Exact Green's functions of generalized Fibonacci lattices

J O U R N A L OF

Journal of Non-Crystalline Solids 153&154 (1993) 439-442 North-Holland

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Exact Green's functions of generalized Fibonacci lattices J.X. Z h o n g , J.R. Y a n a n d J.Q. Y o u Laboratory of Modern Physics, Institute of Science and Technology, Xiangtan Uniuersity, Xiangtan, Hunan 41 1105, China

We study the electronic properties of the generalized Fibonacci lattices generated by the inflation rule ( A , B ) ~ (AB", A) with n > 1. A unified real-space renormalization-group scheme is presented for calculating the exact local Green's function at any given site. Analysis suggests that there are extended states in these lattices. Furthermore, the appearance of the extended states and the smoothness of the local density of states depends on the system parameters, implying that these exists a state transition p h e n o m e n o n similar to that of the incommensurate Harper's model.

I. Introduction Since the remarkable discovery of quasicrystals and the successful fabrication of quasiperiodic superlattices, there has been a growing interest in the theoretical study of the electronic and phonon properties of the one-dimensional (1D) quasiperiodic (Fibonacci) lattice [1,2]. Generalized Fibonacci lattices have been widely studied in recent years [3-5]. In this article, we report recent studies [4] of the local Green's function (LGF) of the generalized Fibonacci lattice generated according to the sequence given by the inflation rule ( A , B ) --* ( A B n, A ) with n > 1, starting from B. These lattices and sequences are termed Bn lattices and Bn sequences, respectively.

2. Renormalization-group approach for LGF We consider the tight-binding model ti46- l + ti+ 10i + 1 + uiOi = EOi,

(1)

t i is the nearest-neighbor hopping integral which takes two values TA and T B arranged in the Bn sequence and u i is the site energy. where

Correspondence to: Dr J.X. Zhong, Laboratory of Modern Physics, Institute of Science and Technology, Xiangtan University, Xiangtan, H u n a n 411105, China.

Our real-space renormalization-group (RSRG) approach [4] for the L G F of the Bn lattice is based on the introduction of 2n + 1 basic renormalization transformations T t, T2 . . . . . T2,+~. These transformations are the decimations of the sites of the B n lattice. T r a n s f o r m a t i o n s Tt, T 2 . . . . . T,+ 1 are represented by ( B ' A n+~, B n A ) - ' * ( A ', B'), ( B n - ' A ' + I B , B n - ~ A B ) - . ( A ', B ' ) . . . . , ( A " +1B ", A B " ) --) ( A ', B ' ), respectively, while the representations of the transformations T n + 2, T n + 3 . . . . . T2n+l are [A n-I (ABn)n+IA, A n I(AB")A] ~ (A', B'), [A'-2(ABn)'+tA2, A n 2 ( A B n ) A 2 ] ~ ( A ' , B'), ... ,[(AB')n+~A ~, ( A B n ) A n] --->( A ' , B'), respectively. For an infinite Bn lattice, when we apply the 2n + 1 basic transformations to it individually, 2n + 1 new Bn lattices are obtained. We find that there is a special site called the key site of type St~ or S~. The feature of the key site of type St3 or S v is that, through transformation Tt or Tn+l, this site remains undecimated and the environment of it in the new lattice is the same as that in the original one. Because of this, the LGF of the key site St3 or S~ in an infinite Bn lattice can be exactly obtained by successive iterations of transformation T~ or T,+I. The 2n sites near the key site are transferred to the key sites of the new lattices by the other 2n transformations. Repeating the same procedure for each new lattice, we

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J.X. Zhong et al.

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/ Green'sfunctions of Fibonacci lattices

can finally transfer any other site to a key site and then calculate the corresponding LGF. Using the above R S R G approach, we have derived a unified R S R G equation for calculating the electronic LGFs of the key sites of the Bn lattice [4]. Numerical results [4,5] of the local density of states (LDOS) indicated that the LDOS of the Bn lattice has smooth parts in some energy regions, which exhibits a much different behavior form the LDOS of the Fibonacci lattice [2]. Using the Drichlet states of the molecules, Sire [4,6] derived some of the extended states corresponding to the smooth part of the LDOS. In the following section, we study the dynamic behavior of the extended states and the LDOS of the Bn lattice.

From eq. (2), one can obtain the wave function at arbitrary site i by successive applications of the transfer matrices. For the infinite Bn lattice, the products of M(v i, t i, ti+ 1) form an infinite matrix string. We find that this matrix string is composed of two e l e m e n t m a t r i c e s M 1 = M ( A B ) M(B)n-IM(BA) and M 2 = M(A) n. From theory of matrices the mth power of the 2 x 2 unimodular matrix M is given by

nm=~'m_,(x)n-~'m_2(x)I,

where x = Tr M is the trace of the matrix M and ~m(X) is the ruth Chebyshev polynomial of the second kind, sin mO ~'m-~(X) sin 0 ' (5) in which 0 = c o s - l ( x / 2 ) if ~'m_l(X) = 0, we have

3. E x t e n d e d

x t= cos--, m

We assume vi takes VA (if ti = ti+ a = TA, or ti = TB, ti+ 1 = TA) and VB (if t i = ti+ 1 = TB, or t i = TA, ti+ 1 = TB). This model can be thought of as the combined model in which the widely studied diagonal and off-diagonal models [3] correspond to (VA=t£VB, T A = T B) and (VA=VB, TA-~ TB), respectively. The matrix M(vi, ti, ti+ 1) is the transfer matrix \

ti,

I xl < 2. Form (5), if

states

Equation (1) can be rewritten as

M(vi,

(4)

ti+l)=

( ( E - vi) //ti+ l 1

-- ti/ti+ 1 0 ]'

(3) which takes one of the following four forms:

M( A) = ( ( E - VA)/TA 1

M(B)=((E-VB)/TB 1

M(AB)=((E-VA)/TA 1

M( BA) = ( ( E - VB)/TBI

-1) 0

'

-l) 0

'

1.

(6)

According to (5) and (6), if ~'n-1[( E -- VB)/TB] = 0 and ~'n_I[(E-VA)/TA]=O, we obtain two groups of energy lw E} 1) = VB + 2TB c o s - - , n

E}2) =

VA + 2TA

lw cos--,

(7)

n

in which l = 1, 2, 3 , . . . , n - 1. It is interesting that energies E} 1) a n d E} 2) make M l = ( - 1 ) t M ( A ) and M 2 = ( - 1)tI, respectively. Therefore the matrix string of the Bn lattice is only composed of M(A) for energy E} 1) and M 1 for E} 2), which is equivalent to the periodic case. For these periodic systems, there are exact extended states obeying ITr M(A)I < 2 and [Tr MI[ < 2, indicating that the states given by E} 1) and E (2) are the extended states of the Bn lattice. Furthermore, the appearance of these states depends on the following existence conditions:

I(E}I)--VA)/TAI<2, E}2)-VATA ~n ( E}2)-VB )-T;

--TB/TA) 0

l = 1, 2, 3 , . . . , m -

(8a)

T'~+T~n-ITA~

'

- T;/TB )"

× ~

TBB

< 2.

(8b)

J.X. Zhong et al. / Green's functions of Fibonacci lattices

One can also associate the nature of the extended states with the invariant of the trace map of the lattice [1,3,4,6]. For the generalized Fibonacci lattice we studied, there is an invariant J which depends on energy E [3]. Extended states correspond to the value J = 0 [1]. It is easy to check that energies given by (7) make the invariant ~.~ of the generalized Fibonacci lattice zero. This further indicates that those states are extended. In addition, since the DOS is given by the cos - ] of the trace, it is locally smooth. Furthermore, since the associated state is extended the LDOS behaves in the same way [4,6].

4. Numerical results and discussion

By examining the LDOSs of the Bn lattices [4,5], we find that E} 1) and E}2) are just those

441

which correspond to the smooth parts of the LDOSs and the appearance of the smooth parts depends on condition (8). For example, we give the LDOSs of the key site S~ of the B3 lattice on the diagonal model in fig. 1. The parameters are chosen to be ( V A = - V B = V, T A = T o = 1). The imaginary part added to the energy and the energy step are both chosen to be 0.001. From (8a), we obtain E (]) = - V + 1, if V < 3 / 2 , and E (]) = - V - 1, if V < 1/2. From (8b), we have E (2)= V + 1, if V < (v~ - 1)/2, or 1/2_< V < f 3 / 2 and E (2)=V-l, if V _ < 1 / 2 or v ~ - / 2 < V < ( v ~ + 1)/2. It is clear in figs. l(a)-(d) that, if V satisfies the conditions for the existence of E ~]) and E (2), there are the smooth parts in the LDOS around the energies E °) and E (z). If not, the LDOS does not exhibit any smooth behavior. Figure 1 show us two types of LDOS, in one of which the LDOS has smooth parts around the energies of the

(a)

(b)

UI

oQ

.2

0

.....

-2.4

I

I

]

-1.2

0

1.2

0 2.4

-3

I -1

-2

I 0

E

I 2

E 1

(c)

(d)

td~

o

°tl .2

-3

I -2

I -t

I 0

E

0

2

-4

I -2

I 0

I 2

E

Fig. 1. Local density of states at the key site of S~ (arbitrary units) of the B3 lattice with various values of site energies. (a) V = 0.1; (b) V = 0.7; (c) V = 1; (d) V = 2.

442

J.X. Zhong et aL / Green's functions of Fibonacci lattices

extended states and in the other the LDOS does not contain any smooth part. The appearance of the extended states and whether or not the LDOS exhibits the smooth behavior in some energy regions depend on the values of the lattice parameters, implying that there exist a state transition phenomenon of the extended states. For the incommensurate Harper's model, Aubry [7] showed that either all states are extended or all states are localized depending on the relative strengths of the potential and the hopping terms, i.e., the appearance of the extended state also depends on the lattice parameters. From this point, we think that the transition phenomenon of the extended states we obtained is similar to that of the Harper's model.

5. Conclusion

We have performed analytical and numerical studies for the Green's function and the dynamic behavior of the density of states of generalized Fibonacci lattice. The most important finding is that there is a state transition phenomenon similar to that of the incommensurate Harper's model. This work has been supported by the National Natural Science Foundation of China.

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