Thermodynamics of ferrimagnetic Ising chainsR. Georges, J. Curely, and Marc Drillon

Citation: Journal of Applied Physics 58, 914 (1985); doi: 10.1063/1.336165 View online: http://dx.doi.org/10.1063/1.336165 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/58/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ferrimagnetism of dilute Ising antiferromagnets Low Temp. Phys. 40, 36 (2014); 10.1063/1.4850534 Ground State and Thermodynamic Properties of the Spin1 Antiferromagnetic Ising Model on a Chain inTransverse Crystal Fields AIP Conf. Proc. 850, 1081 (2006); 10.1063/1.2355079 Thermodynamics of alternate Ising chains of spins 1 and 3/2 with dipolar, biquadratic, and single ion interactions J. Appl. Phys. 81, 4198 (1997); 10.1063/1.365119 The disordered Ising chain: Equivalent formulations for the thermodynamics J. Chem. Phys. 65, 4512 (1976); 10.1063/1.432999 Green's Functions for the Ising Chain J. Math. Phys. 9, 1602 (1968); 10.1063/1.1664489

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Thermodynamics of ferrimagnetic ~sing chains R. Georges and J. Curely Laboratoire de Physique du Solide. Universite de Bordeaux I. 351. cours de [a Liberation. 33405 Talence Cedex. France

Marc Drillon Departement Science des Materiaux. Ecole Nationale Superieure de Chimie. 1. rue Blaise Pascal. 67008 Strasbourg Cedex. France

(Received 1 October 1984; accepted for publication 4 February 1985)

The exact solutions of the so-called ferrimagnetic Ising chain made up of two sublattices (SO,sI) are derived from a transfer matrix method. The short-range ferrimagnetic order occurs when considering different spins and/or different Lande factors on both sublattices. Most of the physical features of interest are shown to be involved in the So = SI = 1 system including an alternation of Lande factors (go,g d and local anisotropies (Ko.K d. Thus, in the limit Ko- 00

stabilizing a Kramers doublet, SZ = ± 1, on even sites, the system behaves like the ferrimagnetic chain (So = 1/2, SI = 1). The susceptibility and magnetization curves are discussed in various situations as a function of the significant parameters.

INTRODUCTiON

In connection with the large coHection of so-called one­dimensional. materials, several theoretical studies have fo­cused, in recent years, on the thermodynamics of regular exchange coupled chains. l

-4 The combination of spin mo­ment operator, treated as a classical or a quantum vector, spin dimensionality related to crystal field effects and com­plexity of the exchange coupling which may involve non­symmetric components leads to a variety of interesting phys­ical situations. Until now, dosed form expressions of the thermodynamic quantities have been derived for the S = 1/ 2 and S = 1 Ising chain,5-7 the S = 112 XY one, 8 and for the classical limit S-oo with arbitrary spin dimensionality.9.10 In the other cases, approximate techniques were required to estimate the behavior in the limit of the infinite chain; we can mention spin-wave theory, II high-temperature series expan­sions. 12- 14 Green's function approaches,15 or numerical cal­culations on limited-length chains applied when an isotropic exchange coupling is assumed. 16-18

Thus. although a lot of work has been spent on the be­havior of regular chains, the investigation of so-caned ferri­magnetic chains made up of two unequal sublattices is a challenging topic that has only recently arisen. In what fol­lows, we will. consider as ferrimagnetic chains one-dimen­sional systems (1D) involving alternating different magnetic moments (due to different spins or/and different Lande fac­tors). In previous studies Dembinski and Wydro,19 Blote,20 and Seyden21 have proposed analytic expressions of the ther­modynamic functions of interest for the classical-quantum ID system. Recently. we have soIved the problem of Heisen­berg quantum chain showing a regular alternation So = 1/2. SI = 1 from the numerical treatment of finite closed chains. 22 In this investigation, the influence of the ratio between Lande factors was emphasized, in particular when these lead to a compensation of the two sub lattices.

Only a few examples of structurally ordered bimetaHic chains have been reported so far. Note the series MM' (ED­T A)6H20 characterized by Beltran et al. where M and M'

stand for divalent transition metals23•24 and the complexes

AMn(S2C202hnH20 (A = Cu,Ni) recently investigated.25

In the former. several situations were emphasized related to some particular combination of the metal ions involved in the exchange-coupled chain. For instance, we can mention Heisenberg ferrimagnetic chain for the [Mn-Ni] complex, Ising ferrimagnetic one for [Co-NiJ, due to the fact that Co(n) ion is highly anisotropic. while [Mn-CuJ is a good candidate for the study of classical-quantum alternating chain. In connection with the situation underlined in CoNi(EDT A)6H20, we report here on thermodynamics of an Ising ferrimagnetic chain of quantum spins having both distinct Lande factors and different local anisotropy con­stants on the two sub lattices.

TRANSFER MATRIX AND PARTITION! FUNCTION

The transfer matrix method2b is particularly convenient for solving exchange-coupled Ising chains with an external magnetic field B applied along the quantization axis. In ad­dition, an uniaxial anisotropy along the same axis may be introduced. The Hamiltonian is expressed as

JY= -JIS~S~+I - IK,,(S~2-S~) " "

(1 )

Here S", gn' and K" are the spin quantum number, the Lan­de factor. and the anisotropy constant (zero-field splitting) at site n. It must be noticed that, with the present notation, positive K" values stabilize the S~ = ± S" states. J refers to the exchange interaction between nearest neighbors only (J < 0 denotes an anti ferromagnetic coupling). For the so­caned ferrimagnetic chain, S",g" , and K" take the val.ue So, go,KoandSI,gl' KI at even and odd sites, respectively. Let us now consider a semi-infinite chain ending at site n, and let Z" be the corresponding partition function. We call Z ~ the con­tribution to Z" arising from aH the eigenstates such that S ~ = 0', and we consider the vector Z" whose 2Sn + 1 com-

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ponents are Z !., Z !.- I, •• Z ,,- s •. When adding one extra spin in position n + 1, it is readily shown that Zit and Zit + I are linked by a linear transformation and, in the specific case of the ferrimagnetic chain, we must distinguish the cases where the extra spin is in odd or even position. For any p index, we put

and

(2)

where the current elements (line index: I; column index: c) of (To) and (TI) may be shown to be

(TO)I.c = exp!,B [J(So + 1 -l)(SI + 1 - c)

+KoS~(l-1){/-3)+goIlBB(So+ 1-1)]1.

(Ttll.c = exp!,B [J(S! + 1 -I )(So + 1 - c)

+K1SW -1)(/- 3) +gIIlBB(SI + 1 - /)] J, (3)

where,B = 1/ kT is the Boltzman factor. Then we obtain

'4.P +2 = (T)Z2p

with

SO(SI02 + XI + ~ ) SIO

so(sio + XI + _1_) SIO

(T)= XO(SIO + XI + _1_) Xo( S I + X I + :1 ) SIO

.l(SI +X I + .l ) 1 ( SI 0 ) - - +XI+-So SI So 0 SI

It turns out that for vanishingB (so = SI = 1), (T) has an evi­dent eigenvalue:

(7)

The two other ones are then immediately available and by labeling as U2(0) and U3(0) the smallest and the largest eigen­value, respectively, we have

S ( S2 )112 U3(0) = - + --p 2 4 '

with

(8)

P = XoXI(01/2 - )/2 r· For nonvanishingB, the largest eigenvalue U3(B) [which ad­mits the limit U3(0) when B vanishes] is a solution of the secular equation

U3-A2!B)U2+AdB)U-Ao(B)=0, (9)

where theA;'s are related to the elements oft T). Forinstance,

915 J. Appl. Phys., Vol. 58, No.2, 15 July 1985

It is well known that, in the limit of very long chains, we can consider an effective partition function per pair of sites which is nothing but the largest eigenvalue of the product matrix (T). It is quite easy, using a computer, toevaluateZas a function of the significant parameters (J, Ko, KI,go,gl' ond B) and to obtain numerical values for the thermodynamic functions of interest such as the specific heat, magnetization, and susceptibility.

THERMODYNAMICS Of THE TWO-SU9LATTICE ISING CHAiN

Ferrimagnetlsm due to different g factors

Most of the physical features of the ferrimagnetic chain are elucidated by the So = S! = 1 system with an alternation of Lande factors as well as anisotropy constants. Fortunate­ly, this problem may be solved exactly without the use of computer so that we shall only deal with this situation. Here­after, we shaH use the following notations:

o = exp(f3J),

xj=exp(-,BKj) (i=O,l),

s; = exp(f3g;IlBB) (i = 0,1).

Then (T) is the following 3 X 3 matrix:

So( S I + X I + :1)

( SI 0 ) Xo - +x l +-o SI

1 ( SI 02

) - -+XI+-So O2 SI

(5)

(6)

Ao(B) and A 2(B) are, respectively, the determinant and the trace of this matrix. It appears from evident symmetry argu­ments that these coefficients are even functions of B. Fur­thermore, they can be expressed as power series in this vari­able, and we have up to second order

AjIB) =AjIO) +ojB 2 +.... (10)

As a result, U3IB) shows the same properties, and we can write

(11) Inserting these expressions into the secular equation, and taking into account the cancellation of the B 2 term, we can express A in terms of the A;(O) and o/s.

We must now take into account that U3(B ) is the parti­tion function per pair of sites, and that the second derivative of its logarithm versus B is closely related to the correspond­ing magnetic susceptibility X. As a result, we readily obtain the zero-field susceptibility Xo as

Xo = ~ 02U ;(0) - °IU3(0) + 0 0 , (12) ,B U3(0) [ 3 U; (0) - 2A2(0) U3(0) + A 1(0)]

where the various coefficients are easily shown to be

Georges, Curely. and Orillon 915

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a) = (,BJiB)2!4X~J[(a - 1)2g2+ + (1 - ~ Yg2_ ]

+(a- ~Y[(XO+XI)(g2+ +g2_ )-2(xJ -xo)g+g-]l.

a2 = (,BJiB)2[(XO +xtl(g2+ +g2_) + 2(x, -xo)g+g_ + 4(a2g2+ +g2_ /a2)],

AI(O)=2x~J[(I-a)2+(1- ~ y]+2(XO+XJl(a- ~ y +(a2

_ :2Y. A2(O) = x~J + 2(xo + XI) + 2( a2 + :2).

with

g ± = ~(g I ± go)·

It is now of particular interest to examine the tempera­ture dependence of X 0 for various combinations of the anisot­ropy constants and g factors.

Actually, we shall consider the normalized function

X T= XoT , n Ji~(io +g1)

(14)

which is more significant and takes uniformly the value 2/3 in the high-temperature limit (this point win be considered in more details in the following discussion).

Let us first look at Fig. 1, where X n T vs k T IIJ I curves are drawn for several values of go, whereas g I takes the con­stant value 2 and Ko and Klare set equal to zero (no local anisotropy). The curve go!gl = 1 corresponds to the classi­cal, S = 1, ID Ising antiferromagnet, for which X n as well as Xn T vanish at absolute zero. Conversely, for go#g) w.e are dealing with a typical ID ferrimagnet for which X nTIS ex­pected to diverge at low temperatures. The minimum of X n T appearing for go not too different from g I is a weB-known feature that is common to various kinds of ferrimagnetic ID systems.22.24

Before discussing the effect of local anisotropy in the go#gl case, it is interesting to come back to the antiferro­magnetic chain. We have reported in Fig. 2Xn T vs kT IIJ I

X.nT

0.3

0.2

O. I

FIG. I. NormalizedXn T product defined by Eqs. (12H141 vs reduced tem· perature for the two·sublattice Ising chain. This figure shows the influence of a distinction between Lande factors on the magnetic behavior; local an· isotropies are set equal to zero on both sites (Ko = K, = 01.

916 J. Appl. Phys., Vol. 58, No.2, 15 July 1985

(13)

curves, in this case, for several values of the parameter Ko! IJ I, whereasK I is taken equal to zero. Forlarge and negative Ko values (actually Ko < - 21J I), Xn T tends towards a con­stant value at low temperatures. This behavior is due to the zero field splitting effect on even sites which lowers the S z = 0 component strongly enough to overcompensate the exchange coupling which would otherwise stabilize SZ = + 1.

A-; a result, the moments on odd sites feel no exchange field (at least in the low-temperature limit) and behave as if they were free. We only observe their contribution to the Curie constant which varies from (g iii B )2/3k at absolute zero to twice that value in the high-temperature limit. Simi­larly, Xn T varies from 1/3 to 2/3 by showing a fiat mini­mum. For smaller negative values of Ko, a similar behavior holds down to temperatures such that the nearest-neighbor correlations are large enough for the exchange field to over­compensate the local anisotropy effect on the odd sites. Then, X n T suddenly falls from a value close to 1/3 down to O. This behavior is more marked when Ko approaches 2J. For positive Ko, the SZ = ± 1 states are favored on even sites, but this does not significantly affect the behavior of Xn T.

Let us now return to the ferrimagnetic system. We have reported in Fig. 3 the X n T = /(kT IIJ I) variations for go! g) = 0.8 and severalvaluesofKoIlJ I, whereas the anisotropy is assumed to be zero on odd sites (KI = 0). We notice that

• for negative and large enough values of Ko (the limiting value

XnT Ko/IJI

0.3

0.2

O. I

0.00 1.5 kT/IJI

FIG. 2. Normalized.t'" T product vs reduced temperature for the antiferro­magnetic Ising chain (Su = S, = 1; go = g ,) with local anisotropy on even sites only.

Georges, Curely, and Drillon 916

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o.oo:-------~--------~------~------~ O. 5 I . 5 kT IIJI

FIG. 3. Normalized X. T product vs reduced temperature for the ferrimag­

netic Ising chain (So = S, = I; go = 0.8 gd with local anisotropy on even sites only.

being of course + 2J as in the antiferromagnetic case), no short-range ordering occurs at absolute zero, because of the absence of exchange field on the odd sites. As a result, we are again dealing with almost free moments and X n T behaves similarly to the antiferromagnetic case; the main difference concerns the low-temperature limit which now differs of course from 1/3. For Ko > - 21J I, strong correlations deve­lop at low temperatures, but instead ofleading to a vanishing X n T near absolute zero, as previously observed in the anti­ferromagnetic case, the difference between go and gl gives rise to a divergence. The results obtained when choosing go larger than gl are not significantly affected as shown in Fig. 4.

Ferrimagnetism due to different spin numbers

As long as no transverse magnetic field is applied, the general expression of Xo may be extended to the So = 1/2,

!

It is not useful to develop this expression to elucidate large SI spin cases, except when specific experimental results would require such a tedious work. In fact, studying the SI = 1 case enables us to emphasize most of the physical features of ferrimagnetic chain. Then the transfer matrix re­duces to

and the largest root is

U = HS + (S2 - 4P)1/2],

with

917 J. Appl. Phys., Vol. 58, No.2, 15 July 1985

XnT

0.5

0.4

0.3

0.2

O. I

0.00 0.5 1.5 kT/IJI

FIG. 4. We consider the ferrimagnetic system described in Fig. 3, but with go= 1.2g,.

SI = 1 ferrimagnetic chain by considering the limit Ko--+ + 00 giving a low lying Kramers doublet on even sites. (Actually, a straightforward renormalization of go should also be taken into account).

Notice that in this case (T) is a 2 X 2 matrix, thus allow­ing for an exact algebraic determination of the largest eigen­value (which, we recall, is the partition function per pair of sites). We are then able to compute the magnetization and other thermodynamic functions for arbitrarily large applied fields. This is of course of interest if some kind of "metamag­netic" behavior is to be expected. Clearly, this possibility is not related to the value of S) which actually may be as large as needed.

With the previously used conventions, and the general expression (3) for the elements of (To) and (T1), we get imme­diately the new transfer matrix:

(15)

1 ( 5

112

) + _ SISO- 112 + _0_ ,

a SI

P= (a - ~ r + X1(SI + 51

1 }a1/2

- a- 1/2)2. (17)

These quantities are the trace and the determinant of (T), respectively.

Let us define by S " S H, P " P H the first and second de­rivatives of Sand P for a magnetic field E, and So, So, S [) , Po, etc ... the corresponding quantities for B = O. It is straight­forward to deduce the magnetic moment for arbitrary field,

M = _1_ (s· + SS· - 2P' )/[S + (S2 _ 4P)1/2] 2/3 (S2 _ 4P)1/2 '

(18) and the initial susceptibility (S 0 = Po = 0),

1 ( SoS" 2P") X = - S" + 0 - 0 /[S + (S2 _ 4P. )1/2] o 2/3 0 (S2_4P)1/2 0 0 0 '

o 0 (19)

Georges, Curely, and Drilion 917

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with

S' = f3J.LB [goXl(SO- '/2 - S~I2)

+ ~ (2g1 +gO{ SO~:12 -S~12SI) I ( SI12 )] + 2; (2g, - go) ~ - so- l12s.) ,

P' = f3J.LBgIX I(a'/2 - a-1!2)2( :1 - SI).

S " VlJ.L B)2 [...2 ( I )...2 ...2 0=--2- XI~O + a + -; (4~1 +~o)

From these results, we can discuss in details the effects ofIocal anisotropy (K. ;':0) on the magnetization curves and the thermal variation of the susceptibility. A phase diagram can be drawn at absolute zero withK/IJ I andg,B IIJ I as the significant parameters. Two possible schemes are obtained corresponding to go>2g1 (full line) and go < 2g1 (dashed line) as shown in Fig. 5.

For go = 2g., a simple metamagnetic behavior is ex­pected for positive KI values. When K. < - IJ I (i.e., when the SZ = 0 state is strongly stabilized on odd sites) a transi­tion occurs with increasing field from a pseudoparamagnetic phase to a fully aligned spin configuration. Actually, the paramagnetic phase is characterized by a zero molecular field on even sites (So = 112).

For negative but small enough local anisotropy con­stant, the paramagnetic phase takes place at intermediate fields, between the anti ferromagnetic one (low fields) and the

t t

1 K./iJI

-- go~29.

---- 90 <29.

FIG. 5. Phase diagram of the two-sublattice Ising chain (I - 1)00 with local anisotropy.

916 J. Appi. Phys., Vol. 56, No.2, 15 July 1965

O.B K,/IJI

0.6

0."

--90 /9,=2

0.2 - - - - 9 0 /9,= 1.6

0.5 1.5 kT/IJI

FIG. 6. NormalizedXn T product defined from Eqs. (14)land (191 vs reduced temperature for the Ising chain (~ - 11 ~ . We consider here the influence of a local anisotropy on the S = 1 sub lattice. The case corresponding to fully compensated (noncompensatedl magnetic moments is displayed in the full line (dashed Iinel.

fully aligned spin configuration. These features are modified when go differs from 2g •.

First of all, the antiferromagnetic phase is replaced by a ferri­magnetic one. Nevertheless, the phase diagram is unchanged for go> 2g I and remains topologically similar for go < 2g I' Several features result from these considerations when ex­amining the curves of magnetic susceptibility (Fig. 6). The curves of the antiferromagnetic system (go = 2g.) are report­ed in the fun line, those of the ferrimagnetic system (go = 1.6 g.l in the dashed one.

In the former, an important point that we wish to em­phasize for KI < - 'J I is the divergence of the susceptibility as liT for T -->0 (x n T becomes a constant). This behavior is a consequence of the absence of molecular fidd on even sites. When K, > -IJ I the divergence disappears (xn T-->O) in which case the behavior of the system converges towards that of an antiferromagnetic Ising chain obtained for K I = O.

When the two sublattices no longer compensate one an­other (go = 1.6 g I on the graph) the susceptibility always di­verges when T -->0 K. The divergence corresponds to a Curie law (xn T reaches a finite value) as long as KI < - :1 I, and becomes more drastic in the opposite case, namely when the zero-field splitting is less than 1'1.

ftntHtt c -J .. ,,+04·<>+<>+ b 1,1,I,f,1 a

2 B/IJI

FIG. 7. Magnetization curvesatkT /1 = 0.1 of the I! - 1100 Ising chain in the fully compensated moment case (go = 2g". Overturned spin configura­tions from the ground state one lal are sketched in the inset on the upper right; (b, and (e) correspond to the first and second stage of magnetization.

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2 .. ~~ .. -;: ..... ..".,.,--"----"'----.. --"----

-Q2 /'; .. ,"'; O. , /-Q2

. .1/ -- 90 /91=2

/; -----90 /9.=1.6

"i'l't'

, .. , ....... "

°o~----~~------------~~----~~~ 2 B/IJI

FIG. 8. Influence of the ratio between Lande factors on the magnetization curves ofthe (! - I)~ Ising chain at kT IIJ I = 0.1.

Finally, whenK t increases to infinity (withK t > 0) thus strongly stabilizing the Kramers doublet S Z = ± 1 on odd sites, the situation corresponds to an S = 1/2 Ising chain showing alternating Lande factors (g) must be renormalized) and the initial susceptibility can be written in closed form:

%0 = /3~~ [(go + 2g))2a l /2 + (go - 2g l )2a -t/Z), (20)

with a given by Eq. (5). Putting go = 2gJ, this reduces to Ising's expression for

the regular antiferromagnetic ID system. Next we focus on the magnetization curves of Figs. 7

and 8, drawn at very low temperature (kT /IJ 1= 0.1). At first, when go = 2g J and for large enough anisotropy

stabilizing theS Z = o value on odd sites (KI < - IJ j), we are dealing with a ground state showing an alternation between spins with no component along the applied field and S = 1/2 spins submitted to vanishing exchange field; a net magneti­zation is expected at absolute zero (Fig. 7).

For values of KI ranging between - IJ I and zero, the S~ = 0 states are available. For weak fields, the alternating structure holds and for large enough ones, we observe a com­plete alignement (see the inset of Fig. 7). Then, the behavior is expected to show something like a double metamagnetic transition in the very low-temperature limit. As soon as go differs from 2g 1 (Fig. 8), a magnetization occurs at low field [M = (g I - ~o),u B] whatever the anisotropy on odd sites due to the noncompensation between the two sublattices. Some kind of metamagnetic effect is then observed when the external field compensates the exchange one. Further, the critical fields shift according to the ratio between Lande fac­tors, likewise the high field magnetization simply given by M = (gl + ~O),uB' Finally, it is to be emphasized that the response of the system to an external magnetic field depends on competing effects as previously discussed from the sus­ceptibility curves. Obviously, the resulting features are only noticeable at very low temperature.

CONCLUSION

In this paper, we have focused on the influence of the anisotropy effects in so-caned ferrimagnetic chains. The use of an Ising-type exchange Hamiltonian allowed, from limit­ed calculations, a rigorous thermodynamic study for various kinds of systems (distinction between spins or Lande factor

919 J. Appl. Phys .. Vol. 58, No.2. 15 July 1985

sublattices). Among the significant results, we notice the presence of

a minimum in the thermal variation of X n T and a divergence at decreasing Twhen assuming noncompensated sublattices. Such a behavior has previously been shown to be an intrinsec property of the ferrimagnetic ID systems,22.24 independent of the dimensionality of the exchange interaction and/or the nature of the spins (quantum or classical).

The Heisenberg model is generally thought to be more appropriate for fitting experimental results. However, this model requires quite different methods to be handled and needs much more expensive computational work. Further, exchange-coupled chains with spins larger than 1/2 fre­quently show pronounced anisotropy in the exchange so that Ising or XY models are required for the analysis of the data. The recently prepared linear complex CoNi (EDT A) 6H20 seems to be a good candidate of alternating (So = 1/2, S I = 1) chain. Due to the high anisotropy of the Co(II) site (gil = 9.8 and gl = 1.4), and the zero-field splitting generally expected for Ni(II) in octahedral surrounding, this complex would be particularly suited to test our model if magnetic studies on single crystal were available. We hope the present study will be an initiation for further experimental investiga­tions. In a forthcoming paper, we shall present results on several type of ferrimagnetic chains involving alternating classical and quantum spins.

ACKNOWLEDGMENT

The authors thank L. J. de Jongh for valuable com-ments.

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