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Anomalous electronic properties of complex crystals: from cluster compounds to quasicrystals
72
Materials Science and Engineering, B19 (1993) 72-76
Anomalous electronic properties of complex crystals: from cluster compounds to quasicrystals S. J. P o o n Department of Physics, University of Vwginia, Charlottesville, VA 22901 (USA)
Abstract It has been reported recently that icosahedral (i-) crystals with perfect quasicrystalline order exhibit semimetallic conductivity that falls below Mott's minimum metallic conductivity. Moreover, their transport properties are anomalous in comparison with disordered systems with low metallic conductivity. Results obtained from these ordered phases based on good metals (A1, Cu) are presented. The anomalies in transport properties and their strong dependence on crystal chemistry have revealed a peculiar bandstructure effect induced by the Fermisurface-Jones-zone-boundaries (FS-JZB) interaction in i-crystals. Central to the FS-JZB interaction criterion of phase stability is the existence of a pseudogap which is enhanced by the global icosahedral symmetry. Further understanding of the i-crystals has been advanced through studies of crystal analogs known as approximants. Several approximant structures are constructed from the b.c.c, packing of large icosahedral clusters. Bandstructure calculations for these cluster compounds are now available. Decagonal-crystal structures constructed from the periodic stacking of two-dimensional quasiperiodic layers known as decagonal crystals are also studied. Comparison between the approximant and amorphous phases has provided insight to the understanding of quasiperiodicity, randomness, and atomic-potential as well as dimensionality effects on electronic properties.
1. Introduction Since the discovery of a metallic phase with icosahedral point group symmetry and no translational symmetry [1], there has been much experimental and theoretical work to investigate the effects of quasicrystalline order on the properties of these icosahedral (i-) materials. Studies of metastable disordered quasicrystals showed that their electron transport properties were metallic glass like [2]. In addition, these studies also revealed the prominence of the Fermi-surface-Jones-zone (FS-JZ) interaction and its importance to the stability of these materials [3]. With the discovery of stable i-phases (A1CuFe and A1CuRu [4, 5]) and decagonal (d-) phases (AICuNi and AICuCo [6, 7]) that are structurally ordered, it became possible to investigate the effects of quasiperiodicity, randomness, atomic potential and dimensionality on the electronic properties. Unusual electronic properties have been observed in the stable ordered i-phases and their crystal analogs known as approximants [8-13]. It is the purpose of this paper to highlight some of these properties and discuss our current understanding of them.
2. Ordering atoms in quasicrystals To date, structural studies [14, 15] have provided convincing evidence for the proposal that the atomic
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decorations of the building blocks in quasicrystals are related to those of the crystal analog phases known as approximants [16-18]. Earlier, two icosahedral clusters found in cubic approximants were considered as possible building blocks for two related i-phase types A1Mn [16-18] and AIMgZn [17-19]. They are the Mackay icosahedral cluster of 54 atoms distributed over three icosahedral shells in A1Mn, and a cluster of 136 atoms distributed over five icosahedral shells in A1MgZn and A1LiCu (Fig. 1). In the cubic approximants, the first cluster neighbors are connected through the three-fold cluster axes parallel to the (111) axis and the second cluster neighbors are connected through the two-fold cluster axes parallel to (100) axes. The 136 atomclusters overlap their first eight neighbors along (111). The existence of icosahedral clusters in i-phases has been demonstrated by the use of high resolution electron microscopy [15]. Quasicrystalline structures have been modeled in terms of the aggregation of polyatomic clusters [20]. Figure 2 illustrates the aggregation of decagonal clusters approximately 40 /k in diameter D in an ordered decagonal quasicrystal m165CulsCo2o [21]. There are only two distances between neighboring clusters, namely D and D/~" (where ~--- 1.618 is the golden mean). However, recent analysis of ordered i-A1CuRu [22] concludes that no local icosahedral clusters are present in the
© 1993-Elsevier Sequoia. All rights reserved
& J. Poon/ Electronic properties of complex crystals
73
[]
a)
b) + c)
d)
e)
Fig. 1. (a) The three atomic shells constituting the Mackay icosahedron found in a-A1MnSi; atomic species that decorate them are (a) AI, (b) Mn, and (c) A1. (b) The five shells-cluster in AILiCu; atomic species that decorate the shell structures are (a) At, Cu, (b) Li and (c) A1, Cu (d) AI, Cu, and (e) Li. (From ref. 20.)
thing about the effect of quasiperiodicity on properties through a comparison study of quasicrystal phases and their crystalline counterparts.
3. Semimetallic i-phase systems 3.1. Ordered i-phases
Fig. 2. Atoms in the decagonal plane of d-Al6sCulsCo20 for the two layers (z = 0 and z = c/2) in each vertical period. The layer at z=c/2 is rotated by 2~r/10 with respect to the z = 0 layer. The two intercluster connections discussed in the text are marked. Circles and squares are for AI and TM (transition metal) respectively; open and filled symbols are for z = 0 and z=c/2 respectively.
quasicrystal structure. However, short-range atomic ordering compatible with the long-range icosahedral symmetry is well defined. Based on the fact that the approximant structures share similar building blocks with the quasicrystals, one may be able to learn some-
The structurally ordered i-phases of AICuFe [9, 11, 12] and AICuRu [13] were the first i-phases to show barely or semimetallic behavior, which is not expected in alloys composed of good metals, i-m165Cu20Rua5 has the lowest 3' value (0.11 mJ (g-at.) -1 K -2) found to date in the i-phases, which is only 10% of the free electron value. The conductivity data 0(T) for i-AICuRu are shown in Fig. 3. The 0(300 K) values are already near the "minimum metallic conductivity" of 200 l-I ~ cm -1 [23], indicating that these phases are near the metal-insulator transition. Low-T resistivity and magnetoresistivity data are explained according to the weak localization theory [13]. Unusual features are seen in the Hall coefficient R H ( T ) and thermopower S(T) (Fig. 3). A large, negative low-TRn is measured, indicating an effective carrier density 2-3 orders of thermopower magnitude lower than the free electron value. In addition, strong temperature dependences of RH are observed. A large value of S as well as unusual behavior in S(T) are also observed, as shown in Fig. 3 reflecting the change in sign of RH(T) for the x = 2 0 sample. These properties are to be compared with those observed in metallic systems that show metallic-glass-like behavior in their transport properties. The latter include tr> 1000 1) -1 cm -1, [RH[<10 -14 cgs, and ]S[<10 ~V K -1 at
S. J. Poon/ Electronic properties of complex crystals
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A1CuFe crystalline approximants, which are crystalline analogs of the i-AI(TM) (TM is transition metal) class are excellent choices for study. In particular, band structure calculations for similar alloys have been performed [25]. The a-A1MnSi samples yield high resistivity values p(300 K)=3100 /~12 cm (0(300 K)=320 f~-i cm -1) and 0(4.2 K)= 160 1~- a c m -1. In addition, a large low-T Hall coefficient and the thermoelectric power as well as anomalous temperature dependences of RH and S are apparent. Preliminary data on the well characterized R-phase A16z.sCu26.sFell [26] are similar to those reported earlier [9]. Thus, unusual electronic behavior is also seen in the approximant phases.
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room temperature as well as weak temperature dependences in them. i-Phase samples of m163.sCu24.sFe]2 show similar unusual behavior [9]. Results on iA165_xCu20Ru15Six where x~< 1 are also shown in Fig. 3. The addition of a small amount of Si to this system causes dramatic changes in the electronic properties. For example, with the addition of as little as 1% Si, RH(T), while still large and negative, is almost temperature independent, and S(T) has become large and positive. The rapid variation in y, m R H ( T ) and S(T) with a small compositional change (about 1 at.%) will be discussed later. Similar behavior of ~r [11, 12, 24] as well as % RH(T), and S(T) [24] has also been seen in i-AICuFe. Structural disorder restores the metallicglass-like behavior in these systems [2, 8].
3.2. Approximant phases In view of the unusual electronic properties observed in the stable i-phases which have quasi-periodic structure, the study of crystalline approximant phases would be useful. The ot-A172.sMn17.4SiloA and R-phase of
Single-grained samples can be obtained in the stable d-phase systems, allowing study of electronic and transport properties in this new class of anisotropic materials. One can thus examine periodicity vs. quasiperiodicity effects on properties in these highly ordered systems (Fig. 2). Anisotropic p, S, RH, and K (thermal conductivity) values have been observed in A1CuCo and A1CoNi d-crystals [27]. Except for the claim of a peculiar scattering mechanism in the measurement of K along the quasicrystalline direction [28], the transport properties show metallic-glass-like behavior. The anisotropy ratio in the resistivity can be as large as 10-20, with p along the quasicrystalline direction being larger (about 300-600 /z12 cm). In order to gain a better understanding of the electronic states parallel to the quasiperiodic plane in a d-crystal, magnetoresistance measurements have recently been carried out with the magnetic field perpendicular to the quasicrystalline (qc) layers (i.e. t ~ with z along the periodic direction). Data on Ap(tl, T)/p(O, T) for currents parallel to z and qc directions are taken. The results are shown in Fig. 4. It is clear that both the magnitude of Ap/p and its field and temperature dependences are similar to those seen in metallic systems in the weakly localized regime [29]. Detailed analysis of these results is underway. 5. Discussion
The unusual electronic properties of ordered i-phases and approximant phases have been ascribed to the Fermi-surface-Jones-zone interaction as well as the strong s~l hybridization in these materials [2, 8]. The role of s-d scattering in i-phases has been discussed [30, 31]. For the i-crystals, the icosahedral point group symmetry leads to almost spherical JZ boundaries, which in turn leads to a large fraction of the Fermi surface
S. J. P o o n / Electronic properties of complex crystals 500 450 400 C O
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in carrier concentration caused by FS-JZ effects, rather than to an increase in the scattering rate [2]. An alternative model for the anomalous conductivities is based on the compartmentalization of i-phase structure [38]. According to this model, the complete envelopment of the conductive icosahedral building blocks by a layered-structure network which is semiconducting leads to a proximity-tunneling mechanism of conductivity. A variety of electronic states, namely those that are localized, critical, and extended has been found in the theoretical treatments of two-dimensional quasi-periodic lattices [25]. Critical eigenstates are those that are neither localized nor extended, but rather decay according to a power law g'(r)~~r -~, where /3<1 (two dimensions). Critical electronic states have also been proposed for icosahedral quasicrystals [39, 40]. From the experimental standpoint, weak-localization and electron-electron interaction effects appear to be able to account for the conductivities as well as magnetoconductivities of quasicrystals studied, including the decagonal crystals and marginally metallic icosahedral crystals. These results suggest that the electronic states are extended rather than critical in the quasiperiodic systems studied so far.
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H (T) Fig. 4. N o r m a l i z e d m a g n e t o r e s i s t i v i t i e s vs. a p p l i e d m a g n e t i c field ( a l o n g p e r i o d i c z d i r e c t i o n ) at v a r i o u s t e m p e r a t u r e s in dAI65Cu15C020. Q C a n d Z d e n o t e d i r e c t i o n s of a p p l i e d c u r r e n t .
being involved in the FS-JZ interaction. Thus, the prominence of the pseudogap as well as the finestructure features in the density of states (DOS) predicted for the approximant phases [25] would be further enhanced in the i-phases. In fact, pseudogaps in iphases have been seen in photoemission experiments [32, 33]. Some of these features may explain the optical measurements of i-A1CuFe [34, 35]. The large S(T) and RH(T) as well as their unusual dependences on temperature point to an effective narrow bandwidth (approximately 0.1 eV) and also a DOS that varies quite rapidly over a range of about 0.02 eV [2]. Further evidence of the latter is found in the i-A1CuRu [13], i-AICuFe [24] and i-AICuRuSi systems where dramatic changes in S(T), RH(T), and p(T) with a small (approximately 1 at.%) compositional variation is observed. Rapid variation of p(T) with composition has also been seen in i-AIPdMn [36]. However, in the disordered phases, the FS-JZ effects are weakened and any rapidly varying features in the DOS would be washed out, even though a weak pseudogap is still present [37]. The propensity toward a metal-insulator transition in the ordered phases is ascribed to the drastic reduction
6. Final remarks In closing, several important questions remain in understanding the physics of quasicrystals. One is still puzzled by the persistence of fine electronic structures (about 0.02--0.1 eV) in the density of states when chemical disorder in the transition-metal sublattice (FeCu, RuCu, PdMn, see Fig. 2) is prevalent. As a result of the disorder, the electron mean free path is comparable with those found in amorphous metals [2]. This also leads to the question of whether this type of disorder would mitigate the quasiperiodic effects on electronic states. Finally, in view of the fact that the effective bandwidth is comparable with the Debye temperatures (approximately 500 K) [9, 13], the study of strong electron-phonon renormalization effects on electron transport is called for.
Acknowledgments This research is supported by the National Science Foundation Contract No. DMR90-15538. I thank Y. He for generating the decagonal-crystal structure shown in Fig. 2.
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References 1 D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Phys. Rev. Lett., 53 (1984) 1951. 2 S. J. Poon, Adv. Phys., 41 (1992) 303. 3 J. L. Wagner, B. D. Biggs and S. J. Poon, Phys. Rev. Lett., 65 (1990) 203. 4 A. P. Tsai, A. Inoue and T. Masumoto, Jpn. J. Appl. Phys., 26 (1987) L1505. 5 A. P. Tsai, A. Inoue, T. Masumoto and N. Kaoka, Jpn. J. Appl. Phys., 27 (1988) L2252. 6 L. X. He, Y. K. Wu and K. H. Kuo, J. Mater Sci. Lett., 7 (1989) 1284. 7 A. P. Tsai, A. Inoue and T. Masumoto, Mater Trans. JIM, 30 (1989) 463. 8 B. D. Biggs, F. S. Pierce and S. J. Poon, Europhys. Lett., 19 (1992) 415. 9 B. D. Biggs, Y. Li and S. J. Poon, Phys. Rev. 13, 43 (1991) 8747. 10 J. L. Wagner, B. D. Biggs, K. M. Wong and S. J. Poon, Phys. Rev. B, 38 (1988) 7436. 11 T. Klein, A. Gozlan, C. Berger, F. Cyrot-Lackmann and Y. Calvayrac, Europhys. Lett., 13 (1990) 129. 12 T. Klein, C. Berger, D. Mayou and F. Cyrot-Lackmann, Phys. Rev. Lea., 66 (1991) 2907. 13 B. D. Biggs, S. J. Poon and N. R. Munirathnam, Phys. Rev. Lea., 65 (1990) 2700. 14 C. Janot and M. de Boissieu, in D. P. DiVincenzo and P. J. Steinhardt (eds.), Quasicrystals: The State of the Art, Directions in Condensed Matter Physics, Vol. 11, World Scientific, Singapore, 1991, p. 57. 15 K. Hiraga, in D. P. DiVincenzo and P. J. Steinhardt (eds.),
Quasicrystals: The State of the Art, Directions in Condensed Matter Physics, Vol. 11, World Scientific, Singapore, 1991, p. 95. V. Elser and C. L. Henley, Phys. Rev. Lett., 55 (1985) 2883. C. L. Henley and V. Elser, Philos. Mag. B, 53 (1986) L59. P. Guyot and M. Audier, Philos. Mag. B, 52 (1985) L15. M. Audier, J. Pannetier, M. Leblanc, C. Janot, J. M. Lang and B. Dubost, Physica B, 153 (1988) 136. 20 P. Guyot, M. Audier and M. de Boissieu, in P. Jena, S. N. Khanna and B. K. Rao (eds.), Proc. NA TO Advanced Research 16 17 18 19
Workshop on Physics and Chemistry of Finite Systems: From
Clusters to Crystals, Richmond, IrA, October8-12, 1991, Kluwer, Dordrecht, 1992, p. 49. 21 S. E. Burkov, personal communications. 22 R. Z. Hu, PhD Dissertation, University of Pennsylvania, 1992. 23 N. F. Mott, Conduction in Non-Crystalline Materials, Oxford University Press, New York, 1987, p. 26. 24 F. S. Pierce, P. A. Bancel, B. D. Biggs and S. J. Poon, unpublished results. 25 T. Fujiwara and H. Tsunetsugu, in D. P. Divincenzo and P. J. Steinhardt (eds.), Quasicrystals: The State of theArt, Directions in Condensed Matter Physics, Vol. 11, World Scientific, Singapore, 1991, p. 343. 26 P. Bancel, preliminary data. 27 K. Kimura and S. Takeuchi, in D. P. Divincenzo and P. J. Steinhardt (eds.), Quasicrystals: The State of the Art, Directions in Condensed Matter Physics, Vol. 11, World Scientific, Singapore, 1991, p. 313. 28 D. L. Zhang, S. C. Cao, Y. P. Wang, L. Lu, X. M. Wang, X. L. M a and K. H. Kuo, Phys. Rev. Lett., 66 (1991) 2778. 29 M. A. Howson and B. L. Gallagher, Phys. Rep., 170 (1988) 265. 30 A. E. Carlsson and R. Phillips, in D. P. Divincinzo and P. J. Steinhardt (eds.), Quasicrystals: The State of the Art, Directions in Condensed Matter Physics, Vol. 11, World Scientific, Singapore, 1991, p. 361. 31 D. Mayou, F. Cyrot-Lackmann, G. Trembly de Laissardiere and T. Klein, unpublished results. 32 E. Belin and Z. Danhkazi, unpublished results. 33 A. Sadoc, E. Belin, Z. Danhkazi and A. M. Flank, unpublished results. 34 C. C. Homes, T. Timusk, X. Wu, Z. Altounian, A. Sahnoune and J. O. Str6m-Olsen, Phys. Rev. Lett., 67 (1991) 2694. 35 S. E. Burkov, T. Timusk and N. W. Ashcroft, unpublished results. 36 P. Lanco, T. Klein, C. Berger, F. Cyrot-Lackmann, G. Fourcaudot and A. Sulpice, Europhys. Lett., 18 (1992) 227. 37 J. Harrier, Phys. Rev. B, 21 (1980) 604. 38 J. C. Phillips and K. M. Rabe, Phys. Rev. Lett., 66 (1991) 923. 39 K. Niizeki and T. Akamatsu, J. Phys. Condens. Matter, 2 (1990) 2759. 40 L. S. Levitov, Europhys. Lett., 7 (1988) 343. L. S. Levitov, unpublished results.
Materials Science and Engineering, B19 (1993) 72-76
Anomalous electronic properties of complex crystals: from cluster compounds to quasicrystals S. J. P o o n Department of Physics, University of Vwginia, Charlottesville, VA 22901 (USA)
Abstract It has been reported recently that icosahedral (i-) crystals with perfect quasicrystalline order exhibit semimetallic conductivity that falls below Mott's minimum metallic conductivity. Moreover, their transport properties are anomalous in comparison with disordered systems with low metallic conductivity. Results obtained from these ordered phases based on good metals (A1, Cu) are presented. The anomalies in transport properties and their strong dependence on crystal chemistry have revealed a peculiar bandstructure effect induced by the Fermisurface-Jones-zone-boundaries (FS-JZB) interaction in i-crystals. Central to the FS-JZB interaction criterion of phase stability is the existence of a pseudogap which is enhanced by the global icosahedral symmetry. Further understanding of the i-crystals has been advanced through studies of crystal analogs known as approximants. Several approximant structures are constructed from the b.c.c, packing of large icosahedral clusters. Bandstructure calculations for these cluster compounds are now available. Decagonal-crystal structures constructed from the periodic stacking of two-dimensional quasiperiodic layers known as decagonal crystals are also studied. Comparison between the approximant and amorphous phases has provided insight to the understanding of quasiperiodicity, randomness, and atomic-potential as well as dimensionality effects on electronic properties.
1. Introduction Since the discovery of a metallic phase with icosahedral point group symmetry and no translational symmetry [1], there has been much experimental and theoretical work to investigate the effects of quasicrystalline order on the properties of these icosahedral (i-) materials. Studies of metastable disordered quasicrystals showed that their electron transport properties were metallic glass like [2]. In addition, these studies also revealed the prominence of the Fermi-surface-Jones-zone (FS-JZ) interaction and its importance to the stability of these materials [3]. With the discovery of stable i-phases (A1CuFe and A1CuRu [4, 5]) and decagonal (d-) phases (AICuNi and AICuCo [6, 7]) that are structurally ordered, it became possible to investigate the effects of quasiperiodicity, randomness, atomic potential and dimensionality on the electronic properties. Unusual electronic properties have been observed in the stable ordered i-phases and their crystal analogs known as approximants [8-13]. It is the purpose of this paper to highlight some of these properties and discuss our current understanding of them.
2. Ordering atoms in quasicrystals To date, structural studies [14, 15] have provided convincing evidence for the proposal that the atomic
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decorations of the building blocks in quasicrystals are related to those of the crystal analog phases known as approximants [16-18]. Earlier, two icosahedral clusters found in cubic approximants were considered as possible building blocks for two related i-phase types A1Mn [16-18] and AIMgZn [17-19]. They are the Mackay icosahedral cluster of 54 atoms distributed over three icosahedral shells in A1Mn, and a cluster of 136 atoms distributed over five icosahedral shells in A1MgZn and A1LiCu (Fig. 1). In the cubic approximants, the first cluster neighbors are connected through the three-fold cluster axes parallel to the (111) axis and the second cluster neighbors are connected through the two-fold cluster axes parallel to (100) axes. The 136 atomclusters overlap their first eight neighbors along (111). The existence of icosahedral clusters in i-phases has been demonstrated by the use of high resolution electron microscopy [15]. Quasicrystalline structures have been modeled in terms of the aggregation of polyatomic clusters [20]. Figure 2 illustrates the aggregation of decagonal clusters approximately 40 /k in diameter D in an ordered decagonal quasicrystal m165CulsCo2o [21]. There are only two distances between neighboring clusters, namely D and D/~" (where ~--- 1.618 is the golden mean). However, recent analysis of ordered i-A1CuRu [22] concludes that no local icosahedral clusters are present in the
© 1993-Elsevier Sequoia. All rights reserved
& J. Poon/ Electronic properties of complex crystals
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[]
a)
b) + c)
d)
e)
Fig. 1. (a) The three atomic shells constituting the Mackay icosahedron found in a-A1MnSi; atomic species that decorate them are (a) AI, (b) Mn, and (c) A1. (b) The five shells-cluster in AILiCu; atomic species that decorate the shell structures are (a) At, Cu, (b) Li and (c) A1, Cu (d) AI, Cu, and (e) Li. (From ref. 20.)
thing about the effect of quasiperiodicity on properties through a comparison study of quasicrystal phases and their crystalline counterparts.
3. Semimetallic i-phase systems 3.1. Ordered i-phases
Fig. 2. Atoms in the decagonal plane of d-Al6sCulsCo20 for the two layers (z = 0 and z = c/2) in each vertical period. The layer at z=c/2 is rotated by 2~r/10 with respect to the z = 0 layer. The two intercluster connections discussed in the text are marked. Circles and squares are for AI and TM (transition metal) respectively; open and filled symbols are for z = 0 and z=c/2 respectively.
quasicrystal structure. However, short-range atomic ordering compatible with the long-range icosahedral symmetry is well defined. Based on the fact that the approximant structures share similar building blocks with the quasicrystals, one may be able to learn some-
The structurally ordered i-phases of AICuFe [9, 11, 12] and AICuRu [13] were the first i-phases to show barely or semimetallic behavior, which is not expected in alloys composed of good metals, i-m165Cu20Rua5 has the lowest 3' value (0.11 mJ (g-at.) -1 K -2) found to date in the i-phases, which is only 10% of the free electron value. The conductivity data 0(T) for i-AICuRu are shown in Fig. 3. The 0(300 K) values are already near the "minimum metallic conductivity" of 200 l-I ~ cm -1 [23], indicating that these phases are near the metal-insulator transition. Low-T resistivity and magnetoresistivity data are explained according to the weak localization theory [13]. Unusual features are seen in the Hall coefficient R H ( T ) and thermopower S(T) (Fig. 3). A large, negative low-TRn is measured, indicating an effective carrier density 2-3 orders of thermopower magnitude lower than the free electron value. In addition, strong temperature dependences of RH are observed. A large value of S as well as unusual behavior in S(T) are also observed, as shown in Fig. 3 reflecting the change in sign of RH(T) for the x = 2 0 sample. These properties are to be compared with those observed in metallic systems that show metallic-glass-like behavior in their transport properties. The latter include tr> 1000 1) -1 cm -1, [RH[<10 -14 cgs, and ]S[<10 ~V K -1 at
S. J. Poon/ Electronic properties of complex crystals
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A1CuFe crystalline approximants, which are crystalline analogs of the i-AI(TM) (TM is transition metal) class are excellent choices for study. In particular, band structure calculations for similar alloys have been performed [25]. The a-A1MnSi samples yield high resistivity values p(300 K)=3100 /~12 cm (0(300 K)=320 f~-i cm -1) and 0(4.2 K)= 160 1~- a c m -1. In addition, a large low-T Hall coefficient and the thermoelectric power as well as anomalous temperature dependences of RH and S are apparent. Preliminary data on the well characterized R-phase A16z.sCu26.sFell [26] are similar to those reported earlier [9]. Thus, unusual electronic behavior is also seen in the approximant phases.
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room temperature as well as weak temperature dependences in them. i-Phase samples of m163.sCu24.sFe]2 show similar unusual behavior [9]. Results on iA165_xCu20Ru15Six where x~< 1 are also shown in Fig. 3. The addition of a small amount of Si to this system causes dramatic changes in the electronic properties. For example, with the addition of as little as 1% Si, RH(T), while still large and negative, is almost temperature independent, and S(T) has become large and positive. The rapid variation in y, m R H ( T ) and S(T) with a small compositional change (about 1 at.%) will be discussed later. Similar behavior of ~r [11, 12, 24] as well as % RH(T), and S(T) [24] has also been seen in i-AICuFe. Structural disorder restores the metallicglass-like behavior in these systems [2, 8].
3.2. Approximant phases In view of the unusual electronic properties observed in the stable i-phases which have quasi-periodic structure, the study of crystalline approximant phases would be useful. The ot-A172.sMn17.4SiloA and R-phase of
Single-grained samples can be obtained in the stable d-phase systems, allowing study of electronic and transport properties in this new class of anisotropic materials. One can thus examine periodicity vs. quasiperiodicity effects on properties in these highly ordered systems (Fig. 2). Anisotropic p, S, RH, and K (thermal conductivity) values have been observed in A1CuCo and A1CoNi d-crystals [27]. Except for the claim of a peculiar scattering mechanism in the measurement of K along the quasicrystalline direction [28], the transport properties show metallic-glass-like behavior. The anisotropy ratio in the resistivity can be as large as 10-20, with p along the quasicrystalline direction being larger (about 300-600 /z12 cm). In order to gain a better understanding of the electronic states parallel to the quasiperiodic plane in a d-crystal, magnetoresistance measurements have recently been carried out with the magnetic field perpendicular to the quasicrystalline (qc) layers (i.e. t ~ with z along the periodic direction). Data on Ap(tl, T)/p(O, T) for currents parallel to z and qc directions are taken. The results are shown in Fig. 4. It is clear that both the magnitude of Ap/p and its field and temperature dependences are similar to those seen in metallic systems in the weakly localized regime [29]. Detailed analysis of these results is underway. 5. Discussion
The unusual electronic properties of ordered i-phases and approximant phases have been ascribed to the Fermi-surface-Jones-zone interaction as well as the strong s~l hybridization in these materials [2, 8]. The role of s-d scattering in i-phases has been discussed [30, 31]. For the i-crystals, the icosahedral point group symmetry leads to almost spherical JZ boundaries, which in turn leads to a large fraction of the Fermi surface
S. J. P o o n / Electronic properties of complex crystals 500 450 400 C O
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in carrier concentration caused by FS-JZ effects, rather than to an increase in the scattering rate [2]. An alternative model for the anomalous conductivities is based on the compartmentalization of i-phase structure [38]. According to this model, the complete envelopment of the conductive icosahedral building blocks by a layered-structure network which is semiconducting leads to a proximity-tunneling mechanism of conductivity. A variety of electronic states, namely those that are localized, critical, and extended has been found in the theoretical treatments of two-dimensional quasi-periodic lattices [25]. Critical eigenstates are those that are neither localized nor extended, but rather decay according to a power law g'(r)~~r -~, where /3<1 (two dimensions). Critical electronic states have also been proposed for icosahedral quasicrystals [39, 40]. From the experimental standpoint, weak-localization and electron-electron interaction effects appear to be able to account for the conductivities as well as magnetoconductivities of quasicrystals studied, including the decagonal crystals and marginally metallic icosahedral crystals. These results suggest that the electronic states are extended rather than critical in the quasiperiodic systems studied so far.
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H (T) Fig. 4. N o r m a l i z e d m a g n e t o r e s i s t i v i t i e s vs. a p p l i e d m a g n e t i c field ( a l o n g p e r i o d i c z d i r e c t i o n ) at v a r i o u s t e m p e r a t u r e s in dAI65Cu15C020. Q C a n d Z d e n o t e d i r e c t i o n s of a p p l i e d c u r r e n t .
being involved in the FS-JZ interaction. Thus, the prominence of the pseudogap as well as the finestructure features in the density of states (DOS) predicted for the approximant phases [25] would be further enhanced in the i-phases. In fact, pseudogaps in iphases have been seen in photoemission experiments [32, 33]. Some of these features may explain the optical measurements of i-A1CuFe [34, 35]. The large S(T) and RH(T) as well as their unusual dependences on temperature point to an effective narrow bandwidth (approximately 0.1 eV) and also a DOS that varies quite rapidly over a range of about 0.02 eV [2]. Further evidence of the latter is found in the i-A1CuRu [13], i-AICuFe [24] and i-AICuRuSi systems where dramatic changes in S(T), RH(T), and p(T) with a small (approximately 1 at.%) compositional variation is observed. Rapid variation of p(T) with composition has also been seen in i-AIPdMn [36]. However, in the disordered phases, the FS-JZ effects are weakened and any rapidly varying features in the DOS would be washed out, even though a weak pseudogap is still present [37]. The propensity toward a metal-insulator transition in the ordered phases is ascribed to the drastic reduction
6. Final remarks In closing, several important questions remain in understanding the physics of quasicrystals. One is still puzzled by the persistence of fine electronic structures (about 0.02--0.1 eV) in the density of states when chemical disorder in the transition-metal sublattice (FeCu, RuCu, PdMn, see Fig. 2) is prevalent. As a result of the disorder, the electron mean free path is comparable with those found in amorphous metals [2]. This also leads to the question of whether this type of disorder would mitigate the quasiperiodic effects on electronic states. Finally, in view of the fact that the effective bandwidth is comparable with the Debye temperatures (approximately 500 K) [9, 13], the study of strong electron-phonon renormalization effects on electron transport is called for.
Acknowledgments This research is supported by the National Science Foundation Contract No. DMR90-15538. I thank Y. He for generating the decagonal-crystal structure shown in Fig. 2.
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