TIAN-SHI Liu and GUO-ZHU WEI: Magnon Modes in Quasi-Periodic Fcrromagnets 513

phys. stat. sol. (b) 174, 513 (1992)

Subject classification: 75.10 and 75.30

Department of Phj>sics, Northeust Univarsity of Technology, Shenyang ')

Magnon Modes in Quasi-Periodic Ferromagnets

TIAN-SHI LIU and GUO-ZHU WEI

Considering ferromagnetic exchange energies J , and J , between layers, which are arranged in quasi-periodic Fibonacci sequence, and the exchange energy J , in a layer, a rescaling approach to magnons in quasi-periodic superlattice ferromagnets is presented. An exact decimation transformation is derived for magnon modes of a- and P-layers in ferromagnets. Iteration of the transformation provides numerial results for the local density of states (LDOS) and the magnetization, when J , + J , = 25, fixed, 1" = J , / J , ( > 1). The band width of LDOS of layer cc is 16SJ,(2 + i)/(l + i), but the band width of LDOS of layer f3 is 24SJ,, the magnetizations of layers cc and p decrease as 1. is increasing for a given temperature, the magnetization oflayer p is always bigger than that oflayer cc.

Eine Resealing-Niherung fur Magnonen in Ferromagneten mit quasiperiodischem Ubergitter wird dargestellt unter Berucksichtigung der ferromagnetischen Austauschenergien J , und J , zwischen den in einer Fibonacci-Folge angeordneten Schichten und der Austauschencrgie J , in einer Schicht. Fur die Magnonenmoden der a- und 0-Schichten wird eine exakte Dezimierungstransformation abgeleitet. Durch Iteration der Transformation erhilt man numerische Werte fur die lokale Zustandsdichte und die Magnetisierung, wenn J , + J , = 2J, konstant ist, 1. = J , / J , > 1. Fur Schicht c( ist die Bandbreite der lokalen Zustandsdichte 16SJ0(2 + i) /( l + i), fur Schicht p 24SJ,; die Magnetisierung nimmt in beiden Schichten bei vorgegebener Temperatur ab, wenn 1. zunimmt, wobei die Magnetisierung der Schicht p immer grooer als die der Schicht cc ist.

1. Introduction

Recently various physical processes have been considered on quasi-crystals in Fibonacci sequences, including the electronic, phonon, and magnetic properties [ 1 to 61. Various methods, both analytic and numerical methods have been employed to find out the nature of the sequences. Among the analytic techniques the most important one is the real-space renormalization group (RSRG) scheme, first introduced by Kohomoto, Kadanoff, and Tang (KKT) [I], it has been used by several authors for studying the nature of electrons and phonons in such systems. The KKT scheme mainly exploits an exact recursion between transfer matrices whose traces exhibit a non-linear dynamical mapping for a 1D model described by a difference equation version of the Schrodinger equation. Ashraff and Stinchcombe [5] give another RSRG scheme, which maps the chain onto itself, thereby maintaining its global quasi-periodic character. This approach had been used in a two magnon 1D Heisenberg chain [7]. Chakrabarti et al. [S, 91 emphasized the importance of local symmetry, developed a rescaling approach to calculate the local density of states (LDOS) of electrons and phonons. For understanding the magnetic properties Xiong [lo] discussed the dispersion relation of a spin wave in a quasi-periodic superlattice. Pang and Pu [l 11 gave the energy structure and magnetic specific heat of a quasi-periodic superlattice by using the transfer matrix method. Other groups discussed ferromagnets of layered structure [12 to 151.

') 110006 Shcnyang, Pcople's Republic of China

514 TIAN-SHI LIU and GUO-ZHU WEI

In our previous work we presented an exact decimation treatment of the quasi-periodic superlattice in order to obtain the Green’s functions of the system, with the resulting rescaling transformation of dynamic variables, and examined the global density of states and the magnetization [ 161. For determining the contributions from different layers, in the present paper we calculate the LDOS and the magnetization of layers c1 and p by using the rescaling method.

2. The Model

We consider a simple cubic ferromagnetic layered superlattice with spin S. Each layer is a 2D square lattice with lattice constant a. The Hamiltonian can be written as

where i, j are the index numbers of layers, v, p are the index numbers of sites belonging to i( j) layer. Nearest-neighbour interations are considered only. In each layer the exchange interaction is J,. Between the layers close to each other, the exchange interaction can be J , or J , (5, < JJ, J , and J, are arranged according to a Fibonacci sequence. In the x-z plane the structure is shown in Fig. 1.

If we consider the difference between J , on the left and J , on the right and the opposite case, each of the atoms must be in any one of three different nearest-neighbour surroundings. We shall adopt the notation of the site energy in layer i, ci (i = a, p, y, cf. Fig. 2). The Green’s function

satisfies the equation of motion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x ..... ; 1 I I 1 ..... ..... f I .....

...... + .....

-..-. * I ..... ...... 1 I I .....

I -... I I I .....

...... .....

I I ! 1 1 ! ! ! ! . . . . . . . . . . . . . . . . . . . . . .

* . . . . . . . . . * . . . . . . . . .

Fig. 1. The quasi-periodic layered ferromagnetic structure in x-z plane

Magnon Modes in Quasi-Periodic Ferromagnefs 515

-.....- A-O-C-A+A~-C-~A-~---O--A--- Fig. 2. The Fibonacci sequence, whet.& o is a site in a-type layer, A a site in 0-type layer, a site in y-type layer

Because there is translation invariance in the x-y plane, we can perform the Fourier transformation

L3 GYf(w) = ; Fij(w, k ) exp (ik . (R , - R,)) , (4)

where N is the number of sites in the x-y plane, k = (k,,k,) and R = (R,,R,) are the wave vector and coordinate vector in the x-y plane, respectively. If using random phase approximation, taking any cc-layer as a reference layer, from (3) we obtained

. . . . . . . . . . . . . . . . . ,

where

F i j = F i j ( o , k ) ,

E , = 4J1s + 4JoS(2 - cos k,a - cos k , ~ ) ,

E P = E, = 2(51 + J z ) S + 4JoS(2 - cos k,a - cos k,a) ,

t1 = -2SJ1,

t, = -2SJz.

From (6) we can see that both layer p and layer y are the same because E~ = E.,.

3. Rescaling Method and Results

The matrix element Fij can be obtained from ( 5 ) using a rescaling or decimation transformation [7]. The decimation transformation corresponds to a reduction in the number of degrees of freedom and provides the accompanying transformation of parameters, and the original system can be related to a simpler one, or in the case of scale invariant systems, to a reduced version of itself. Such a method is able to deduce an exact transformation of E~ and tl(t2) under a length scaling factor b. When iterated, such a transformation yields a system with a fraction of the original number of degrees of freedom and with renormalized parameters E: and ti(&). In the present case the rescaling factor b = t = (1 + fi)/2. The decimation is achieved by removing appropriate sites following the rule I: AB -+ A , A + B' or rule 11: BA +. A', A -+ B. If i is one of the sites removed in the decimation having site energy ii = E,,, the renormalized bond joining site i - 1 to site i + 1 is t' and the renormalized energy of site i & 1 includes contributions cif and t:+ l/(o - E ~ ) . The transformation deci-

516 TIAN-SHI LIU and GUO-ZHU WEI

mations associated with a length rescaling factor b = z according rules 1 and 11 are given by

and

respectively. For obtaining the LDOS of each layer, we can use the various combinations of rules I and 11 to calculate the local Green’s functions. Because layer p is the same as layer y, in the present case we only need to calculate G,, and G,, by using the transform I-- I for layer CI and the transform 11-1 for layer p, respectively.

The LDOS e i (o) is related to the imaginary part of the local Green’s function G,,(w). By using (4),

where i = CI, p.

The local magnetization can be obtained by

where fl = l/k,T and M , is the magnetization at T = 0 K. When taking J , + J , = 2J,, 1. = J,/J, , we calculated the LDOS of layer CI and

p by the above decimation procedure. The results are shown in Fig. 3, where the solid (dashed) curves corresponds respectively, to the LDOS of layer ~ ( 0 ) and parts a), b), c), and d) correspond to i = 3,9,19,99, respectively. From Fig. 3 we can see that the deviations of LDOS in a quasi-periodic system from that in a 3D periodic system depend on various i-values. The band width of layer M. is a function of 2, it is represented as 16SJ,,(3. + 2)/ (1 + d), and decreases as increases. The band width of layer p is the same as the one in a 3D periodic system with exchange energy Jo. When i + m, the LDOS show the obvious character of a single layer for layer ‘x and the character of a double layer for layer p. respectively. It is easy to infer that as i is bigger, the contributions to the global density of states come mainly from the layers between which the exchange constant is J,, in agreement with the expectation in [16]. ’The curves of reduced magnetization M / M , versus k,,T/4SJo are shown in Fig.4. where the curves a, b, c, d correspond to i. = 3, 9, 19,

Magnon Modes in Quasi-Periodic Ferromagnets 517

1

I v) 0 0 -I

0

'I I b d

0 2 4 6 8 0 2 4 6 8 E / 4 S J ,

Fig. 3. The local density of states (LDOS). J, + J, = 2J0, i = J2/Jl. A = a) 3, b) 9, c) 19, d) 99; solid (dashed) curves represent LDOS in a(P) layer

and 99, respectively. We notice that for a given temperature 7; both magnetizations of layers M and p decrease as 1, increases, but the magnetization of layer 0 is always h g e r than that of layer a, because J , > J1. When measuring the average magnetization of the system, the contributions from the 0-like layers dominate in agreement with the results of [16].

4. Summary

We have studied a ferromagnet with layered structure, in each layer the exchange constant is J,. Both different exchange constants between nearest layers, J , and J,, are arranged according to a Fibonacci sequence. An exact decimation approach for obtaining the local Green's functions has been presented, the exact results for the LDOS of layers M and in the quasi-periodic superlattice are obtained. The numerical procedure based on real-space

TIAN-SHI LIU and GUO-ZHU WEI: Magnon Modes in Wasi-Periodic Ferromagnets

1

0 0 1

kBT/4SJo - Fig. 4. The magnetization M / M , vs. k,T, in a) @-layer and b) @-layer. 1 = (a) 3, (b) 9, (c) 19, (d) 99, respectively

rescaling methods provides a very direct way of obtaining information about the properties of the quasi-periodic superlattice spectrum. We have calculated the LDOS and the reduced magnetization of layers M and p for various 2 = J, /J , ( > l), when J , + J , = 25, fixed. We found the band width of LDOS of layer 01 as 16SJ0(2 + l)/(l + i), but the band width of the LDOS of layer p is 24SJ,, the magnetizations of layers 01 and p decrease as 1, increases for a given temperature, the magnetization of layer p is always bigger than that of layer a. The LDOS and the local magnetization of different layers are quite different for various A. The case A < 1 will be discussed elsewhere.

References

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(Received April 30, 1992)