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Electronic structures of Cd-based quasicrystals and approximants
Journal of Alloys and Compounds 342 (2002) 343–347
L
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Electronic structures of Cd-based quasicrystals and approximants a, b Y. Ishii *, T. Fujiwara b
a Department of Physics, Chuo University, Kasuga, Tokyo 112 -8551, Japan Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113 -8654, Japan
Abstract Electronic structures of cubic Cd 6 M (M5Yb, Mg and Y), which are approximant crystals of newly discovered Cd-based quasicrystals, are studied theoretically. It is found that stabilization due to alloying is obtained if M is an element with low-lying unoccupied d states. This leads to the conclusion that the cohesion of the Cd-based compounds is certainly due to the hybridization of the d states with a wide sp band. Cd 6 M is composed of clusters of icosahedral shape, as are most approximant crystals, but a core of the cluster is of non-icosahedral symmetry. It is found that the atoms in the non-icosahedral core are rather weakly bound and do not play an important role in stabilizing the cluster and its packing. Stability of Zn 17 Sc 3 , which has a similar structure to Cd 6 M but has no non-icosahedral core, is understood in the same context as Cd 6 M. 2002 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystals; Electronic band structure; Crystal binding and equation of state
1. Introduction The discovery of new materials has been stimulating further developments in our knowledge of quasicrystals (QC). Recently stable icosahedral phases have been discovered in binary Cd–Yb and Cd–Ca alloys [1]. These are the first stable binary icosahedral QC. The quasicrystalline phases are identified as unknown phases in the phase diagrams, Cd 5.7 Yb and Cd 17 Ca 3 . At a composition very close to the QC, cubic crystalline phases, Cd 6 M (M5Yb, Ca), are obtained [2,3]. Although isostructural cubic alloys are also obtained for other systems with M5Sr, Y and most rare-earth elements [4], the icosahedral QC is realized only for M5Yb and Ca. A similar cubic phase is also found in Zn–Sc alloy, Zn 17 Sc 3 [5]. Very recently, Kaneko et al. have discovered a new QC phase near cubic Zn 17 Sc 3 by substituting Mg for a few percent of Zn [6]. The electronic structures of QC have been studied for many kinds of approximant crystals [7]. It has been recognized that a pseudogap in the density of states (DOS) is a universal feature of QC and contributes to stabilizing such novel materials. The origin of the pseudogap is believed to be due to the Brillouin-zone (BZ)-Fermi-sphere (FS) interaction (the Hume-Rothery mechanism). In the *Corresponding author. Tel.: 181-3-3817-1780; fax: 181-3-38171792. E-mail address: [email protected] (Y. Ishii).
Hume-Rothery mechanism, a strong interference of electronic waves at k and k 1 G induces the pseudogap near the Fermi level where G is the reciprocal lattice vector giving strong Bragg scattering and satisfying uGu¯2k F . In fact, a diameter of the FS calculated from the electron density is very close to the (222100) and (311111) Bragg scatterings for the icosahedral Cd–Yb. However, Ishii and Fujiwara have emphasized the importance of the sp-d hybridization for the Cd-based compounds, which do not involve transition elements [8]. In this paper, we present studies on the electronic structures of the cubic Cd 6 M and Zn 17 Sc 3 crystals in order to understand the stability mechanism of the new family of QC. Although Mg forms a continuous solid solution with Cd and the cubic Cd 6 Mg is hypothetical, we calculate the electronic structure to examine the roles of unoccupied d states. We shall also discuss the roles of the cluster core with non-icosahedral symmetry found in Cd 6 M.
2. Models Cd 6 Yb is a body centered cubic crystal with space group ] Im3, which contains 168 atoms in a cubic cell with lattice ˚ [2]. The structure of Cd 6 M is parameter a515.638 A composed of clusters of icosahedral shape, as are the most approximant crystals. The atomic cluster of icosahedral shape is placed at the corner and the body-center of a cubic
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unit cell. The core of the cluster is an atomic shell of non-icosahedral symmetry. Four Cd atoms are placed at vertices of a small cube with occupancy probability 0.5. Because the Cd atom is not small enough to occupy neighboring vertices, the central shell of Cd atoms may be of tetrahedral shape. The tetrahedral symmetry is subgroup symmetry of the icosahedral one and the symmetry axes of the tetrahedral core coincide with those of outer icosahedral shells. Therefore, we presume that the tetrahedral core is very probable. The second and third atomic shells are a dodecahedron of 20 Cd atoms and an icosahedron of 12 Yb atoms, respectively. The fourth shell is a Cd icosidodecahedron obtained by placing 30 Cd atoms on the edge of the Yb icosahedron. A few more Cd atoms are placed between the clusters. The structure of cubic Zn 17 Sc 3 is very similar to Cd 6 Yb but the cluster core with non-icosahedral symmetry is ˚ [5], missing. The lattice constant for Zn 17 Sc 3 is 13.852 A which is smaller than that for Cd 6 Yb by |10%, but the atomic radii of Zn and Sc are also smaller than those for Cd and Yb by |10%. Therefore there is large vacancy at the core of the atomic cluster in Zn 17 Sc 3 structure. To avoid the fractional occupation of Cd atoms in the electronic structure calculations, a tetrahedral cluster of four Cd atoms, instead of a cubic cluster, is placed at the corner and the body-center of the cubic unit cell. Depending on the orientation of the tetrahedral clusters, we have two possible structures under the cubic symmetry. One is obtained if the tetrahedral cores at the corner and the body-center are in the same orientation, and then its space group symmetry is I23. In the other structure, the tetrahedral core at the body-center is related to that at the corner by inversion. Then the space group symmetry is ] Pn3. Such symmetry has recently been proposed for cubic Cd–Ce [9]. Since the tetrahedral core is missing, the ] symmetry group for Zn 17 Sc 3 is Im3. In this article, we ] present the electronic structures of Cd 6 M with I23 or Pn3 and Zn 17 Sc 3 . The hypothetical Cd 6 Mg is investigated and ˚ which is the its lattice parameter is assumed to be 15.22 A, optimal one obtained from the calculation. Calculation is done with the tight-binding linear muffin-tin orbitals (TBLMTO) method in the atomic-sphere approximation (ASA) [10].
3. Results and discussion
3.1. Roles of low-lying d states The total and partial p- and d-DOS for Cd 6 Yb with I23 symmetry are shown in Fig. 1. The Cd-4d band at about 20.8 Ryd does not contribute to cohesion. Another narrow band just below the Fermi level is attributed to the Yb-4f states. The total width of the 4f band is at most 0.01 Ryd and a crystalline field at the Yb site seems very small. Since the narrow 4f band is almost filled, Yb is divalent as
Fig. 1. The total and partial (p- and d-) DOS for Cd 6 Yb with I23.
are Ca and Mg. This is consistent with measurements of magnetic susceptibility [2]. A shallow dip in the DOS is seen between the 4f band and an unoccupied peak at 0.0–0.1 Ryd, which is made from the Yb-5d states. The occupied states below the dip are predominantly made from the Cd-5p states except for the narrow 4f band. This is qualitatively different from the case of pure Cd metal, in which the s and p states contribute equally to the states near the Fermi Level. We speculate that hybridization of the Cd-5p and Yb-5d states makes the bonding orbitals below the Fermi level leading to the dip (or the pseudogap) in the DOS. The total and partial DOS for Cd 6 Mg are shown in Fig. 2. One can see that the shallow dip near the Fermi level vanishes for Cd 6 Mg. Since Mg has no low-lying unoccupied d states near the Fermi level, we can say that hybridization of the d states near the Fermi level is essential for the dip formation in the Cd-based compounds. This is very different from the other QC without transition elements such as Al–Li–Cu and Zn–Mg–Y, where the pseudogap does not vanish even if Cu in Al–Li–Cu and Y in Zn–Mg–Y are replaced with the elements without the d states, Al and Mg, respectively [11,12]. Although the Fermi level of Cd 6 Yb is pinned at the shoulder of the occupied band, not at the minimum of the
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interference effect associated with the strong peaks in the structure factor is not of primary importance for the stability of the Cd-based compounds.
3.2. Roles of non-icosahedral core The cluster core with non-icosahedral symmetry is one of the characteristics of the cubic Cd 6 M. To study the role of such a core structure on the stability of the compounds, we first investigate the effect of orientation of the tetrahedral clusters by calculating the electronic structures of ] the cubic Cd 6 Yb with Pn3 symmetry. The total DOS is shown in Fig. 3. One can see no significant change at all in the DOS. The cohesive energy is calculated as 1.68 eV/ atom, which is the same as that for Cd 6 Yb with I23 symmetry within numerical accuracy. Although we have not checked a system with the tetrahedral core in arbitrary orientation, because calculation is more demanding if assumption of the cubic symmetry is removed, we presume that the energy difference due to the orientation of the tetrahedral core is very small. The structure of Zn 17 Sc 3 is almost similar to that of the cubic Cd 6 M but has no tetrahedral core at the corner and the body-center of the cubic cell. To check differences caused by the tetrahedral core, it is more convenient and
Fig. 2. The total and partial (p- and d-) DOS for Cd 6 Mg with I23.
DOS, a reasonable amount of energetic stabilization is obtained for the cubic Cd 6 M compounds. In fact the cohesive energies at a fixed lattice constant are calculated for Cd 6 Yb as 1.68 eV/ atom. This is because the occupied states just below the dip are the bonding orbitals, which yield chemical stabilization due to alloying. On the other hand, the cohesive energy for Cd 6 Mg is 1.47 eV/ atom even at the optimal lattice parameter. This value is similar to that for pure Cd (1.48 eV/ atom for f.c.c. with a54.45 ˚ and smaller by |15% than those for Cd 6 Yb with the A) low-lying d band near the Fermi level. Therefore the hybridization of the d state with a wide sp band certainly contributes to stability of the Cd 6 M compounds. The diameter of the FS is calculated from the electron ˚ 21 , which is very close to the density as 2k F 52.75 A (222100) and (311111) Bragg scatterings for the icosahedral Cd–Yb. This seems to support the HumeRothery mechanism, in which the BZ–FS interaction induces the pseudogap near the Fermi energy. However, there is neither distinct dip (or pseudogap) in the DOS nor additional stabilization due to alloying for isostructural and isovalent Cd 6 Mg, which has a similar structure factor to the other Cd 6 M compounds. In other words, the pseudogap due to the Hume-Rothery mechanism is at most what we see in the DOS of Cd 6 Mg. Thus we conclude that the
] Fig. 3. The total and partial (p- and d-) DOS for Cd 6 Yb with Pn3.
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direct to compare systems with the same valence. The total and partial DOS for Cd 6 Y (I23) and Zn 17 Sc 3 are shown in Figs. 4 and 5. The pseudogap is observed again at the Fermi level in both of the systems. Splitting of the p- and d-symmetric components below and above the pseudogap is similar to that in the cubic Cd 6 Yb. Thus the sp-d hybridization is also important in stability of Zn 17 Sc 3 . Since Y and Sc are trivalent elements, the Fermi energy is shifted towards a minimum of the DOS and we obtain larger cohesive energies, 2.15 and 2.48 eV/ atom for Cd 6 Y and Zn 17 Sc 3 , respectively. The qualitative difference in the DOS is a small midgap peak just above the Fermi level for Cd 6 Y (marked by an arrow in Fig. 4). Such an unoccupied peak is also seen in the DOS for Cd 6 Yb and is more remarkable in the local DOS at the tetrahedral core sites in Cd 6 Yb [8]. One possibility concerning the stability may be that the midgap states are counterparts of the bonding states just below the Fermi level. Although we have not checked the bonding character of the midgap states thoroughly for the moment, we do not think the midgap states are anti-bonding because enhancement of the bonding states and sufficient cohesion are obtained for Zn 17 Sc 3 , for which the midgap peak is missing. Finally we estimate the formation energy of atomic
Fig. 5. The total and partial (p- and d-) DOS for Zn 17 Sc 3 .
vacancy at the cluster core. We calculate the electronic structures of hypothetical cubic Cd–Yb crystal without the tetrahedral core and estimate the vacancy formation energy from the total energy difference. The result is 0.45 eV/ atom, which is very small in comparison with the cohesive energy. This implies that the atoms in the tetrahedral core are weakly bound and do not play a significant role in stabilizing the compound. The picture of the weakly bound tetrahedral core may be consistent with experimental observations such as ambiguity in the atomic sites in the cluster core and large temperature effects [2,9].
4. Concluding remarks
Fig. 4. The total and partial (p- and d-) DOS for Cd 6 Y with I23.
We have studied the electronic structures of the cubic Cd 6 M approximant crystals of the newly discovered binary QC. It is found that stabilization due to alloying is obtained if M is an element with low-lying unoccupied d states. This leads to the conclusion that the cohesion of the Cd-based compounds is certainly due to the hybridization of the d states with a wide sp band. Stability of Zn 17 Sc 3 is also understood in the same context as the Cd-based alloys. It is reasonable to believe that the same mechanism works also in QC because the local atomic structure would be
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similar. The roles of the non-icosahedral core of the cluster have been investigated by checking the total energy difference due to orientation of the tetrahedral core and the electronic structure of the cubic Zn 17 Sc 3 , which has large vacancy instead of the non-icosahedral core at the center of the cluster. All of these checks indicate that the atoms in the non-icosahedral core are rather weakly bound and do not play an important role in stabilizing the cluster and its packing. Recently Tamura et al. [13] have reported that a linear coefficient of the specific heat varying with temperature is extremely large for the icosahedral Cd–Yb (2.87 mJ / mol K 2 ). According to the present calculation, the band contribution is |0.8 mJ / mol K 2 , which is a bit larger than the conventional QC but still reasonable. It is known that a two-level system leads to the specific heat varying linearly with temperature in glassy materials, where the temperature coefficient is typically of the order of 0.01 mJ / mol K 2 [14]. It is not evident if the weakly bound tetrahedral core behaves like the two-level system but we expect that the density of the tetrahedral core is much more than that of the two-level systems in glassy materials. It would be interesting to check the possibility that motion of the tetrahedral cores contributes to the low-temperature specific heat. In the present paper, we have only discussed the electronic mechanism for stability of Cd-based alloys. By calculating the electronic structures of the cubic Cd 6 M with other divalent elements M, we can show that similar electronic structures and cohesive energy are obtained for M=Ca and Sr [8]. In spite of similarities, however, only Cd–Yb and Cd–Ca form QC. For long-range quasiperiodic packing of the clusters, matching of the atomic size may be crucial [8].
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Acknowledgements The authors would like to thank A.P. Tsai for valuable discussions. This work is partly supported by Core Research in Environmental Science and Technology, Japan Science and Technology Corporation.
References [1] A.P. Tsai, J.Q. Guo, E. Abe, H. Takakura, T.J. Sato, Nature 408 (2000) 537. [2] A. Palenzona, J. Less-Common Metals 25 (1971) 367. [3] G. Bruzzone, Gazz. Chim. Ital. 102 (1972) 234. [4] T.B. Massalski, H. Okamoto, P.R. Subramanian, K. Karcprzak, Binary Alloy Phase Diagrams, 2nd Edition, ASM International, Ohio, 1990. [5] P. Villars, Pearson’s Handbook, Desk Edition, ASM International, Ohio, 1997. [6] Y. Kaneko, Y. Arichika, T. Ishimasa, Phil. Mag. Lett. 81 (2001) 777. [7] T. Fujiwara, in: Z.M. Stadnik (Ed.), Physical Properties of Quasicrystals, Springer, Heidelberg, 1998, p. 169. [8] Y. Ishii, T. Fujiwara, Phys. Rev. Lett. 87 (2001) 206408. ¨ [9] M. Armbruster, S. Lidin, J. Alloys Comp. 307 (2000) 141. ¨ [10] O.K. Andersen, O. Jepsen, D. Glotzel, in: F. Bassani, F. Fumi, M.P. Tosi (Eds.), Highlights in Condensed Matter Theory, North-Holland, New York, 1985, p. 59. [11] T. Fujiwara, T. Yokokawa, Phys. Rev. Lett. 66 (1991) 333. [12] K. Oshio, Y. Ishii, J. Alloys Comp. 342 (2002) 402. [13] R. Tamura, Y. Murao, S. Takeuchi, K. Tokiwa, T. Watanabe, T.J. Sato, A.P. Tsai, Jpn. J. Appl. Phys. 40 (2001) L912. [14] P.W. Anderson, B.I. Halperin, C.M. Varma, Phil. Mag. 25 (1972) 1.
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www.elsevier.com / locate / jallcom
Electronic structures of Cd-based quasicrystals and approximants a, b Y. Ishii *, T. Fujiwara b
a Department of Physics, Chuo University, Kasuga, Tokyo 112 -8551, Japan Department of Applied Physics, University of Tokyo, Hongo, Tokyo 113 -8654, Japan
Abstract Electronic structures of cubic Cd 6 M (M5Yb, Mg and Y), which are approximant crystals of newly discovered Cd-based quasicrystals, are studied theoretically. It is found that stabilization due to alloying is obtained if M is an element with low-lying unoccupied d states. This leads to the conclusion that the cohesion of the Cd-based compounds is certainly due to the hybridization of the d states with a wide sp band. Cd 6 M is composed of clusters of icosahedral shape, as are most approximant crystals, but a core of the cluster is of non-icosahedral symmetry. It is found that the atoms in the non-icosahedral core are rather weakly bound and do not play an important role in stabilizing the cluster and its packing. Stability of Zn 17 Sc 3 , which has a similar structure to Cd 6 M but has no non-icosahedral core, is understood in the same context as Cd 6 M. 2002 Elsevier Science B.V. All rights reserved. Keywords: Quasicrystals; Electronic band structure; Crystal binding and equation of state
1. Introduction The discovery of new materials has been stimulating further developments in our knowledge of quasicrystals (QC). Recently stable icosahedral phases have been discovered in binary Cd–Yb and Cd–Ca alloys [1]. These are the first stable binary icosahedral QC. The quasicrystalline phases are identified as unknown phases in the phase diagrams, Cd 5.7 Yb and Cd 17 Ca 3 . At a composition very close to the QC, cubic crystalline phases, Cd 6 M (M5Yb, Ca), are obtained [2,3]. Although isostructural cubic alloys are also obtained for other systems with M5Sr, Y and most rare-earth elements [4], the icosahedral QC is realized only for M5Yb and Ca. A similar cubic phase is also found in Zn–Sc alloy, Zn 17 Sc 3 [5]. Very recently, Kaneko et al. have discovered a new QC phase near cubic Zn 17 Sc 3 by substituting Mg for a few percent of Zn [6]. The electronic structures of QC have been studied for many kinds of approximant crystals [7]. It has been recognized that a pseudogap in the density of states (DOS) is a universal feature of QC and contributes to stabilizing such novel materials. The origin of the pseudogap is believed to be due to the Brillouin-zone (BZ)-Fermi-sphere (FS) interaction (the Hume-Rothery mechanism). In the *Corresponding author. Tel.: 181-3-3817-1780; fax: 181-3-38171792. E-mail address: [email protected] (Y. Ishii).
Hume-Rothery mechanism, a strong interference of electronic waves at k and k 1 G induces the pseudogap near the Fermi level where G is the reciprocal lattice vector giving strong Bragg scattering and satisfying uGu¯2k F . In fact, a diameter of the FS calculated from the electron density is very close to the (222100) and (311111) Bragg scatterings for the icosahedral Cd–Yb. However, Ishii and Fujiwara have emphasized the importance of the sp-d hybridization for the Cd-based compounds, which do not involve transition elements [8]. In this paper, we present studies on the electronic structures of the cubic Cd 6 M and Zn 17 Sc 3 crystals in order to understand the stability mechanism of the new family of QC. Although Mg forms a continuous solid solution with Cd and the cubic Cd 6 Mg is hypothetical, we calculate the electronic structure to examine the roles of unoccupied d states. We shall also discuss the roles of the cluster core with non-icosahedral symmetry found in Cd 6 M.
2. Models Cd 6 Yb is a body centered cubic crystal with space group ] Im3, which contains 168 atoms in a cubic cell with lattice ˚ [2]. The structure of Cd 6 M is parameter a515.638 A composed of clusters of icosahedral shape, as are the most approximant crystals. The atomic cluster of icosahedral shape is placed at the corner and the body-center of a cubic
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unit cell. The core of the cluster is an atomic shell of non-icosahedral symmetry. Four Cd atoms are placed at vertices of a small cube with occupancy probability 0.5. Because the Cd atom is not small enough to occupy neighboring vertices, the central shell of Cd atoms may be of tetrahedral shape. The tetrahedral symmetry is subgroup symmetry of the icosahedral one and the symmetry axes of the tetrahedral core coincide with those of outer icosahedral shells. Therefore, we presume that the tetrahedral core is very probable. The second and third atomic shells are a dodecahedron of 20 Cd atoms and an icosahedron of 12 Yb atoms, respectively. The fourth shell is a Cd icosidodecahedron obtained by placing 30 Cd atoms on the edge of the Yb icosahedron. A few more Cd atoms are placed between the clusters. The structure of cubic Zn 17 Sc 3 is very similar to Cd 6 Yb but the cluster core with non-icosahedral symmetry is ˚ [5], missing. The lattice constant for Zn 17 Sc 3 is 13.852 A which is smaller than that for Cd 6 Yb by |10%, but the atomic radii of Zn and Sc are also smaller than those for Cd and Yb by |10%. Therefore there is large vacancy at the core of the atomic cluster in Zn 17 Sc 3 structure. To avoid the fractional occupation of Cd atoms in the electronic structure calculations, a tetrahedral cluster of four Cd atoms, instead of a cubic cluster, is placed at the corner and the body-center of the cubic unit cell. Depending on the orientation of the tetrahedral clusters, we have two possible structures under the cubic symmetry. One is obtained if the tetrahedral cores at the corner and the body-center are in the same orientation, and then its space group symmetry is I23. In the other structure, the tetrahedral core at the body-center is related to that at the corner by inversion. Then the space group symmetry is ] Pn3. Such symmetry has recently been proposed for cubic Cd–Ce [9]. Since the tetrahedral core is missing, the ] symmetry group for Zn 17 Sc 3 is Im3. In this article, we ] present the electronic structures of Cd 6 M with I23 or Pn3 and Zn 17 Sc 3 . The hypothetical Cd 6 Mg is investigated and ˚ which is the its lattice parameter is assumed to be 15.22 A, optimal one obtained from the calculation. Calculation is done with the tight-binding linear muffin-tin orbitals (TBLMTO) method in the atomic-sphere approximation (ASA) [10].
3. Results and discussion
3.1. Roles of low-lying d states The total and partial p- and d-DOS for Cd 6 Yb with I23 symmetry are shown in Fig. 1. The Cd-4d band at about 20.8 Ryd does not contribute to cohesion. Another narrow band just below the Fermi level is attributed to the Yb-4f states. The total width of the 4f band is at most 0.01 Ryd and a crystalline field at the Yb site seems very small. Since the narrow 4f band is almost filled, Yb is divalent as
Fig. 1. The total and partial (p- and d-) DOS for Cd 6 Yb with I23.
are Ca and Mg. This is consistent with measurements of magnetic susceptibility [2]. A shallow dip in the DOS is seen between the 4f band and an unoccupied peak at 0.0–0.1 Ryd, which is made from the Yb-5d states. The occupied states below the dip are predominantly made from the Cd-5p states except for the narrow 4f band. This is qualitatively different from the case of pure Cd metal, in which the s and p states contribute equally to the states near the Fermi Level. We speculate that hybridization of the Cd-5p and Yb-5d states makes the bonding orbitals below the Fermi level leading to the dip (or the pseudogap) in the DOS. The total and partial DOS for Cd 6 Mg are shown in Fig. 2. One can see that the shallow dip near the Fermi level vanishes for Cd 6 Mg. Since Mg has no low-lying unoccupied d states near the Fermi level, we can say that hybridization of the d states near the Fermi level is essential for the dip formation in the Cd-based compounds. This is very different from the other QC without transition elements such as Al–Li–Cu and Zn–Mg–Y, where the pseudogap does not vanish even if Cu in Al–Li–Cu and Y in Zn–Mg–Y are replaced with the elements without the d states, Al and Mg, respectively [11,12]. Although the Fermi level of Cd 6 Yb is pinned at the shoulder of the occupied band, not at the minimum of the
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interference effect associated with the strong peaks in the structure factor is not of primary importance for the stability of the Cd-based compounds.
3.2. Roles of non-icosahedral core The cluster core with non-icosahedral symmetry is one of the characteristics of the cubic Cd 6 M. To study the role of such a core structure on the stability of the compounds, we first investigate the effect of orientation of the tetrahedral clusters by calculating the electronic structures of ] the cubic Cd 6 Yb with Pn3 symmetry. The total DOS is shown in Fig. 3. One can see no significant change at all in the DOS. The cohesive energy is calculated as 1.68 eV/ atom, which is the same as that for Cd 6 Yb with I23 symmetry within numerical accuracy. Although we have not checked a system with the tetrahedral core in arbitrary orientation, because calculation is more demanding if assumption of the cubic symmetry is removed, we presume that the energy difference due to the orientation of the tetrahedral core is very small. The structure of Zn 17 Sc 3 is almost similar to that of the cubic Cd 6 M but has no tetrahedral core at the corner and the body-center of the cubic cell. To check differences caused by the tetrahedral core, it is more convenient and
Fig. 2. The total and partial (p- and d-) DOS for Cd 6 Mg with I23.
DOS, a reasonable amount of energetic stabilization is obtained for the cubic Cd 6 M compounds. In fact the cohesive energies at a fixed lattice constant are calculated for Cd 6 Yb as 1.68 eV/ atom. This is because the occupied states just below the dip are the bonding orbitals, which yield chemical stabilization due to alloying. On the other hand, the cohesive energy for Cd 6 Mg is 1.47 eV/ atom even at the optimal lattice parameter. This value is similar to that for pure Cd (1.48 eV/ atom for f.c.c. with a54.45 ˚ and smaller by |15% than those for Cd 6 Yb with the A) low-lying d band near the Fermi level. Therefore the hybridization of the d state with a wide sp band certainly contributes to stability of the Cd 6 M compounds. The diameter of the FS is calculated from the electron ˚ 21 , which is very close to the density as 2k F 52.75 A (222100) and (311111) Bragg scatterings for the icosahedral Cd–Yb. This seems to support the HumeRothery mechanism, in which the BZ–FS interaction induces the pseudogap near the Fermi energy. However, there is neither distinct dip (or pseudogap) in the DOS nor additional stabilization due to alloying for isostructural and isovalent Cd 6 Mg, which has a similar structure factor to the other Cd 6 M compounds. In other words, the pseudogap due to the Hume-Rothery mechanism is at most what we see in the DOS of Cd 6 Mg. Thus we conclude that the
] Fig. 3. The total and partial (p- and d-) DOS for Cd 6 Yb with Pn3.
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direct to compare systems with the same valence. The total and partial DOS for Cd 6 Y (I23) and Zn 17 Sc 3 are shown in Figs. 4 and 5. The pseudogap is observed again at the Fermi level in both of the systems. Splitting of the p- and d-symmetric components below and above the pseudogap is similar to that in the cubic Cd 6 Yb. Thus the sp-d hybridization is also important in stability of Zn 17 Sc 3 . Since Y and Sc are trivalent elements, the Fermi energy is shifted towards a minimum of the DOS and we obtain larger cohesive energies, 2.15 and 2.48 eV/ atom for Cd 6 Y and Zn 17 Sc 3 , respectively. The qualitative difference in the DOS is a small midgap peak just above the Fermi level for Cd 6 Y (marked by an arrow in Fig. 4). Such an unoccupied peak is also seen in the DOS for Cd 6 Yb and is more remarkable in the local DOS at the tetrahedral core sites in Cd 6 Yb [8]. One possibility concerning the stability may be that the midgap states are counterparts of the bonding states just below the Fermi level. Although we have not checked the bonding character of the midgap states thoroughly for the moment, we do not think the midgap states are anti-bonding because enhancement of the bonding states and sufficient cohesion are obtained for Zn 17 Sc 3 , for which the midgap peak is missing. Finally we estimate the formation energy of atomic
Fig. 5. The total and partial (p- and d-) DOS for Zn 17 Sc 3 .
vacancy at the cluster core. We calculate the electronic structures of hypothetical cubic Cd–Yb crystal without the tetrahedral core and estimate the vacancy formation energy from the total energy difference. The result is 0.45 eV/ atom, which is very small in comparison with the cohesive energy. This implies that the atoms in the tetrahedral core are weakly bound and do not play a significant role in stabilizing the compound. The picture of the weakly bound tetrahedral core may be consistent with experimental observations such as ambiguity in the atomic sites in the cluster core and large temperature effects [2,9].
4. Concluding remarks
Fig. 4. The total and partial (p- and d-) DOS for Cd 6 Y with I23.
We have studied the electronic structures of the cubic Cd 6 M approximant crystals of the newly discovered binary QC. It is found that stabilization due to alloying is obtained if M is an element with low-lying unoccupied d states. This leads to the conclusion that the cohesion of the Cd-based compounds is certainly due to the hybridization of the d states with a wide sp band. Stability of Zn 17 Sc 3 is also understood in the same context as the Cd-based alloys. It is reasonable to believe that the same mechanism works also in QC because the local atomic structure would be
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similar. The roles of the non-icosahedral core of the cluster have been investigated by checking the total energy difference due to orientation of the tetrahedral core and the electronic structure of the cubic Zn 17 Sc 3 , which has large vacancy instead of the non-icosahedral core at the center of the cluster. All of these checks indicate that the atoms in the non-icosahedral core are rather weakly bound and do not play an important role in stabilizing the cluster and its packing. Recently Tamura et al. [13] have reported that a linear coefficient of the specific heat varying with temperature is extremely large for the icosahedral Cd–Yb (2.87 mJ / mol K 2 ). According to the present calculation, the band contribution is |0.8 mJ / mol K 2 , which is a bit larger than the conventional QC but still reasonable. It is known that a two-level system leads to the specific heat varying linearly with temperature in glassy materials, where the temperature coefficient is typically of the order of 0.01 mJ / mol K 2 [14]. It is not evident if the weakly bound tetrahedral core behaves like the two-level system but we expect that the density of the tetrahedral core is much more than that of the two-level systems in glassy materials. It would be interesting to check the possibility that motion of the tetrahedral cores contributes to the low-temperature specific heat. In the present paper, we have only discussed the electronic mechanism for stability of Cd-based alloys. By calculating the electronic structures of the cubic Cd 6 M with other divalent elements M, we can show that similar electronic structures and cohesive energy are obtained for M=Ca and Sr [8]. In spite of similarities, however, only Cd–Yb and Cd–Ca form QC. For long-range quasiperiodic packing of the clusters, matching of the atomic size may be crucial [8].
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Acknowledgements The authors would like to thank A.P. Tsai for valuable discussions. This work is partly supported by Core Research in Environmental Science and Technology, Japan Science and Technology Corporation.
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