has an enough space for an A1 atom but actually it remains vacant. This must be

p

<

understood from the view point of stability. We calculated the selfconsistent electronic structure again in

< ~6

~ 2

a fictitious system with A1 atoms occupying the MI centers. Overall feature has not altered except the fol-

v

lowings: The p-DOS on A1 (MI center) is enhanced at

0 s

E ~ - 0 . 6 ,-- - 0 . 8 R y d and the s-DOS on A1 (MI center) at E "~ - 0 . S R y d with resonating s-DOS in the same en-

2

c~

ergy range on Ala, A1/31 and Alfl2(1). Figure 3 shows the local DOS on MI center, with and without an A1

0 & -I.0

........ Energy

0 (Ryd)

0

0.5

~

atom , respectively. The local DOS on Al (MI center)

N

vanishes in the Fermi energy region as the original vacant MI center. The total valence charge on A1 atoms

F I G U R E 2b Projected DOS and integrated DOS of Ala(1) site.

at the MI center is reduced to about 2.he/atom. The remaining excess valence charge would be distributed on the surrounding neighbours at around the Fermi

T h e total density of states (DOS) and integrated

energy and we could not find a gain of the cohesive en-

density of states (IDOS) are shown in Fig.1. The Fermi

ergy. Therefore, the MI center could be a stabilization

energy EF is 0.115 Ryd shown by the vertical line, the

center for an atom of the group I or II but not for A1,

main peak at E -~ 0.0 ,,~ 0.1Ryd comes from the Mn

and is left vacant.

3d-states and the deep tail of the DOS from the Al 3s-

A17 and A16 atoms were replaced by vacant atoms

state. The DOS consists of a set of spiky peaks and

and then a role of glue atoms A17 and Al6 were studied.

847

T. Fujiwara/ Electronic structure in three-dimensional quasicrystals

Total DOS is seen in Fig.4. The total valence charge on each atom does not change much but the local DOS in the region E< -0.5Ryd is highly reduced, which causes a large reduction of cohesive energy. Several narrow peaks appear in the whole energy range, which corresponds to the almost localized states within each MI. These facts suggest that the large amount of electrons extends outside MI through the glue atoms rather

3x103

8:(102

r..)

g

~

than direct connection of MI's and MI's are not unique constituent pieces, especially for the valence electrons of deep energy levels.

11[ LI,L, , \

'1' I

~

£1.0

Energy

(Ryd)

0

0.5

o~

g FIGURE 4 Total DOS and integrated DOS of A1Mn alloy without glue atoms.

~o r)

MI without Al

~ 4 ~

N -1.0

M.Kohmoto, B.Sutherland, and Tang, Phys.Rev. B35 (1987) 1020 ; P.A.Kalugin, A.Yu.Kitaev, and L.S.Levitov, Zh.Eksp.Theor.Fiz. 91 (1986) 692 [Sov.Phys. - JETP 64 (1986) 410]. Energy (R.yd)

0

0.5

FIGURE 3 Projected DOS and integrated DOS of MI(1)-center with and without an A1 atom. 4. CONCLUDING REMARKS We have discussed electronic structures in threedimensional Penrose lattice and icosahedra (A1Mn) alloy by using the minimal rational approximation. The quasicrystals have not the translational symmetry as well as the amorphous or liquid systems though the former have the long range bond orientational order. Therefore, the Bloch's theorem is not useful here in a rigorous sense. We have the LMTO-Recursion method which is only existing method of real space band structure calculation designed for random systems, l° but it is not applicable to quasi-periodic systems. The most proper method should be developed further in the future.

T.Fujiwara, M.Kohmoto and T.Tokihiro, to be published in Phys.Rev. B. H.Tsunetsugu, T.Fujiwara, K.Ueda, and T.Tokihiro, J.Phys.Soc.Jpn. 55 (1986) 1420. T.Fujiwara, M.Arai, T.Tokihiro, and M.Kohmoto Phys.Rev. B37 (1988) 2797; M.Arai, T.Tokihiro T.Fujiwara, and M.Kohmoto, Phys.Rev. B38 (1988) 1621; T.Tokihiro, T.Fujiwara, and M.Arai, Phys.Rev. B38 (1988) 5981; T.Fujiwara, T.Tokihiro and M.Arai, Materials Science Forum 22-24 (1987) 459. 5 M.Krajci and T.Fujiwara, Phys.Rev.B38 (1989) 12903. 6 O.K.Andersen, Phys.Rev.B12 (1975) 3060. 7 T.Fujiwara, Phys.Rev.B40 (1975) in press. 8 J.E.Socolar and P.J.Steinhardt, Phys.Rev.B34 (1986) 617. 9 V.Elser and C.L.Henley, Phys.Rev.Lett. 55 (1985) 2883. 10 T.Fujiwara, J.Non-Crystal. 1039.

Solids 61-62 (1984)