VOLUME 28, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 1 MAY 1972

N. Tzoar and J . I . Gersten, to be published. 6F. D. Tappert and C. M. Varma, Phys. Rev. Lett.

25, 1108 (1970).

7For InSb we take m*=me/609 c* = l . l l x i 0 8 cm/sec, ex = 16, andT=77°K. For PbTe, m*=0.06me (average), c*=0.57xl08 cm/sec, 6^=32, and T=4°K.

Localized Magnetic Excitations in Substitutional^ Disordered Antiferromagnets

E. N. Economou Department of Physics and Center for Advanced Studies, University of Virginia, Charlottesville, Virginia 22901

(Received 27 December 1971; revised manuscript received 14 February 1972)

It is shown that the theory of localization developed by Economou and Cohen together with the coherent-potential approximation can be used to describe the magnetic excita­tions in subs tit utionally disordered antiferromagnets. The results are in good agree­ment with recently published experimental data.

Recently Buyers et al} studied the character of magnetic excitations in single crystals of the sys­tems KCo^Mn^Fg and CoxMn1_3CF2 by neutron in­elastic scattering. It is shown here that their ex­perimental data are in quantitative agreement with results obtained from first-principles-mod­el calculations based on the theory of localization developed by Economou and Cohen2 and on the co­herent-potential approximation.3 To the best of the author's knowledge, this is the first system for which such an agreement has been obtained.

The basic result of the theory of localization2'4

is the existence of a localization function L(E) such that to every value of the energy E satisfy­ing L{E)>1, there correspond extended (or prop­agating) eigenstates. On the other hand if L{E) <1, either there exist no eigenstates of energy E or they are localized. At the critical energies Ec, termed mobility edges, such that L(EC) =1, the character of the states changes abruptly from lo­calized to propagating. Such a behavior was hy­pothesized by Mott5 and by Cohen, Fritzecke, and Ovshinsky6 and is known as the Mott-CFO model. Explicit, though extremely complicated, expres­sions for L(E) have been obtained2'4 which cor­rectly predict that for a periodic system all the states are extended.

In order to estimate L(E), some rather drastic approximation is needed. It has been shown2 that within the framework of any single site approxi­mation3 a function E(E) can be found such that E(E)<1 implies L(E)<1. Moreover, the function E(E) is related to and no more difficult to calcu­late than the average density of states. Thus F{E) can be used as an estimate of L{E). The er ror is always in the direction of underestimating the ef­fects of the randomness and increases with them.

The energies Ec*, where F(EC*) = 1, lie in regions of localized states.

The general theory outlined above has been ap­plied7 to a simplified model for a random binary alloy of the type AJB1.X. The density of states and the function F(E) have been calculated for this model within the coherent-potential approxi­mation which is the best single site approxima­tion.3

The problem of magnons in substitutional^ dis­ordered antiferromagnets can be handled within the framework of a two-sublattice model Hamil-tonian. Besides the concentration x and the lat­tice structure, there are six parameters charac­terizing this Hamiltonian: the anisotropy ener­gies ojA and o)£, and the spin-spin coupling ma­trix elements VA = 2JAASA, VB = 2JBBSB, VAB

= 2JABSB, and VBA-2JBASA, where J is the ex­change integral taken as positive, S is the spin, and the subscripts A and B denote the Co and Mn atoms, respectively. (The quantity % has been taken equal to unity.) Starting from this Hamil­tonian and neglecting randomness in the off-diag­onal matrix elements, one can show that the problem of magnons in substitutional^ disordered antiferromagnets can be reduced to that of the bi­nary alloy presented in the previous paragraph. Details of this rather elaborate reduction will be presented elsewhere.

To perform the numerical work, one has to choose the parameters. Four of them (ooA, ooB, yA* VB) w e r e chosen by requiring that the band edges predicted by the theory coincide with the experimental ones for the cases of pure-A and pure-5 crystals. The other two were fixed from the requirement that the frequencies of the local modes for A or B impurities should coincide with

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VOLUME 28, NUMBER 18 P H Y S I C A L R E V I E W LETTERS 1 MAY 1972

^ o k. ^r U~ K

^ ki ^ ^ O* <o

c* <o

1.0

0.8

0.6

0.4

0 ?

0.0 0 1 2 3 4 5 6 7 8

FREQUENCY (THz)

FIG. 1. Theoretical results for the trajectories of band edges (solid curves) and the trajectories of the frequencies Ec* (dot-dashed curves), as a function of the concentration x for KCo^Mnj-^Fs. The trajectories of JEC* are outer bounds for the trajectories of Ec sep­arating the localized states (shaded area) from the ex­tended states (white areas within the bands). Bars rep­resent experimental results (Ref. 1) for the position of the bands. The elimination for theoretical curves at low frequencies indicates that the theory is unreliable in this region.

the observed1 positions (or the ones calculated using molecular field theory1). Thus the parame­ters were fixed by forcing the theory to agree with experiment for x =0 and x = 1. The parame­ters obtained this way were consistent with esti­mates based on molecular field theory.8 In any case the theory can be checked by comparing it with experiments performed at any intermediate concentration.

The results for KCo^Mn^Fg are presented in Fig. 1. The theoretical calculations for the t ra­jectories of the band edges (solid curves), and the trajectories of the energies Ec*, where F(EC*) = 1 (dot-dashed curves) were plotted as functions of the concentration #. According to the general the­ory, the regions between the trajectories of the band edges and the trajectories of Ec* consist en­tirely of localized states (shaded areas) although the latter always extend beyond the trajectories of £ c*. Cutting the diagram by horizontal lines, x ~ const, one obtains the positions of the two sub-bands as well as the nature of the eigenstates within each sub-band. The eigenenergies close to zero are obtained as a difference of two large, almost equal numbers. Thus any er ror is magni­fied in the low-energy region and the theory be­comes unreliable there; to indicate this explicit­ly, the lowest part of the band edge and Ec* t ra­jectories have been eliminated from Fig. 1 as well as from Fig. 2.

The experimental results for the positions of the sub-bands are shown as bars in Fig. 1 (x

= 20%, x = 71%). In general the agreement between theory and experiment is good for the upper band and fair for the lower band. The basic defect of the theory is that it overestimates the width of the lower sub-band for low concentrations up to x = 60%, This defect is directly linked with the omission of the off-diagonal randomness. It would be extremely interesting to compare future theories incorporating off-diagonal randomness with the present experimental results, which seem sensitive enough to serve as checks of such generalized theories. It should be noted that a proper incorporation of the off-diagonal random­ness is a necessary step in the evolution of the theories of disordered systems.

When the degree of randomness 6 — °°, the prob­lem of finding if the states are extended reduces to that of classical percolation theory, namely, to the problem of finding if paths exist consisting of A (or B) atoms only and extending to infinity.7

The percentage of extended states in the A (B) sub-band is then equal to the percolation probabil­ity P(x) [P(l -x)] of finding such A (B) paths. P(x) is zero for x <xc where xc, the critical con­centration for percolation, depends on the lattice (#c»30% for a simple cubic lattice). The binary-alloy calculations demonstrated that one is al­ready very close to the limit 6 -* °o when 6 ^ 3 . For the upper band this condition is satisfied and consequently one expects all the states of the up­per sub-band to be localized for x<xc*3Q%, as indicated in Fig. 1. On the other hand, for the lower band the condition 6£ 3 is definitely violat­ed in the concentration region (x £ 70%) where classical percolation would predict all states lo­calized. Thus the critical concentration for the disappearance of extended states from the B sub-band should be (if it exists at all) considerably higher than 70%.

By looking at the k dependence of the bands, Buyers et al} were able to determine that the states in both bands are basically extended for x = 71%, and that they are extended for the lower and localized for the upper band at 20%. These experimental findings are in agreement with the results of the present theory, as can be seen from Fig. 1. It seems that the theory predicts a higher localization in the tails of the lower sub-band a t # =71%. This is probably due to the omis­sion of the off-diagonal randomness in combina­tion with the sensitivity of the present theory in this region.

It would be very important if one could verify experimentally the existence of the critical con-

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V O L U M E 28, N U M B E R 18 PHYSICAL REVIEW LETTERS 1 M A Y 1972

1.0

0.8

0.6

0.4

0.2

0.0

M

If -

_

i _ _ i

i T ^

1 1

K ' i

! L i L _

-

-

-

Hk

1 2 3

FREQUENCY (THz)

FIG. 2. Same as for F ig . 1, but for Co3CMn1„xF2.

centrations at which all the states become local­ized, i.e., the band undergoes the so-called An­derson transition.2

To this effect, experiments similar to that of Ref. 1 should be performed below and above x «30% to check if a rather drastic change in the k dependence of the upper sub-band occurs. A sim­ilar possibility exists for the lower sub-band for x~S0%.

In Fig. 2 the results for CoxMn1>xF2 are pre­sented in a way identical to the one already dis­cussed. Regarding the positions of the bands, the agreement with experiment is very good for the upper band and fair for the lower. It seems that the theory is consistent with experiment regard­ing the nature of the eigenstates of the lower band for x = 70%.

The nature of the eigenstates in the upper sub-band deserves some further discussion. For the present case, the parameters VA, VB, VAB, and VBA, determined as was explained above, are VA

*VB*VBA~0.3 THz and VAB*0.5 THz. This means that the states in the upper band (for x ^ 70%) are associated not with A atoms but with A-B pairs. It is the A-B bond which introduces the large matrix elements VAB responsible for the high frequencies of the upper sub-band for x % 70%. Thus, in the limit 6 - <», the existence of extended states in the upper sub-band depends on the existence of A-B-A-B* • • paths extending to infinity. The probability of existence of such paths can be shown to be equal to P(x)P(l ~x), where P is the percolation probability for one of the two magnetic sublattices. Hence, for the present case, P(x)P(l - x) is zero when x < 30% or x > 70%. For x < 30%, 6 > 3 and the predictions of classical percolation theory are almost exact. On the other hand, for x > 70%, .6 < 1.87 and the classical percolation theory becomes progres­sively invalid. Hence the nature of the eigen­states is as shown9 in Fig. 2. This behavior is

consistent with the observed character of the up­per band at x = 70%. It would be interesting to check experimentally whether or not a higher per­centage of the states in the upper band becomes extended as one lowers x to about 50%.

Buyers et al} proposed a simple empirical formula for the localization function L(E), which was partly successful in predicting roughly the nature of the states in each sub-band. By examin­ing the convergence of the perturbation theory for the self-energy2 A0, one can obtain, after us­ing some very drastic approximations, Eq. (1) of Ref. 1. This very illuminating derivation togeth­er with details of the present theory will be pre­sented elsewhere.

Finally, it should be pointed out that other sys­tems, most notably optical phonons in disordered mixed crystals, are expected to exhibit a behav­ior qualitatively similar to that of antiferromag-netic magnons discussed here.

In summary, it was shown here that results of a first-principles-model calculation employing the coherent-potential approximation are in good agreement with experimental data describing the nature of magnetic excitations in disordered anti-ferromagnets. Several predictions of the theory were presented awaiting experimental check.

I wish to acknowledge helpful discussions with K. L. Ngai and J. Ruvalds.

2W. J . L. B u y e r s , T. M. Holden, E . C. Svensson, R. A. Cowley, and R. W. H. Stevenson, Phys . Rev. Let t . 27, 1442 (1971).

2 E. N. Economou and M. H. Cohen, P h y s . Rev. Let t . 25 , 1445 (1970), and Phys . Rev. B _5, 2931 (1972).

B . Velicky, S. Ki rkpat r ick , and H. Eh ren re i ch , P h y s . Rev. 175, 747 (1968); D. W. Taylor , Phys . Rev. 156, 1017 (1967); P . Soven, P h y s . Rev. 156, 809 (1967).

^E. N. Economou, to be published. 5N. F . Mott, Advan. P h y s . 16, 49 (1967). 6M. H. Cohen, H. F r i t z s c h e , and S. R. Ovshinsky,

P h y s . Rev. Let t . 22, 1065 (1969). 7E. N. Economou, S. Kirkpat r ick , M. H. Cohen, and

J . E g g a r t e r , Phys . Rev. Let t . 25 , 520 (1970). 8 E. C. Svensson, W. J . L. B u y e r s , T. M, Holden,

R. A. Cowley, and R. W. H. Stevenson, Can. J . P h y s . 47 , 1983 (1969). ~ T h i s behavior of the t r a j ec to r i e s of the mobil i ty ed­ges is not ref lected in the t r a j ec to r i e s of a>c*, s ince the whole effect is due to off-diagonal r andomness which has been neglected in de termining Ec*. It would be ve ry in te res t ing to check if effective medium t h e o ­r i e s including off-diagonal r andomness can predic t such a behavior for the p re sen t s y s t e m . Theore t ica l work along this di rect ion is p resen t ly in p r o g r e s s .

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