Approach to saturation of the magnetization of dilute AuFe alloys

Solid State Communications, Vol. 26, pp. 91-94. 0 Pergamon Press Ltd. 1978. Printed in Great Britain

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APPROACH TO SATURATION OF THE MAGNETIZATION OF DILUTE AuFe ALLOYS* F.W. Smith and J.C. Liu Department of Physics, The City College of the City University of New York, New York, NY 1003 1, U.S.A. (Received 6 September 1977 by H. SuhZ)

We present a reinterpretation of our recent measurements of the magnetic properties of some dilute AuFe alloys. We find that the observed approach to saturation of the magnetization for these AuFe alloys can be understood if both single-impurity (Kondo) effects and effects due to interactions between impurities via the Ruder-man-Kittel-Kasuya-Yosida (RKKY) interaction, V(r) = (Fe cos 2kFr)/r3, are properly included in the analysis. The analysis yields for the strength of the RKKY interaction Ve = (1.1 f 0.3) x 1O-36ergcm 3, for the s-d exchange parameter 1Jl = (1.9 f 0.3) eV, and for the Kondo temperature TK = (0.8 f 0.1) K. We conclude that mean free path effects do not significantly influence the observed approach to saturation of the magnetization for the AuFe alloys studied. IN A RECENT communication [ 11 we presented some results of measurements of the magnetic properties of a series of dilute AuFe alloys which were interpreted as giving evidence for “self-damping” of the RudermanKittel-Kasuya-Yosida (RKKY) interaction [2], Y(r) = [(V, cos 2kFr)/r3 ] , between the Fe impurities. In particular, from an analysis of the approach to saturation of the magnetization M which took into account only the effects of the RKKY interaction, we concluded that V,-,, the strength of the RKKY interaction, decreased rapidly as a function of Fe concentration n. We noted, however, that in order to explain the observed decrease of Ve in terms of mean free path effects (“self-damping”), it was necessary to assume values of mean free path 1which were an order of magnitude smaller than values obtained from residual resistance measurements. Since then, a theoretical calculation [31 has indicated the way in which the approach to saturation of the magnetization is influenced by single-impurity (Kondo) effects. We have therefore reanalyzed our AuFe results to determine if single-impurity effects contributed to the measured magnetization. We present here the results of this analysis, which indicate that both single-impurity (Kondo) effects and impurityimpurity (RKKY) effects contribute to the observed approach to saturation of the magnetization for the AuFe alloys studied. We also conclude that mean free path effects (“self-damping”) do not significantly influence the magnetization of these AuFe alloys. * Research supported (in part) by the PSC-BHE Research Award Program of the City University of New York.

For the analysis of our AuFe magnetization data, we make use of two recent theoretical predictions for the approach to saturation of the magnetization in a dilute magnetic alloy. Schotte and Schotte [3] have considered single-impurity (Kondo) effects, using a “resonance level” model which has a resonance of Lorentzian shape and width A centered at EF. The density of states in this model is given by p(e) = A/tr(e’ + A’). Here A is approximately kB TK , where TK is the Kondo temperature. In the limit T + 0 their model yields M = (2ngpBS/n) arctan (gCcBH/A),which for gpBH 3 A reduces to M = ngpBS(l-

2Al~gcc,H),

(1)

where S is the spin of the magnetic impurity. The “resonance level” model thus predicts a l/H approach to saturation for M, with a slope 2AlrrgccBwhich is independent of concentration n. Larkin et al. [4] have investigated the effect of the

RKKY interaction on the thermodynamic functions of a dilute magnetic alloy. They predict, for gpBH % kBT and nV,, that M = rig/.@@@@ - 2(2S -I- l)nV0/3gpBH].

(2)

The slope of M vs l/H (as H + -) is thus predicted to be proportional to n V, . To determine if our magnetization data for these dilute AuFe alloys can be represented by either of the above expressions, we have plotted out data as M vs l/H. l/H+O(H+m), We obtain Msat =ng@inthelimit along with the slope of M vs l/H which we denote as Ho. We have found that our data fit the expression M = Msat[ 1 - (H&I)] , where H,, is a function of both

SATURATION

92

OF THE MAGNETIZATION

OF DILUTE AuFe ALLOYS

Vol. 26, No. 2

Table I. Magnetic properties of AuFe alloys

na

Cb ( lob6 emu K/g)

@pm Fe)

M ’ ( lzt3 emu G/g)

Ho (n 9Old (lo3 G)

AkBd (lo3 G/K)

18

0.140 f 0.006

42

0.292 ?r0.020

0.8 f 0.1

2.93 f 0.05

3.6 * 0.4

3.1 * 0.3

94

0.549 f 0.020

0.9 f. 0.1

5.82 + 0.10

4.9 f 0.3

2.8 f 0.3

169

0.99 f 0.02

0.9 f 0.15

10.9 f 0.2

4.9 * 0.2

2.8 + 0.3

242

1.51 f 0.02

1.0 f 0.15

15.5 f 0.3

7.1 + 0.2

2.7 + 0.3

572

3.73 * 0.04

1.1 * 0.2

33.7 f 0.7

8.0 f 0.2

1.9 f 0.2

1160

7.58 f 0.10

1.25 f. 0.2

65.0 f 2.0

12.5 + 0.3

0.9 f 0.2

2225

16.2 f 0.5

1.7 + 0.25

6050

60.6 f 1.2

2.7 * 0.3

a Concentration

n of Fe impurities

b Measured Curie constant x(T) = C/(T + 6). ’ Measured saturation

(in parts per million) as determined

C and Curie-Weiss

magnetization

temperature

determined

d Parameters derived from magnetization

200

400 n

600 bm

Fig. 1. Ho@, 0) as a function dilute AuFe alloys.

800

0 determined

from extrapolation

from fit of susceptibility

data at low T to

of measured M vs l/H to l/H = 0.

data (see text).

AuFe

0

from measured resistance ratios.

1000

1200

Fe)

of Fe concentration

for

n and T. The values of Mat which we obtain are listed in Table 1 (along with values of the Curie constant C and Curie-Weiss temperature 0 determined from a fit of our susceptibility data [5] to a Curie-Weiss law, x = C/(T + 0)). Values for the spin S for these AuFe alloys have been given previously [ 11. We note that it has not been possible to obtain Mm+,by this procedure for the 18 ppm Fe alloy or for the 2225 and

6050 ppm Fe alloys. For the latter two alloys, the condition gpBZ-Z% n PO was not satisfied. Similar lack of saturation of M has been noted for higher concentration ZnMn alloys [6] . We have plotted our measured values of Ho@, T), obtained as described above, vs temperature T for the six AuFe alloys with 42 < n < 1160 ppm Fe. As observed for several other dilute magnetic alloy systems [7], we have found that Ho is linear in temperature for T < 4.2 K, and can be represented by Ho(n, T) = Ak,T + Ho@, 0). Values of Ak, and Ho@, 0) obtained from straight lines fitted to the data are listed in Table 1. If the “resonance level” model [8] were appropriate for these AuFe alloys, then Ho@, 0) would be equal to 2AlngpB, independent of n, whereas the model of Larkin et al. [4] predicts that Ho@, 0) is proportional to n. To further test these two models against our data we have plotted H(n, 0) vs n in Fig. 1. In our previous analysis [ 11 of these AuFe alloys, we simply set Ho@, 0) = Bn V. , with B = 2(2+S + 1)/3gpB by analogy with equation (2). As noted before, the values of PO so obtained decreased rapidly as n increased. We note from Fig. 1 that Ho(n, 0) is linear in n, but

Vol. 26, No. 2

SATURATION OF THE MAGNETIZATION OF DILUTE AuFe ALLOYS

approaches a non-zero value, He(0,0) = (3.7 + 0.5) x 10’ G, as n + 0. We suggest that the intercept &(O, 0) represents the contribution from single impurity effects (n + 0) and we apply the “resonance level” model [3] by setting He (0,O) = 2A/rrgpB = 2kB TK/ngpB . From our result for He(0, 0), we obtain TK = (0.8 f 0.1) K, a value which is somewhat larger than previous estimates of TK for AuFe [B-l I], 0.1-0.46 K. We attribute the increase of He@, 0) with increasing n to impurity-impurity (RKKY) effects and, following equation (2), we equate the slope of He (n, 0) vs n to 2(= + 1) Fe/3gpB. Using the experimentally determined slope, (8.2 + 2.5) G/ppm Fe, along with g = 2 andS= 1.2,we find Fe =(I.1 +0.3)x 10-36ergcm3 for AuFe. We note that V,,/k, = (4.8 f 1.3) K/at .% Fe for these dilute AuFe alloys. From the expression [2] V0 = 3~~5~/167rnE~, where z = 1 is the host valence, n = 0.5955 x 10” cme3 is the concentration of conduction electrons, and EF = 5.5 1 eV is the Fermi energy, we obtain for AuFe IJI = (1.9 f 0.3)eV for the s-d exchange parameter. Thus mean free path effects (“self-damping”) are not responsible for the dependence of the parameter Ho@, 0) on Fe concentration n. Instead, we have found that our results for the approach to saturation of the magnetization for these dilute AuFe alloys can be represented by M = Msat [ 1 -Ho@, T)/H] , where H&r, T) is composed of three separate terms: H&z, T) = AkBT + 2kBTK/mgpB + 2(2S + l)nV, /3g&#.

(3)

The second term is associated with single Fe atoms and is consistent with the theory of Schotte and Schotte [3], and the third term characterizes Fe-Fe interactions as predicted theoretically by Larkin et al. [4] . The first term, AkB T, has been observed previously for several dilute magnetic alloy systems [7], and has no theoretical explanation to date. Thus, we have shown that single

93

impurity (Kondo) effects and impurity-impurity (RKKy) effects exist simultaneously in these dilute AuFe alloys, and that both contribute to the approach to saturation of the magnetization [ 121. We note that the magnetic susceptibility measured for these AuFe alloys, x = C/(T + e), apparently also exhibits both Kondo and RKKY effects. From Table 1 and a plot of 0 vs n, we have found that our measured values for f3(except for the 18 ppm Fe alloy) can be represented by e(n) = Be + em, where Be = (0.85 f O.l)K and 8r = (3.3 f 0.4)K/at.% Fe. To within experimental error, we can identify t!10with TK and 0 1 with V0/kB , as determined above. Thus our results for the measured susceptibility can be represented by x=C/TewhereT,-,=T+Be+Brn.Thethreeterms appearing in T,,(n, T) are similar in form and relative magnitude to the three terms appearing in HO(n, T), equation (3). In fact, we have approximately that gpBHo = kBTO. Tholence [9] has also noted that e(n) = B,, + 0 1n for dilute AuFe alloys. For 8 0 (which he identified as TK) Tholence obtained 0.46 K while for f3i , he obtained (2.9 f 0.2) K/at.% Fe, in reasonable agreement with our results. A similar concentration dependence for e(n) has been noted previously for dilute ZnMnalloys [13],withB,, =0.25Kande1 = 10.9 K/at.% Mn. The observation that the magnetic properties of these dilute AuFe alloys are influenced by both the Kondo effect and RKKY effects is consistent with previous studies of AuFe. Studies of magnetic susceptibility [B, 9, 14, IS] resistivity [16, 171 and Mossbauer effect [ 181 have all indicated that both the Kondo effect and RKKY effects can be observed in dilute AuFe alloys, thus making this alloy system a good candidate in which to study both effects. Acknowledgement - We wish to thank M.P. Sara&k

for useful discussions.

REFERENCES 1.

LIU J.C. & SMITH F.W., Solid Stare Commun. 17,595 (1975).

2.

RUDERMAN M.A. & KITTEL C., P&s. Rev. 96,99 (1954); KASUYA T., Progr. 7’heor. Phys. (Kyoro) 16, 45 (1956); YOSIDA K., Phys. Rev. 106,893 (1957).

3.

SCHOTTE K.D. & SCHOTTE U., Phys. Lerr. SSA, 38 (1975).

4.

LARKIN A.I. & KHMEL’NITSKII D.E.,Zh. Eksp. Teor. Fiz. 58,1789 (1970) [Sov. Phys. JE7P 31,958 (1970)] ; LARKIN A.I., MEL’NIKOV V.I. & KHMEL’NITSKII D.E., Zh. Eksp. Teor. Fiz. 60,846 (197 1) [Sov. Phys. JETP 33,458 (1971)].

5.

LIU J.C., Unpublished

6.

SMITH F.W., Phys. Rev. BlO, 2980 (1974).

thesis, The City College of the City University

of New York (1975).

94

SATURATION

OF THE MAGNETIZATION

7.

SMITH F.W., Phys. Rev. B14,241 (1976).

8.

THOLENCE J.L. & TOURNIER

9.

THOLENCE J.L., Unpublished

OF DILUTE AuFe ALLOYS

Vol. 26, No. 2,

(1976) and references therein; SMITH F.W., Phys. Rev. Left. 36,122l R., J. Phys. Paris 32, Cl-21 1 (1971).

thesis, Grenoble (1973).

10.

LORAM J.W., GRASSIE A.D.C. & SWALLOW G.A., Phys. Rev. B2,2760

11.

LORAM J.W., WHALL T.E. & FORD P.J., Phys. Rev. B2,857

12.

Similar results have recently been reported for dilute MoFe alloys, where the same analysis as used above yielded I’,-, = (3.8 + 0.1) x 10e3’ ergcm3, IJI = (0.8 + O.l)eV and TK - 0.4K for dilute MoFe alloys. SMITH F.W. & SARACHIK M.P., Phys. Rev. B16,4142 (1977).

13.

SMITH F.W., Phys. Rev. BlO, 2034 (1974).

14.

CLAUS H., Phys. Rev. B5,1134

(1972).

15.

THOLENCE J.L. & TOURNIER

R.,J. Phys. Paris 35, C4-229 (1974).

16.

FORD P.J., WHALL T.E. & LORAM J.W., Phys. Rev. B2, 1547 (1970).

17.

SCHILLING J.S., FORD P.J., LARSEN U. & MYDOSH J.A., Phys. Rev. B14,4368 SCHILLING J.S., FORD P.J. & MYDOSH J.A., Physica 86-88B, 846 (1977).

18.

STEINER P., BELOSERSKIJ Commun. 14, 157 (1974).

(1970).

(1970).

G.N., GUMPRECHT D., ZDROJEWSKI

(1976); LARSEN U.,

W.V. & HUFNER S., Solid State