Magnetization and susceptibility of disordered alloy with competitive interactions

Magnetization and susceptibility of disordered alloy with competitive interactions

S K G H A T A K

Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur 721 302, India

MS received 15 January 1984

Abstract. A model calculation of magnetization and susceptibility of disordered alloy (ApB~ _p) where both A and B represent the magnetic atoms is presented. It is based on the cluster-variational method where interactions within the clusters of all possible configurations are treated exactly and the rest of the interaction is replaced by an effective variational field.

The frustration effect is introduced taking the exchange interactions JAB or JBB or both to be antiferromagnetie whereas the exchange JAA is ferromagnetic. The results are qualitatively similar to the observed behaviour of moment and susceptibility in some metallic glasses. The critical concentration for ferromagnetic state is determined in the presence of competitive interactions.

Keywords. Disordered magnetic alloy; lsing model; cluster-variational method; frustration;

metallic glass.

1. Introduction

Recent studies in a m o r p h o u s alloys, particularly in transition metal-based glasses, have revealed that the magnetic p h e n o m e n a occurring in these systems are c o m p l e x a n d rich.

The iron-based metallic glass (eg. FecNl -c, N represents glass-former like P, C, Si, B or c o m b i n a t i o n o f them, a n d c lies a r o u n d 75 to 80 ~o is ferromagnetic with well-defined transition and its m o m e n t saturates to its full value at low t e m p e r a t u r e T). But such n o r m a l b e h a v i o u r o f the m o m e n t is significantly altered when some o f the Fe a t o m s are replaced by a n o t h e r transition metal a t o m like Mn, Ni, Cr etc. F o r example, the magnetization in (Fe-Cr)75N25 glass passes t h r o u g h a b r o a d m a x i m u m as T i s lowered below the transition t e m p e r a t u r e Tc but ferromagnetic state persists d o w n to lowest t e m p e r a t u r e studied (Yeshrun et al 1981). O n the other hand, the f e r r o m a g n e t i s m which a p p e a r s at higher Tel collapses at lower t e m p e r a t u r e in (Fe-Ni)75N25 (Bhagat et al 1980; Bhagat 1981; S a l a m o n et al 1981) a n d in (Fe-Mn)TsNz5 (Salamon et al 1981;

M a n h e i m e r et al 1982). T h e collapse o f ferromagnetic state is m a r k e d by vanishing o f s p o n t a n e o u s m o m e n t a n d sharp peak in a.c. susceptibility at second critical t e m p e r a - ture T¢2. T h e disappearance o f f e r r o m a g n e t i s m has also been observed in crystalline AuFe (Crane a n d Claus 1980, 1981; K l e i m a n et al 1982). T h e c o m m o n aspect o f these systems is the disorder which gives rise to a distribution o f the exchange interaction.

Due to the distribution, some o f the magnetic ' b o n d s ' in ferromagnetic alloy can be a n t i f e r r o m a g n e t i c in character. The presence o f such ' w r o n g b o n d s ' m a y n o t be felt at high T, b u t at low T their influence is i m p o r t a n t as some o f the non-aligned magnetic configurations are favoured c o m p a r e d to saturated configuration. These are b o r n e 421

422 S K Ghatak

through the theoretical calculation on a model with gaussian distribution of exchange (Sherrington and Kirkpatrick 1975). The meanfield results point out that the stability o f ferromagnetic state at low T depends on the ratio o f first a n d second moments o f exchange distribution (Kirkpatrick and Sherrington 1978).

As short range atomic order prevails in these systems a n d it is more appropriate to take this into account in the model calculation. This can be d o n e in the framework o f cluster-variational approach (Oguchi 1976; G h a t a k 1978). In this paper, the effect of 'wrong bonds' or the 'lYustration' on magnetic properties is investigated theoretically assuming a simple model for the above systems. The metallic glasses can be represented by a general formula (ApB x -p)cN~ -~ where A and B are two kinds o f magnetic atoms with respective concentration pc and (1 - p ) c , and N being metalloids with concentra- tion ( 1 - c ) . The metalloids stabilize the amorphous phase and reduce moment of transition metal atom due to p-d hybridization (also partly responsible for exchange distribution). Therefore, so far as magnetic interactions are concerned, the above alloy can be modelled as a binary alloy ApB1 _p with A and B having some effective moment.

Taking this simple picture, the alloy is treated using the cluster-variational method described in §2. The frustration effect is invoked taking either (A-B) or (B-B) or both bonds to be antiferromagnetic in nature. The results and discussions are given in §§3 and 4 respectively.

2. Model and approximate free energy

As the metalloids do not carry any moment the metallic glass can be viewed as a binary alloy ApB~ _p o f two magnetic components A and B. It is also assumed that all magnetic interactions are localized in character and pure-A system is ferromagnetic. The addition o f B-atom introduces the 'frustration' effect into the system. We further assume that the magnetic interaction can be described by Ising Hamiltonian.

H = - 1~, J,jS,=Sjz - h E Siz, (1)

i, j i

where Siz ( + 1) is the Ising spin at the ith site and h is the external magnetic field. The exchange interaction Jo takes three values JAA, JBB and JAB for magnetic bonds A-A, A- B and B-B respectively. The free energy o f the system is given by

F = - k T [In Tr exp ( -/~H)]a,, (2)

where I---lay indicates the average overall possible configurations and fl = ( k T ) - 1 Defining V = H - Ho where Ho is a suitable chosen Hamiltonian whose eigenstates are known, F can be expressed as

F = Fo - kT[ln ( e x p ( - f l V ) ) ] a v (3)

where

Fo = - kT[ln Tr exp ( - flHo)] and (- - -

is ensemble average over the states of H o which is taken as

H 0 = - ~ I i S i z . i

The approximation involves the decomposition o f the system into small 'building

blocks' and use o f / a s variational parameter so that the variational f o r m o f f r ~ energy (Oguchi 1976; G h a t a k 1978)

L

F~ = Fo - k r ~ [ln (exp(--flV,)>]a~.

(4)

l = l

Here V~ is the lth division o f Vwhich is divided into L number o f blocks. With decrease of/_,, F~ tends to exact value. In order to make the calculation tractable the block o f four atoms is taken as 'building block' (figure 1). The configuration average is taken assuming two kinds o f atoms appearing randomly in the block. The variational free energy is found to be (Ghatak 1978)

"[{.

}

+ pZq2

{4 In

+

)}]

1

Zo = X~ cosh(4Y) + 4 cosh (210 + 2 + X ~ 4,

Z , = X 2 X 2 cosh (4Y) + (2 + X ; 2 Xc 2 + X 2 X c 2) cosh (2Y) + 2 + X~- 2 X c 2, Z2 = Xc 4 cosh (4Y) + 4 cosh (2}I) + 2 + X c 4,

Z~ = X a X Bx2c cosh (4IO + 2(X aX~ 1 + X ; 1 XB)cos h (2 Y) + X~ 1 X ; 1 X c 2 + XAXBXc 2 + X ~ X ~ I X 2 c,

X a = exp(flJaa/2); XB = exp(flJBB/2); Xc = exp(flXaB/2), f l [ ! h + l i ( 1 - 2 / n ) J and q = 1 - p .

Y =

F r o m the expressions o f Zo and Z1, the expressions for Z4 a n d Z3 can be obtained replacing X a by X B. Here small n is the first coordination number. In ferromagnetic state the effective I i is site-independent and its value is determined from the minimization o f F~

dFd~I

=

p4 4p3q 4pq3

! ,, II i

- t . . . . 4 - I q "~ .. . . i"

2 p2q 2 4 p2q2 q4

Figure 1. Schematic representation of building blocks of four atoms• Solid, dashed and dot- dashed lines represent respectively Jaa, JaB and Jsa. The probabilities of different configurations are given below the diagrams.

424 S K Ghatak

This results in a self-consistent equation o f I a n d the a p p r o x i m a t e free energy then becomes

Fo = fv (Io), (7)

where Io is the non-trivial solution o f (6) which minimizes Fv. The reduced magnetization (a) per atom is given by

= tanh (fllo), (8)

and the susceptibility

X = [fl sech2 (fllo)] (Olo/c3h). (9)

E q u a t i o n (6) is solved numerically for two different concentrations taking either JAB, dna o r b o t h to be antiferromagnetic in nature. In the following, the parameters are redefined as T = kT/JAa; ~t = JAB/Jaa and A = JBa/JAA and all the results are for n = 8.

3. Magnetization and susceptibility

3.1 Concentration p = 0.5

3.1a p = 0-5: T h e magnetic properties o f such alloy are strongly influenced by the presence o f 'wrong' magnetic bonds. T h e results for two typical cases o f alloy with p = 0-5 are presented here. In the first case, only interatomic exchange JAB is taken to be antiferromagnetic with ~t = -0-1 and small Jss (A = ~1). The thermal variation o f m o m e n t is shown in figure 2. The ferromagnetism i.e. non-zero spontaneous m o m e n t appears at a critical temperature Tcx and instead o f remaining at this state up to T = 0°K it disappears at lower transition temperature To2. As the magnitude o f ct increases the difference between Tel and T~2 decreases and the ferromagnetism disappears completely with sufficient magnitude of'~c This critical value o f ~ depends on A. This suggests an existence o f tricritical point. The maximum value o f spontaneous m o m e n t which varies with 0t and A is less than unity. The rise o f m o m e n t below T < T~I shows flattening which is characteristic o f disordered system (Handrich 1969; Jones and Yates 1975;

Simpson and Brambly 1972; G h a t a k 1978). The rate o f fall o f m o m e n t around T,2 is

A= 0.1 I

0.5

0.,5 1.0

T/Tel

Figure 2. The reduced magnetization for p = 0-5 is plotted against reduced temperature for

a = -0-1, A = 0-1 and different values of magnetic fields. The transition temperature T,l = 0"748JAA.

much sharper than that around Tel. This shows that the fluctuation around To2 which is due to frustration is much stronger than thermal fluctuation around T~. The collapse of ferromagnetic phase has been observed in (Fe-Mn):sN25 glass (Salamon et al 1981;

Manheimer et al 1982) where (Fe-Mn) magnetic bond is considered to be antifer- romagnetic in character.

In the presence of magnetic field the moment increases more rapidly around T¢2 than that at higher transition temperature (figure 2). The maximum o f moment is shifted to the lower temperature side by the magnetic field. The result also shows that it is difficult to saturate at low T which seems to be the characteristic of a frustrated system. For complete alignment of non-aligned moment the Zeeman energy should be much larger than the energy associated with 'wrong' bonds.

The magnetic susceptibility in the presence of a small field has been obtained from (9) and is plotted in figure 3. It shows the two-peak structure. With increasing field, tbo

3 0 0

2 0 0

o l L I L-~--~

0.10 0-15 0.20 0.6

Temp.

a = - 0 . 1

&= 0.1

• , ( 1 )

... ( 2 ) . . . { 3 )

- - - - ( 4 )

----t5)

0.8 1.0

Figure 3. Magnetic susceptibility Z for p = 0-5 as a function T for different magnetic fields.

The numbers I to 5 refer to field h = 10- 4, 5 x 1 0 - 4, 1 0 - 3, 3 x 10- 3 and 5 x 1 0 - a respectively.

All are in units of JaA.

0 . 7 9

0 . 7 7

0 - 7 5

p = O , 5 A : 0 . 1

I I ! l I I

2 ,4 6

( x lO ~' )

0 . 1 7

0 . 1 5

O . t 3 * _ ~

0 . 1 1

Figure 4. Plot of peak temperatures T~*I and T~* 2 against square root of magnetic field.

426 S K Ghatak

peak appearing at high temperature (/"1) shifts towards the higher temperature region and that at low temperature (T* 2 ) shifts towards the lower temperature region. The increase o f T*I is linearly proportional to the square root o f magnetic field (h) (figure 4) a n d the extrapolated value is Tel = 0-747 which is the same as obtained from moment behaviour (figure 2). It is to be noted that this value is much smaller than the paramagnetic Curie temperature 0p = [p2 -k 2pqg + q2A] ]J.4 A which is 0-9 for this case.

This result is the consequence of the duster approximation. Similarly, the decrease of T*2 is also a linear function o f x / ~ (figure 4) and again the extrapolated value gives transition temperature To2 = 0-167 (figure 2). A similar field dependence o f T*2 has been reported in metallic glass (Manheimer et al 1982). In AuFe (Kleiman et al 1982), T*I follows such field dependence but T*2 cannot be fitted with square-root field variation. The peak susceptibility z(h, T*) when plotted in double logarithmic scale against field h, gives a straight line (figure 5). This indicates power-law behaviour o f the form

Z (h, T* 1 ) ,-~ h (t - t)

The exponent for height transition is estimated to be c~ = 2.85. Similar plot for X (h, T* 2 ) against h also yields a straight line (figure 5) with 6 = 3-27. A larger value o f ~ at T*2 indicates the sharper response o f the system. Such power-law behaviour has been reported in a number o f disordered alloys (Manheimer et al 1982; Kleimann et al 1982).

In AuFe alloy (Kleimann et al 1982) data at lower transition appear to be not at par with such power-law behaviour.

The calculations are then carried out taking both JAB and JsB negative. The results are shown in figure 6 and are very similar to the above case. It is to be noted that the magnitude o f both bonds are small compared to JAA. The ferromagnetism appearing at higher temperature collapses at lower transition temperature. The susceptibility also shows two-peak structure and the shift of peak temperature also follows square root dependence on field. When JBB is only negative, ([J~BI < JAA), the ferromagnetic state, persists with smaller moment and the qualitative behaviour is similar to the concentrated case discussed below.

3.2 Concentration p = 0.7.

F o r concentrated alloy p = 0.7, the magnetization is given in figure 7 for different values o f ct = - 0 . 1 , - 0 - 2 and - 0 - 5 for A = 1. It is seen that with small antiferromagnetic interaction the m o m e n t drops down at low temperature and does not saturate to its full value. The ferromagnetism persists down t o T = 0. The maximum

°

x _5

P - 0-5

I I I I

- 8 - 6 - 4 - 2

tn h

6

4

2.

0

Figure 5. Peak susceptibility plotted against magnetic field on double-logarithmic scale.

3 0 0

2 0 0 - -

1elf

I 0-1(

0 - 5

Temp

p = 0 - 5 tx = ~ = - 0 - 0 5

- - ( 1 ) 0 - 5 ... 12.1

. . . . 1 3 1

o- ^{- - - -} 14)

- - - - ( 5 1

0 . 2 0 0 - 6 0 0 - 8 0 1 - 0 0

T e m p .

Figure 6. Magnetic susceptibility ~ as a function temperature. Inset shows thermal variation of magnetization for different fields at ~ = A = - i f 0 5 .

1 . 0

0 - 5

l

I I I I I i I I ~ \,,d

0 - 5 1.o

T / T c

Figure 7. Corresponding curve for p = 0 - 7 f o r A = 1 a n d a = - 0 - 1 , - 0 . 2 a n d - 0 - 5 . The dotted lines refer to magnetization for A = 1 a n d , , = - 0 - 2 in the presence o f field h .

value o f spontaneous moment decreases and the maximum is shifted towards the higher temperature region as the magnitude of JAB increases. The effect o f the magnetic field is pronounced at low temperature as shown in figure 7 (dotted line). The maximum moves towards lower temperature region and at sufficiently strong field it disappears. This is in qualitative agreement with the observation in (Fe-Cr) glass (Yeshrun e t al 1981). Similar qualitative behaviour is found for A < 0. The susceptibility behaviour and variation of T c with ~ and A will be discussed elsewhere.

3.3 Ferromagnetism at T = 0

In view ofthe above results, the existence o f ferromagnetic state at T = 0 in the presence of antiferromagnetic bonds of smaller magnitude is examined from the limiting

428 S K Ghatak

behaviour o f equations (6) to (8). F o r JAB < 0 but [JAs[ < JAA and JaB > 0, the system remains ferromagnetic for concentration p > Pc where Pc is the solution o f 1 + 3p~

- 3pc = n/[4(n - 2)] and Pc = 2/3 for n = 8, when both JAB and Jan are negative and their magnitudes are < JAA, the equation for Pc becomes

(i) p~ = n/[4(n - 2)] for I JAB [ > I JBB I

(ii) p2(2 --Pc + 2p~) = 3n/[4(n -- 2)] for JAB = JaB and (iii) p2(1 - p c + p 2) = n / I n ( n - 2 ) ] for

IJABI < IJBBI"

The solutions for n = 8 are respectively Pc = 0.69, 0-67 and 0-66. These values indicate the smaller influence of JBn than that o f JAB which is due to the presenc¢ o f more number o f ( A - B ) bonds than (B-B) bonds. When JBn is only negative and IJnBI > JAB, the equation for Pc, is 2pc - 2Pc + 2Pc - Pc = 2 3 4 n/[4(n -- 2)] and Pc = 0.2 for n = 8. For [JBBI

< JAn, there is no Pc i.e. ferromagnetic state continues up to T---- 0.

4. Discussions

The above results clearly demonstrate the strong influence o f wrong bonds in ferromagnetic state. In all the cases, the susceptibility Z ~ ( T - 0p)- 1 at high T ( T ~> To) with positive paramagnetic curie point O r = (~ x J--) where J - = [ p + 2 p ( 1 - p ) ~ t + (1--p)2A]JAA. Noting that the probability distribution o f exchange P ( J ) = p 2 6 ( j -JAA) + 2p(1 - - p ) ~ ( J - JAB)+ (1 - - p ) 2 6 ( J --JBB); the mean value or the first moment o f distribution is J. This means that the thermal fluctuations at high temperature wash out the subtlety o f frustration effect and the system behaves as if it has exchange interaction o f strength J. As the temperature is lowered, the strongly magnetic state evolving below Tc~ is ferromagnetic, and the neighbouring spins coupled by fer- romagnetic exchange form the percolation chain. Subsequent evolution o f magnetic state with zero-moment at lower temperature (To2) can be due to either the formation o f compensated spin state, or 'freezing' of individual spin or realigned spin structure with zero-moment. On cooling below Tel there is a gradual inclusion of'loose' spin or small cluster o f spins with percolation chain via antiferromagnetic bonds and this causes a decrease o f moment due to compensation effect leading to zero moment at To2.

Alternatively, there can be complete breakdown o f percolation chain due to competit- ive exchange and the spins 'freeze' out individually. Lastly, there can be total realignment o f the spins in the percolation chain such that it becomes a structure with zero h o m o g e n o u s moment (in this case inhomogenous moment o f non-zero wavevec- tor can exist). But it is difficult to conceive such a structure in the presence o f small antiferromagnetic interaction. Moreover, large field sensitivity around To2 seems to favour the first two pictures. The enhanced response can be understood noting that the compensation effect and the spin freezing can be suppressed by a magnetic field as it favours ferromagnetic orientation o f spins. It is expected that wavevector-dependent susceptibility will be different in two cases. But considering the results o f concentrated case (p = 0-7) where ferromagnetic state prevails down to T = 0°K the compensated picture is more plausible as complete breakdown o f percolation chain is ruled out owing to finite spontaneous moment. Hence, assuming the same physical process to operate in both cases it can be suggested that competitive exchange leads to a state with partial or full compensation of percolation chain. Similar views were also put forward

earlier (Kleiman et al 1982). A point about the method is in order. The configuration averaging pr .ocedure used here takes care of different exchange configuration in more detail than the mean-field method with continuous exchange distribution (Sherrington and Kirkpatrick 1975; Kirkpatrick and Sherrington 1978). In the latter method, the configurations, where all exchange bonds are either positive, negative or zero, are considered and the possibility of antiferro-bond in a configuration with predominant ferro-bonds does not appear. But such configurations are important at low temperature for moment reduction and their effects have been included in this procedure.

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