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Electronic energy spectrum of a decagonal covering quasicrystal
Physics Letters A 319 (2003) 395–400 www.elsevier.com/locate/pla
Electronic energy spectrum of a decagonal covering quasicrystal Xiujun Fu ∗ , Zhilin Hou, Youyan Liu Department of Physics, South China University of Technology, Guangzhou 510640, China Received 27 July 2003; received in revised form 10 October 2003; accepted 17 October 2003 Communicated by R. Wu
Abstract We study the structure of the electronic energy spectrum of a two-dimensional quasicrystal based on the cluster covering model. The energy levels are analyzed by the use of the perturbation theory in the framework of the tight-binding Hamiltonian. The integrated density of states is obtained by investigating the nearest neighbor configurations of the decagonal covering pattern. 2003 Elsevier B.V. All rights reserved. PACS: 71.23.Ft; 73.20.At; 61.44.Br Keywords: Quasicrystals; Energy spectrum; Cluster covering
1. Introduction Quasicrystals have been of interest for both physicists and mathematicians since the first experimental discovery of the diffraction pattern with fivefold symmetry in an Al–Mn alloy [1]. In order to understand the atomic structure of the quasicrystalline phase, many theoretical models have been proposed, of which the Penrose tiling [2] is considered to be the most suitable one. The Penrose pattern has been successfully employed to explain many physical properties of quasicrystals. To construct a Penrose tiling, it needs at least two distinct tiles and neighboring tiles must obey strict matching rules. This is a rather complicated proce* Corresponding author.
E-mail address: [email protected] (X. Fu). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.10.041
dure compared with the unit cell picture of the periodic crystals. Yet it has been found from the experimental data that there exist many overlapped decagonal clusters in the AlCuCo and AlNiCo quasicrystals [3] which disagree with the edge-to-edge touch of tiles in the Penrose matching scheme. In view of the above fact, Gummelt has proposed a cluster covering approach [4] to describe the twodimensional quasicrystals. In the covering scheme, the quasiperiodic structure is generated from a single decorated decagon similar to the unit cell of a crystal. The difference is that the decagons cover each other (overlap) according to certain covering rules. Since the cluster covering theory is an entirely new concept for quasicrystals, it has received much attention in recent years [5–15]. Previous work on covering model has mainly concentrated on the structure and formation of quasicrystals and very few have devoted to the physical proper-
396
X. Fu et al. / Physics Letters A 319 (2003) 395–400
Fig. 1. The Gummelt decagon and five covering types between two decagons.
ties. Although there is a correspondence between the Penrose tiling and the decagonal covering pattern, the atomic distribution in the two models can be very different, which will lead to different physical properties. In this Letter, we study the electronic energy spectrum of a two-dimensional quasicrystal model based on the cluster covering theory. In Section 2 we present the model first. A tight-binding Hamiltonian is constructed in the framework of a decagonal covering pattern. The atoms are divided into five sets according to the nearest neighbor configurations. In Section 3 the energy spectrum is analyzed by the use of the perturbation theory and is compared with the numerical result. In Section 4 the integrated density of states (IDOS) is obtained and expressed by the concentration of different atoms. Finally a summary is given in Section 5.
2. The model The decagonal covering structure was first proposed by Gummelt [4] who used a decorated decagon as the “quasi-unit-cell” (Fig. 1). The decagon is divided into white and gray regions. The overlap between any two decagons is required to obey the covering rules: the area of the overlapped part is greater than the smaller gray region and the color of both decagons in the overlapped region agrees. Thus there are five allowed covering types between two decagons, which are shown in Fig. 1, four of type A and one of type B. In a perfect decagonal covering structure all the decagons must be fully surrounded by its neighbors, which means that all the edges of the decagon are covered by the interior of other decagons. The resulting nearest neighbor configurations can be of nine cases [4,5,15], named as G1 to G9, which are shown in Fig. 2. The covering types around the central decagon are also indicated in the figure. The tight-binding Hamiltonian can be established on the decagonal covering pattern. Suppose each decagon stands for an atom (or a cluster of atoms) with
Fig. 2. The nine nearest neighbor configurations and the covering types around the central decagon.
site energy εi and the hopping integral between site i and site j is tij , then the Hamiltonian is expressed as εi |ii| + tij |ij |, H= (1) i
i,j
where |i is the Wannier state. In the present work, we assign the hopping integral tij = t for site i and site j being nearest neighbors and tij = 0 otherwise. The site energy is determined in the following way. It can be seen from Fig. 2 that the environment of a decagon can be one of the nine cases if the nearest neighbors (covering decagons) are considered only. We divide the decagons into five sets according to the number and types of their coverings. For configurations G1 and G2, the central decagons have four type A coverings and are regarded as set 1. For G3 and G5 there are also four coverings but three of type A and one of type B, so we put them into set 2. G4 and G6 are surrounded by four type A and one
X. Fu et al. / Physics Letters A 319 (2003) 395–400
Fig. 3. A decagonal covering pattern.
397
Fig. 4. A two-dimensional quasilattice with five types of atom developed from Fig. 3.
type B coverings and are defined as set 3. G7 and G8 belong to set 4 which have three type A and two type B coverings. The last set is G9 which is covered by six neighbors, four of type A and two of type B. According to the above definition, a decagonal covering pattern (Fig. 3) corresponds to a two-dimensional quasilattice with five types of atom (Fig. 4). The atoms are, respectively, labeled as 1, 2, 3, 4, and 5 in Fig. 4. The nearest neighbors are connected by a bond, which represents the nonzero hopping. The site energy can accordingly take five values, 1, 2, 3, 4, and 5 in arbitrary unit.
3. The energy spectrum Based on the tight-binding Hamiltonian (1) with nearest neighbor interactions, we have calculated the energy spectrum by numerical diagonalization method for the decagonal covering model. The studied system contains 3286 sites distributed in a circle area of radius 60 in the unit of the decagon edge length. The free boundary conditions are used. The results are shown in Fig. 5. It can be seen that when the hopping integral t = 0 the spectrum is merely the five “bare” atomic energy levels E = 1, 2, 3, 4, 5, respectively.
Fig. 5. The electronic energy spectrum of the decagonal covering quasilattice. The 12 main subbands are indicated for t = −0.1.
With the increase of the hopping integral t, the above degenerate energy levels will shift and split. The spectral structure can be studied by the perturbation approach, which has been successfully used in the one-dimensional and two-dimensional quasicrystals [16–19]. To perform a perturbation analysis, we divide the Hamiltonian (1) into two parts εi |ii|, H1 = t |ij |, H0 = (2) i
i,j
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X. Fu et al. / Physics Letters A 319 (2003) 395–400
where H0 is the unperturbed Hamiltonian and H1 is the perturbation term when |t| is small. In the absence of the perturbation term H1 , the eigen-energy is the site energy for each atom and the wave function is simply the Wannier state. When H1 is taken into account the degenerate perturbation theory is employed to calculate the first-order correction Ei(1) for energy Ei(0) , which is determined by the secular equation (1) deti|H1 |j − Ei δij = 0. (3) Now we start to study the energy level splitting rules using the degenerate perturbation method. Let us first treat the type 1 atoms. It can be seen from Fig. 4 that there exist two clusters of atom 1 under the nearest neighbor interaction, the single atom and five-atom cluster. In the former case the eigen-energy (1) keeps unchanged, so E1 = 0. In the latter case, the secular equation is t 0 0 t − E − E t 0 0 t (4) t − E t 0 = 0. 0 0 t − E t 0 t 0 0 t − E √ (1) Three eigen-values are obtained: E1 = 2t, ( 5 + √ 1)t/2, −( 5 + 1)t/2. Therefore, the energy level E = 1 will split into four subbands when t = 0. The nearest neighbors of the type 2 atoms constitute diatomic clusters only. It is easy to get the splitting levels E2(1) = ±t. So this energy level has two subbands. The atomic clusters of type 3 and 4 are the same as that of type 2 atoms and each level will split into two subbands in the first order correction. The type 5 atoms constitute a five-atom cluster only and all the five atoms in the cluster have interactions each other. The secular equation is t t t t − E − E t t t t (5) t − E t t = 0. t t t − E t t t t t t − E By solving Eq. (5) we obtain two subbands, the non-degenerate one E5(1) = 4t and the four-fold (1) degenerate one E5 = −t.
Fig. 6. The integrated density of states. The 12 steps occur at the gaps. Their height can be labeled by two numbers related to τ (see Table 1).
Summarizing the above results, we have totally 12 subbands under the first order perturbation approximation. It is confirmed by the numerical diagonalization result, which is shown in Fig. 5. The 12 main subbands are respectively noted as 1, 2, . . . , 12 from left to right in Fig. 5. It should be pointed out that because we use the free boundary conditions, some unexpected atomic clusters show up at the boundary and result in additional energy levels, which can be distinguished by the first order perturbation calculation. These levels are very few and have been removed in Fig. 5. Since only the first order correction is analytically performed, further splitting of the sublevels can be seen in the numerical result and the position of the energy levels shifts from the analytical results when |t| gets large.
4. The integrated density of states Now we turn to study the IDOS defined by I (E) = −∞ ρ(E ) dE , where ρ(E ) is the density of states. As the energy spectrum consists of 12 main subbands, staircases will occur at the gaps in the IDOS plot as shown in Fig. 6. In an infinite decagonal covering pattern the concentrations of the nine types of configurations can be expressed √ in terms of the powers of the golden mean τ = ( 5 − 1)/2 [15]. Let Qi (i = 1, 2, . . . , 9) be the concentration of the configuE
X. Fu et al. / Physics Letters A 319 (2003) 395–400
399
Table 1 Occupation probability of energy subbands and integrated density of states above the corresponding subbands Number
Energy
1 2 3 4 5 6 7 8 9 10 11 12
ε1 + 2t ε1 + τ t ε1 ε1 − t/τ ε2 + t ε2 − t ε3 + t ε3 − t ε4 + t ε4 − t ε5 + 4t ε5 − t
Occupation probability −0.8 + 1.4τ −1.6 + 2.8τ 2 − 3τ −1.6 + 2.8τ −3 + 5τ −3 + 5τ 5 − 8τ 5 − 8τ 5 − 8τ 5 − 8τ −2.2 + 3.6τ −8.8 + 14.4τ
ration Gi, we have Q1 = τ 2 ,
Q2 = τ 5 ,
Q3 = τ 5 ,
Q4 = τ 6 ,
Q5 = τ 5 ,
Q6 = τ 6 ,
Q7 = τ 6 ,
Q8 = τ 6 ,
Q9 = τ 5 + τ 7 .
(6)
The occurrence probabilities of the five types of atoms are obtained from (6)
IDOS −0.8 + 1.4τ −2.4 + 4.2τ −0.4 + 1.2τ −2 + 4τ −5 + 9τ −8 + 14τ −3 + 6τ 2 − 2τ 7 − 10τ 12 − 18τ 9.8 − 14.4τ 1
= 0.06525 = 0.19574 = 0.34164 = 0.47214 = 0.56231 = 0.65248 = 0.70820 = 0.76393 = 0.81966 = 0.87593 = 0.90031
by (m + nτ ), where m and n are integers. In the twodimensional Penrose tilings, it has been shown that the IDOS takes values in the frequency module of configurations [23], which is of the form (m + nτ )/10. In the present decagonal covering structure, we see that the IDOS equals to (m + nτ )/5. It is expected that if we perform higher-order perturbation calculation, the other values of IDOS can be observed.
P1 = Q1 + Q2 = τ 2 + τ 5 = −2 + 4τ, P2 = Q3 + Q5 = 2τ 5 = −6 + 10τ,
5. Summary
P3 = Q4 + Q6 = τ 4 + τ 5 = −1 + 2τ, P4 = Q7 + Q8 = 2τ 6 = 10 − 16τ, P5 = Q9 = τ + τ = −11 + 18τ. 5
7
(7)
In the above equations we have used the recursion relations [20] τ n = (−1)n (Fn−1 − Fn τ ),
n1
(8)
to simplify the expression. Here Fn is the Fibonacci number with F0 = 0, F1 = 1 and Fn = Fn−1 + Fn−2 for n 2. Referring to the degeneracy of energy levels in the first order perturbation approximation, the occupation probabilities of the 12 subbands can be obtained, and the stair height of the IDOS just above the corresponding levels is easy to get. The results are listed in Table 1. In the one-dimensional Fibonacci quasilattice, the energy spectrum has a Cantor-like set and the IDOS appears to have a Devil’s-staircase structure. This feature can be described by the gap-labeling rules [20– 22]. The IDOS corresponding to each gap is expressed
We have performed the analytical and numerical calculations for the structure of electronic energy spectrum in the decagonal covering model. We obtain the 12 main subbands by the use of the perturbation approach. From the concentrations of the different decagonal configurations and the degeneracy of the energy levels, the stair height of IDOS is derived and is expressed by two numbers related to the golden mean τ . The result explains the splitting of the degenerate energies and the band structural characteristics. Although it is only a model constructed on the cluster covering pattern, this work should improve our understanding to the physical properties of quasicrystals based on the cluster covering theory.
Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No.
400
X. Fu et al. / Physics Letters A 319 (2003) 395–400
90103027 and Guangdong Provincial Natural Science Foundation of China under Grant No. 013009.
References [1] D. Schechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [2] R. Penrose, Bull. Inst. Math. Appl. 10 (1974) 266. [3] S.E. Burkov, Phys. Rev. Lett. 67 (1991) 614, and references therein. [4] P. Gummelt, Geom. Dedicata 62 (1996) 1. [5] H.-C. Jeong, P.J. Steinhardt, Phys. Rev. B 55 (1997) 3520. [6] J.S. Wu, X.Z. Li, K.H. Kuo, Philos. Mag. Lett. 77 (1998) 359. [7] Y. Yan, S.J. Pennycook, A.-P. Tsai, Phys. Rev. Lett. 81 (1998) 5145. [8] S.I. Ben-Abraham, F. Gähler, Phys. Rev. B 60 (1999) 860. [9] P. Kramer, J. Phys. A: Math. Gen. 32 (1999) 5781.
[10] E. Cockayne, Mater. Sci. Eng. A 294–296 (2000) 224. [11] P. Gummelt, C. Bandt, Mater. Sci. Eng. A 294–296 (2000) 250. [12] E. Abe, K. Saitoh, H. Takakura, A.P. Tsai, P.J. Steinhardt, H.-C. Jeong, Phys. Rev. Lett. 84 (2000) 4609. [13] E.A. Lord, S. Ranganathan, U.D. Kulkarni, Philos. Mag. A 81 (2001) 2645. [14] F. Gähler, M. Reichert, J. Alloys Compd. 342 (2002) 180. [15] X. Fu, Z. Hou, Y. Liu, Solid State Commun. 128 (2003) 75. [16] P. Ma, Y. Liu, Phys. Rev. B 39 (1989) 10658. [17] D. Barache, J.M. Luck, Phys. Rev. B 49 (1994) 15004. [18] P. Tong, Phys. Rev. B 53 (1996) 1795. [19] P. Tong, Phys. Rev. B 54 (1996) 16540. [20] Y. Liu, X. Fu, W. Deng, S. Wang, Phys. Rev. B 46 (1992) 9216. [21] J.M. Luck, D. Petritis, J. Stat. Phys. 42 (1986) 289. [22] J.A. Ashraff, J.M. Luck, R.B. Stinchcombe, Phys. Rev. B 41 (1990) 4314. [23] J. Bellissard, E. Contensou, A. Legrand, C. R. Acad. Sci. I— Math. 326 (1998) 197.
Electronic energy spectrum of a decagonal covering quasicrystal Xiujun Fu ∗ , Zhilin Hou, Youyan Liu Department of Physics, South China University of Technology, Guangzhou 510640, China Received 27 July 2003; received in revised form 10 October 2003; accepted 17 October 2003 Communicated by R. Wu
Abstract We study the structure of the electronic energy spectrum of a two-dimensional quasicrystal based on the cluster covering model. The energy levels are analyzed by the use of the perturbation theory in the framework of the tight-binding Hamiltonian. The integrated density of states is obtained by investigating the nearest neighbor configurations of the decagonal covering pattern. 2003 Elsevier B.V. All rights reserved. PACS: 71.23.Ft; 73.20.At; 61.44.Br Keywords: Quasicrystals; Energy spectrum; Cluster covering
1. Introduction Quasicrystals have been of interest for both physicists and mathematicians since the first experimental discovery of the diffraction pattern with fivefold symmetry in an Al–Mn alloy [1]. In order to understand the atomic structure of the quasicrystalline phase, many theoretical models have been proposed, of which the Penrose tiling [2] is considered to be the most suitable one. The Penrose pattern has been successfully employed to explain many physical properties of quasicrystals. To construct a Penrose tiling, it needs at least two distinct tiles and neighboring tiles must obey strict matching rules. This is a rather complicated proce* Corresponding author.
E-mail address: [email protected] (X. Fu). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.10.041
dure compared with the unit cell picture of the periodic crystals. Yet it has been found from the experimental data that there exist many overlapped decagonal clusters in the AlCuCo and AlNiCo quasicrystals [3] which disagree with the edge-to-edge touch of tiles in the Penrose matching scheme. In view of the above fact, Gummelt has proposed a cluster covering approach [4] to describe the twodimensional quasicrystals. In the covering scheme, the quasiperiodic structure is generated from a single decorated decagon similar to the unit cell of a crystal. The difference is that the decagons cover each other (overlap) according to certain covering rules. Since the cluster covering theory is an entirely new concept for quasicrystals, it has received much attention in recent years [5–15]. Previous work on covering model has mainly concentrated on the structure and formation of quasicrystals and very few have devoted to the physical proper-
396
X. Fu et al. / Physics Letters A 319 (2003) 395–400
Fig. 1. The Gummelt decagon and five covering types between two decagons.
ties. Although there is a correspondence between the Penrose tiling and the decagonal covering pattern, the atomic distribution in the two models can be very different, which will lead to different physical properties. In this Letter, we study the electronic energy spectrum of a two-dimensional quasicrystal model based on the cluster covering theory. In Section 2 we present the model first. A tight-binding Hamiltonian is constructed in the framework of a decagonal covering pattern. The atoms are divided into five sets according to the nearest neighbor configurations. In Section 3 the energy spectrum is analyzed by the use of the perturbation theory and is compared with the numerical result. In Section 4 the integrated density of states (IDOS) is obtained and expressed by the concentration of different atoms. Finally a summary is given in Section 5.
2. The model The decagonal covering structure was first proposed by Gummelt [4] who used a decorated decagon as the “quasi-unit-cell” (Fig. 1). The decagon is divided into white and gray regions. The overlap between any two decagons is required to obey the covering rules: the area of the overlapped part is greater than the smaller gray region and the color of both decagons in the overlapped region agrees. Thus there are five allowed covering types between two decagons, which are shown in Fig. 1, four of type A and one of type B. In a perfect decagonal covering structure all the decagons must be fully surrounded by its neighbors, which means that all the edges of the decagon are covered by the interior of other decagons. The resulting nearest neighbor configurations can be of nine cases [4,5,15], named as G1 to G9, which are shown in Fig. 2. The covering types around the central decagon are also indicated in the figure. The tight-binding Hamiltonian can be established on the decagonal covering pattern. Suppose each decagon stands for an atom (or a cluster of atoms) with
Fig. 2. The nine nearest neighbor configurations and the covering types around the central decagon.
site energy εi and the hopping integral between site i and site j is tij , then the Hamiltonian is expressed as εi |ii| + tij |ij |, H= (1) i
i,j
where |i is the Wannier state. In the present work, we assign the hopping integral tij = t for site i and site j being nearest neighbors and tij = 0 otherwise. The site energy is determined in the following way. It can be seen from Fig. 2 that the environment of a decagon can be one of the nine cases if the nearest neighbors (covering decagons) are considered only. We divide the decagons into five sets according to the number and types of their coverings. For configurations G1 and G2, the central decagons have four type A coverings and are regarded as set 1. For G3 and G5 there are also four coverings but three of type A and one of type B, so we put them into set 2. G4 and G6 are surrounded by four type A and one
X. Fu et al. / Physics Letters A 319 (2003) 395–400
Fig. 3. A decagonal covering pattern.
397
Fig. 4. A two-dimensional quasilattice with five types of atom developed from Fig. 3.
type B coverings and are defined as set 3. G7 and G8 belong to set 4 which have three type A and two type B coverings. The last set is G9 which is covered by six neighbors, four of type A and two of type B. According to the above definition, a decagonal covering pattern (Fig. 3) corresponds to a two-dimensional quasilattice with five types of atom (Fig. 4). The atoms are, respectively, labeled as 1, 2, 3, 4, and 5 in Fig. 4. The nearest neighbors are connected by a bond, which represents the nonzero hopping. The site energy can accordingly take five values, 1, 2, 3, 4, and 5 in arbitrary unit.
3. The energy spectrum Based on the tight-binding Hamiltonian (1) with nearest neighbor interactions, we have calculated the energy spectrum by numerical diagonalization method for the decagonal covering model. The studied system contains 3286 sites distributed in a circle area of radius 60 in the unit of the decagon edge length. The free boundary conditions are used. The results are shown in Fig. 5. It can be seen that when the hopping integral t = 0 the spectrum is merely the five “bare” atomic energy levels E = 1, 2, 3, 4, 5, respectively.
Fig. 5. The electronic energy spectrum of the decagonal covering quasilattice. The 12 main subbands are indicated for t = −0.1.
With the increase of the hopping integral t, the above degenerate energy levels will shift and split. The spectral structure can be studied by the perturbation approach, which has been successfully used in the one-dimensional and two-dimensional quasicrystals [16–19]. To perform a perturbation analysis, we divide the Hamiltonian (1) into two parts εi |ii|, H1 = t |ij |, H0 = (2) i
i,j
398
X. Fu et al. / Physics Letters A 319 (2003) 395–400
where H0 is the unperturbed Hamiltonian and H1 is the perturbation term when |t| is small. In the absence of the perturbation term H1 , the eigen-energy is the site energy for each atom and the wave function is simply the Wannier state. When H1 is taken into account the degenerate perturbation theory is employed to calculate the first-order correction Ei(1) for energy Ei(0) , which is determined by the secular equation (1) deti|H1 |j − Ei δij = 0. (3) Now we start to study the energy level splitting rules using the degenerate perturbation method. Let us first treat the type 1 atoms. It can be seen from Fig. 4 that there exist two clusters of atom 1 under the nearest neighbor interaction, the single atom and five-atom cluster. In the former case the eigen-energy (1) keeps unchanged, so E1 = 0. In the latter case, the secular equation is t 0 0 t − E − E t 0 0 t (4) t − E t 0 = 0. 0 0 t − E t 0 t 0 0 t − E √ (1) Three eigen-values are obtained: E1 = 2t, ( 5 + √ 1)t/2, −( 5 + 1)t/2. Therefore, the energy level E = 1 will split into four subbands when t = 0. The nearest neighbors of the type 2 atoms constitute diatomic clusters only. It is easy to get the splitting levels E2(1) = ±t. So this energy level has two subbands. The atomic clusters of type 3 and 4 are the same as that of type 2 atoms and each level will split into two subbands in the first order correction. The type 5 atoms constitute a five-atom cluster only and all the five atoms in the cluster have interactions each other. The secular equation is t t t t − E − E t t t t (5) t − E t t = 0. t t t − E t t t t t t − E By solving Eq. (5) we obtain two subbands, the non-degenerate one E5(1) = 4t and the four-fold (1) degenerate one E5 = −t.
Fig. 6. The integrated density of states. The 12 steps occur at the gaps. Their height can be labeled by two numbers related to τ (see Table 1).
Summarizing the above results, we have totally 12 subbands under the first order perturbation approximation. It is confirmed by the numerical diagonalization result, which is shown in Fig. 5. The 12 main subbands are respectively noted as 1, 2, . . . , 12 from left to right in Fig. 5. It should be pointed out that because we use the free boundary conditions, some unexpected atomic clusters show up at the boundary and result in additional energy levels, which can be distinguished by the first order perturbation calculation. These levels are very few and have been removed in Fig. 5. Since only the first order correction is analytically performed, further splitting of the sublevels can be seen in the numerical result and the position of the energy levels shifts from the analytical results when |t| gets large.
4. The integrated density of states Now we turn to study the IDOS defined by I (E) = −∞ ρ(E ) dE , where ρ(E ) is the density of states. As the energy spectrum consists of 12 main subbands, staircases will occur at the gaps in the IDOS plot as shown in Fig. 6. In an infinite decagonal covering pattern the concentrations of the nine types of configurations can be expressed √ in terms of the powers of the golden mean τ = ( 5 − 1)/2 [15]. Let Qi (i = 1, 2, . . . , 9) be the concentration of the configuE
X. Fu et al. / Physics Letters A 319 (2003) 395–400
399
Table 1 Occupation probability of energy subbands and integrated density of states above the corresponding subbands Number
Energy
1 2 3 4 5 6 7 8 9 10 11 12
ε1 + 2t ε1 + τ t ε1 ε1 − t/τ ε2 + t ε2 − t ε3 + t ε3 − t ε4 + t ε4 − t ε5 + 4t ε5 − t
Occupation probability −0.8 + 1.4τ −1.6 + 2.8τ 2 − 3τ −1.6 + 2.8τ −3 + 5τ −3 + 5τ 5 − 8τ 5 − 8τ 5 − 8τ 5 − 8τ −2.2 + 3.6τ −8.8 + 14.4τ
ration Gi, we have Q1 = τ 2 ,
Q2 = τ 5 ,
Q3 = τ 5 ,
Q4 = τ 6 ,
Q5 = τ 5 ,
Q6 = τ 6 ,
Q7 = τ 6 ,
Q8 = τ 6 ,
Q9 = τ 5 + τ 7 .
(6)
The occurrence probabilities of the five types of atoms are obtained from (6)
IDOS −0.8 + 1.4τ −2.4 + 4.2τ −0.4 + 1.2τ −2 + 4τ −5 + 9τ −8 + 14τ −3 + 6τ 2 − 2τ 7 − 10τ 12 − 18τ 9.8 − 14.4τ 1
= 0.06525 = 0.19574 = 0.34164 = 0.47214 = 0.56231 = 0.65248 = 0.70820 = 0.76393 = 0.81966 = 0.87593 = 0.90031
by (m + nτ ), where m and n are integers. In the twodimensional Penrose tilings, it has been shown that the IDOS takes values in the frequency module of configurations [23], which is of the form (m + nτ )/10. In the present decagonal covering structure, we see that the IDOS equals to (m + nτ )/5. It is expected that if we perform higher-order perturbation calculation, the other values of IDOS can be observed.
P1 = Q1 + Q2 = τ 2 + τ 5 = −2 + 4τ, P2 = Q3 + Q5 = 2τ 5 = −6 + 10τ,
5. Summary
P3 = Q4 + Q6 = τ 4 + τ 5 = −1 + 2τ, P4 = Q7 + Q8 = 2τ 6 = 10 − 16τ, P5 = Q9 = τ + τ = −11 + 18τ. 5
7
(7)
In the above equations we have used the recursion relations [20] τ n = (−1)n (Fn−1 − Fn τ ),
n1
(8)
to simplify the expression. Here Fn is the Fibonacci number with F0 = 0, F1 = 1 and Fn = Fn−1 + Fn−2 for n 2. Referring to the degeneracy of energy levels in the first order perturbation approximation, the occupation probabilities of the 12 subbands can be obtained, and the stair height of the IDOS just above the corresponding levels is easy to get. The results are listed in Table 1. In the one-dimensional Fibonacci quasilattice, the energy spectrum has a Cantor-like set and the IDOS appears to have a Devil’s-staircase structure. This feature can be described by the gap-labeling rules [20– 22]. The IDOS corresponding to each gap is expressed
We have performed the analytical and numerical calculations for the structure of electronic energy spectrum in the decagonal covering model. We obtain the 12 main subbands by the use of the perturbation approach. From the concentrations of the different decagonal configurations and the degeneracy of the energy levels, the stair height of IDOS is derived and is expressed by two numbers related to the golden mean τ . The result explains the splitting of the degenerate energies and the band structural characteristics. Although it is only a model constructed on the cluster covering pattern, this work should improve our understanding to the physical properties of quasicrystals based on the cluster covering theory.
Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No.
400
X. Fu et al. / Physics Letters A 319 (2003) 395–400
90103027 and Guangdong Provincial Natural Science Foundation of China under Grant No. 013009.
References [1] D. Schechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [2] R. Penrose, Bull. Inst. Math. Appl. 10 (1974) 266. [3] S.E. Burkov, Phys. Rev. Lett. 67 (1991) 614, and references therein. [4] P. Gummelt, Geom. Dedicata 62 (1996) 1. [5] H.-C. Jeong, P.J. Steinhardt, Phys. Rev. B 55 (1997) 3520. [6] J.S. Wu, X.Z. Li, K.H. Kuo, Philos. Mag. Lett. 77 (1998) 359. [7] Y. Yan, S.J. Pennycook, A.-P. Tsai, Phys. Rev. Lett. 81 (1998) 5145. [8] S.I. Ben-Abraham, F. Gähler, Phys. Rev. B 60 (1999) 860. [9] P. Kramer, J. Phys. A: Math. Gen. 32 (1999) 5781.
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