J. ELECHTA: On SCCF Theory of Magnetic Properties of 1-D Binary Alloys 151

phys. stat. sol. (b) 133, 151 (1986)

Subject classification: 75.10

Department of Physics, University of Leedsl)

On SCCF Theory of Magnetic Properties of 1 -D Binary Alloys (Magnon Spectrum, Magnetization)z)

BY J. ~ ~ K ~ I T A

A detailed SCCF calculation of the magnon densities of states of 1-D binary alloys, both with statis- tical short range order and purely random ones, is presented and the results are compared with the electronic densities of states of those alloys. Also the demonstration of the higher spatial correla- tions (HSC) in the shapes of the magnon densities of states is discussed. It is shown that for some constitutions of 1-D binary alloys the long range magnetic order is likely to exist and results for the temperature dependence of the magnetization of those binary alloys is also given.

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1. Introduction

The SCCF method has been shown to be the best method of calculation of the density of states of disordered materials in existence [l]. It has been applied successfully to the transport properties of those materials [2, 31. The core of that method is same for all the types of elementary excitations of the solid in question [4] and that provides a unique common ground for a comparison between their properties.

In this work the method is applied to investigation of the magnetic properties of 1-D binary alloys studied recently also by different authors [5]. At first there are presented the shapes of the densities of states of magnons.

2. Theory

The equation of motion of the magnon component of the elementary excitations of magnetic materials can be written in the form [6, 71

neigh.

Here cu is the magnon frequency, CT is the magnetization per a spin, Jij is the magnetic exchange interaction, Gij is the Green function of the problem, i, j are the indeces of the locations of those two spins in focus. The equation of motion (1) can be easily

1) 18, Lidgett Hill, Leeds 8, LS 8 lPE, Great Britain. 2) Preliminary presented a t the 21th Annual Solid State Physics conference of the Institute of

Physics held in Southampton in December 1984.

152 J. SLECHTA

modified into the form suitable for the application of the SCCF method [a, 81. Namely it can be written into the form

c KijGjdE) = 4 z , (2 ) j

where Kii = Et = E - C r J i k

near. neigh.

and K , =nJ, for i $; j

2nw E = - with

fs and

(3)

The Green function Gij(E) was calculated by the SCCF method 141.

3. Numerical Results and Discussion

For the sake of continuity of this work with the previous works on application of the SCCF method to 1-D binary alloys [2,4, 8 to 101 the results presented here were obtained for a similar model of 1-D binary alloys and values of the parameters involved as in the preceding works mentioned. Namely: 01 = 1 - the strength of the A-A magnetic exchange interaction, B = 5 - the strength of the A-B magnetic interaction and y = 2 - the strength of the B-B magnetic interaction. The only larger difference between the magnetic and electronic cases is in the form of the diagonal term Kii in (2) given by (3) which plays here the role of the “atomic” energies in the electronic case 141. That difference proves responsible for the special features of the magnon densities of states in relation to the electronic ones.

The densities of states of magnons G(E) acquired by SCCF method for selected values of the parameters of relative concentration x and the sticking parameter E

[ 2 to 41 are given in Fig. 1 a to c. At the first glance they are in various aspects notice- ably different to the electronic densities of states of similar binary alloys 14, 9, lo]. Kamely :

(i) they show less clearly defined separate bands of atoms A and B, relative to the electronic ones; due to the term (3) in (2). Because of: that term we have here three diagonal “pseudoatom” energies, El = a/2 = 0.5, E, = f3/2 = 2.5, and E3 = y / 2 = 1. These formally play the role of the centres of three “bands”. However none of those “pseudoatomic” energies plays its role in any of all the representative M-testing clusters [3,4], consistently. Nor there can be ascribed to i t clear values of the off- diagonal elements which define the width of the relevant bands. That is one of the largest differences between the magnon and electronic cases.

(ii) They show only one dominant peak within the interval (O. l ) , in this case. It actually consists of two close peaks; a t E = 0.1 and E = 0.3. Off those two peaks the one a t E = 0.3 is higher for x = 0.1 ; the peak a t E = 0.1 is higher for x = 0.9. This existence of the main peak of the density of states of magnons a t E x 0 agrees with the findings of different authors [7, 111. A second side peaks develops only in the case of the purely random alloys; a t E = 2.3. It is most intensive for x = 0.1.

(iii) They are much more slowly convergent towards the crystalline densities of states of magnons that is for x ---* 0, or x + 1 (100% of atoms A). In the electronic case,

On SCCF Theory of Magnetic Properties of 1-D Binary Alloys 153

g:I1IJ(i--i x=o9

2

0 7L=fP=-4 2 7 !3A 0

7

Q.! 0 7 2 3

r , I 1 I 1 x= 09 v = + 4

i

t I n i -7 0 7 2 3

E170~zeVl - Fig. 1. The concentration dependence of the shape of the magnon densities of states for a ) E = 0.1, separative case; b) E = 0.5, the separative and alternative cases are equal here; and c) E = 0.1, alternative case. x is the relative concentration parameter 0 5 x 5 1

unlike for magnons, strong van Hove singularities [12] showed themselves already for x = 0.1 and x = 0.9 [4,9]. Here they are nonexistent, yet.

(iv) They change less profoundly with the changes of the values of the parameters 5 and E which control the higher spatial correlations (HSC) [3].

Because of the fast convergence of SCCF [4, 9, 101 the results for the shapes of the magnon densities of states of the 1-D binary alloys presented above are close to the true shapes of those densities in real (pseudo) I-D binary alloys and serve for a set of reference densities of states of magnons for that type of alloys. By comparison the experimentally acquired magnon densities of states with a spanning set of computed ones, for a set of parameters x and E, one can determine those parameters in the given material and thus HSC, especially when simultaneously supplemented by different studies of such a kind [ Z , 3, 13 to 151.

T a b l e 1 The value of the basic characteristics of the I-D purely random binary alloys [14, 171

2 Jay T:: TE/Jav f T M a

0.1 1.57 3.14 2 0.12 0.038 0.3 2.45 4.89 1.97 0.15 0.031 0.5 2.91 5.81 1.99 0.13 0.025 0.7 3.08 6.14 1.99 0.12 0.02 0.9 2.4 4.7 1.95 0.08 0.017

J,, average exchange interaction; T,, average values of the local Curie temperatures T:f f~~ relative fluctuations of T:f; a magnetic enhance factor.

1-54 J. SLECHTA Table 2 The values of the basic characteristics of 1-D binary alloys with statistical short range order [ 14, 171

0.1 alt. 4.02 9.15 2.27 0.17 0.018 0.3 alt. 4.03 7.97 1.97 0.21 0.028 0.5 2.23 4.42 1.98 0.28 0.063 0.3 sep. 2.23 4.42 1.98 0.19 0.042 0.1 sep. 1.84 3.65 1.98 0.18 0.049

Jav average exchange interaction; Tav average values of the local Curie temperature TF; relative fluctuations of T F ; a magnetic enhance factor; and for x = 0.5 (50%

concentration of both the types of the atoms). For different concentrations the values of those parameters can be obtained approximately from those given here by a simple scaling.

fTbl

We shall progress further by evaluation of the possibility of the existence of the long range magnetic order (LRMO) in the alloys studied here. According to the Stoner criterium [16] such LRMO can exist only when there is fulfilled the inequality

where V is the value of the (average) exchange interaction, and E, is the Fermi level. Let us give here, first, in Table 1 and 2, the values of the basic characteristics of mag- netic alloys [14, 171. By utilizing the results for the values of E, given in [2] and the values of G(E,) which can be extracted from [4], there are given, in Table 3, the values of l/@(E,) for the three types of compositions of the binary alloys considered in [2], namely: (i) M-M - both atoms A and B are metallic, (ii) M-I - atoms A are meballic while atoms B are those of an insulator, (iii) I-M when atoms A are of an insulator and atoms B are metallic. By the combination of the results in Tables 1, 2 and 3 there can be obtained the information about the cases when there is possible LRMO in 1-D binary alloys. These are marked by "Y", in Table 3. The opposite cases are marked by "N" there. From that there can be seen that the proofs of lack of LRMO in I-D magnetic systems given in [lS, 191 are not strictly conclusive for the disordered materials for within them there had been used: a) the concept of k-space not valid for disordered matter ([4], appendix 2) ; b) Born-Karman conditions, also not very suitable, although less principly, for the non-crystalline state.

Because of the possibility of the existence of LRMO in 1-D binary alloys there is meaningful1 to investigate also the temperature dependence of the magnetization per a spin o(T) on the bases of the modified form of the formula in [6, 71

v > l/G(EF) 9 ( 7 )

m

Here B = l/k,T, where k, is the Boltzmann constant ,(E) = Im C"(E). From Table 1 and 2 there can be seen that the values of fThZ differ little for all the types of the alloys considered and so the shapes of o(T) can be expected to be nearly the same for all of them, as a consequence of [17]. That really was confirmed by computation. In Fig. 2 there is given a sample o(T) for those alloys studied here. There, similarily to the transport properties of 1-D binary alloys in [2, 31, holds the scaling properties of the temperature and energy scale. Namely when the energy unit in Fig. 1 a to c is

On SCCF Theory of Magnetic Properties of 1-D Binary Alloys 155

Table 3 The concentration dependence of the values of 1 /5(E~) for the three types of the composition of the 1-D binary alloys (see the text) for various values of E

X M-M M-I I-M

a) E = 0.1, separative case 0.1 6.25N 100 N 0.3 2.7 N 50 N 0.5 3.03 N 14.3 N 0.7 3.03 Y 14.3 N 0.9 1.67 Y 1.9 N

b) E = 0.3 separative case 0.1 3.03 N 4.55 N 0.3 6.25 N 33.33 N 0.5 12.50 N 12.5 N 0.7 3.85 Y 5.26 N 0.9 11.11 N 2.86 Y

C) E = 0.5 0.1 5.56 N 16.67 N 0.3 3.23 N 33.33 N 0.5 7.69 N 12.5 N 0.7 12.5 N 16.67 Pi 0.9 6.25 N 11.11 N

d) E = 0.3 alternative case 0.1 8.33 N 5.26 N 0.3 3.85 N 25 N 0.5 25 N 14.29 N 0.7 5 N 25 N 0.9 2.78 Y 4 Y

e) E = 0.1, alternative case 0.1 9.09 N 20 N 0.3 5.26 N 25 N 0.5 11.76 N 11.76 N 0.7 12.5 N 11.76 N 0.9 4.35 N 8.33 N

f ) purely random case 0.1 5.55N 100 N 0.3 5.55 N 16.6 ?i 0.5 6.G7 N 20 x 0.7 10 N 3.7 N 0.9 11.11 N 6.25 N

12.50 N 7.14 N

12.5 N 33.33 N 50 N

3.23 N 33.33 N 14.29 N 12.5 N 16.67 N

12.5 N 10 N 11.11 N 33.33 N 50 N

11.11 N 12.5 N 14.29 N 33.33 N 11.11 N

9.09 N 10.53 N 11.11 K 16.67 N 50 N

10 N 11.11 N 33.3 N 33.3 x 9.09 N

In the table there is denoted the case when the long range magnetic order is possible, see (7), by the mark “Y” there, the opposite case by the mark “N”.

eV the temperature unit in Fig. 2 is 100 K. Due to that the temperature range in Fig.2 can be adjusted to any given value of the average interaction Jav. We add still that the results for o(T) presented above can be interpreted as to be constructed within the basis of the vectors yi [17] the directions of which are determined to satisfy the balance of the local interactions of every spin, for the absolute temperature T = 0, for the given disorder. Within such an interpretation a(T) presented here is valid for any

156 J. SLECHTA: On SCCF Theory of Magnetic Properties of I-D Binary Alloys

b

Fig. 2. The representative example of the tem- perature dependence of the magnetization of a spin of the alloys studied in this work t7-I

kind of LRMO of 1-D binary alloys, like ferromagnetic, or antiferromagnetic, or a inore general one [17], with the same value of fTY as in the cases above.

Acknowledgements

The results were computed on the author’s Micro Sinclair ZX Spectrum with 48KRAM. He would like to thank the Benevolent Fund of the Institute of Physics, and Dr. L. Cohen, the executive secretary of the Institute of Physics, for their financial and nioral support of himself; and the Brotherton Library, especially Mr. D. Cox, its librarian, for their hospitality.

References

[1] J. ~ L E C H T A , phys. stat. sol. (b) 120, 329 (1983). [2] J. SLECHTA, J. Phys. C 12, 1819 (1979). 131 J. SLECHTA, phys. stat. sol. (b) 127, 403 (1985). [4] J. SLECHTA, J. Phys. C 10, 2047 (1977). 151 D. LJ. MIRJANI~, Z.BUNDALO, B. S. ToBrC, and J. P. SETRAJEIC, phys. stat. sol. (b) 128, 151

[6] S. V. TYABLIKOV, Methods inQuantum Theory of Magnetism, Plenum Press, New York 1967. [7] C. C.&~ONTGOMERY, J. I. KRUGELS, and R. &I. STUBBS, Phys. Rev. Letters 25, 669 (1970). [S] J. SLECHTA, J. Phys. F 4, 1148 (1974). [9] J. SLECHTA, phys. stat. sol. (b) 71, K165 (1975).

[ lo] J. ~ L E C H T A , phys. stat. sol. (b) 104, K143 (1981). 1111 F. GOEDSCHE, phys. stat. sol. (b) 44, 191 (1971). [12] L.vaa HOVE, Phys. Rev. 89, 1189 (1953). [13) J. @.ECHTA, 5. non-crystall. Solids 22, 345 (1976). [14] J. SLECHTA, J. Magnetism magnetic Mater. 22, 113 (1980). [I51 J. SLECHTA, J. Physique 42, C4-229 (1981). [IS] R. BROUT, Phase Transitions, W. A. Benjamin, Inc., New York 1965. 1171 J. SLECHTA, phys. stat. sol. (b) 67, 595 (1975); 70, 531 (1975); 123, K133 (1984). [18] N. D. MERMIN and H. WAGNER, Phys. Rev. Letters 17, 1133 (1966). 1191 S. ROBASZKIEWICZ and R. MICNAR, phys. stat. sol. (b) 73, K35 (1976).

(1985).

(Received July 5, 1985)