High-field low-temperature magnetisation curves of anisotropic ferrimagnets

Journal of Magnetism and Magnetic Materials 323 (2011) 1068–1082

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High-field low-temperature magnetisation curves of anisotropic ferrimagnets M.D. Kuz’min Leibniz-Institut f¨ ur Festk¨ orper- und Werkstoffforschung, Postfach 270116, D-01171 Dresden, Germany

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 November 2010 Received in revised form 26 November 2010 Available online 19 December 2010

Possible shapes of high-field magnetisation curves of two-sublattice ferrimagnets are analysed within a model with two anisotropy constants for the subdominant sublattice. The model is relevant to 3d–4f intermetallic compounds. For the purpose of classification the magnetisation curves are regarded as ordered sequences of horizontal, slanted and vertical sections, without reference to quantitative details. Every shape type corresponds to a domain in the plane of relative anisotropy parameters k1 and k2. The k1k2 diagram proves rather complicated; however, in the limiting cases of very weak or very strong anisotropy (in relation to intersublattice exchange) it takes the same universal form characteristic of simple ferromagnets. & 2010 Elsevier B.V. All rights reserved.

Keywords: High-field magnetisation Ferrimagnet Magnetic anisotropy

1. Introduction Ferrimagnets are solids comprising two (or more) non-equivalent ordered magnetic sublattices that in the absence of magnetic field prefer to orientate themselves antiparallelly. The application of a sufficiently strong magnetic field will force a ferrimagnet, like any other system, into a ferromagnetic state, all the constituent magnetic moments eventually becoming parallel to the field and thus, to each other. Although the general course of this transformation is rather obvious, the details feature great diversity and are therefore interesting. Even in the simplest case of an isotropic two-sublattice ferrimagnet the magnetisation curve is piece-wise linear, with two horizontal and one sloping portions [1–3]. The magnetisation curve of a real ferrimagnet may contain several horizontal stretches, where the magnetisation hardly depends on magnetic field, alternating with growing parts and nearly vertical sections (jumps). The latter are associated with field-induced first-order transitions, while abrupt but finite changes of slope (kinks) are indicative of phase transitions of second order. For a recent survey of experimental results (see [4]). ¨ It was admitted already in Schlomann’s [2] pioneering work that the complete neglect of magnetic anisotropy was an over-simplification. Since then many attempts to allow for the anisotropy have been undertaken, see [5] for a review. The problem of computing the magnetisation curve from known anisotropy energy and intersublattice exchange is long since solved. Our task in this work is of another kind. In practice anisotropy energy is rarely known a priori and magnetisation is only measured up to a certain maximum field rather than to full saturation. To find anisotropy constants compatible with the observed part of the magnetisation curve and to reconstruct the rest of it – these are typical tasks facing those who interpret high-field data. In order to facilitate their work, we would like to compile a reasonably complete catalogue of possible patterns of magnetisation curves of ferrimagnets. We have to admit, it would be impracticable to allow for the anisotropy in the most general way because of the excessive intricacy of the result. This work is an attempt to find a compromise between two extremes – between rigorous account and full neglect of the anisotropy. In the next section a model of a two-sublattice anisotropic ferrimagnet is introduced which, despite being an enormous simplification of reality, still has relevance to a wide range of magnetic materials and permits an exhaustive and easily surveyed presentation of all possible (within the model) shapes of magnetisation curves. A large amount of material presented in a compact form may be hard to digest at once. So before tackling the model in its general form, we shall take a run-up in Section 3 by considering a special case of K2 ¼ 0. This is a simplified example permitting a largely analytical treatment and a clear depiction of the result. The compact description required for the subsequent presentation of the general case, will be first introduced in Section 3 as an alternative description of the readily understandable special case. The general problem will then be treated in Section 4, followed by a discussion (Section 5) and a conclusion (Section 6).

2. The model This work is limited to a two-sublattice ferrimagnet in an applied magnetic field. We further restrict ourselves to a special case of T¼0 and regard both sublattice magnetisations, M1 and M2, as saturated, i.e. 9M1,29  const., irrespective of the field. The model system is E-mail address: [email protected] 0304-8853/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2010.12.009

M.D. Kuz’min / Journal of Magnetism and Magnetic Materials 323 (2011) 1068–1082

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described by the following thermodynamic potential:

F ¼ lM 1 UM 2 M 1 UHM 2 UH þEa :

ð1Þ

Here l is an intersublattice exchange constant, l 40, H is the applied magnetic field and Ea the magnetic anisotropy energy. Without loss of generality it may be postulated that M1 4M2. A yet further simplification of the model consists in neglecting the anisotropy of the dominant sublattice (#1). This rather special assumption appears justified for iron- and cobalt-rich intermetallic compounds with heavy 4f elements, where the dominant sublattice is that of the 3d metal. Let a and b be the angles between the sublattice moments, M1 and M2, respectively, and the magnetic field H (see Fig. 1). The thermodynamic potential (1) can be rewritten as follows:

F ¼ lM1 M2 cosða þ bÞM1 H cos aM2 H cos b þ Ea ,

ð2Þ

Ea ¼ K1 sin2 b þ K2 sin4 b:

ð3Þ

with

Here K1 and K2 are anisotropy constants of the subdominant sublattice (#2). According to the assumption made, the anisotropy energy Ea depends only on the angle b. Two further assumptions are implicit in Eq. (3): (i) the magnetic field is applied along a high-symmetry crystal direction and (ii) sixthand higher-order terms in Ea are negligible. The latter has been assumed for the sake of simplicity, in order to limit the number of model parameters. Particularly, in 4f magnets there is no physical reason for an a priori neglect of sixth-order anisotropy terms at T¼ 0 [6]. Note that the use of Eq. (3) does not imply that the system has a uniaxial symmetry or that the symmetry axis is parallel to the magnetic field. Rather, the sublattice moments, as they rotate under the field, prefer to stay within a certain crystallographic plane. We take this to be the plane of Fig. 1. According as the specific symmetry may be, the constants K1 and K2 in Eq. (3) would generally be linear combinations of conventional anisotropy constants. Given that the field is applied along a high-symmetry direction, Eq. (3) is nearly always applicable. One could think of just two exceptions: (i) when the crystal is triclinic, with no high-symmetry directions at all, or (ii) when the moments rotate about a six-fold symmetry axis, i.e. in the basal plane of a hexagonal crystal; then the omission of the sixth-order term from Ea is inadmissible. As against that, Eq. (3) does apply if the moments rotate around a fourth-fold axis, provided one takes K2 ¼–K1. Further consideration can be conveniently carried out in terms of the following dimensionless quantities:

j¼

F H M2 K1 K2 , h¼ , m¼ , k1 ¼ , k2 ¼ : lM1 M1 lM12 lM12 lM12

ð4Þ

The thermodynamic potential (2, 3) is rewritten as follows:

jða, bÞ ¼ m cosða þ bÞhcos amh cos b þ k1 sin2 b þ k2 sin4 b:

ð5Þ

Here the dimensionless magnetic field h is an external parameter, whereas the sublattice orientation angles a and b are the system’s internal parameters. This means, a and b are to be determined from the condition of thermodynamic equilibrium. The necessary conditions of minimum of j(a, b) are given by m sinða þ bÞ þ h sin a ¼ 0,

ð6Þ

m sinða þ bÞ þ mh sin b þ2k1 sin b cos b þ4k2 sin3 b cos b ¼ 0:

ð7Þ

Knowing a and b for a given h, one readily finds the magnetisation in the direction of applied magnetic field: MH ¼ M1 cos a þ M2 cos b:

ð8Þ

Our main task in this work is to classify magnetisation curves according to their shape, which obviously does not depend on normalisation. So it is convenient to introduce instead of Eq. (8) a dimensionless magnetisation

s¼

cos a þm cos b : 1m

ð9Þ

According to this definition, s is normalised to unity in a weak magnetic field applied along an easy magnetisation direction, when a-0 and b-p as h-0. Thus, the calculation of a dimensionless magnetisation curve, s(h), consists in solving the simultaneous equations (6) and (7) for a and b and setting the solution into Eq. (9). The shape of the curve depends on three model parameters, k1, k2 and m (0om o1). Since a general analytical solution of the problem is impossible, an exhaustive analysis of the shapes of s(h) will be carried out for all k1 and k2, but limited to m¼0.5, taken in the middle of the interval. Such a strategy relies on the assumption that small deviations of m from the chosen representative value do not lead to qualitative changes in the main k1k2 diagram (Fig. 7), only the boundary lines shift somewhat. In all cases when these lines can be described analytically for any m, the corresponding expressions will be presented in order to illustrate the dependence on m.

M1 α

H

β M2 Fig. 1. Mutual orientation of the sublattice moments, M1 and M2, and applied magnetic field H.

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The numerical minimisation of the function j(a, b) (Eq. (5)), presents no major difficulties. The angle b is regarded as an independent variable, 0 r b r p. The other unknown angle is readily expressed from Eq. (6):

a ¼ arctan

m sin b : hm cos b

ð10Þ

The problem is thus reduced to minimising a function of one variable on a finite interval. In order to make the presentation more digestible, the treatment of the full problem will be postponed until Section 4. First, in the next section, a special case of K2 ¼ 0, or k2 ¼0, will be considered. In this case the main results can be presented in analytical form, as well as graphically.

3. A special case of K2 ¼ 0 In this case the shape of the magnetisation curve is determined by two dimensionless parameters, m and k1. The former is defined as a ratio of the sublattice moments (4), where it has been assumed that M1 4M2. Therefore, m is a real number between 0 and 1. By contrast, there is no a priori restriction on either magnitude or sign of k1. The magnitude of k1 is a measure of intensity of the anisotropy energy (in relation to intersublattice exchange). If k1 is positive, the direction of the magnetic field is an easy magnetisation direction in the crystal. A negative k1 means that the field is applied perpendicularly to the easy direction. The function j(a, b) (Eq. (5)), can reach its minimum either at the boundary or in the interior of the square where it is defined (see Fig. 2). In the former case it can be readily demonstrated that the minimum cannot be at an inner point of a side of the square, but only at a vertex. (Thus, e.g., near the left-hand side of the square j(a, b) decreases towards the interior, (@j/@a)a ¼ 0 ¼–m sin b o0, provided that 0 o b o p, etc.) Furthermore, from simple stability arguments it follows that the angle a between the larger sublattice moment M1 and the field H must be acute (as indicated by hatching in Fig. 2). Therefore, it suffices to check the two vertices on the left-hand side of the square for a minimum of j(a, b). In total, there are three possibilities. 1. The minimum of j(a, b) is at the point a ¼ 0, b ¼ p: ferrimagnetism. The energy at the minimum is given by

jferri ¼ mhþ mh,

ð11Þ

and the reduced magnetisation s equals identically unity. 2. The minimum is reached at a ¼ b ¼ 0: a ferromagnetic state is stable. The energy of the system equals

jferro ¼ mhmh,

ð12Þ

the magnetisation being

s

1þ m : 1m

ð13Þ

From comparison of Eqs. (11) and (12) it is clear that the ferromagnetic state is more favourable in a strong, and the ferrimagnetic state in a weak magnetic field. This is hardly surprising given that the system under study is a ferrimagnet. In the ferromagnetic state the magnetisation takes its highest value possible within the model. Consequently, if the ferromagnetic state is reached at a certain finite field, this state will remain stable at any higher field. (A quasi-stationary process where magnetisation would decrease with magnetic field is impossible.) The only physically admissible alternative might be asymptotic growth of magnetisation, when the upper bound (13) would not be attained at any finite h. This possibility has been purposely excluded from the model by restricting it to high-symmetry field directions. Thus, the high-field part of a magnetisation curve in our model has to be a horizontal straight line. 3. The minimum of the function j(a, b) is reached at an inner point of the square: a canted phase is stable. The equilibrium values of the angles a and b are determined from the general equations (6) and (7). For k2 ¼0 the magnetisation curve can be presented in a parametric form. Let us introduce an auxiliary quantity: t¼

2cos b : h

ð14Þ

Eliminate the angle a from simultaneous equations (6) and (7), then substitute th/2 for cos b in the result. In this way h is readily expressed as a function of the running parameter t: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 1ðm þ k1 tÞ2 h¼ : ð15Þ m þ k1 t 1mt

Fig. 2. Domain of the function j(a, b) defined by Eq. (5). For stability it is prerequisite that the angle a be acute, as indicated by the hatching.

M.D. Kuz’min / Journal of Magnetism and Magnetic Materials 323 (2011) 1068–1082

The expression for reduced magnetisation (9) is rewritten as follows: h  k1 k1 2  1þ t t : s¼ 1m m 2

1071

ð16Þ

Eqs. (15) and (16) provide a parametric description of the magnetisation curve s vs h, o rather, of its sloping part. In total, a magnetisation curve may consist of elements of three kinds. (A) Horizontal sections with s 1 or s (1+ m)/(1–m), corresponding to the ferri- or ferromagnetic phase, respectively. (B) Sloping (growing) parts associated with intervals of stability of the canted phase. These are described by the parametric equations (15) and (16). (C) Vertical sections (jumps). They do not correspond to any particular stable phase, but rather to a first-order transition between two phases. Theoretically, the magnetisation curve has a discontinuity at this point. In practice, the growth of magnetisation proceeds at a large but finite rate and may be accompanied by hysteresis. Further complications during first-order transitions may involve splitting of the rare earth sublattice into two or more non-collinear sublattices [11,12] or its continuous re-magnetisation [13]. Such effects cannot be described within the model of Section 2 and are beyond the scope of this work. Herein first-order transitions will be depicted simply as vertical segments.

For the sake of completeness one should also mention second-order phase transitions, which are manifested by abrupt changes of slope (kinks) at the joints of sections of type A and B. As an illustration of the above, Fig. 3 displays five representative magnetisation curves for m¼0.5 and the values of k1 indicated in the drawings. A common feature of all the curves is a horizontal portion on the right, where s 3. As expected, the ferromagnetic phase is always stable in the high-field region, whatever the anisotropy. If k1 o0 the low-field section of the curves is inclined. This section starts at the origin and corresponds to the canted phase. Such behaviour is what one would intuitively expect in a situation where the field is applied perpendicularly to the easy axis. At large negative k1 the growth of s continues up to the transition into the ferromagnetic phase (saturation), which is a phase transition of second order. At small negative k1 the growth of the magnetisation is interrupted momentarily at s ¼1, where a horizontal segment is observed. This segment

Fig. 3. Representative magnetisation curves for k2 ¼ 0 and m ¼0.5.

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corresponds to an interval of stability of the collinear ferrimagnetic phase. It is characteristic of this phase that M1mmH and M2mkH, as distinct from the ferrimagnetic structure stable in an infinitesimally small field, where M1,2?H. The ferrimagnetic stretch of the curve is limited at both ends with second-order transition points. The critical field of the transition at the low-field end of the ferrimagnetic segment is known as anisotropy field. At large negative k1 the notion of anisotropy field becomes meaningless, as the horizontal segment vanishes. If k1 40, i.e. the field is applied in the easy direction, the ferrimagnetic phase is always stable within a certain finite interval of fields next to h¼0. Accordingly, the magnetisation curves in Fig. 3c, d and e start with a horizontal stretch at s ¼1. The details of further growth and approach to saturation at s ¼3 depend on the magnitude of k1. If k1 is very small (Fig. 3c), a continuous growth via the canted phase takes place, just like in the isotropic case [1–3]. In the beginning and in the end of the growing part of the curve second-order transitions occur. At large positive k1 the system goes over from ferri- to ferromagnetism in a single step (Fig. 3e); the changeover is a first-order phase transition known as metamagnetic transition. At intermediate values of k1 both a first- and a second-order transitions take place, as well as continuous magnetisation growth in between (Fig. 3d). All this rather intricate information for various k1 can be presented as a single phase diagram in the plane k1–h, or rather, in its upper semi-plane (see Fig. 4). The ferromagnetic phase occupies the upper, high-field part of the diagram. The domains of the ferrimagnetic and canted phases are situated in the low-field part of the diagram, on the right and on the left, respectively. A given system has a fixed k1; according as the external parameter h varies, the system’s locus in the phase diagram moves along a vertical line. As it does so, it may cross a phase boundary; then a phase transition will take place. In particular, a first-order transition occurs when a dashed line is crossed, crossing a solid line means a second-order transition. Magnetisation curves of type a (see Fig. 3a) are observed in systems located to the left of point V in Fig. 4. If a system is situated in the second quadrant to the right of point V, its magnetisation curve will be shaped as in Fig. 3b. Systems in the first quadrant have magnetisation curves of types c, d or e, depending on whether the locus is to the left of point C1, between C1 and C2, or to the right of C2. Let us investigate the position of the phase boundaries in Fig. 4 quantitatively. To locate a second-order transition between the canted and the collinear ferrimagnetic phases, assume that the angle b differs from p infinitesimally little, b ¼ p–Z. Then a is a small quantity too; by Eq. (10) it can be expanded in odd powers of Z. Substituting this expansion for a and p–Z for b in Eq. (7), we present the latter as Landau’s expansion: aZ þ bZ3 þ    ¼ 0,

ð17Þ

where   1 1 þ 2k1 , a ¼ mh hþ m

ð18Þ

1 m2 h2 k1 : b ¼  aþ 6 2ðh þ mÞ3

ð19Þ

According to Landau [7], in order for a second-order transition to take place it is necessary that a¼ 0 and b40. Hence, setting Eq. (18) to zero, one gets an expression describing the arc OVC1 in Fig. 4:   mh 1 1 : ð20Þ k1 ¼ 2 h þm Thus, OVC1 is an arc of a hyperbola. The coordinates of the vertex V are easily found from the condition @k1/qh ¼0: pffiffiffiffiffi hV ¼ mm, pffiffiffiffiffi kV ¼ 12mð1 mÞ2 :

ð21Þ ð22Þ

Fig. 4. Phase diagram in the k1h plane containing domains of the ferrimagnetic (mk), forced ferromagnetic (mm) and canted (-%) phases. Dashed and solid curves are first- and second-order transition lines, respectively.

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The tricritical point C1 is where the second-order transition becomes a first-order one. The necessary conditions of the change are: a ¼b¼0. Setting Eqs. (18) and (19) to zero and eliminating k1, one arrives at a cubic equation in h: h3 þ ð3m1Þh2 þ3mðm1Þhþ m2 ðm1Þ ¼ 0:

ð23Þ

Since m o1, there is a single change of sign in the sequence of coefficients of Eq. (23). By Descartes’ rule of signs, this equation has one positive solution. This solution has to be the ordinate of the point C1: " # 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 þ 9m27m2 : ð24Þ 1þ 3mcos arccos hC1 ¼ m þ 3 3 3 2ð1 þ3mÞ3=2 The abscissa of the same point, kC1 , is obtained by substituting Eq. (24) for h in Eq. (20). The boundary between the ferromagnetic and the canted phases (the upper solid curve in Fig. 4) is found in a similar fashion. Now both a and b should be regarded as small angles. The result is like Eq. (20), only the of sign of m is changed:   mh 1 1 : ð25Þ k1 ¼ 2 hm Thus, the upper solid curve in Fig. 4 is a hyperbola too. The dashed line separating the ferri- and ferromagnetic phases in Fig. 4 is a horizontal line at h¼1. This follows immediately when the energies of the two phases are equated, Eqs. (11) and (12). Setting h¼1 in Eq. (25), one obtains the abscissa of the critical point C2:

kC2 ¼

m2 : 2ð1mÞ

ð26Þ

Finally, we did not succeed in finding an analytic expression for the first-order transition line between the ferrimagnetic and the canted phases. The dashed arc C1C2 in Fig. 4 was calculated numerically. Fig. 4 contains information sufficient for sketching the magnetisation curve of a system with arbitrary k1, yet it is limited to m¼0.5. Now we would like to find out, what happens when m is changed. A straightforward way to answer this question would be to produce a series of drawings similar to Fig. 4 for all possible m between 0 and 1. However, we prefer to take a different approach – we shall present the result of this case study in another, more compact form. This will enable us to give a short answer to the question about the dependence on m. Moreover, the new graphic representation will prove useful for the analysis of the general case (k2 a0) in subsequent sections. The price of the extra compactness will be the loss of some information as compared with the old representation (Fig. 4). The latter contains full information about the critical fields, which suffices for exact reproduction – as opposed to mere sketching – of the magnetisation curves (except their sloping parts). This information is not indispensable for our task here, which is to classify the shapes of the magnetisation curves as ordered sequences of constituent elements A, B and C, without regard to quantitative details. For instance, it does not make much difference that the curves with k1 ¼0.3 and with k1 ¼0.5 are truly identical, whereas the ones with k1 ¼–0.3 and with k1 ¼–0.5 differ only in the value of the saturation field. To us the former two are of type e (or ACA) and the latter two of type a (BA). All five possibilities for m ¼0.5 are depicted by a single one-dimensional diagram (Fig. 5). Unlike Fig. 4, Fig. 5 is not a phase diagram. The five domains in Fig. 5 are occupied by distinct curve patterns rather than phases. When a domain boundary is crossed, a change of shape takes place, not a phase transition. The boundaries are positioned at k1 ¼ kV, 0, kC1 , kC2 , as defined by Eqs. (20), (22), (24) and (26). The dependence on m is therefore known. It is displayed in Fig. 6. One can appreciate that the boundaries do not cross inside the unit interval. Thus, according as k1 grows at any fixed m, 0 omo1, the system goes through the same sequence of five shape types as the one shown in Fig. 3.

Fig. 5. Partition of the k1 axis among the five curve types shown in Fig. 3.

0.2

0.1 κ1

e d c

0

b a

-0.1

0

0.2

0.4

0.6

0.8

1

m Fig. 6. A fragment of the mk1 plane partitioned among the curve types of Fig. 3.

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The main advantage of Fig. 5 is the possibility to add a second dimension, k2, necessary for the presentation of the general case. This will be done in the next section. 4. General case, j2 a0 Let us now add a vertical axis k2 to Fig. 5 and lift the restriction k2 ¼0 from the model, leaving however m¼0.5. The final result of the mainly numerical study is a diagram shown in Fig. 7. The entire plane k1k2 is split into 16 domains labelled with lower-case letters from a to o. (Two small domains in the central part of the diagram are not labelled but rather highlighted, blue for b and pink for p.) The corresponding types of magnetisation curves are sketched in Fig. 8. As expected, the k1 axis cuts through five domains containing the already familiar curve types, a to e, cf. Fig. 3. In the upper half-plane there are five more domains with new types of curves, f to j. Further six domains, k to p, lie in the lower halfplane. On the whole, Fig. 7 is a rather involved diagram with many lines, some of which are straight, but most are not. In the following subsections we shall attempt at classifying the patterns of magnetisation curves and give a systematic description of the lines in Fig. 7. 4.1. Classification by initial state In all magnetisation curves the final, high-field state is the same – ferromagnetic. But the initial state can be three-fold: (99) easy magnetisation direction is parallel to the applied field, s-1 as h-0; (?) easy direction is perpendicular to the field, s-0; (+) easy direction makes an acute angle with the field, s-s0, 0 o s0 o1. On this principle, all the curves in Fig. 8 can be divided into three classes. Types c, d, e, f, m and n belong to class ‘‘99’’, types a, b, k, l, o and p to class ‘‘?’’, types g, h, i and j to ‘‘+’’. Accordingly, the k1k2 plane splits into three sectors corresponding to the three initial states (see Fig. 9). The boundaries of the sectors are straight lines described by the equations

0.3

0.2 h

0.1

g

f

κ2

i j

a

0

c

e

d d

o -0.1

n

-0.3 -0.3

I

k

-0.2

-0.2

-0.1

0

0.1

m

0.2

0.3

0.4

κ1 Fig. 7. Partition of the k1k2 plane among the curve types sketched in Fig. 8. Highlighted are domains b (blue) and p (pink). (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)

a

b

c

d

e

f

g

h

i

j

k

l

m

n

o

p

Fig. 8. 16 possible patterns of magnetisation curves (schematic).

M.D. Kuz’min / Journal of Magnetism and Magnetic Materials 323 (2011) 1068–1082

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κ2

κ1

Fig. 9. Partition of the k1k2 plane among three ground-state structures in the absence of magnetic field.

k1 ¼0, k1 + k2 ¼0 and k1 + 2k2 ¼0. These lines are respectively identical with lines cj/fg, cp/do/lm and bj/ai in Fig. 7. Thus, the state of the system at h¼0 appears to be determined essentially by the ratio k1/k2. Indeed, the minimisation of the thermodynamic potential (5) at h¼ 0 consists in finding b that minimises the anisotropy energy, k1 sin2 b + k2 sin4 b, and taking a ¼ p–b. The anisotropy energy with two anisotropy constants was first minimised by Smit [8], who also obtained a diagram similar to the one displayed in Fig. 9. It is clear from dimensionality considerations that the optimal b depends on the ratio of the anisotropy constants.

4.2. Classification by approach to saturation Although all the magnetisation curves in Fig. 8 end up saturating in a strong magnetic field, the ways to approach the saturation may vary. It can be either a continuous growth with a kink in the end or a jump to saturation. (The third option – an asymptotic approach to saturation – was purposely excluded from the model by restricting the orientation of the field to high-symmetry directions in the crystal.) This opens up a possibility to classify the curve shapes on the principle of approach to saturation, kink-wise versus step-wise. Such a division is meaningful only within the classes ‘‘99’’ and ‘‘?’’ because all curves of the class ‘‘+’’ (types g, h, i and j) always saturate with a kink.

4.2.1. Initial state ‘‘?’’: conversion of saturation kink into a jump, line ak/lo To investigate the behaviour of the magnetisation near saturation, the angles a and b are treated as infinitesimals. By means of Eq. (10) a is presented as an expansion in odd powers of b (regarded as an independent variable):

a¼

m mhðhþ mÞ 3 mhðh3 þ 11mh2 þ11m2 h þ m3 Þ 5 b b þ b þ : hm 120ðhmÞ5 6ðhmÞ3

ð27Þ

This expression is then used to transform Eq. (7), expanding its left-hand side in a power series: 3

5

ab þbb þ cb þ    ¼ 0,

ð28Þ

where  a ¼ mh 1

 1 þ2k1 , hm

1 m2 h2 b ¼  aþ k1 þ 4k2 , 6 2ðhmÞ3 c¼

1 1 3m3 h3 a b 3k2 : 30 4 8ðhmÞ5

ð29Þ

ð30Þ

ð31Þ

According to Landau [7], in order for the saturation to proceed via a second-order phase transition (kink-wise), it is necessary that a¼ 0 and b40. If b o0, the transition will be of first order (a jump). Thus, the conditions a¼ b¼0 (and c 40) determine a place where the saturation kink in the magnetisation curves turns into a jump, i.e. the line ak/lo in the k1k2 plane (Fig. 7). (In this notation ak stands for the borderline between the domains a and k, ak/lo denotes a union of ak and its continuation lo.) Setting Eqs. (29) and (30) to zero results in   mh 1 1 , ð32Þ k1 ¼ 2 hm 1 4

k2 ¼ k1 

m2 h2 8ðhmÞ3

:

ð33Þ

Here h is the saturation field, which plays the role of a running parameter. Within the domains a and o the magnetisation saturates by way of a kink. Towards the border with k and l the kink gradually sharpens up, eventually becoming a discontinuity. This happens on the line ak/lo. The difference between the types a and o, as well as between k and l consists in the presence or absence of other features in the magnetisation curves, which does not interfere with the way the saturation is reached (for more details see Section 4.3.4). Therefore, lines ak/lo and ao/kl simply cross each other.

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σ

1

1 2 3

4 h

0

σ

1 0

4 3 2 1 h

Fig. 10. Evolution of a magnetisation curve of type d towards type e: (a) as k1 grows at constant k2 and (b) as k2 grows at constant k1.

4.2.2. Initial state ‘‘99’’: absorption of saturation kink by a jump Within the class ‘‘99’’ there are curve types with a kink at saturation (c, d, f and n), as well as those with a jump (e and m). The transition from the former to the latter takes place on the lines de and mn and proceeds differently than in the class ‘‘?’’. Namely, the saturation kink does not sharpen up to become a jump. Rather, it is absorbed by an already present discontinuity approaching from the low-field side. Fig. 10a shows schematically how a magnetisation curve evolves when the locus in Fig. 7 moves within the domain d towards the border with e, i.e. as k1 grows at constant k2. As the locus reaches the line de, the kink disappears, it is ’absorbed’ by the jump. Such a magnetisation curve (#4 in Fig. 10a) belongs already to type e. In a similar fashion a curve of type n transforms to type m when the saturation kink is absorbed by the jump next to it. It is easy to demonstrate that the boundary de in Fig. 7 is a vertical straight line defined by

k1 ¼

m2 : 2ð1mÞ

ð34Þ

Indeed, on and to the right of the boundary de a metamagnetic transition ferrimagnetism–ferromagnetism takes place. At the transition point the energies of the two phases, (11) and (12), must equal each other. Hence the threshold field of the metamagnetic transition equals unity. Just left of the line de the magnetisation curve has an infinitesimally narrow chamfer at the point of saturation. On the chamfer the necessary conditions of equilibrium (6) and (7) must be fulfilled, a and b being nonzero infinitesimals. Rewriting Eqs. (6) and (7) at h¼1 to terms linear in a and b: ð1mÞamb ¼ 0,

ð35Þ

ma þ2k1 b ¼ 0,

ð36Þ

and demanding the existence of a non-trivial solution, one arrives at Eq. (34). Thus, in order for a metamagnetic transition ferrimagnetism– ferromagnetism to take place, it is necessary that the following inequality be satisfied:

k1 Z

m2 : 2ð1mÞ

ð37Þ

If the possibility of a large negative k2 can be excluded, the condition (37) is also a sufficient one. Interestingly, the parameter k2 does not itself enter in the inequality (37). Moreover, allowance for higher-order terms in the anisotropy energy (3), such as K3 sin6b, would not affect the condition (37), as long as large negative values of K3 are avoided. We did not succeed in finding an analytic description for the line mn. The curve for m¼0.5, presented in Fig. 7, was computed numerically. It is worth noting that there is a general tendency towards metamagnetism on moving away from the origin in the entire sector ‘‘99’’ of the k1k2 plane (cf. Figs. 7 and 9). The domains d and f are open from above, i.e. a magnetisation curve of type d or f will formally always remain as such when k1 is fixed and k2 is increased. However, its shape will differ ever less from type e. That is, the shape type e will be approached asymptotically as k2-N. This process is illustrated in Fig. 10b: according as k2 grows at constant k1, the jump gains in prominence while the kink becomes ever more obtuse, disappearing in the limit k2-N. To prove this point, let us evaluate the differential susceptibility qs/qh immediately to the left of the kink. Like in Section 4.2.1, the angles a and b are regarded as small quantities, b being an independent variable and a expressed by Eq. (27). Up to terms in b2, the reduced magnetisation (9) is as follows: " # 1þ m m m  1þ s¼ b2 , ð38Þ 1m 2ð1mÞ ðhmÞ2 where a b

b2 ¼  ,

ð39Þ

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1077

as determined in the first non-vanishing approximation from Eq. (28). Taking a derivative of Eq. (38) with respect to h at the transition point, i.e. at a¼ 0, or by Eq. (29) at sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 þm k1 1 þm k1 2   þ h¼ þ 2k1 , ð40Þ 2 2 m m one finds finally @s @h

h i2 1 þm=ðhmÞ2 m2 ¼ , kink 1m m2 h2 =ðhmÞ3 2k1 þ 8k2

9

ð41Þ

where Eq. (30) has been used. By Eq. (40), the critical field of the kink h does not change as k2-N at k1 ¼const. At the same time, by Eq. (41), the differential susceptibility left of the kink tends to zero. Thus, according as k2 grows, the magnetisation curve evolves as shown in Fig. 10b: the saturation kink becomes ever less pronounced and the type d curve tends asymptotically to type e. The evolution of a type f curve proceeds in a similar manner: the two slanted sections slope ever more gently while the vertical section becomes ever more prominent, the overall shape gradually approaching that of type e. Thus, moving away from the origin leads to metamagnetic behaviour throughout sector ‘‘99’’ of the k1k2 plane. The only exception is a narrow strip near the diagonal of the fourth quadrant, where the two-step shape of type m persists even in the strong-anisotropy limit. 4.2.3. Merger of two jumps as an alternative way of going over to metamagnetism As stated in the previous subsection, a narrow strip near the diagonal of the fourth quadrant (Fig. 7) features two-step (type m) magnetisation curves. On moving along the strip away from the origin, the shape of the magnetisation curve gradually ’freezes’, both jumps remaining in place. However, if one moves across the strip m towards the border with domain e, the sloping section between the two jumps narrows down and disappears as soon as the line em is reached. The two jumps merge here. Let us describe the boundary em quantitatively. Just before the line (on the m side) the magnetisation curve has an infinitesimally narrow sloping section between the two jumps. Within this section a canted phase with certain a and b is stable and the necessary conditions of equilibrium (6) and (7) are fulfilled. At the same time, the thermodynamic potentials of the canted, ferri- and ferromagnetic phases, Eqs. (5), (11) and (12), equal each other. In particular, it follows from the equality of (11) and (12) that the sloping section is positioned at h¼1. Now set h¼1 into Eqs. (6) and (7), as well as into the equality condition for the energies (5) and (11): m sinða þ bÞ þ sin a ¼ 0,

ð42Þ 3

m sinða þ bÞ þ m sin b þ2k1 sin b cos b þ4k2 sin b cos b ¼ 0,

ð43Þ

m cosða þ bÞcos am cos b þ k1 sin2 b þ k2 sin4 b ¼ 1:

ð44Þ

Solve simultaneous linear equations (43) and (44) for k1 and k2:

k1 ¼ k2 ¼

4ð1cos aÞðmcos bÞ þ 3 sin bðsin am sin bÞ 2 sin2 b cos b 2ð1cos aÞðcos bmÞ þ sin bðm sin bsin aÞ 2 sin4 b cos b

,

,

ð45Þ

ð46Þ

and Eq. (42) for a:

a ¼ arctan

msin b : 1m cos b

ð47Þ

Eqs. (45)–(47) provide a parametric description of the boundary line em in Fig. 7. The parameter b runs from a certain small positive number to p/2. The upper bound of the interval corresponds to the limit of strong anisotropy, k1-N. In this limit, a ¼arctan m and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 þ k2 ¼ 1þ m2 1: ð48Þ Thus, away from the origin the boundary em asymptotically approaches a straight line parallel to the diagonal of the fourth quadrant. As can be appreciated from Fig. 7, the boundary does not deviate much from its asymptote at any point. 4.3. Classification by behaviour at s ¼1 The magnetisation curves can be further classified according to their behaviour near another special value of magnetisation, s ¼ 1 (or M¼Ms, Ms being the spontaneous magnetisation). There are several possibilities in this respect. It may be that at the point whose ordinate is s ¼1 the magnetisation experiences no anomaly at all, just monotonic growth. This happens when the magnetic field is not parallel to the easy axis and the anisotropy is strong (curve types a, h, i and k in Fig. 8). However, in most cases the magnetisation curve contains a horizontal section at s ¼1. It corresponds to a range of stability of the collinear ferrimagnetic phase. Approach to and departure from this section can be considered separately. This can proceed as a second-order phase transition (a kink in the curve) or as a first-order one (a jump). 4.3.1. Transformation of a kink into a jump: lines cd/op and bp On approach to the stability range of the ferrimagnetic phase the angle a becomes small, while b tends to p. Let b ¼ p–Z, where Z is an infinitesimal regarded as an independent variable. By Eq. (10) a is expanded in a series in odd powers of Z. Upon a substitution of this series

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for a and of p–Z for b in Eq. (7), the latter is presented as Landau’s expansion: aZ þ bZ3 þ cZ5 þ    ¼ 0,

ð49Þ

where   1 1 þ 2k1 , a ¼ mh hþ m

ð50Þ

1 m2 h2 b ¼  aþ k1 þ4k2 , 6 2ðh þ mÞ3

ð51Þ

c¼

1 1 3m3 h3 a bþ 3k2 : 30 4 8ðh þ mÞ5

ð52Þ

Note that Eqs. (50)–(52) differ from Eqs. (29)–(31) only in the sign of m. As expected, Eq. (19) is a special case of k2 ¼0 in Eq. (51). Since Eq. (50) does not contain k2, it is identical to Eq. (18). A change from a second-order transition at s ¼1 to a first-order one may be associated with several boundary lines in Fig. 7. If the transformation of a kink into a jump takes place on the high-field side of the horizontal stretch of the curve, one deals with a conversion of curve types c or p into d or o, respectively. If the same happens on the low-field side of the horizontal section, then it is a change from type b to type p. Mathematically, the boundaries between all these pairs of domains are described by the same conditions: a¼b ¼0, c 40. Setting Eqs. (50) and (51) to zero, one obtains   mh 1 1 , ð53Þ k1 ¼ 2 h þm 1 4

k2 ¼ k1 

m2 h2 8ðh þ mÞ3

:

ð54Þ

Here h is a parameter running between zero and a certain maximum value determined from the following equation: m2 h2 ðh þ mÞ5

þ

mh ðhþ mÞ3

þ

1 ¼ 1: hþ m

ð55Þ

Thus, hmax ¼ 0.793 for m ¼0.5. Eq. (55) is obtained from the conditions a¼b¼ c¼0 and corresponds to a point in the first quadrant where the line cd splits into two lines, cf and df. Beyond that point the discontinuity in the magnetisation curve emerges not at s ¼1, but rather at a slightly higher ordinate. As the parameter h increases from zero, the point in the k1k2 plane described by Eqs. (53) and (54) moves away from the origin along the line bp in the third quadrant. The limit of small h corresponds to small k1 and k2, i.e. to weak anisotropy or strong exchange. In this limit the system can be regarded as a one-sublattice magnet. The fact that the line bp has a slope of 1/4 near the origin is directly related to the wellknown behaviour of one-sublattice magnets; there the line K2/K1 ¼1/4 in the third quadrant separates the domains of continuous (with a kink) and step-wise saturation [9]. The first-order transition taking place in the latter case is often referred to as first-order magnetisation process (FOMP) [10]). Thus, at small h Eqs. (53) and (54) describe a line separating the domains p and b, which correspond to magnetisation curves with and without FOMP. As h increases further, the point defined by Eqs. (53) and (54) reaches its left-most position in the k1k2 plane and starts moving to the right. The boundary line bp goes over to op. The coordinates of the turning point in the curve are determined by the condition qk1/qh¼0, pffiffiffiffiffi whence h ¼ mm and pffiffiffiffiffi k1 ¼ 12mð1 mÞ2 , ð56Þ pffiffiffiffiffi

k2 ¼ 18mð1mÞð1 mÞ:

ð57Þ

Since the equation defining the dependence k1(h), Eq. (53), is identical to Eq. (20), the abscissa of the left-most point in the curve bp equals kV (22), i.e. that point is situated exactly underneath the point (kV,0), where the boundary ab crosses the abscissa axis. This is consistent with the fact that ab is a vertical straight line, as demonstrated in Section 4.3.3. pffiffiffiffiffi At h 4 mm Eqs. (53) and (54) describe a boundary (op/cd) whose meaning is different to that of bp. On the line op/cd a transformation of a kink into a jump takes place on the high-field side of the horizontal section at s ¼1. As regards the low-field side, here a FOMP is observed everywhere below the diagonal of the fourth quadrant (i.e. on the line op). The threshold field of the FOMP decreases on approach to the diagonal and vanishes at the crossing-point. Beyond the crossing-point the line op continues as line cd without any major modification. One can therefore speak of a single aggregate boundary line op/cd. 4.3.2. Absorption of a kink by a jump: line df The line cd ends at a triple point defined by Eq. (55). There the boundary splits into two, cf and df. Beyond the triple point the transition from type c to type d proceeds in two stages: first a discontinuity appears inside the sloping section, at s 41, which constitutes a change from c to f. At this stage both kinks, at s ¼1 and at s ¼(1+m)/(1–m), are preserved. Then the discontinuity grows higher, moves closer to the kink at s ¼1 and finally absorbs it (a transition from f to d). Let us consider the latter transformation in some more detail. It must be confessed, we did not succeed in finding an analytic description for the boundary df at arbitrary m. The curve shown in Fig. 7 is a result of a numeric calculation with m ¼0.5. Yet it is easy to demonstrate that the boundary df possesses a vertical asymptote situated at

k1 ¼

m2 : 2ð1 þmÞ

ð58Þ

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1079

It was demonstrated in Section 4.2.2 that the shape of the magnetisation curves on both sides of the border df tends to that of type e as

k2-N. This has for a consequence that the threshold field of the jump becomes unity in the same limit. On the other hand, just left of the boundary df the magnetisation curve is shaped as type f, with a kink to the left of, and infinitesimally close to h ¼1. At the kink a second-order phase transition takes place, therefore, the leading coefficient of Landau (50) must vanish. Consequently, k1 must satisfy Eq. (53) with h ¼ 1, i.e. Eq. (58). 4.3.3. Annihilation of two kinks: line ab/ij/gh At certain values of the anisotropy parameters the magnetisation curve can develop an N-shaped anomaly near s ¼1 (see e.g. Fig. 3b). It is clear that the falling portion of the curve corresponds to a range of magnetic field where the canted phase is unstable. It is in fact the collinear ferrimagnetic structure that is stable here, accordingly, s 1 as shown with a solid line in Fig. 3b. The parameters can be changed in such a manner that the N-shaped anomaly will decrease and eventually disappear. The horizontal stretch of the curve will shrink into nonexistence; the kinks on its both sides will run into each other and vanish, annihilate. Mathematically the point of annihilation is expressed by the equality of the critical fields of both kinks. These are determined by the zeros of the leading Landau’s coefficient (50), i.e. by Eq. (53). The latter is equivalent to a quadratic equation in h. It is essential that the parameter k2 does not enter in either Eq. (50) or (53), therefore they are perfectly identical to Eqs. (18) and (20). As demonstrated in Section 3 for k2 ¼0, pffiffiffiffiffi the two kinks annihilate at point V of Fig. 4, i.e. at k1 ¼ kV ¼ 12mð1 mÞ2 , cf. Eq. (22). This condition should hold irrespective of k2. Therefore, the boundary between the domains a and b, i and j, or g and h, that is where the kinks annihilate, is a vertical straight line in the k1k2 plane (see Fig. 7). The distinction between three curve types on each side of the boundary (e.g. between a, i and h) is associated with additional features unrelated to the annihilation of kinks at s ¼1. 4.3.4. Annihilation of two jumps: line ao/kl A magnetisation curve may have a horizontal section at s ¼1 delimited by two discontinuities, cf. types l and o in Fig. 8. By changing the parameters k1 and k2 these discontinuities can be made approach each other and annihilate. In such an event type l goes over to type k, whereas type o turns into a. On the border line where the transformation takes place the following four conditions must be satisfied: two conditions of equilibrium for the canted phase, Eqs. (6) and (7), the condition s ¼1, or by Eq. (9): cos a þm cos b ¼ 1m,

ð59Þ

and finally, that the energies of the canted (5) and ferrimagnetic (11) phases be equal: m cosða þ bÞhcos amh cos b þ k1 sin2 b þ k2 sin4 b ¼ mh þmh:

ð60Þ

Now simultaneous linear equations (7) and (60) can be solved for k1 and k2,

k1 ¼ k2 ¼

mhð1 þ 4cos b þ 3cos2 bÞ4m cos b½1 þcosða þ bÞm sin b sinða þ bÞ4hcos bð1cos aÞ 2 sin2 b cos b 2m cos b½1 þ cosða þ bÞ þ m sin b sinða þ bÞ þ 2h cosbð1cos aÞmhð1þ cos bÞ2 2 sin4 b cos b

,

,

ð61Þ

ð62Þ

whereas the quantities a and h are expressed from Eqs. (59) and (6):

a ¼ arccosð1mm cos bÞ, h¼

m sinða þ bÞ : sin a

ð63Þ ð64Þ

The aggregate of Eqs. (61)–(64) provides a parametric description of the line ao/kl in Fig. 7. The parameter b runs between p/2 and p. The upper bound of the interval corresponds to a ¼0 and h¼hV, where hV is given by Eq. (21). In this limit Eqs. (61) and (62) go over to (56) and (57), respectively, i.e. the curve ends at a point where it merges with the lines ab, bp and op. The lower bound of the interval, b ¼ p/2, corresponds to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the limit of strong anisotropy, k1-N. In this limit, a ¼arccos(1–m), h ¼ ð1mÞ m=ð2mÞ, and the sum of Eqs. (61) and (62) remains finite: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 þ k2 ¼ m mð2mÞm: ð65Þ Thus, away from the origin the boundary between the domains k and l approaches an asymptote parallel to the diagonal of the fourth quadrant. 4.4. Emergence of discontinuities inside sloping sections 4.4.1. Lines cf/gj/hi and dn Sometimes discontinuities appear inside sloping parts of the magnetisation curves at arbitrary s, 1o s o(1+ m)/(1–m). This involves a first-order transition that is not a transition between a canted and a higher-symmetry phase, but rather between two kinds of canted phase that differ only quantitatively. In such a discontinuity disappears at a certain critical point, it does disappear completely, rather than turning into a kink associated with a second-order transition. Thus, a discontinuity can emerge within the sloping section of a type c curve, which thereby becomes a curve of type f. Similarly, a curve of type j or i transforms into type g or h, respectively. If a type d curve acquires a second jump inside the slanted part, it turns to type n. Mathematical description of such transformations is difficult because the critical values of the angles a and b are not known a priori. An approximate description would not be very useful for either line cf or dn; both have to arrive accurately at the respective triple points, to meet several other lines. Therefore, the lines dn and cf/gj/hi in Fig. 7 were computed numerically. As regards the latter, one can describe analytically its asymptotic behaviour in the second quadrant, where it goes away to infinity. It is important to realise that in the limit of strong anisotropy, k1-–N, the discontinuity appears as the system tries to ‘‘jump over’’ a hard direction at b ¼ p/2. This is the critical value of the angle b. By Eqs. (6) and (7), at the critical point sin a ¼m and h2 ¼1–m2. A standard procedure follows: b is set to p/2+ Z, Z being an infinitesimal; a is

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expanded in powers of Z by means of Eq. (10); a and b are substituted into Eq. (7), whose left-hand side is then presented as a power expansion in Z: #   " 1 m2 h2 2 k 4 k Z þ    ¼ 0: ð66Þ mh 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 2 m2 þh2 ðm2 þ h2 Þ3=2 Eq. (66) defines implicitly a dependence of h(Z), which in conjunction with s(Z) provides a parametric description of the magnetisation curve. At the point where an S-shaped anomaly emerges, the following must hold: dh @ ðl:h:s:Þ=@Z  ¼ Z ¼ 0 ¼ 0: dZ Z ¼ 0 @ ðl:h:s:Þ=@h

9

ð67Þ

Consequently, the partial derivative of the left-hand side (l.h.s.) of Eq. (66) with respect to Z must be nil at Z ¼0. Equating the square bracket of Eq. (66) to zero and substituting 1–m2 for h2 therein, we obtain an equation of a straight line in the k1k2 plane:

k1 þ2k2 ¼ 12m2 ð1m2 Þ:

ð68Þ

Eq. (68) describes the asymptote of the boundary cf/gj/hi in the limit k1-–N. It runs parallel to the line ai/bj. It is clear from Fig. 7 that the deviation of the boundary cf/gj/hi from its asymptote in the second quadrant is insignificant. 4.4.2. Two different ways to go over from type n to type d If one considers a magnetisation curve of type n with two jumps and two sloping sections, two obvious ways of converting it into a type d curve come to mind: either the upper discontinuity can be closed (a merger of the two sloping sections) or the lower sloping section can be removed (a merger of both discontinuities). The former is a process considered in the previous subsection; it takes place on the boundary dn where its slope is gentle. Interestingly, the latter option is realised as well. Namely, in the right-hand part of the boundary dn, which is a continuation of the line em and has a significant negative slope. That the two ways of changing from n to d are physically distinct is confirmed by the fact that respective parts of the boundary do not merge smoothly, but rather make a kink. Unfortunately, neither part of the boundary dn can be computed but numerically. 4.4.3. Locating the merging point of lines dn and mn According to Section 4.2.1, the line ak/lo is where the saturation kink in the magnetisation curve becomes a jump. In order for this to happen, it is necessary that the two leading Landau’s coefficients (29) and (30) be nil and the third one (31) be positive. On moving further to the right in Fig. 7 the line lo splits into two. Accordingly, the transformation of the kink into a jump proceeds in two stages: first (on line dn) a discontinuity appears at a certain intermediate magnetisation, rather than at saturation; then the discontinuity grows and (on line mn) absorbs the saturation kink. The transition from the one- to the two-stage scenario thus involves a displacement of the critical magnetisation of the discontinuity from the maximum, (1+m)/(1–m), to a smaller value. Mathematically, it is necessary that the third Landau’s coefficient (31) vanish, the first two coefficients being nil as well. Setting to zero Eqs. (29)–(31), one obtains an equation for the critical field: m2 h2 ðhmÞ5



mh ðhmÞ3

þ

1 ¼ 1: hm

ð69Þ

Note that a substitution of –m for m in Eq. (69) leads to Eq. (55), which describes the triple point in the first quadrant. At m¼ 0.5 the solution of Eq. (69) is h¼ 1.391. Eqs. (32) and (33) then yield the coordinates of the sought point: k1 ¼0.0425, k2 ¼  0.0748. It turns out that the merging point of lines dn and mn lies to the left of the diagonal of the fourth quadrant. This means that the k1k2 diagram contains a further very narrow triangular domain q, indiscernible in Fig. 7. In principle, the magnetisation curves within that domain are distinct from all other types shown in Fig. 8. However in practice, because of the extreme narrowness of domain q, they can be hardly distinguished from those of type l or o.The position of the merging point of lines dn and mn is very sensitive to m. Thus, at m4(O5 1)/2 ¼0.618 this point lies in the third quadrant. At mo(21O105 105)/250¼0.441 it finds itself to the right of the diagonal of the fourth quadrant, so that the k1k2 diagram contains strictly no domain q.

5. Discussion Fig. 7 can predict the shape of the magnetisation curve for arbitrary k1 and k2, but only for one value of m, m¼ 0.5. A direct approach to answering the question about the dependence on m would be to produce many such diagrams for values of m covering the interval from 0 to 1 with sufficient density. Without analytic expressions for all the lines, this would be a laborious task going beyond the scope of the present work. Here we would rather try to give a qualitative answer. We have learned in Section 3 that as k1 grows at k2 ¼ 0, the system goes through the same sequence of five curve patterns, a to e, irrespective of m (see Fig. 6). These correspond to the five domains cut through by the abscissa axis in Fig. 7. A number of further obvious facts can be stated, which hold for any m, 0omo1. The boundary ab/ij/gh always lies to the left of cj/fg (which coincides with the ordinate axis). The latter is always situated to the left of the vertical asymptote of the boundary df, which is in turn left of the line de. Further, the asymptote of the boundary cf/gj/hi is always higher than and parallel to the line ai/bj. The slope of both lines equals  1/2, regardless of m. Finally, direct calculations demonstrate that the triple point cdf always lies in the first quadrant left of the vertical asymptote of the boundary df and below the sloping asymptote of cf/gj/hi. Summarising, the topology of the upper half of the k1k2 diagram does not change with m. The same cannot be said about the lower half of the diagram. In Section 4.4.3 we have seen that the position of the merging point of lines dn and mn depends sensitively on m. At m40.441 the spit of the domain n protrudes through the diagonal of the fourth quadrant and forms, strictly speaking, a separate domain q (too narrow to discern in Fig. 7). At m 40.618 the spit is situated in the third quadrant and another

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1081

κ

j b

c

κ

e

κ

p

κ

h a k m l

Fig. 11. The k1k2 diagram of Fig. 7 in the limits of (a) weak anisotropy, k1,2-0 and (b) strong anisotropy, 9k1,29-N.

new domain is formed. We are far from being able to give a detailed account here. Suffice it to say: mutual positions of several other lines depend essentially on m, so that the topology of the lower half of the k1k2 diagram is not preserved. As against that, the necessary condition of metamagnetism (i.e. of a field-induced transition directly from ferri- to ferromagnetism) proves a robust criterion. The inequality (37) contains only k1, but not k2. Moreover, if the anisotropy energy (3) was appended with a term in sin6 b, the extra anisotropy constant would not enter in (37) either. It should be emphasised, though, that the quantity K1 in Eq. (3) is generally a linear combination of conventional anisotropy constants depending on the crystal symmetry and orientation of applied magnetic field. Generally speaking, (37) is only a necessary condition of metamagnetism. It can be upgraded to a sufficient condition if one can exclude large negative values of k2 (and/or of higher-order anisotropy constants), so as to guarantee that the locus of the system in Fig. 7 remains above the line em. On the whole, Fig. 7 presents a rather confusing picture, even if one stays with m¼0.5. It is therefore of interest to consider simpler special cases. Such is, e.g., the limiting case of weak anisotropy, k1,2-0. Zooming in on the origin in Fig. 7 leads eventually to a diagram displayed in Fig. 11a. Here all the boundaries are straight lines, their slopes being independent of m. Comparing Figs. 11a and 9, one can notice in the former an extra line in the third quadrant. Smit’s original diagram (Fig. 9) does not contain this line, which divides sector ‘‘?’’ into two, with and without FOMP (domains p and b, respectively). Studying one-sublattice ferromagnets, Melville et al. [9] found for this extra boundary a slope of 1/4. Beyond the one-sublattice approximation the line bp has a slight downward curvature (not visible in Fig. 11a), while the other three boundaries are perfectly straight. Let us now consider the case of a very strong anisotropy, 9k1,29*1. This time we zoom out of Fig. 7. As discussed in Section 4.2.2, in the entire sector ‘‘99’’ the shape of the magnetisation curves tends to type e on moving away from the origin. That is to say, lines de and df fade away as k2-N, even though they formally extend to infinity. Similarly, line ai fades away at infinity, as magnetisation curves of type i tend to type a. It is immaterial that the new boundary between sectors a and h ( ¼line hi in Fig. 7) does not go through the origin – the finite displacement is not visible on the infinitely large scale of Fig. 11b. Likewise, domains l and m are only seen as a thin double line in Fig. 11b, although they have a finite width and maintain their distinct character at infinity. Finally, for the boundary ak one obtains an equation of a straight line in the spirit of Melville et al. [9], k2 ¼ 14k1 , by taking the limit h-N in Eq. (33). It is remarkable that the k1k2 diagrams obtained in the limits of weak and strong anisotropy (Fig. 11a and b) are similar. Namely, in both cases it is Smit’s diagram (Fig. 9) appended with Melville’s line in the third quadrant. Smit’s three boundaries (but not the one of Melville et al.) are also present as true straight lines in the exact k1k2 diagram (Fig. 7). The Smit–Melville diagram is characteristic of one-sublattice magnets, yet it is for different reasons that the one-sublattice approximation applies in the two limiting cases. In the weak-anisotropy (strong-exchange) limit the two sublattices always maintain their antiparallel mutual orientation. In the strong-anisotropy (weakexchange) limit the system becomes a superposition of two non-interacting ferromagnets, an isotropic and an anisotropic one. The former plays the role of a background whereas the latter, endowed with magnetisation M2 and anisotropy energy (3), is responsible for the peculiar anisotropic effects. Since the slopes of the boundaries in the Smit–Melville diagram are universal constants independent of magnetisation, one comes to the paradoxical result that the weak- and strong-anisotropy limits of Fig. 7 are similar and independent of m.

6. Conclusion High-field magnetisation curves of two-sublattice ferrimagnets have been analysed from the point of view of their shape. In the adopted model the magnetic anisotropy is associated with the subdominant sublattice only and limited to second- and fourth-order terms. A special case where the sublattice moments are related as 1:2 has been investigated in some detail. In that case as many as 16 different curve types are possible. Which one is realised depends on the strength of the anisotropy in relation to the intersublattice exchange as well as on the orientation of the magnetic field with respect to the crystal axes. (Only high-symmetry orientations have been considered.) Out of all possible curve patterns, four are preserved in the limit of weak anisotropy: b, c, j and p. Four other shape types (a, e, h and k) are characteristic of the strong-anisotropy case; two further types, l and m, are less probable albeit not impossible. Remarkably, in both limiting cases the possible curve types are distributed in a similar and simple way in the parameter plane of relative anisotropy constants k1 and k2 (Fig. 11a and b). In both cases the k1k2 diagram bears similarity to the Smit–Melville diagram characteristic of one-sublattice ferromagnets.

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