Mn

Journal of Magnetism and Magnetic Materials 128 (1993) 47-57 North-Holland

Ad"

Magnetic properties of Co-rich amorphous alloys containing Cr/Mn A. D a s a n d A . K . M a j u m d a r Department of Physics, Indian Insatute of Technology, Kanpur 208 016, Uttar Pradesh, India

Received 16 November 1992; in revised form 3 March 1993

The dc magnetization has been measured between 20 and 300 K in the glassy alloy series Fe5Co50Ni17_xCrxB16Si12 (0 < x < 15 at%), and in FesCo50Mn17B16Sit2 and Fe7.sCo3L2Ni24Mnx5B14Si8 glasses. The variation of the magnetic moment with addition of Cr/Mn has been discussed on the basis of the virtual bound states. The temperature dependence of magnetization follows the Bloch T 3/2 law for the Cr-containing samples. In samples containing Mn, we find contributions of the form T 5/2 in addition to the T 3/2. The spin-wave stiffness constant varies linearly with the Curie temperature.

1. Introduction T h e role of disorder on the various aspects of magnetization continues to be an area of considerable interest. A m o r p h o u s materials are used as a tool to investigate the effects of the lack of long-range structural order on the fundamental magnetic properties. A comparison of data on crystalline and amorphous materials appears to suggest that the absence of long-range order has very little effect on the magnetic m o m e n t (/~), Curie t e m p e r a t u r e (T¢) and saturation magnetostriction (A s) [1]. They are found to be m o r e sensitive to the various types of short-range order. The addition of metalloids to Co-, Ni- and Fe-based metallic glasses brings about a change in the behavior of the magnetic m o m e n t differently in these three alloy systems [2]. T h e variation of the magnetic m o m e n t with the substitution of one transition metal by another falls on a curve equivalent to the Slater-Pauling curve for

the crystalline Fe, Co and Ni alloys [3]. The interpretation was in terms of the charge-transfer model [1,4], which is actually a special case of the rigid-band model [5]. These models are observed to hold for small concentrations of impurities that weakly perturb the periodic potential of the matrix (I Z I = 1, where Z is the chemical valence) [6], while it fails for I Z I>_ 2. Experiments on C080_xTxB20 glasses (T = Fe, Mn, Cr, V) [6] show that Friedel's virtual bound state approximation [7] holds good for Cr- and V-containing alloys and is able to account for the sharp fall in the m o m e n t with the addition of these impurities. The t e m p e r a t u r e dependence of magnetization ( M ( T ) ) is explained largely in terms of the spin-wave excitations which, at low temperatures, follow the well-known Bloch equation [8]: M ( T ) = M(O)[1 - B T 3/2 - C T 5/2 - . . .

].

T h e T 3/2 t e r m originates from the harmonic (q2) t e r m and the T 5/2 t e r m from the (q4) term in the

Correspondence to: Dr A.IC Majumdar, Indian Institute of

Technology, Low-Temperature Laboratory, Department of Physics, P.O.-I.I.T. Kanpur 208 016, India. Tel: + 512 250151, ext. 2965/2294; e-mail [email protected].

spin-wave dispersion relation. The existence of spin-waves in amorphous materials has been confirmed directly from magnetic inelastic neutron scattering measurements. The values of the spin-

0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

48

A. Das, A.K. Majumdar / Magneticproperties of Co-rich amorphous alloys

wave stiffness constant D, obtained from the magnetization measurements, do not differ from those obtained from the neutron scattering measurements in non-Invar metallic glasses [9]. However, in Fe-rich glasses the difference is found to be as much as a factor of 2 [9,10], which shows that modes of excitation other than the spinwaves, i.e. Stoner single-particle-type excitations also contribute to the demagnetization process. Babic et al. [11] found the presence of Stoner terms together with the spin-wave terms from the magnetization data of FexNi80_xB18Si 2 and Fe8oB20. It is observed that the magnetization versus temperature ( M ( T ) ) in C o - F e - B - S i alloys [12] are well described by the Bloch's T 3/2 law, and additional terms in the form of T 5/2 are unnecessary at temperatures below 300 K. The spin-wave stiffness constant, obtained from bulk magnetization measurements, is found to increase with increasing Fe concentration and Tc. On replacing Co with Ni [13] the magnetic moment and Tc are found to decrease. The M ( T ) data follows the T 3/2 law. The values of D ( T ) vary linearly with Tc. The magnitude of D in both alloy systems is found to be nearly half of that reported for crystalline Co. On replacing Ni with C r / M n in an alloy system with composition very close to those described above, we find several interesting features in the temperature dependence of resistance [14], magnetoresistance [15], and critical exponent [16] behaviour. Low-field ac susceptibility measurements on these samples indicate that Mn-containing samples undergo a transition to a reentrant spin-glass state below ~ 30 K [15]. However, on application of dc magnetic fields the indication of this state disappears. We report here the magnetic properties of these alloys with an aim to study the effect of substitution of Ni by Cr and Mn on (i) the magnetic moment and (ii) the temperature dependence of magnetization, i.e. if the thermal demagnetization is only through the T 3/2 term or additional terms due to anharmonic terms in the spin-wave dispersion relation, temperature dependence of spin-wave stiffness constant a n d / o r Stoner single-particle excitations are required to explain the results. The dependence

of the spin-wave stiffness constant on Tc is also investigated. The influence of Cr and Mn on the magnetization process has received considerable attention in Fe- and Co-rich glasses [11,17-19] and crystalline materials [20,21]. The magnetic moment, the Curie temperature, and the average hyperfine field of the alloy decrease on addition of Cr and Mn but with a difference. While substitution with very small amounts of Cr produces a drop in the moment, it is found that in the case of Mn it actually increases. This behavior is found to be common in both Fe- and Co-rich glasses. However, recent M6ssbauer study on a-Co-Fe-B-Si alloys shows that there exists a striking difference in the hyperfine parameters between Mn doped Fe- and Co-rich alloys [22]. The M ( T ) study of Fe-Cr-containing glasses shows that the T a/2 law holds good in these alloys [20,21,23]. Additionally, the presence of Stoner excitations, in the form of a T 2 term, is detected and is found to increase with the increase in Cr. 2. Theory Experimental evidences on amorphous ferromagnetic materials suggest that long-wavelength spin-wave excitations follow the same dispersion relation as crystalline ferromagnets, namely: E(q) = glxaHin t + Dq 2 + Eq 4 + . . . ,

(1)

where glzBHint < < Dq 2 denotes an energy gap in the presence of an effective internal field Hint arising from the applied field, the anisotropy field, and the spin-wave demagnetizing field. D is the spin-wave stiffness constant and E is the constant of proportionality for the q4 term. At low temperatures, the change in magnetization (in emu/g) due to spin-wave excitations, according to the Heisenberg model, is given by [8,24]: AM(T) M(O)

M(T) -M(O) M(O) = BZ(3/2, Tg/T)T 3/2 + CZ(5/2, Tg/T)T 5/2

(2)

A. Das, A.K Majumdar / Magneticproperties of Co-rich amorphous alloys

When eq. (9) is substituted in eq. (2) it gives

= B3/2Z(3/2, T g / T ) ( T / T c ) 3/2

+ C5/2Z(5/2, T g / T ) ( T / T c ) 5/2, (3)

aM(r) M(O)

AT3~2( 1 _ D1T2 _ D2TS/2 ) - 3 / 2 T5/2.

where Tg is the gap temperature given by gp.Bnint/ka. The internal field Hint is determined from the relation: Hin t = napplie d - 4 " r r N M ( 0 ) + H A,

(4)

where N is the demagnetization factor and H A is the anisotropy field. The functions Z(3/2, Tg/T) and Z(5/2, T J T ) are defined as [24]: Z ~-, ~-

= 2.612 n-1 ~ n-3/2

expl- T ) '

(5)

and Z

,

1.341

E n=l

n-5/2 exp

. (6)

The coefficients B and C are related to the spin-wave stiffness constant D and the average mean:square range (r 2) of the exchange interaction by:

kB[2"612gl~B] 2/3, M(O)pB

49

(7)

O(=O(O)) = ~

T j (10) A binomial expansion of the first term gives a T 7/2 a n d / o r a T* term. Thus the deviation from the T 3/2 dependence may show up as a T 5/2 term due to the anharmonic t e r m ( q 4 ) in the spin-wave dispersion relation, and a T 7/2 a n d / o r a T 4 term depending on the temperature dependence of D. Besides the spin-wave excitations, there exist Stoner single-particle excitations which also contribute to the low-temperature demagnetization of a ferromagnet. According to the approach of itinerant electron or band ferromagnet, the paramagnetic densities of states for spin-up and spindown electrons are displaced in energy with respect to each other. This displacement is assumed to be proportional to the spontaneous magnetization and, in the low-temperature limit, the decrease in magnetization with temperature is the result of excitation of electrons from the spin-up to spin-down bands. The general expression for the single-particle excitation is given by:

and

= A T a exp( - k---~) , (r 2) = 1.948

B '

(8)

where p is the density. In the above relation the ratio of the coefficients C / B gives the range of interaction. The above relations assume that D is temperature independent. When the number of magnons excited is large, interactions between them (magnon-magnon) lead to a T 5/2 term in D in the localized model [25,26]. The interaction between spin-waves and itinerant electrons leads to a T 2 term, and magnon-magnon interactions lead to a T 5/2 term in the itinerant electron model [27,28]. Thus in the low-temperature limit, D = D(O) (1 - D,T 2 - D2T5/2).

(11)

(9)

where A is the energy gap between the top of the full sub-band and the Fermi level. This expression reduces to AM

(with a = 2, A = 0) for weak ferromagnets where holes exist in both the sub-bands as in the case of Fe, for example, and

13, (with a = 3/2, A ~ 0) for strong ferromagnets where one of the sub-bands is completely filled

50

A. Das, A.IC Majumdar / Magnetic properties of Co-rich amorphous alloys 400

and holes exist in only the minority band as in the case of Co and Ni, for example. At low temperatures, when deviation from saturation magnetization at 0 K is small, the excitations from the spin-wave and single particle are nearly independent and the thermal demagnetization is given by the sum of the terms arising from both the contributions, i.e., A M = [AM]s w + [AM]s P.

300 Y

i...~200

(Feo.07Coo.7Nio.z~-~Cr~)72B~eSi12 1 O0 0.00

I

I

0.05

a

I

I

0.10

I

I

0.15

I

0.20

0.25

x (at.

(14) 0.80

E 3. Experiment

(Feo.o7Coo.7Nio.z~-xCr~)72BleS}12

O o

(b)

"~m 0.60

The samples studied are (a) Fe5Co50Ni17_xCrx Bz6Sil2 (x = 0, 5, 10 and 15, designated as A1, A2, A3 and A4), (b) Fe5fosoMnl7B16Sil2 (m5), and (c) FeT.sCoal.2Ni24Mn15B14Si8 (B5). The amorphous nature and the chemical composition of the samples were verified by X-ray measurements and EDAX analysis, respectively. The dc magnetization measurements were done between 20 and 300 K in a magnetic field up to 10 kOe using a vibrating-sample magnetometer (PAR, 155), a closed-cycle helium refrigerator (CTI) and Varian 15" electromagnet. A 100 l~ precalibrated platinum resistance thermometer, placed very close to the sample, was used to monitor the temperature.

4. Results and discussion

4.1. Curie temperature (Tc) and magnetic moment The Curie temperatures of samples A1-A5 and B5 were measured by low-field ac susceptibility (Xac) and dc magnetization methods. The values obtained from these measurements (for some of these samples) agree within + 0.5 K and are between 170 and 400 K. The highest value of 400 K is obtained in sample A1 and the value of Tc decreases with the addition of Cr or Mn. Figure l(a) shows the nature of the decrease in Tc. Initially, the decrease in Tc is much more rapid than with the later addition of Cr. On an average, the decrease is about 10 K / a t % Cr. This trend is

0.40 O.Oq

0.05

o. I0

O. 15

0.20

0.25

x (at. Fig. 1. (a) Variation of Curie temperature (Tc) with Cr concentration. (b) Variation of magnetic moment per atom with Cr concentration.

similar to that observed in F e - N i - C r - B - S i glasses [29]. The rate of decrease in Tc is much more rapid, about 20 K / a t % TE (where TE = V, Mn, Cr, and Mo) in Fe-based glasses containing Cr [30] and V, Mn, Cr and Mo [7,31]. The present value is also much smaller than the rate of decrease of about 97 K / a t % Cr observed in Co80_xCrxB20 glasses [8]. The sharp fall in Tc with the addition of Cr implies a loss of magnetic exchange interactions, which may be due to the antiferromagnetic coupling of Cr atoms. The complete replacement of Ni with Mn in this series (sample A5) does not, however, lower the Tc as much as observed in the case of A4, where Cr almost fully replaces Ni. The magnetization at 0 K, M(0) of the samples is obtained from the extrapolation to 0 K of the M(10 kOe, T) data from which the magnetic moment per atom (~) is calculated. The variation of the magnetic moment with the addition of Cr is very similar to that of Tc and is shown in fig. l(b). The decrease is rapid at low concentrations and is slower at higher concentrations. However, due to the lack of data points at low concentra-

A. Das, A.K. Majumdar / Magneticproperties of Co-rich amorphous alloys

tions, little emphasis could be placed on this observation and therefore, while calculating the d T c / d x and d ~ / d x , single straight lines passing through majority of data points are considered. The values of Tc and the magnetic moment are given in table 1. With the addition of Cr, the rate of decrease in ~ with Cr concentration (x) is approximately - 1 / z B / a t % Cr. If it is assumed that B and Si do not contribute to the moment of the sample and only the transition elements do and then the magnetic moment/transition element atom is calculated, there too the value of dl.t/dx comes to - 1/zB/at% Cr. Contrary to the behavior observed in samples with Cr, the replacement of Ni with Mn raises the magnetic moment significantly. Similar behavior for the variation of the magnetic moment with Mn and Cr has been observed in Cos0_xTxB20 ( T = Fe, V, Cr, Mn) glasses [7,17]. It is found that the magnetic moment of Co80_xTxB20 glasses increases up to 6 at% Mn and then it decreases [17]. These results are explained on the basis of the idea of the virtual bound states (VBS) [7]. The increase in the magnetic moment with the additional of small amounts of Mn appears to be a feature encountered only in amorphous systems. Magnetic moment variation in crystalline C o - M n system shows a monotonic decrease in the moment from 1.7/za for Mn = 0 to 1.34/z a for Mn = 8 at% [32]. A qualitative discussion of the above results are given on the basis of the VBS model. Accord-

51

ing to this model, a fivefold-degenerate 3dr virtual bound state is lifted out of the 3d t band near the impurity due to its repulsive potential. If the majority spin 3dr VBS remains below the Fermi level (EF) , then the solute affects the moment because of the difference in populations of 3d~ band relative to that of the host. In this case, an equation of the form: (14)

ZCftB,

/£ = jl~matrix --

is obeyed, where Z is the valence difference between the solute and the host. If the potential of the solute is sufficiently repulsive, then the majority spin VBS moves above the E F and five 3d r electrons are transferred to 3d~ states, thereby reducing the average magnetic moment by 10/~a in addition to the valence difference. Thus, in this approximation, ft =/'~matrix- ( Z d- 10)c/.~B.

(15)

Therefore, in this case ~ = dl~/dc = - ( Z + 10). The parameter - ( ~ + Z ) is a good indicator of the position of the majority band. If - (~ + Z) -- 0, eq. (14) is followed; in other words, the majority 3d r band lies below the Fermi level. If -(~: + Z ) = 10, then eq. (15) is followed where distinct VBS appears above E F. When - (~ + Z ) = 5, which is the case in Cr-containing samples, it is suggested that the Fermi level E F intersects the 3d r VBS. In the present samples, if Z is taken as ~- - 3.5 (Cr in the Co-Ni system) and g = - 1 , then - ( ~

Table 1 Composition, Curie t e m p e r a t u r e (Tc), magnetization at 0 K (M(0)), spin-wave stiffness constant (D(0)), average magnetic m o m e n t per transition metal atom (g), and the coefficients B3/2 and Cs/2 (eq. 3) for samples A 1 - A 5 and B5

A1 A2 A3 A4 A5 B5 Fe Co Ni

Tc

D(O)

D(O)/TC

(K)

(meV ~2)

(meV A 2 / K )

400 267 222.5 174 300 370 1043 1394 631

101.5 74.8 55.7 46.0 95.2 97.8 286 397

0.24 0.28 0.25 0.26 0.32 0.26 0.27 0.25 0.63

B3/2 0.48 0.51 0.65 0.66 0.26 0.35 0.12 0.17 0.12

C5/2 0.02 0.42 0.65 0.04 . 0.15

C5/2/B3/2 0.04 1.60 1.86 0.33 .

. 1.25

M(0) (emu/g)

( / ~ B / T M atom)

59 48 45 41 77 75 -

0.62 0.50 0.47 0.42 0.80 -

. -

-

52

A. Das, A.K. Majumdar / Magnetic properties of Co-rich amorphous alloys

+ Z ) = 4.5, which is roughly the same as obtained in CrxCo80_xB20 system [7]. This implies, in the above model, that the 3d ~ VBS intersects the E F and thus explains the observed fall in the magnetic moment with the addition of Cr. However, a direct calculation of Z from eq. (15) yields {d/~/dx = - 1 = - ( Z + 10)} Z = - 9 . This result is in disagreement with the expected value of = - 3.5. The increase in the moment of the alloy, on replacing Ni by Mn, shows, using the above model, that the majority spin band is filled and below the E F there are minority spin holes only. Due to the non-availability of alloys with lower Mn concentration, one cannot really say if the maximum in the moment versus concentration curve has been reached or not. Dobrzynski et al. [13] studied the variation of magnetic moment in (Co0.93_yNiyFe0.07)75B15Si10 alloys. They found that with the increase in y the average magnetic moment per transition metal atom (/zTM) decreases. Our sample A1 (Fe0.07 Co0.7Ni0.23)72B16Si12 is very near to their composition with y = 0.23. But the values of the magnetic moment and Tc reported for y = 0.23 alloy are about 0.9/z B and 500 K, respectively, and are not in good agreement with 0.62/z B and 400 K, respectively, for A1. This disagreement may probably be due to the higher content of metalloids in A1. However, if an alloy is chosen which has a Tc = 4 0 0 K, i.e. for y = 0 . 3 5 , its /zxM~0.68 is comparable with that of A1. From the MSssbauer studies on the same system it is reported that for alloys with y < 0.4, the magnetic moment of iron is ( 1 . 7 3 _ 0.03)/~ B. This is used along with the magnetization data to infer that ]2,Ni 0.12/x B. Compton scattering and hyperfine interaction studies [33] by the same group reveal that the earlier assumption of Ni carrying zero moment is not true. Using these values, we make an attempt to calculate the magnetic moments of Co and Cr in our alloys. In A1, since Xcr = 0, subtracting the contributions of Fe and Ni using the above values, we find the contribution of Co to be 0.67~u. With the knowledge of the moments of Fe, Co and Ni, we calculate the moment of Cr in A2, A3 and A4. We assume in this calculation that the fall in the moment with the addition of Cr is due solely to the moment of Cr and in the process the =

moments of Fe, Co, and Ni do not change. Under this assumption, the calculation shows that Cr has to be attributed with - 1 . 5 , - 1 . 3 4 and -0.85/x B in A2, A3 and A4, respectively. The negative sign means that the moment on Cr is of opposite sign to those of Fe, Co, and Ni. From similar arguments extended to F e - N i - C r - B - S i alloys [29], - 4 / z B is attributed to Cr atoms. Hasegawa and Kanamori [34], based on a study of F e - C r system in the coherent potential approximation (CPA), show that Cr atoms carry about -0.75/~ B in the low Cr concentration and then the moment progressively decreases and becomes zero at 50 at% Cr. The study also shows that, under the same conditions, even the Fe moment decreases. The theoretical results are in good agreement with the neutron diffraction data. Thus our assumption that the moments in Fe, Co and Ni remain constant may not be quite justified. 4.2. Temperature dependence o f magnetization The magnetization data taken at H = 10 kOe are analyzed in terms of the spin-wave and other contributions to the thermal demagnetization of samples A 1 - A 5 and B5. The analysis is extended up to T - 0 . 5 Tc as it is well known that in amorphous ferromagnets the range of spin-wave analysis can be extended to temperatures as high as 0.8 Tc. A least-squares fitting method was used to differentiate between the various fits. We find that in samples containing Cr (A2, A3 and A4), a fit to eq. (2) containing both T 3/2 and T 5/2 terms yields a negative coefficient for the T 5/2 term. The negative coefficient seems to be unphysical. The X 2 value does not change appreciably with the inclusion of the T s/2 term. Similarly, inclusions of T 4 or T 7/2 and T 2 terms, in addition to the T 3/2, result in negative coefficients for the higher terms. This shows that M ( T ) of these alloys, within the experimental resolution of the present measurements, is best described by a T 3/2 term arising from the q2 term in the spin-wave dispersion relation and the T 5/2 term is negligibly small. Figure 2 shows the plot of M ( T ) / M ( O ) versus T / T c of these alloys. The solid lines are the best fits to the data. The coefficient B of this term is observed to increase

A. Das, A . I ( Majumdar / Magnetic properties o f Co-rich amorphous alloys

with the increase in Cr. The spin-wave stiffness constant D(0) is obtained from the coefficient B (eq. (7)) and is observed to decrease with the increase in the concentration of Cr. The value of D ( O ) / T c remains nearly constant at (0.265 + 0.015) meV ~,2/K (table 1). In sample A1 (without Cr), the inclusion of a T 5/2 term, in addition to the T 3/2, does not change the X 2 value appreciably but gives a T 5/z term which is of the same sign as the T 3/2 term but of magnitude 104 times smaller (table 2). The coefficient obtained is, at least, an order of magnitude smaller than that observed in F e - B - C alloys [35]. A comparison of the value of D(0), obtained from the T 3/2 term, with an alloy of similar composition [13] having a T c = 400 K, shows that our result is in agreement with theirs of 100 meV ,~2. We have also attempted to observe the effect of the inclusion of other terms in the fit. The addition of another term due to single-particle excitations would involve too many parameters, so we have tried to see the effect of the other terms in place of the T 5/2 term. A very marginal improvement in X 2 value is found when, apart from the T 3 / 2 term, another term due to the single-particle excitations is incorporated, with 1.00

.

~

H = 10 kOe

0.95

0 A2

~o.9o I---

0.85

0.80 o.lo

0.20

0.30

0.40

o.5o

T/Tc Fig. 2. Normalized magnetization M ( T ) / M ( O ) versus T ~ T c of samples A2, A3 and A4. T h e curve of A4 is shifted along the M-axis for clarity. T h e solid curves are fits of the experimental data to eq. (2).

53

Table 2 Fit of A M / M ( O ) ( = M ( T ) - M(O)/M(O)) to various equations for samples A1, A5 and B5 (a) (b) (c)

- B T 3/2 - B T 3/2 - C T 5/2 - B T 3/2 - C T 3/2 e x p ( - A / k B T ) B (K - 3 / 2 )

C (K -3/2 or (K-S/2)

Sample A1 (a) 6.1 x 10 - 5 (b) 6.0 × 10-5 (c) 6.0 × 10 -5

6.3 × 10-9 2.6 x 10 - 5

Sample A 5 (a) 8.7×10 -5 (b) 5.1 x 10 -5 (c) 7 . 6 x 1 0 -5

2.7 X 10 - 7 3.3×10 -4

Sample B5 (a) 9.1 x 10 - 4 (b) 5.0X10 - 5 (c) 1.3X10 -5

2.5×10 -7 1.1×10 -4

1

__ (Y/(fit)

- Y/(data))

A/k u (K)

X 2a

270

7.3 × 10-7 7.2× 10 - 7 7.2X 10 - 7

400

3.2 x 10 - 6 4.3 × 10-7 3.1 X 10 - 7

71

1 . 0 x 10 -5 6.0×10 -7 1.1x10 -7

2

A #= 0 (eq. (13)), i.e., T 3/2 e x p ( - A / k B T ) . In this analysis, A is taken as a parameter and is obtained from the least-squares fit. The inclusion of this term does not affect the coefficients of the T 3/2 term and gives a value of A / k a = 270 K. The coefficient of the single-particle term is only a factor of 2 small than that of the T 3/2 term (table 2). The value of A / k a obtained for Ni, from the saturation magnetization data, varies between 156 and 743 K [36] and the coefficient of the term between (0.5 and 6.2) x 10 -5 K - 3 / 2 . A similar analysis on FexNi80_xBlaSi 2 alloys shows that the Stoner gap A / k a varies between 20 and 60 K. We find that, in the case of A1, the choice whether the deviation from the T 3/2 term is to be attributed to the T 5/2 term or to the Stoner single-particle excitations is difficult to make from the X 2 values alone. However, we choose the fit with the T 5/2 term as it gives consistent results for all the samples as shown below. In contrast to the behavior observed in A1, in A5 a term in addition to the T 3/2, of the form T 5/2 decreases the )(2 values by an order of magnitude, thus showing a clear necessity for an

A. Das, A.I~ Majumdar / Magneticproperties of Co-rich amorphous alloys

54 1.00

kOe

I.-,,0.95 0

" ~ 0.90 At

0.85 0.0!

O. 15

0.25

0.35

0.45

T/Tc Fig. 3. Normalized magnetization M(T)/M(O) versus T~ Tc of samples A1, A5 and B5. The solid curves are fits of the experimental data to eq. (2).

additional term. The coefficient of the T 5/2 is smaller than that of the T 3/2 term by a factor of 100. Figure 3 shows plots of M ( T ) / M ( O ) versus T / T c for samples A1, A5 and BS. The continuous line is the best-fit curve and the coefficients are given in table 2. The lowest X2 is again, as in A1, for the fit with the exponential term. The value of A / k B obtained in this case is 400 K and the value of the coefficient is larger (by a factor of 4) than the coefficient of the spin-wave term (T3/2). The value of D ( O ) / T c, obtained from this fit, is in agreement with those obtained for samples A1-A4. Again, as in A1, we have chosen only the T 5/2 term. The marginal improvement of X 2 with the inclusion of the single-particle excitation may be due to the greater number of parameters, or it may actually be present. But in the absence of D values obtained independently from other measurements we do not interpret our data on the lines of the Stoner excitations. Similarly, in B5 a T 5/2 term, in addition to the T 3/2, is found to be necessary to obtain a good fit. The coefficient of the additional term is two orders of magnitude smaller than that of the T 3/2 term. Here too, the exponential term when included gives the best X 2 value. However, the

value of A / k B obtained is ( ~ 71 K) considerably smaller than those obtained for samples A1 and A5. We also find that in this sample the coefficient of the spin-wave term is small, giving rise to a large value of D(0) and the ratio of D ( O ) / T c is also considerably larger than those in the other set of samples. Another feature found in both samples A5 and B5, is that a two-term fit with A - B T 2 gives a good fit with comparable X 2 values. Addition of a T 3/2 term to this results in only a marginal improvement of X 2. Puznaik et al. [37] found that, in Co-rich C o - S i - B alloys, a fit of the form A - B T 3/2 - C T 2 best describes their results. However, they suggested that the origin of the T 2 term is not due to the single-particle excitation. They noted that the Co-based alloys are best described as strong ferromagnets and the T 2 term in their study is associated with the small value of the mean-square dispersion A of the exchange integral J~j. To summarize, we find that the magnetization of (i) Cr-containing samples (A1-A4) is described by the T 3/2 term alone, and (ii) A1, A5 and B5 has contributions also from a second term of the form T 5/2 in addition to the T 3/2 term. It is seen that to compare various values of the coefficients B and C with widely varying Tc's, it is better to write them in the form B3/2 - - B T 3/2 and C5/2 = CT~/2 (eq. (3)). We find that B3/2 varies between 0.26 and 0.66. The value of B3/2 varies with Tc in different ways in the Cr- and Mn-containing samples. It decreases with decreasing Tc in Mn-containing samples, in agreement with the observation of Kaul [38], but in Cr-containing samples it increases. Since the values are varying over such a wide range it is difficult to compare them with the theoretical calculations [39] based on the Heisenberg model of localized spins. A similar large variation in the values of B3/2 has also been reported in C o - B - S i alloys [40]. The theoretically calculated values for B3/2 a r e 0.587 and 0.512 for S = 1 / 2 and S = 1, respectively. It was found that ferromagnetic glasses are much better described by this model than the corresponding crystalline alloys. As in other amorphous ferromagnets the values o f B3/2 is considerably large as compared to the value of about 0.12 found in crystalline Fe, Co and Ni.

A. Das, A.K Majumdar / Magneticproperties of Co-rich amorphous alloys

o~

< >

55

120

tion of T o c a n be represented by an empirical relation:

100

D(O) = D O + m r c ,

80

Q)

E 80 * A4

,.-.,40 0 C~ 20

0

I

0

I

IO0

I

I

20{3

i

I

300

I

400

Tc(K) Fig. 4. Variation of spin-wave stiffness constant ( D ) with Curie t e m p e r a t u r e (Tc).

where D O and m are the intercept and the slope, respectively, of the straight line. Extensive data on F e - N i amorphous alloys having Tc down to temperatures ~ 30 K show that D O# 0 [38]. It is o2 found to be --- (27 + 2) meV A . Luborsky et al. [43] found three sets of data that give a linear dependence between D and Tc corresponding to (1) F e - B - X , (2) F e - N i - B - P and (3) C o - X in increasing order of D. For (1) extrapolation to zero D occurs at ,-, 380 K, for (2) at about 200 K, and for (3) at about 0 K. Similar observations have also been made in C o - B - S i alloys [40]. These extrapolations, however, suffer from the drawback that they are from very large values of Tc ~ 400 K, as pointed out by Kaul [38]. For a nearest-neighbor cubic ferromagnet the Heisenberg model gives the expression for Tc as

[81: Tc = [2S(S + 1 ) / 3 k a ] zJ, The magnitude of C5/2 of A5 and B5 is about an order of magnitude larger than those reported in Fe- and Ni-rich metallic glasses [41,38,42]. The theoretically predicted values are found to be 0.156 and 0.195 for S = 1 / 2 and S = 1, respectively. Since the value of C is very small, the error is also relatively large. The ratio of C / B gives the range of the exchange interaction (eq. 8) (rE) 1/2. It is found that in A1 the range is of the order of I A, whereas in A5 and B5 it is about 7.5 ,~. Magnetization study on crystalline N i - F e - C r alloys [21] suggests that the introduction of Cr suppresses the anharmonic term (T5/2). We find that the coefficient of this term is very small in A1. With the substitution of Cr (A2-A4) this coefficient may be further suppressed and this might explain the observation of only the T 3/2 term in these samples. Figure 4 shows the plot of D(0) versus Tc. There exists a correlation between D(0) and Tc in that D(0) is found to decrease with decreasing Tc. The plot shows that a straight line drawn through the data points will also pass through the origin. Kaul [38] suggested that D(0), as a func-

where J is the exchange coupling constant between nearest-neighbor pairs, S is the localized atomic spin, and z is the number of nearest neighbors. D is related to J by D = Szja2/3,

where a is the nearest-neighbor distance. D, then, is written as:

o

O = [kBa2/2(S

+ 1)]T c.

The above expression shows that D versus Tc should be a straight line that passes through the origin. Kaul [38] calculated the values of the slope as 0.14 and 0.187 meV .~2 K-1 for S = 1 and S = 1/2, respectively using a = 2.55 A. We obtain the value of the slope ~ 0.25 meV ~2 K-~, which is somewhat higher than the above values. In summary, the Heisenberg model explains the temperature dependence of magnetization of the present series of alloys. The samples with Cr are adequately described by the T 3/2 term alone. The coefficient of this term increases with the increase in Cr. The sample without Cr and the

56

A. Dus, A.K. Majumdar / Magnetic properties of Co-rich amorphous alloys

one containing Mn show the presence of a T 5/2 term. The values of D ( O ) / T c vary between 0.2 and 0.3 meV ~2 K-1. The plot of D(0) versus Tc shows a straight line that also passes through the origin, as predicted by the Heisenberg model.

and Technology, Government of India, is gratefully acknowledged.

References 5. Conclusions The magnetic moment and the Tc decrease identically with the addition of Cr. The variation of the magnetic moment with the addition of Cr and Mn is discussed on the basis of the VBS model. We find that the rate of change of magnetic moment with Cr concentration ¢ = d l z / d x = . - 1 . The fall in /~, however, could not be described by the 'virtual bound state' model which gives ~ =/x - ( Z + 10)c~B, where Z is the valence difference between the solute and the host. We obtain Z = - 9 whereas actually it is about - 3 . 5 . On utilizing the values of /~Fe and tzNi obtained by Dobrzynski et al. [13,33] on a composition comparable with A1, we find values of ~cr = - 1.5, - 1.34 and - 0.85/z B in samples A2, A3 and A4, respectively. The variation of ~Cr is in qualitative agreement with the theoretical study of Hasegawa and Kanamori [34]. Spin-wave analysis is extended up to 0.5Tc. Bloch's T 3/2 law adequately describes the temperature dependence of magnetization in Cr-containing samples. The calculated spin-wave stiffness constant D decreases with the increase in Cr. In A5 and B5, additional terms of the form T 5/2 a r e found to be necessary. The results of the fit on including the Stoner single-particle excitations are discussed. The value of D scales with Tc and the ratio D / T c ~ 0.27 meV ~k2/K. When the D ( T ) curve is extrapolated to Tc = 0, within experimental error, it is found to pass through the origin. This result is in agreement with the Heisenberg model.

Acknowledgements We thank Dr I. Nagy of the Hungarian Academy of Science, Budapest, for providing the samples. Financial assistance from project No. S P / S 2 / M - 4 5 / 8 9 of the D e p a r t m e n t of Science

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