A. FREUDENHAMMER: On the RKKY Interaction in Spin Glass Systems 579

phys. stat. sol. (b) 86, 579 (1978)

Subject classification: 13 and 18; 21.1

hboratorium fiir Festktkperphysik der Cesamthochschule Duisburg, Fachbereich 6 l)

On the RKKY Interaction in Spin Glass Systems BY

A. FREUDENHAMMER

The dependence of the critical temperature T, of a spin glass on the Fermi-momentum kF is calculated numerically. A strong dependence of T, on kp is found. A proposal is made that this dependence should be measured experimentally in order to obtain information about the basic interaction in spin glass systems.

Die Abhiingigkeit der kritischen Spinglas-Temperatur T , vom Fermi-Impuls kp wird numerisch berechnet. Eine erhebliche Abhiingigkeit der Spinglas-Temperatur T, von wird gefunden. Um bessere Kenntnisse iiber die grundlegende Wechselwirkung in Spingliisern zu erhalten, werden weitere experimentelle Untersuchungen vorgeschlagen.

1. Introduction I n the last few years spin glass systems have been investigated to a large extent

( [ l to lo]). These systems of magnetic moments distributed randomly on a metallic host show a phase-transition-like behaviour, as measured by Cannella and Mydosh [2]. It is assumed that the indirect RKKY interaction could serve as the basic interaction in the case of low concentrations of the magnetic moments because of the alternating sign of this interaction. Other interactions, for intances the dipole interaction of the magnetic moments, have also been considered (for example by Smith [ll]). We think that there is a need for more experimental work in order to decide on what kind of interaction spin glass systems, for example CuMn, AgMn, and AuMn, are based. A pos- sible way of obtaining information about the kind of interaction occurring in a par- ticular spin glass is described in the following account. We have chosen the Ising model analogous to the numerical calculations of Binder and Schroder [l]. But unlike Binder and Schroder we took RKKY interaction.

2. The Spin Glass Model As our model we take a three-dimensional f.c.c. lattice substituting the host atoms,

e.g. copper, ranging from a percentage of 0.3 to 100 atyo (100% is a hypothetical case, but as will be seen, it is important). The interaction of the spins is taken to be of RKKY form :

J, depends on a and the nearest neighbour distance a as J, a aa - a ( 2 4

and is taken to be negative. The length is measured in units of the nearest neighbour distance a and a = 2k@.

The RKKY interaction was developed with rotationally invariant states of the magnetic impurity. Caroli and Blandin [ 121 and Campbell and Blandin [ 131 took into

I) 4100 Duisburg, BRD.

580 A. FREUDENHAMMER

account the angular structure of the impurity spin (e.g. orbital quantum number 1 = 2) with the result

where JA depends on a) S (spin of the magnetic impurity) and a as JA oc l/(a2 - a - S . (S + 1)) (2b)

with tp being a phase shift of the scattered electron to be determined by comparison with the experiment. We consider Ising spins with the interaction ( la)

N H = - + C J(r i - r,) S&, (3)

i, j

where Si = +1 and the number of spins N = 216. Convergence was examined using N about 3300 spins.

The collective behaviour of our system will be treated more conveniently with the diagonal form of the interaction matrix (1)

where we have introduced the eigenvectors pik) and eigenvalues rewrite the Hamiltonian (3) with the representation (4) and get

of (Jij). We may

The eigenvalues and eigenvectors of (4) were calculated numerically by standard pro- cedures. Thus, the thermodynamic behaviour can be obtained from the knowledge of i lk and p‘,“).

I n order to get further insight into the nature of the representation (5) we point out that the mean-field approximation (MFA) leads to the following equation for the internal fields

N Hi = ,Z Jij - S * B,(/9h2SH,) , (6)

3 = 1

where Bs is the Brillouin function. Linearizing this equation we obtain

Thus the transition temperature T, is given [la] by T, = A2S(S + 1) AN/3k,) where 2, is the largest eigenvalue of (4). The dependence of the magnitude of the eigenvectors on the temperature can be calculated from (6) with the approximation of Bs up to the third order.

3. Numerical Results

Numerical calculations for the largest eigenvalue A N and lowest eigenvalue A, are performed for N = 216 spins distributed randomly on a f.c.c. lattice with a concen- tration of 3.7 at%. Fig. 1 shows the dependence of I N and I , on a(= 2k,a) where we have set the interaction between nearest neighbours to zero. However 1, and 1, show the results when the nearest neighbour interaction is also taken to be RKKY inter- action. As can be seen, the dependence is quite different in the region of a = 6.94; this value of a corresponds to CuMn, AgMn, and AuMg.

On the RKKY Interaction in Spin Glass Systems 581

Fig. 1 Fig. 1

Fig. 1. A,, iN are the largest and Al, the lowest eigenvalues of the interaction matrix (Jij) for concentration c = 0.037 and N = 216 spins. is computed with RKKY interaction for all distances on a f.c.c. lattice, whereas for the computation of A the interaction is zero for the nearest distances a

Fig. 2. AN, A$ are the largest and A,, 1: the lowest eigenvalues of the interaction matrix (Jij) for concentration c = 0.037 and N = 216 spins. Full curve: The calculations are shown with zero interaction for nearest neighbours and RKKY otherwise. Dashed curve : The interaction between nearest-neighbour spins is the direct exchange D and RKKY for larger distances than a (D/J, = 1)

We show in Section 4 a possible way to increase the Fermi momentum k, and to decide whether RKKY is valid or not in this spin glass system. The quasiperiodic behaviour of the eigenvalues is also obtained for larger values of u. Calculations are graphed in Fig. 2 for u = 20 to u = 32 and a concentration of c = 0.037 of magnetic moments, here we took the nearest neighbour interaction to be equal to zero in order to get a lower limit in the variation of A(u) (solid curve). The broken curves are cal- culations for values DIJ, = l, where D is the constant of direct exchange between nearest neighbours with distance a ; for larger distances the interaction is taken to be RKKY. As expected the oscillations are much smaller with additional direct exchange interactions [ 141. From Fig. 1 and 2 we can draw the conclusion that the dependence of the eigenvalues on u is quasiperiodic and that the interaction of the nearest neigh- bour is very important. A similar dependence was obtained for the inner eigenvalues by Freudenhammer [ 141 for a large variety of concentrations of the impurities. A fur- ther remark should be made on the work of Souletie and Tournier [15]. They argued that the period of the sine function has no effect on the results provided its wavelength remains small. Therefore we took u from 5 to 32 and showed the influence of the lattice on the results for a large interval of concentrations. The error bars in Fig. 1 are about 10%. To get information on the convergence of N -P 00, we calculate the quantity

where c is the concentration of the magnetic moments on the lattice points (the sum- mation in (8) is taken over all randomly occupied lattice points). The quantity (8) is equal to the sum of the squares of all eigenvalues of (4) divided by cN. We showed in [la] that the inner eigenvalues yield a similar quasiperiodic behaviour as AN and A. Before taking larger values for N in (8) we consider the case N = 216 and c = 0.037; for these values Fig. 3a shows similar oscillation as a function of u as does the curve AN in Fig. 1. In order to show that (8) has only a small concentration dependence we

582 A. FREUDENHAMMER

Fig. 3 Fig. 4

Fig. 3. The quantity C J$(cN) as a function of the Fermi momentum k~ (a = 2 k ~ a ) for (8)

c = 0.037 and N = 216, (b) c = 0.003 and N = 216, (c) c = 1.0 and N = 3375

Fig. 4. The quantity C &/(cN) as a function of the Fermi momentum kF (a = 2 k e ) for (a) c = = 0.037 and N = 3320, (b) c = 0.003 and N = 3311, (c) c = 1.0 and N = 3375

lowered c in Fig. 3b (c = 0.003) while N ( = 216) is kept constant. Because of the larger fluctuations for smaller concentrations the statistical error becomes larger. A good approximation to the case N -+ 00 was calculated in Fig. 3c with N = 3375 and the hypothetical case c = 1.

This hypothetical case is the limit of quantity (8) for N --f 0 0 ; it can be calculated analytically. Because of the reason mentioned above, the interaction of the nearest neighbours was set equal to zero for the calculations of the curves of Fig. 3 and 4. In order to calculate (8) analytically we define random variables ti, i = 1, ... N for each lattice point. The E1 are defined as being statistically independent. Hence we get a random variable 7,

M I I nf \

i , j=l i = l (9)

where M is the total number of 1attice:points of our f.c.c. lattice. Because of the statis- tical independence we obtain for the mean value, to a good approximation,

or

M

where x' means summation over all lattice points of the f.c.c. lattice. Thus, the quan- tity (11) is independent of the concentration of the impurities. The linear dependence of i j on the concentration has been verified numerically [14]. But in order to calculate (11) we do not need to solve the eigenvalue problem (7) numerically which is limited by the size of the computer memory, whereas the calculation of (11) is a question of computer time only.

From Fig. 3 c for c = 1 and Fig. 3a for c = 0.037 we conclude that the curve 3a is only slightly different from the hypothetical crystal (c = 1). This means that the convergence of (8) with N = 216 can be considered satisfactory and because of the similar behaviour of all eigenvalues the calculations of the curves shown in Fig. 1 should be a good approximation.

For comparison we drew the values of c = 1 and N = 3375, curve 4c. There is a remarkable difference between the curve 4a and 4c because of the smaller surface effect in the case of lower concentration of the magnetic moments. The curves of Fig. 4 however are not only interesting from convergence considerations. According to Edwards and Anderson [3] and Fischer [4] the quantity (10) can be related in MFA to

On the RKKY Interaction in Spin Glass Systems 583

the spin glass temperature !?,,

The sum is taken over all occupied lattice points j . f$ is independent of the index i; thus we can write

here i and j are taken over all occupied lattice points.

netic impurities on each lattice point. From (13) we obtain equation (10) if we take the same random distribution of mag-

4. Discussion In Fig. 1 to 4 we show that a large variation of the eigenvalues and hence of the

critical temperature as a function of Fermi momentum will occur if the RKKY inter- action (1) is taken to be the basic interaction between the impurity spins. This varia- tion will remain even in the limit of small concentrations of impurity. In accordance with Ahmed and Hicks [16], who measured magnetic correlations in CuMn (containing up to loat% Mn and using neutron polarization analysis), the nearest neighbour interaction is very important as can be seen in Fig. 1.

According to the theory of Edward and Anderson [2] we can take equation (13) in order to define the critical temperature T,. Thus EA theory combined with RKKY interaction also yields a quasiperiodic behaviour of T, as a function of the Fermi momentum.

In order to obtain information about the interactions in spin glasses, two things are necessary:

a) In the case of nearest neighbour distances between the magnetic impurities comparable to the atom diameter more theoretical work is needed (the calculations of Malmstrom et al. [ 171 seem to yield some disagreements with experimental results, for instance in the case of dilute CuMn).

b) Experimental investigations should be carried out with continuously varying Fermi momentum, i.e. of a = 2k@. All measurements done until now were taken on Heusler alloys with discrete values of a. Similar measurements to those performed by Webster and Ramadan [18] on Heusler (L2,) alloys varying a should be performed with small concentrations of magnetic impurities in the regime of spin glasses. (An- other possibility could be CuMn Sb(C1,) alloys as published by Endo [19].)

For large n.n. distances between magnetic impurities as in the case of L2, and c1b alloys the RKKY interaction, ( la) or the double resonance model, equation ( l b ) , should be a good approximation (Malmstrom et al. [17]). Thus, taking T,, S, a, and k, from measurements we could calculate the quantity

In the case of RKKY interaction T:(a) in (14) should coincide with curve &(a) of Fig. 1 according to (7).

On the other hand, if the double resonance model is valid, we calculate

TB**(a) = T,(a) - Lx - aa . (15)

Again we have to compare Tt* from (15) with xN of Fig. 1. However according to Webster and Ramadan [ 181 the result should be shifted by cp = 150" i.e. by a,, = 2.62. 38 physica (b) 86/2

584 A. FREUDENHAMMER: On the RKKY Interaction in Spin Glass Systems

Guntherod [20] showed in the case of metglass that an increase of the Fermi energy is possible. The alloy of e.g. copper and germanium yields a maximum increase of Fermi momentum by a factor of 1.1. This is the factor that can be calculated according to the Hume-Rothery rule because copper yields one and germanium four conduction electrons per atom; a further increase causes a phase transition to a b.c.c. lattice - this is confirmed by the phase diagram of Hansen [21].

Measurements of the lattice constant a and the Fermi momentum k, are possible in order to get the right U-value.

A possible second method would be to take hydrogen in order to increase the Fermi energy.

A variation of u should not be assumed when putting the crystal under pressure because k,. is va,rying with l/a. But the factor (2) should increase with ae according to (2a) or with

Hence, experiments should be done on Cul-,-vGe,Mnv (0 5 x 5 0.1, y < 0.05) alloys in order to obtain informat,ion on the question of interaction between spins in spin glasses.

Acknowledgements

We would like to thank Prof. Heber for many helpful discussions and Mrs. Moritz for some numerical calculations. The work was performed at the Rechenzentrum Duisburg and Koln. We are most grateful for their support.

according to (2b).

References

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[lo] J. L. TEOLENCE and R. TOURNIER, J. Physique 36, C 4 229 (1974). [ll] D. A. SMITE, J. Phys. F 6, 2148 (1975). [12] B. CAROLI and A. BLANDIN, J. Phys. Chem. Solids 27, 503 (1966). [13] I. A. CAMBPELL and A. BLANDIN, J. Magnetism magnetic Mater. 6, 97 (1975). [14] A. FREUDENHAMMER, J. Magnetism magnetic Mater. 6, 97 (1977). [15] J. SOULETIE and R. TOWNIER, J. low-Temp. Phys. 1,95 (1969). [l6] N. AHMED and T. J. HICKS, J. Phys. F 6, 2168 (1975). [17] G. MALMSTROM, D. J. W. GELDART and C. BLOMBERQ, J. Phys. F 6,1953 (1976). [lS] P. J. WEBSTEE and M. R. I. RAMADAN, J. Magnetism and magnetic Mater. 6,51 (1977). [19] K. ENDO, J. Phys. SOC. Japan 29,643 (1970). [ZO] H.-J. GUNTEERODT, Adv. Solid State Phys. 17 (1977). [21] M. HANSEN, Constitution of Binary Alloys, McGraw-Hill, London 1958.

(Received December 28, 1977)