Materials Science and Engineering, B14 (1992) L5-L9 L5

Letter

L o c a l i z a t i o n in h y b r i d i z e d F i b o n a c c i quas i - c rys ta l s

K. Hirose

Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo- ku, Tokyo, 113 (Japan)

D.Y.K. Ko Department of Theoretical Physics, University of Oxford, Oxford OX1 3NP (UK)

T. Hatakeyama Kawasaki Steel Corporation, Technical Research Division, LS1 Research Center, 1 Kawasaki-cho, Chiba 260 (Japan)

H. Kamimura

Department of Applied Physics, Science University of Tokyo, 1-3 Kagurazaka, Shin]uku-ku, Tokyo 162 (Japan)

(Received January 31, 1992)

Abstract

The combined effects of interband scattering and long-range quasi-periodicity are clarified in a two-band tight binding model in which a narrow "d band" lies in a wide "s band". In the present model the quasi-periodicity is introduced by arranging two atomic species in a one-dimensional Fibonacci sequence. It is shown that, whilst the wavefunction is strictly speaking critical, the physically meaningful effect of the s-d scattering with the quasi-periodic potential is to produce a localization of the electron. The relevance to the nature of the electronic states in quasi-crystals is briefly discussed.

1. Introduction

Since the realization [1] that the quasi-crystalline phase, discovered by Shechtman et al. [2], may be thermodynamically stable, and with the success in the artificial growth by Merlin et al. [3] of one-dimensional quasi-periodic superlattices, the past few years have seen a surge of effort in examining the properties of these systems (see for example ref. 4). Recently, as high quality samples are becoming available, much attention has been focusing on their electronic behavior [5-10], in particular since the long-range quasi-periodicity allows the electronic states to exhibit a number of

exotic properties [11-13]. These include, for example, fractal or chaotic wavefunctions which are strictly neither localized nor extended, and singular contin- uous energy spectra which, in an infinite one-dimen- sional system, reduce to everywhere-dense energy gaps.

Experimentally, Yamaguchi et al. [5] have observed similar energy spectra in one-dimensional Fibonacci superlattices. Their results also show that the transport properties of the associated electronic states lie between those of the periodic and random systems. In higher dimensions, the situation is more complicated, with many of the general trends remaining to be fully explained. There is now experimental evidence [7] that the quasi-crystalline phase is made stable by a Fermi surface-Jones zone interaction [14-16], with a pseudogap opening at the Fermi level to lower the total energy. The linear muffin-tin orbital (LMTO) energy band calculations by Fujiwara and Yokokawa [15, 16] on the similar crystalline a phase [17] of AIMn etc. revealed that this may be achieved as a result of an (s, p)-d orbital interaction, such that the resulting hybridization produces the desired gap.

As for the transport properties, the conductivity of the quasi-crystalline materials is found to be abnor- mally low. Indeed, Biggs et al. [8] have obtained values for the conductivity below the characteristic minimum metallic value in AICuRu. Mizutani and coworkers [6, 9] have found that the conductivity for the quasi- crystalline phase lies below that of the amorphous phase. At the same time detailed studies have revealed that the electronic specific heat is below the free- electron value. Hall measurements [8] show that the sign of the Hall coefficient may even change with temperature. Clearly, a combination of band structure and localization effects seem to be at work and, in order to understand these and other results fully, a detailed knowledge of the electronic structure at the Fermi surface is important.

In terms of theory, however, the combination of band hybridization effects, which involve interorbital interactions at short distances, and the long-range quasi-periodicity has meant that not much progress has been possible in this direction. What has been possible has been restricted to either calculations on small clusters [18] or similar crystalline systems [15, 16, 19], or one-orbital tight-binding calculations in Fibonnaci

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and Penrose systems [20-24]. The combined effect of interband hybridization and quasi-periodicity on the electronic states remains largely unknown. In addition, Sokoloff [25] has pointed out that properties such as the abnormally high resistivity cannot occur simply from single scattering from a quasi-periodic potential. This gives a zero resistivity in the same way as for a periodic system. The effect thus lies beyond the appli- cability of simple pseudopotential approaches [14]. In order to assess the situation in the experimental systems, it is crucial to gain at least a qualitative under- standing of the combined effects of long-range quasi- periodicity and interband scattering on the electronic states.

As a step towards this, we present here an exact diagonalization calculation of a two-band tight-binding system, with the parameters chosen to mimic a narrow d band lying inside a wide s band (Fig. 1 ). Partly owing to computational restrictions, and partly to keep the physics clear, we have restricted our calculation to one dimension. The quasi-periodicity effect is included by introducing two atomic species, A and B, which are arranged in a Fibonacci sequence. The transfer between the orbitals on neighbouring sites is taken to be independent of the atomic species• The model is therefore an extended version of the more common on-site or diagonal model• A renormalization group analysis by Chakrabarti et al. [26] has shown that the energy spectrum remains a Cantor set. This is because of the inherent quasi-periodic symmetry of the system and, mathematically speaking, the wavefunction has to respect this and remain "critical". Our results show that, however, s-d scattering in a quasi-periodic poten- tial will tend to localize the "critical" wavefunction. Since the electron coherence length is finite, as far as the physical properties are concerned, the effect will be to produce a localized state.

2. Calculations

The starting hamiltonian is given by

n=E E 2 E (1) i a s,d <i /> a,[J=s,d

The Roman indices denote the site, and the Greek indices denote the corresponding orbitals. For a given Fibonacci generation, the hamiltonian is diagonalized using periodic boundary conditions. The resulting band structure for the crystalline case, with VA. ~ = 0.30, V~.~ = - 0.30, VA, d = 2.55, V~. d = 2.35, t~, = 3.00, tdd = 1.00 and t~d = 1.73 is shown in the inset of Fig. 2(a). We find that the energy dispersion on the edge of the gap is reduced almost to zero when tsd 2 equals t j d d.

The density of states (Fig. 2(a)) shows a similar struc- ture to the pseudogap discussed by Fujiwara and Yoko- kawa [15, 16], with the d band splitting to give two large contributions on either side of the gap. The dens- ity of states for the quasi-periodic situation (Fig. 2(b)), corresponding to the 14th Fibonacci generation, shows that the gap remains well defined. Other subsidiary gaps open up in the energy spectrum due to the quasi- periodicity. The situation for the random case (Fig. 2(c)) shows that unlike the quasi-periodic situation there exists a band tail into the gap region, as expected for amorphous materials. The implication is that the density of states for a Fermi level lying in the gap may actually be higher for the random situation.

The almost zero dispersion of the hybridized d state on either side of the band gap (inset in Fig. 2(a)) is fundamental to the development of localization behavior in the quasi-periodic system. In terms of perturbation theory starting from the scattering of Bloch electrons, the quasi-periodic situation may be developed as the scattering of an electron state with

E(k)-

O- f ' l

(a/ 0 k

Vd

A B A A B . . . .~ .

. . . . . ~

ts~.. ' . . . "" " .... ~" V S . . . .b. . . -- m

Fig. 1. (a) The dispersion relation for the periodic system and (b) the schematic energy diagram of the two-band on-site model with two atomic species, A and B, arranged in a Fibonacci sequence.

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0.04 ~ (a) (b) (c)

.02

I I

I I I I I I I I I I I I I I ' " 1 I I I -0.5 tO 2.5 4.90. 5 tO 2.5 4.00. 5 tO 2.5 4.0

ENERGY ENERGY ENERGY Fig. 2. The density of states in the vicinity of the hybridized gap for (a) a periodic system, (b) a 14-generation Fibonacci system (610 atoms), and (c) a random system. The results for (c) have been averaged over ten samples. The inset in (a) shows the band structure in the vicinity of the hybridized gap for the periodic situation.

wavevector k and energy E(k) off Bragg planes into a new state with {vavevector k + K and energy E(k + K). K is the wavevector corresponding to the appropriate Bragg plane. As the size of the quasi-crystal increases, there is a Bragg plane at every wavevector K. Each electron state may therefore scatter off every other electronic state. This, together with the almost fiat energy dispersion such that the energies E(k) and E(k + K) are nearly degenerate for almost all K, will result in strong multiple scattering for the electrons near the edge of the gap and lead to their localization.

In order to examine the localization behavior of the wavefunctions, we have calculated the inverse partici- pation ratio

~'~i ~)i 4 p=(x,7)i2) 2 (2)

Whereas it is true that a full multifractal analysis of the wavefunction requires the use of more general defini- tions, for the case of the localization properties the present form is sufficient. In the limit of an extended state, the inverse participation ratio p approaches zero as N-1, where N is the number of sites. For a state totally localized on one site, p = 1. The results for a 14- generation Fibonacci system are shown in Fig. 3. The most salient feature is the peaks in the inverse partici- pation ratio at energies corresponding to the edge of the gap (region (1)). This indicates a localization of the wavefunction at these energies. Throughout the rest of the energy spectrum (regions (2)) the inverse participa- tion ratio is surprisingly uniform, with a small value indicating a non-localized state. The dependence of p on the system size is shown in the inset. We find that, whereas the average value for the non-localized regions (regions (2)) falls significantly with increasing system

0.3 l_ ° . oooooooS l

I 0.2 ± 3 5 7 9 11 13 15

GENERAllON A

|o.1

' i ' .-0.5 5 4.O ENERGY

Fig. 3. The inverse participation ratio p for a 14-generation Fibonacci system as a function of energy. The corresponding dependence on the system size is shown in the inset for region (1) denoting the largest inverse participation ratio and region (2) denoting the average value of p in the energy region from - 0.5 to 1.0 and from 2.5 to 4.0. The line 1IN is included as reference.

size, the value for the peak region is hardly changed. This is consistent with what is expected from a local- ized state, since extending the boundary should have no effect.

The localization character is best observed by examining the wavefunction directly. This is shown in Fig. 4, where its amplitude in real space for the state with the largest inverse participation ratio in the Fibonacci system is plotted. The result shows that the electron is clearly concentrated in a very narrow region, with secondary peaks occurring at increasing distances apart. That is, over 90% of the amplitude is concentrated at the central peak alone, and the ampli- tude of the secondary peaks is some 10-2 times smaller

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0.5" 0.005

j ° o 610

Atomic Sites

0 , 1, , , l i , I 1 1",

0 610 Atomic Sites

Fig. 4. Wavefunction amplitude corresponding to region (1)in Fig. 3. The inset shows a typical wavefunction at the regions (2) whose energy is E = 0.01434215.

than the value at the central peak, with the regions in between having almost zero amplitude. The separation between the successive secondary peaks increases by a factor of about r 3, where r is the golden mean. These small secondary peaks are the remnants of the quasi- periodicity of the lattice, which, mathematically speak- ing, implies that the wavefunction is critical. The important point is, however, that mathematical state- ments of criticality alone do not determine the elec- tronic properties. As seen above, the separation between successive secondary peaks increases rapidly so that, within a finite area determined by, for example, the inelastic mean free path, the electronic state is essentially localized.

It is important to note that it is only the combined effect of the s-d interaction and the quasi-periodicity which results in the localization of the wavefunction. In fact, Aoki [27] found some time ago that, with just on- site hybridization between the two orbitals, localization is not observed in a random system. It is therefore the intersite scattering of the two orbitals which is respon- sible for the localization effect, as postulated by Sokoloff [25]. In the regions where this hybridization is weak, or where the energy dispersion is strong such that strong multiple scattering off the quasi-periodic Bragg planes is not possible, the wavefunction shows the usual critical behavior (see the inset in Fig. 4). There are no signs of localization, or even partial localization, to within any region. This is, in fact, the same as the situation expected from a single-band Fibonacci model [28]. Thus the hybridization does produce a rather significant effect on the nature of the electronic state in the system.

We have thus found that, at least in one dimension, localization is enhanced in a manner similar to weak localization by s-d multiple scattering in a quasi- crystal. Although our model is strictly one dimensional, with only one particular quasi-periodicity, the physical origin of the effect is general. It involves only the multiple scattering of nearly degenerate states, produced as a result of hybridization between the s and the d orbitals, off a quasi-periodic potential. This suggests that in three dimensions a similar effect may occur, with the combination of hybridization and multiple scattering leading to the weak localization of electronic states around a pseudogap. Since, as Fujiwara and Yokokawa [15, 16] have shown, locating the Fermi level in such a pseudogap (formed by (s, p)-d hybridization) may stabilize the quasi-crystal, the localization of the electronic states around the pseudogap region will lead to abnormal transport properties. This may account for the abnormally high resistivity, which increases with the quality of the samples, as well as negative dependence of the resis- tivity on temperature. In addition, the gap is well defined in the quasi-crystal, whereas in the amorphous samples the band-tailing effect introduces states into the gap region. Thus the random material may produce a higher density of states at the Fermi level. Corre- spondingly a lower electronic specific heat might be expected, as has been observed experimentally.

3. Conclusion

In view of the strong effects of hybridization and quasi-periodicity on the electron localization behavior in one dimension that we have found, it would be interesting to consider a real situation in three dimensions, where band overlap effects must also be considered.

We are indebted to Professors M. Tsukada, H. Aoki and R. Saito, Drs. A. A. Yamagnchi, T. Tada and T. Nakayama for stimulating discussions. We would like to thank Professor U. Mizutani for showing us a number of experimental data and for useful suggest- ions. The numerical calculations were carried out at the Tokyo University Computer Center. This work was partially supported by a Grant-in-Aid from the Ministry of Education, Science and Culture of Japan. In addition, one of the authors (D. Y. K. Ko) would like to acknowledge the Japan Society for the Promotion of Science for a postdoctoral fellowship.

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