inflation rules, band structure, and localization of electronic states

8/2/2019 Inflation Rules, Band Structure, And Localization of Electronic States

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PHYSICAL REVIEW B VOLUME 39, NUMBER 14 15MAY 1989-IInflation rules, band structure, and localization of electronic statesin a two-dimensional Penrose lattice

Penghui MaDepartment ofPhysics, Jinan Uniuersity, Guangzhou, Guangdong, People's Republic ofChinaYouyan LiuCenter of Theoretical Physics, Chinese Center ofAduanced Science and Technology (World Laboratory),Beijing, People's Republic ofChinaand Department ofPhysics, South China Institute of Technology, Guangzhou, Guangdong, People's Republic ofChina(Received 6 July 1988; revised manuscript received 30 November 1988)

We study the center model of Penrose lattices with a fivefold rotational symmetric axis in theframework of a tight-binding Hamiltonian. The inflation (or production) rules of "finite Penrosepatterns" generated by repeated application of delation and rescaling are found, which show adefinite hierarchical structure of the finite Penrose patterns. A similarity transformation is intro-duced to reduce the Hamiltonian, and the degeneracies of eigenstates are analytically determined.It is found that two-thirds of the energy states are doubly degenerate and the remaining one-thirdnondegenerate, The energy spectra for finite Penrose patterns of the first six generations and thedensity of states for three samples are presented. The Household and improved Dean method forsolving energy eigenvalues and eigenstates are used to examine the localization of electronic states.By use of several different methods and criteria, three kinds of wave-function behavior (extended,localized, and intermediate states) are clearly observed.

I. INTRODUCTiONThe experimental discovery by Schechtman et al. ' ofan Al-Mn alloy which exhibits a dift'raction pattern withpeaks showing icosahedral symmetry has led towidespread interest in systems with a quasiperiodic struc-ture. To date, a large amount of accumulated experimen-tal facts and theoretical analyses has already proved the

existence of quasicrystals with noncrystallographic sym-metry. ' Levine, Steinhardt, and Socolar ' have pro-posed a three-dimensional Penrose tiling as the basicstructure to explain how icosahedral symmetry can coex-ist with long-range order. Their theory is based on gen-eralizations of a quasiperiodic tiling of the two-dimensional plane which was extensively studied by Pen-rose" and by de Bruijn. 'Electronic and phonon modes on one-dimensional (1D}Fibonacci quasicrystals have been studied by severalgroups. ' ' Compare with those of their 1D counter-parts, the properties of the 2D Penrose quasilattices-though some results on the electronic and vibrationalspectra have been given in recent papers re farfrom being fully understood and a more detailed study isstill necessary. Because of the complexity of the ques-tion, except for a few particular electronic states whichhave been analytically studied by Sutherland, byKohmoto, and by Fujiwara et a/. , it is in general notpossible to obtain an exact solution of the electronicproperties for the whole spectrum. Therefore an exten-sive numerical investigation has been carried out to ex-tend our understanding of their systems.In this paper we study the two-dimensional pentagonal

quasicrystals which can be generated via the generalizeddual method from a periodic pentagrid. ' The corre-sponding tilings are the original case studied by Pen-rose." de Bruijn' has proved that in this Penrose tilingthere exist eight kinds of vertices (or "arrowed rhombuspatterns") of which two have fivefold symmetry (S andS5, in de Bruijn's notation). We take the "core" of the"infinite Sun pattern" generated by S or S5 as the seedto generate "finite Penrose patterns" with a fivefold rota-tional symmetric axis. %'ithout loss of generality, we cantake this kind of finite Penrose patterns as being represen-tative of all Penrose lattices, since they belong to thesame "local isomorphism class," and so have the sameelectronic properties.In the study of two-dimensional quasicrystals, there aretwo kinds of models, the center and vertex models, basedon the tight-binding Hamiltonian,

H =yz, ~t ) & t ~+ty'~t ) &1~,where ~i ) is the ith Wannier state. For the Vertex model,the atoms occupy the vertices of the tiles and the elec-trons hop only along the edges. Another model is thecenter model, in which the s-type atomic orbitals are lo-cated on the rhombuses (both the 72'-108 "fat" rhombusand the 36 -144 "thin" rhombus) and the site energy Z;is a constant and has been chosen to be zero. The elec-trons can transfer only from one orbital to another onelocated on adjacent rhombuses and the transfer-matrixelement is assumed to be constant (t =1). Therefore,the Hamiltonian simply is

39 9904 1989 The American Physical Society


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39 INFLATION RULES, BAND STRUCTURE, AND LOCALIZATION. . . 9905II='li&& jl .

The topologic construction of Penrose lattices is veryimportant for explaining the structure of energy spec-trum, the localization of electronic states, and the spatialdistribution of the wave function. Therefore, we firstcarefully investigated the evolution of the lattices gen-erated by repeated application of the debating and rescal-ing method. ' We take the "core" of the infinite Sunpattern as the seed, which is the smallest Penrose tilingwith fivefold rotational symmetric axis and with both"fat" and "thin" rhombuses. Following the terminologyof Fibonacci sequence, we call the original seed (the"Sun") first generation and name the second generation"ruby. " Penrose patterns of the first six generations areshown in Fig. 1. From the plotted tilings of successivegenerations, we can see the following interesting con-structive rules.(1) The innermost (centric) cores of the tilings for suc-cessive generations are alternately the "Sun" or "ruby. "(2) The central part of the ith-generation tiling is exact-

ly the (i)-th-generation tiling, which is surrounded byall subsequent-generation tilings from the first to the(i)-th generation. As an analog, in some sense, thisexhibits the same behavior as the Fibonacci sequence, forwhich the ith-generation Fibonacci chain can be ex-pressed asF;; 2+F; 3+F' 4+ +F3+F2+F2+F), (3)

I

In this paper we concentrate on the study of the centermodel of two-dimensional Penrose lattices. In Sec. II, wefirst briefiy discuss the infiation (or production) rules andcharacteristics of the construction of lattices under study;then, a similarity transformation is introduced whichreduces the Hamiltonian into a block-diagonal form.This reduction of the Hamiltonian saves much computa-tion time on the one hand; and on the other hand, it al-low us to draw some conclusions analytically on the de-generacy of the electronic states. In this section, we alsoinvestigate the energy spectra of the finite Penrose pat-terns of the first six generations and the density of statesfor some samples. In Sec. III, the localization of elec-tronic wave functions is examined under di6'erent ap-proaches. Three kinds of wave-function behavior besidesthe surface state (i.e., the extended state, localized state,and intermediate state) are clearly observed. This kind ofcoexistence, which is characteristic of quasiperio diestructures, is brieAy discussed in the summary.II. INFLATION (OR PRODUCTION) RULES,DEGENERACY, AND ENERGY SPECTRA

Sun Ruby

FIG. 1. The finite Penrose patterns of the first six genera-tions, evolving into a Penrose lattice with fivefold rotationalsymmetric axis. The "Sun" and "ruby" are named for the first-and second-generation finite Penrose patterns, which are alter-nately the "cores" of the successive-generation Penrose pat-terns. The grid parameters are r =0.4 and g, r; =2. .where the F, is B for odd i, or A for even i.(3) If at any location of the ith-generation tiling thereexists a "Sun," then in the succeeding generations this"Sun" will "expand" in place according to evolutive rules(1) and (2) specified above.(4) There are overlaps among di6'erent-generation til-ings, i.e., a "cell" may be common to several expanding"Sun" patterns.The above-mentioned constructive rules show the ex-istence of the hierarchical structure in these lattices,which results in the unusual electronic properties of the20 Penrose lattices. When comparing with other typesof Penrose lattices, we may say that the lattices whichcontain a fivefold rotational symmetric axis are the bestexample to show this typical hierarchical structure.Now we turn to discuss the possible reduction of theHamiltonian due to the fivefold rotational symmetry. Wedivide the Penrose lattice into five identical wedge sublat-tices. For the center model with tight-binding Hamil-tonian discussed earlier, the inter-sublattice interactionexists only between adjacent sublattices. Assume that thePenrose lattice has X rhombuses and if we label therhombuses of five identical sublattices in the same waythen the Hamiltonian can be written as

hT0N/5 xN/5 h

0/5 xN/5 /5 xN/50/5 xN/5 N/5 xN/5

/5 xN/5 (4)0/5 xN/5 N/5 xN/5 h

0/5 xN/5 N/5 xN/5


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9906 PENQHUI MA AND YOUYAN LIU 39where 2, h, and h are X/5 XX/5 real square matrices, A is the Hamiltonian for a sublattice, h is the hopping matrixfor adjacent sublattices, and h is the transposed matrix of h. We introduce a transformation matrix P as the following:I I I I I/5 /5 /5 /5

s&I A&IEOI E I a~IeoI e I e2I

e,I s2Ic3I c4Ic3I c4Ic3I c4Ic.4~I c.~I

(5)

where Iz&s is an N/5 XN/5 unit matrix and ek = exp(k X 2@i /5). Then we have 0/5 XN/5 /5 XN/5 N!5XN/5 /5 XN/50 0 I I/5 /5 /5 /5 /5I I/5 XN/5/5 XN/5 0N/5 XN/5

0/5 XN/5 -N/5 XN/5 /5 XN/500/5 XN/5 /5 XN/50N/5XN/5 /5 XN/5 /5 XN/50 N/5 XN/5'/5 XN /5 /5 XN /5 /5 XN/5 /5 XN/50 0

E,IsLIs,I

eg e4iI

e2I e3Ic.zI c.3Ic2I e.3I3c2I c3I4

a~I =H'P,

where H =PHP is a block-diagonal matrix and itsblock submatrices, respectively, areAo=A+h +h,

+~,h +~&hA2 =A+c2h +czh,A3 =A+a. 3h +C.3h,A43 +S4h +C4h .

(6)

Consequently, the eigenenergies ofH are the sum of those0 A2 . , A4. Becausec,=c4 and cz=c3,

therefore we haveA, =34 and

(7)

On the other hand, A,.'s are Hermitian, therefore 3 i andA4 have an identical energy spectrum, as have A2 and3 3 but A 0 has an independent energy spectrum. Here,we can conclude that for Penrose lattices tvith a ftvefoldrotational symmetric axis, except for the accidental highdegeneracy for a few energy eigenstates such as E=2,turbo thirds of the tight -binding ener-gy states of their spectraare doubly degenerate and one third of the -energy statesare nondegenerate.Now we proceed to determine the eigenvector for thewhole system from the subeigenvector which correspondsto submatrix A;. It is easy to prove that PP =5IN,where IN is an 1VXX unit matrix. Because H'=PHPif X' is an eigenvector of H', then the correspondingeigenvector of H is X=,P X'. Generally, if X' is theeigenvector of A ., which corresponds to the eigenenergyE, then the corresponding eigenvector X(j) for the wholesystem can be constructed by subeigenvector X'. It iseasy to prove that

4X(j)=,' 83 si,JXJ, (9)where 6 means direct sum, and Ez= exp(k X2~i/5).

I

The integer k can be used to classify the wave functionsinto five categories labeled by k =0, 1, 2, 3, 4, respective-ly. But for the doubly degenerate states the X(j) is4X(j)=,' e (Ek~XJ+eq'Xi*), (10)

where j and I mark the submatrices which have the sameeigenenergy. Based on the above analytical study, theprocedure of computation is that we first calculate theeigenenergies of a submatrix and their correspondingsubeigenvectors, then use the formulas (9) and (10) to cal-culate the eigenvectors X(j). In practice, we use the nu-merical diagonalization method to calculate the electron-ic spectra of the finite Penrose patterns up to the sixthgeneration. The results are shown in Fig. 2.As we know, the "Sun" (first-generation finite Penrosepattern) consists of ten rhombuses (see Fig. 1); thereforeit has ten eigenstates, two-thirds of which (i.e., fourstates) are doubly degenerate and the rest (i.e., the' IIIIIIIIIIII IIIII&%NIIIIIIIIIIIIIIISIUIIIIIIIIIQIIII&WIIIUIIII' IIIIIII IIIIIIIIIIII I IIIIIIIIIIIIII IIIIIII IIIII IIII UIIIIIIIUII II IIIIII I III IIIIIIIIIIIIII II I l ldll' "l l II I III I I III I IIIIII I IIIII I III IIII llil IIIII I III II I III III I I llll il

I I IIIII I

FIG. 2. Energy spectra for finite Penrose patterns of the firstsix generations. , in the generation-versus-eigenenergy plane.Roughly, two-thirds of the energy eigenstates are doubly degen-erate and the remaining one-third are nondegenerate. The high-ly degenerate state E=2 appears after the third generation, inwhich the local configurations of rhombuses supporting thestate have become available.


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39 INFLATION RULES, BAND STRUCTURE, AND LOCALIZATION. . . 9907D.9"D.7"

0.3D. 1 ~ ~

FIG. 3. The integrated density of states {IDOS), normalizedto unity, as a function of energy for the sixth-generation finitePenrose pattern with X=1140. The jump of IDOS at the F =2shows the high degeneracy of this state.

remaining two states) are nondegenerate. The energyspectrum of the first-generation Anite Penrose patterndisplays these six eigenenergies. The energy spectrum ofthe "ruby" (the second-generation finite Penrose pattern)has same distribution of degeneracy. But from the thirdgeneration onward, a few eigenstates with higher degen-eracy appear. The most remarkable one is the peculiarstate at E=2, which has been analytically studied byFujiwara et al. and by Ninomiya. This state isdisplayed in Fig. 3, which shows the integrated density ofstates (IDOS) for the sixth-generation finite Penrose pat-tern with N=1140. In Fig. 3 we can see that at theE =2 the IDOS has a big jurnp, which implies the state ishighly degenerate. According to our computation, thedegeneracy of this state is roughly 7% of the total num-ber of states, which is in agreement with the calculationof Tsunetsugu et al. but much higher than the lowerbound 0.429 k estimated by Semba and Ninomiya.This degenerate state E=2 does not exist in the Sun andruby (the first- and second-generation finite Penrose pat-terns) because this state definitely corresponds to someconfiguration of rhombuses which exists only from thethird generation onward.An interesting question is whether or not the high de-

O. 90

0.70C11

~ 0 ~50M

O.$0'P

O. 1.0-& .OO & .00

O. 90

0,70CO

O.50,0.$00.10

& .00 3.00 E/t

FIG. 4. Upper figure is the quasilattice with grid parametersr =0.5 and g, r; =2.5. Below one with r .=1.67 andr; =8.35. T.hey belong to diff'erent local isomorphism classesand both of them are different from the case shown in Fig. 1.

FIG. 5. The integrated density of states {IDOS) as a functionof energy. {a) is for the quasilattice with r =0.5 and {b) for thequasilattice with r = 1.67. The corresponding lattices have beenshown in Fig. 4. {a) shows a small jump of IDOS at E =0, butthere is not any remarkable jump in {b).


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9908 PENGHUI MA AND YOUYAN LIU 39generacy of an eigenstate depends on the local topologi-cal structure of the lattice. For this purpose, we have cal-culated the IDOS of other two quasilattices, which areshown in Fig. 4. The grid parameter of one is r =0.5,

r, .=2.5 and the other one is r =1.67, g, r; =8..35. 'These two lattices and the one studied above (r =0.4,r, =2) have the same orientational order and quasi-periodic translational order but belong to different localisomorphism classes, and so they have different local to-pological structure. Figure 5 shows that the quasilatticewith r = 1.67 has no highly degenerate state, but for thequasilattice with r =0.5 the IDOS has a jump at E =0,for which the corresponding states are thus highly degen-erate. Therefore, a reasonable conclusion is that the highdegeneracy for some peculiar states is dependent on thelocal topological structure, even when the orientationaland translational orders are fixed.III. LOCALIZATION OF THE ELECTRONIC STATE

For examining the localization of electronic states ofPenrose lattices, we first use the Household method totransform the original Hamiltonian into a tridiago-nal matrix, then use the eigen vector Dy-Wu-Wonglawatnugool (EDWW) method (improved Deanmethod) to calculate the eigenenergies, eigenvectors, gen-eralized first moment (GFM) and generalized second mo-ment (GSM), and inverse participation ratio (IPR) undera rigid-boundary condition.Because the finite Penrose patterns being studied are"circle-like" and have rotational symmetry, the wavefunctions exhibit the same kind of symmetry. In thiscase, it is natural to choose r, the distance from the i'thrhombus to rotational-symmetry center, as its positionparameter. In other words, the polar coordinate systemis used.In the framework of tight-binding model, the jth nor-malized eigenfunction can be expressed as

As an analog, it is worthwhile recalling some resultsfor one-dimensional quasiperiodic systems. In the frame-work of the single-particle model, the electronic states ina one-dimensional periodic system are all extended, butare all localized if the system is disordered. There is ageneral expectation that since quasiperiodic systems arebetween the periodic and disordered ones, the wave func-tions are neither localized nor extended. A typical spatialdistribution of the wave function is decaying practicallyto zero and recovering to large values then decayingagain alternately. The term "critical" or "intermediate"is usually used to describe these kind of eigenstates whichcorrespond to a singular continuous spectrum. But Liuand Riklund' have numerically shown that in Fibonacciquasicrystals the extended states, localized states, and in-termediate states coexist. Furthermore, Liu also showsthat for the Aubry model of one-dimensional incommen-surate systems at the critical point the localization of thewave function depends on the wave vector and energy ei-genvalues. The coexistence of the eigenstates withdifferent localization also does hold. This coexistence, infact, is characteristic of quasiperiodic systems. In ouropinion, the quasiperiodic systems by themselves form anindependent universal class. They have their own unusu-al physical properties. The periodic, quasiperiodic, anddisordered systems stand like the three legs of a tripod.Simply considering them as intermediate systems betweenthe periodic and disordered ones may be not suitable.Therefore, a single term "critical state" is not enough todescribe the electronic properties of 1D, and also of 2Dand 3D, quasiperiodic systems. For the two-dimensionalquasicrystals, Kohmoto and Sutherland have shownthe existence of the localized states and critical states.Fujiwara also proved that, for the model studied in thispaper, some states with special parameters are extendedand some are localized. Because it is not possible to ob-tain an exact solution for whole spectrum, except forsome peculiar states the electron properties are far frombeing fully understood. A more extensive numerical in-vestigation of the localization of the electronic states isstill necessary.

where the B, is the amplitude of the normalized wavefunction in the ith rhombus, of which the distance to therotational center is r;. Then the inverse participation ra-tio (IPR) is defined as'(12)

For the center model, the generalized first moment(GFM) is defined asM,= (r)= gr, IB, (13)

The generalized first moment gives the average radiusof the wave function (r). The IPR and GSM, fromdifferent aspects, measure the localization of the wavefunction. The IPR is a measure of the reciprocal ("in-verse") of the number of rhombuses occupied by the wavefunction. ' Generally, the smaller the IPR is, the moreextended the state is. For a Penrose lattice with X sites,the IPR of an extended state would be very closed to1/N, but for a localized state the IPR is of order 0.1 to0.01. The IPR values of the intermediate states are in themiddle of them, which responds to the unusual behaviorof the wave function, decaying practically to zero and re-covering to a large amplitude then decaying again alter-nately. The GSM is a measure of the extension of thewave function in the radial distribution. For the boundstate in a one-dimensional infinite potential well, we caneasily prove that the second moment equals 0.28, which isa characteristic number to distinguish the extended statesfrom other states. This means that the state with thesecond moment greater than 0.28 is a reasonable candi-date of the extended state. As a analog of the studied

and the generalized second moment (GSM), which givesthe effective width of wave functions, is defined as

2 1/2(14)


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39 INFLATION RULES, BAND STRUCTURE, AND LOCALIZATION. . . 9909(circle-like) finite Penrose patterns with rigid boundary,in the Appendix we have calculated the first ten boundstates for an infinite circle potential well and found thatthe averaged second moment equals 0.28 too. Evidently,for a localized state the GSM is a small number roughlyunder 0.1, which responds to the small extension of thewave function. The candidates of intermediate states arethose states with GSM from 0.1 to 0.28. If we numerical-ly calculate the GSM and IPR for all states of a Penrosesystem, then a wide quasicontinuous distribution of thespectra implies the coexistence of the three kinds of elec-tronic states. The GSM and IPR spectra also allow us toeasily pick a state with definite localizatio~ to show thespatial distribution of its wave function. Figures 7 and 8,which show the spatial distribution of three kinds ofwave functions, are made in this way.Figures 68 show the num. erical results of the House-hold and EDWW method for a Penrose lattice withN =1140. In Fig. 6(a) we plotted the generalized firstmoment (average radius ( r ) ) versus eigenenergy, wherewe can see that the averaged radius of wave functionsvaries extensively from 0.1 to 0.98, but the majority is be-tween 0.5 to 0.8. In Fig. 6(b) we can see a large variationin the generalized second moment, which suggests thecoexistence of three kinds of wave functions: extended,localized, and intermediate (critical). The same con-clusion can be drawn from Fig. 6(c) which shows the IPRversus eigenenergy. The distribution of the GSM andIPR displays another difference between the quasicrystalsand disordered systems. It is well known that for disor-dered systems the states in the middle part of an energyband are more extended than those in the edge of theband and the localization of electrons is symmetric in the

energy spectrum to the E =0. But in the quasicrystalsthe situation is different. In the Fig. 6(b) we can' see thatthe extended states with GSM greater than 0.28 locate inthe energy region with E (2. Its amount roughly is 6%of the total number of states. As a general trend, withthe increase of energy the GSM slowly and slightly de-crease, which means the electrons tend to become morelocalized. In the Fig. 6(c) the IPR shows same trend,which can be seen by the increase of IPR and by the ap-pearance of more and more peaks with increasing energy.This kind of trend in localization of electrons is absent inthe disordered systems. On the other hand, we can seethat the majority of states are intermediate states andthey distribute over the whole spectrum. The remarkablehighly degenerate state E =2 appears in Fig. 6 as a dot-ted line. Figure 6(a) shows these degenerate states exten-sively spread over the system and Figs. 6(b) and 6(c)show that they have difFerent localization, but almost allof them are intermediate states. This point suggests to usthat the degenerate states E=2 may not only be locatedon a very finite string of rhombuses, but have a certainextension over the system.

O. B0.6

UO. i

~ ~ 1 ~~4, "~~ r ~ l. ~ 4L ~

~ ~1 ~ ~

tr ~

~ ~

~ ~~ ~ ~

~ g

~ \~ r ~~ ~ ~ ) ~

~ f ~ lr ~~ ~

0.2 (a)0.

4A 0.2l12 ~ ~W {b)

rZr

~ ~-~ .~. .EQg ~~ ~ ~

~ p

g l~ ~ ~g

~ ' i ~~ r g~ ~ ~ ~~ ~

~ ~~ ~

~e ~

S

~ ~ '0&.l' ~4

0.050..lK60.025.. (c) r ~ ~~ ~ ~ ~Sag:: keep'v) ~r+gQ~V4 Q~%C+4'f40-2 -1 0 1 2

FIG. 6. The generalized first moment, generalized secondmoment, and the IPR vs eigenenergy for the sixth-generationfinite Penrose pattern (N =1140) are shown, respectively. (a)shows that the averaged radius of the wave functions spread ex-tensively. (b) and (c) show that the localizations of wave func-tions are very different from each other. The trend, namely thatthe energy increases the more localized the states are, can beclearly seen.FIG. 7. %'ave functions with different localization for thesixth-generation finite Penrose pattern (N = 1140). (a) is a local-ized state, (b) is an intermediate state, and (c) is an extendedstate.


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9910 PENGHUI MA AND YOUYAN LIU 39ties of electronic states. The rotational symmetry ofthese finite Penrose patterns results in the degeneracy ofthe energy eigenstates. The reduction of the Hamiltonianleads to reduced computation time and also to an esti-mate of the degeneracy of the eigenstates. The analyticalcalculation shows that two-thirds of the tight-binding en-ergy states are doubly degenerate and the remaining one-third nondegenerate. The question of localization hasbeen addressed with several approaches: the generalizedfirst and second moments, the inverse participation ratio,and the spatial distribution of the wave functions. Withall these approaches, we arrive at the coexistence of theeigenstates with varying degrees of localization, i.e., ex-tended, localized, and intermediate states. In ouropinion, this coexistence is characteristic of quasiperiodicsystems and independent of the dimensionality of the sys-tem. In disordered systems, all states in a one-dimensional system are localized, but in three-dimensional systems there are mobility edges whichseparate the extended states from the localized states.D =2 is a critical dimensionality. Generally, for higherdimensionality the electronic states would be more ex-tended. But this does not appear to be the rule for quasi-periodic systems. As we have shown, in one- and two-dimensional quasicrystals three kinds of wave-functionbehavior (extended, localized, and intermediate) coexist.We may expect that this kind of coexistence would existin three-dimensional systems as well. That is because thelocalization of electronic states strongly depends on thetopological structure of the system. The quasicrystals,which are a very rich variety of topological structures,are the right systems to support this coexistence. Astudy for three-dimensional systems is now under way.

FIG. 8. Same as Fig. 7, but (a) shows a surface state, and (b)is the typical case of the ring state. (c) shows the ground statewhich should be regarded as an extended state.

ACKNOWI. EDGMENTSThe work of one of us is supported in part by theFoundation of Zhongshan University AdvancedResearch Center.

More solid evidence of the coexistence of the differentlocalization is the spatial distributions of the wave func-tions themselves. In Fig. 7 we show the localized, extend-ed, and intermediate state which is charactered by thelarge fluctuation of the wave function. These states alsodisplay the fivefold symmetry, and some kind of quasi-periodicity. It is evident that different localizations of theeigenstates coexist for the finite Penrose patterns understudy. In Fig. 8, we show the surface state, ground state,and the ring state which is typical for Penrose latticeswith a rotational symmetric axis and for the E =2 statewhich has been analytically studied by Fujiwara et al.

IV. BRIEFSUMMARYIn summary we have presented the inflation (or pro-duction) rules of Penrose lattices with a fivefold rotation-al symmetric axis. From the evolution of finite Penrosepatterns with the successive generations, a hierarchicalstructure is shown, which is responsible for the quasi-periodicity, self-similarity, and other particular proper-

APPENDIX: GKNKRAI. IZED SECOND MOMENTThe Schrodinger equation of particle in a two-dimensional infinite circle potential well is

(A l)Using polar coordinates %(r)=R(r)y(0) the radial eigen-function R (r) satisfies the ordinary differential equation

d Rr +r +(r k )R=Odr (A2)For the studied case the n is an integer and k =2mE/A .Letting x =kr, we have

dR dRx +x +(x )R =0.dx (A3)This is the integral-order Bessel equation.Assume the radius of potential well ra=1, then wehave the boundary condition R (x =k)=0. The solutionof the question is the positive-integral-order Bessel func-


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39 INFLATION RULES, BAND STRUCTURE, AND LOCALIZATION. . . 9911tions:

R (x)=cJ(x), n ~ 0 (A4)TABLE I. Parameters for the first ten eigenfunctions.

GFMwhere c is a constant. Because the radial wave functionshave to satisfy the boundary condition at the x =k, theeigenenergies are determined by k =x. x is the nthzero of the Bessel function.It is easy to prove that the generalized first moment& r & (GFM) and the generalized second moment b. (GSM)are as follows:

& r &= f x J(x)dx f xJ(x)dx,X~ 0 0

2.404 825.392 528.752 6711.664015.041017.944521.326324.226227.610630.5085

0.4240 590.4668 500.5178 380.4856 660.5122 170.4913570.5093 480.4939 760.5076 050.4954 48

0.2180570.2964 710.3537 190.3126740.3476 530.3390 140.3443 800.3241 240.3422 340.3262 50

0.1955290.2804 930.2925 110.2771 330.2920 390.3123 810.2914 560.2830 390.2909 830.2842 30

&r'&= ', f "x'J'(x)dx f "xJ'(x)dx,x (A5)&=(&r'&&r &')' '

We have numerically calculated the above parameters

for the first ten eigenfunctions and list them in Table I.The averaged second moment b =0.279 978 is almost ex-actly the same as the accurate value 0.28 for the one-dimensional infinite potential well.

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inflation rules, band structure, and localization of electronic states

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