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Chapter 5 First-order magnetic processes
chapter 5 FIRST-ORDER MAGNETIC PROCESSES
G. ASTI Physics Department University of Parma 43100 Parma Italy
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Phenomenology of first-order magnetization processes in terms of anisotropy constants. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Uniaxial symmetry . . . . . . . . . . . . . . . . . . . . . . . 2.3. Cases of trigonal, hexagonal and tetragonal symmetry . . . . . . . . . . . 2.4. Cubic symmetry . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Mechanisms responsible for high-order anisotropy constants . . . . . . . . . 2.6. Domain wall processes at the transition point . . . . . . . . . . . . . . 3. Processes involving competition between exchange and anisotropy . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Role of exchange at transition points . . . . . . . . . . . . . . . . . 3.2.1. Multisublattice magnetic systems with uniaxial anisotropy . . . . . . . . 3.2.2. Field-induced spin reorientation transitions. Critical fields . . . . . . . . 3.2.3. Boundary conditions for first-order transitions . . . . . . . . . . . 3.2.4. The two-sublattice system . . . . . . . . . . . . . . . . . . 3.2.5. Linear regime and magnetic transitions in the canted phase . . . . . . . 3.3. The small-angle canting model . . . . . . . . . . . . . . . . . . . 3.3.1. General remarks . . . . . . . . . . . . . . . . . . . . . 3.3.2. The two-sublattice model . . . . . . . . . . . . . . . . . . 3.3.3. Some applications . . . . . . . . . . . . . . . . . . . . . 3.4. Hexagonal ferrites having block-angled and spiral structures. Role of antisymmetric exchange . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Considerations on the experimental methods . . . . . . . . . . . . . . . . 4.1. Measurements in high magnetic fields . . . . . . . . . . . . . . . . 4.2. The singular-point detection technique . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
398
399 402 402 402 414 419 421 423 427 427 428 428 430 432 433 440 447 447 448 453 454 455 455 457 462
1. General
The magnetization curve of a magnetically ordered crystal, ferro- or ferrimagnetic, can, under special circumstances, show anomalous discontinuities that appear as sudden increases of the magnetization with increasing magnetic field. In these conditions, the crystal actually undergoes a first-order magnetic transition, with characteristics that are, in general, strongly dependent on the relative orientation of the applied magnetic field direction and the crystal axes. The observed phenomenon is an irreversible process of a different nature than the ordinary magnetization reversal, which always occurs when the applied magnetic field intensity is equal to the coercive field; a fact that is reflected in the well-known symmetry of the hysteresis loop of all ferromagnets (by the general term 'ferromagnet', we shall refer hereafter to both types of material, properly called ferromagnetic or ferrimagnetic, when the magnetic order is considered to be irrelevant for the phenomenon under study). These anomalous magnetization jumps, being first-order transitions, also ought to be accompanied by hysteresis loops, which would essentially be localized in the first quadrant of the magnetization curve, but, in general, the hysteresis around the transition happens to be very small and can only be revealed with difficulty. The observed jump in the magnetization curve can be due to a rotation of the magnetization vector, to a change of the modulus of the magnetization vectors, or, in complex systems, to a change of the magnetic order. In the first case, we deal with changes in the magnetic state of the system that can effectively be ascribed to its magnetic anisotropy, while in the second case, we are, in most cases, faced with direct manifestations of microscopic effects, such as metamagnetic transitions that can be the consequences of particular crystal-field effects, and/or the effect of the magnetic field on the energy-band structure. The third case involves effects of the type of spin-flop transitions, with drastic changes in magnetic structures. In the present chapter we will not consider the second type of transitions, which has been treated in the reviews by Wohlfarth (1980, 1983), Buschow (1980) and Khan and Melville (1983). Some aspects of the third type of phenomena, i.e., the case of anisotropic ferrimagnets, will be examined in section 3.2. In section 2 we deal with the first type of field-induced first-order transition, hereafter referred to as FOMP [first-order magnetization process (Asti and Bolzoni 1980)]. In this case, 399
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G. ASTI
the magnetic anisotropy of the system by itself allows the existence of two inequivalent minima of the free energy that correspond to two distinct directions of the magnetization vector. The effect of the external field is to produce a relative shift between the energies of the two states involved in the transition. When the two states reach the same energy the transition can start, provided a mechanism exists that allows the nucleation and growth of the new phase. It is likely that this nucleation occurs inside the domain wall (see section 2.6). When the changing parameter is temperature, the system can make the transition in zero external field, making a sudden rotation of the easy-axis direction. This type of transition is usually referred to as a spin reorientation transition (SRT), and is essentially the same phenomenon as FOMP. In fact, in two-dimensional phase diagrams, energy versus H or T, the transition is represented by lines that in general can be crossed by changing either T or H. FOMPs in cubic crystals have been considered in the frame of the problem of phase transitions and critical phenomena (Mukamel et al. 1976, Cullen and Callen 1984) as examples of the Potts model (see section 2.4). However, the widest application of the concepts developed around the FOMP transition is indeed in studies of the anisotropic properties of multisublattice magnetic systems, especially rare earth (R) intermetallic compounds. In practice, it is very common to find this type of behaviour in these materials. The study of FOMP and the measurement of critical parameters, on one hand, provide a great deal of information about the magnetic anisotropy and, on the other hand, the interpretation of observed FOMPs represents a formidable test for a theory of magnetic anisotropy in these systems. Moreover, one has to consider the inherent precision and proper reliability of data about critical parameters, as compared to best fit approach to saturation and similar procedures. A phenomenological description of FOMP in terms of the anisotropy constants or the anisotropy coefficients shows that these transitions require high-order terms in the expansion of the free energy. Crystal-field theory applied to single ions does not in many cases justify the existence of such high-order terms. This fact by itself indicates how important it is to take the competing interactions typical of multisublattice systems into account, as, e.g., the competition between anisotropy of different sublattices (Cullen and Callen 1985). In this respect, the exchange interaction also plays an important role, especially for the case of ferrimagnetic order (Sarkis and Callen 1982). Indeed, the presence in R intermetallic compounds of exchange of the same order of magnitude as the anisotropy energy implies deviations from the collinear magnetic order. Even for small canting angles, the free energy of the system is strongly modified, and the macroscopic anisotropy properties are influenced in various ways. Hence, the classical phenomenological description is inadequate, sometimes leading to wrong conclusions. For instance, non-monotonic dependences of the anisotropy constant in pseudobinaries, such as (Y, Nd)Cos, (Y, Gd)Cos, and (Sm,Nd)Cos, have erroneously been considered as evidence of the non single-ion origin of the anisotropy energy (Ermolenko 1979). A convenient approach to the problem, which applies to the majority of cases, is to treat the
FIRST-ORDER MAGNETICPROCESSES
401
small-angle canting (SAC) between sublattice moments as a first-order perturbation (Rinaldi and Pareti 1979). The advantage of this model (hereafter referred as the SAC model) is that the system can again in many respects be treated as an ordinary ferromagnet having certain effective anisotropy constants K1, g2,.... It means that the effect of canting of the sublattice moments can be merely accounted for by using proper expressions for the effective anisotropy constants; these include the exchange-interaction parameters and reduce to the ordinary algebraic addition of the anisotropy constants of the individual sublattices when the anisotropy is negligible with respect to the exchange-energy density, a condition that corresponds to normal collinear magnetic order. Actually, the K's obtained are not rigorously constant because they depend on the intensity of the magnetic field, but the model in many cases allows to get a simple physical picture of the mechanisms responsible for the observed phenomena. As a matter of fact, it gives an explanation of the existence in these intermetallic compounds of effective anisotropy constants of high order. Certainly, canting effects play a role in the presence of FOMP and non-linear effects in the magnetization curves. As mentioned above, at high magnetic fields, pronounced transitions of the spin-flop type are possible for ferrimagnetically ordered RE intermetaUic compounds. They most frequently involve direct competition between magnetostatic energy and exchange. With increasing fields, magnetic configurations with large angles between the magnetic moments can be energetically favorable. A particular type of first-order transition of this category, named first-order moment reorientations (FOMR), was observed by Radwanski et al. (1985) and extensively studied in easy-plane ferrimagnetic compounds of the R2T17 type (here T is a transition metal). In this case, the applied field indeed lies in the basal plane and the six-fold symmetry plays a role in allowing non-collinear moment configurations to become energetically favorable. Computed curves for ferrimagnetic systems with magnetic anisotropy, presenting transitions between different magnetic configurations have also been described by other authors (Sannikov and Perekalina 1969, Kazakov and Litvinenko 1978, Rozenfeld 1978, Gr6ssinger and Liedl 1981, Ermolenko 1982, Radwanski 1986). A certain analogy exists with the phase diagrams of antiferromagnets that show spin-flop transitions. Classic treatments on angled spin structures but without anisotropy (Clark and Callen 1968), show continuous transitions between various structures. Among the peculiar effects found, are a temperature-independent susceptibility (Clark and Callen 1968) and an effective decoupling between the sublattices in the canted phase (Meyer 1964, Boucher et al. 1970, Acquarone and Asti 1975). A theoretical model (Acquarone 1981) of an antiferromagnetic chain including anisotropic and antisymmetric exchange, as well as anisotropy terms up to sixth order, explains the observed phenomena in the so-called 'angled block structures' in complex ferrimagnetic oxides of hexagonal and trigonal structure. Certain compositions show field-induced transitions and unusual effects, because the system behaves like an effective two-sublattice canted antiferromagnet, so as to give rise to various possible configurations: angled, conical helix or flat spiral.
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G. ASTI
2. Phenomenology of first-order magnetization processes in terms of anisotropy constants 2.1. Introduction
The anisotropy energy of a ferromagnetic crystal can be expressed as a power series of the direction cosines of the magnetic moment with respect to the crystal axes. So the expansion has different forms depending on the symmetry of the crystal. The coefficients of the various terms are the anisotropy constants and, in general, a satisfactory description of the magnetization curve is obtained by limiting the expansion to the first few terms. Under normal conditions, this curve is continuous in the region preceding saturation, where the magnetization process takes place by reversible rotation of the magnetization vector M s . However, when the anisotropy constants fall within certain ranges corresponding to some specific conditions, irreversible rotations of M S are possible, implying first-order transitions between inequivalent magnetization states. In fact, depending on the values of the anisotropy constants, the system may possess two or more inequivalent minima of free energy E for different magnetization directions; so it is evident that an applied magnetic field H of suitable intensity and orientation can induce transitions between these minima. Actually, the addition of a magnetostatic term modifies the total energy surface in such a way that the two energy levels become equal. The existence of two minima of E in zero field is not a necessary condition: the effect of the magnetostatic energy component is in certain circumstances enough to produce an additional minimum besides the absolute one (easy direction). In principle, in the case of F O M P we expect to observe minor hysteresis loops around the critical field, for which the two states have equal energy. However, it is difficult to observe hysteresis in F O M P even at low temperatures (Melville et al. 1976, Ermolenko and Rozhda 1978). The reasons probably lie in the very low coercivity inherent in this type of magnetization process (see section 2.6). 2.2. Uniaxial s y m m e t r y
A very special but important case is that of uniaxial symmetry. In this case the free energy of the system can be expressed as E -- K 1 sin e 0 + K 2 sin 4 0 + K 3 sin 6 0 + . . . .
H M s cos(0 - q0,
(2.1)
where K1, Kz,/(3,... , are the anisotropy constants and 0 and ~p are the orientation angles of the magnetization vector M s and the magnetic field H with respect to the symmetry axis c, respectively, As a first step, it is important to know the phase diagram of easy and hard directions. For reasons of symmetry, the c axis (0 = 0) and the basal plane (0 = ½It) are always points of extremum. With the anisotropy energy expanded up to the sixth order, we can have, at most,
FIRST-ORDER MAGNETIC PROCESSES
403
two additional extrema along conical directions at angles given by sin 0c = ( [ - K 2 ++-( K 2 - 3 K1K3) 1/2]/3K 3 }1/2.
(2.2)
Let C ÷ and C - be the cones associated with the + and - sign in eq. (2.2). It turns out that C ÷ is always a minimum while C - is always a maximum, so that only C ÷ can be an easy direction. A convenient representation of the diagram of the easy directions and the other extrema is representation in terms of reduced anisotropy constants x = K 2 / K 1 and y -- K 3 / K 1. These two variables are sufficient to distinguish all the possible cases. In fig. 1 the phase diagram for the two cases K i > 0 (upper) and K 1 < 0 (lower) is shown. All the information concerning the easy directions and the other extrema are contained in a special symbol that marks every different region. It simulates a polar type of energy representation indicating existing extrema by concave (minimum) and convex tips (maxima). Vertical and horizontal stems refer to the symmetry axis and the basal plane, respectively. The left-hand and right-hand oblique stems indicate the C - and C ÷ cones, respectively. The absolute minimum (easy direction) is indicated by filling
I
4
¥
L, ×
-2 Fig. 1. Magnetic phase diagram of the uniaxial ferromagnet: (Top) K~ > 0; (Bottom) K 1 < 0. T h e coordinates are x = Kz/K ~ and y = K3/K~. For the explanation of the symbols see text (Asti and Bolzoni 1980).
404
G. ASTI
of the tip. The boundary lines of the various regions in the phase diagram of fig. 1 are specified by the equations labelled by the same letters in table 1. The determination of the conditions for the existence of FOMP requires the analysis of the magnetization curve and its dependence of the values of the anisotropy constants, for different directions of the magnetic field. For most purposes it is enough to consider the cases of H parallel or perpendicular to the c axis, hereafter indicated as the A-case and P-case, respectively, where A denotes axial while P stands for planar. The general case is relevant when considering the effects of FOMP in polycrystalline materials (see section 4). Here, it suffices to mention that a peculiar phenomenon is only observed for H lying not in a symmetry direction; under certain conditions of anisotropy constants and field orientations, an additional transition is possible, so that one observes a double FOMP, consisting of an ordinary and extraordinary FOMP (Asti and Bolzoni 1985). The equilibrium equation is OE/O0 = 0, where E is given by eq. (2.1). For the P-case (~ = ½) it turns out to be, h = 2m(1 + 2 x m a + 3 y m 4 ) K 1 / l K l [ ,
(2.3)
where m = M / M s is the reduced magnetization, h = 2H/IHAII the reduced field and HA1 = 2K1/Ms, the c axis anisotropy field. Using eq. (2.3) and the condition that the total energy is equal in the two phases, we obtain the equation for the critical magnetization, (3m 2 + 2m + 1)x + (5m 4 + 4m 3 + 3 m 2 + 2m + 1)y + 1 = 0.
(2.4)
For the A-case, the equilibrium equations corresponding to eqs. (2.3) and (2.4) are h = - 2 m [ 1 + 2x(1 - m 2) + 3y(1 -
m2)2]KJ[KI(1+ 2x + 3y)[,
(1 + m)(1 - 3 m ) x + (1 + m)(1 - m2)(1 - 5 m ) y + 1 = 0,
and
(2.3') (2.4')
TABLE 1 Equations of the boundary lines of the various regions in the phase diagram (fig. 1) and in the F O M P diagram (figs. 3 and 4). The lines are labelled by a letter that appears in figs. 1 and 3. The couples of curves conjugated in the K ~ R transformation [eqs. (2.6)-(2.8)] appear on the same line in the table. Curve
Label
Conjugated curve
Label
3y+2x+l=0 x+y+l=0 15y + 6 x + 1 = 0
1 m n
K I = 0 , i.e., x+y+l=0 4x- 1 =0
1'
4y - x 2=0
o
3 y 2 + 2xy - x z + 4y =0
3y-
p
3y-x
q
3x z + 8xy + 12y z - 5y = 0
r
x4+
x 2 =0
5y - 3x z = 0
x,y---~
n' o'
2 =0 q'
Eq. (2.11) in the text
x3y - 8xay - 3 6 x y a -
27y 3 +
16y 2 = 0
r'
FIRST-ORDER MAGNETIC PROCESSES
405
where h = 2 H / I H A 2 ] with HA2 = -2(K~ + 2 K 2 + 3 K 3 ) / M ~ , the anisotropy field along the basal plane. There is a transformation, called the K*+ R transformation, of the anisotropy constants into conjugate quantities that allows to transfer immediately all the results obtained for the P-case to the A-case and vice versa, according to the following symmetrical dual correspondence,
basal plane
c axis
K1 K2 K3
(2.5)
R1 R2 R3
The K ~ R transformation is obtained by imposing on the energy expressions the same formal dependence on the reduced magnetization m, for the A and P cases. So one obtains the following linear transformation, RI=-Ka-2K2-3K
3,
R2=K2+3K
3,
3.
(2.6)
/£3 = - R 3 .
(2.7)
R3=-K
The inverse transformation obviously has the same form, K 1 = - R 1 - 2R 2 - 3R 3 ,
K2 = R 2 + 3R 3 ,
In terms of reduced variables it becomes, 2= (-x-
3y)/(1 + 2 x + 3 y ) ,
x = (-2-
337)/(1 + 22 + 337),
fi = y / ( 1 + 2 x +
3y), (2.8)
y = 37/(1 + 2£ + 337),
where 2 = R 2 / R 1 and 37= R 3 / R 1. One can easily recognize that the obtained correspondence is a projective transformation, i.e., a homology having the center in the point C ( - 1 , 0) and the axis being the line with equation 2x + 3y = 0, with an invariant equal to - 1 , i.e., the transformation is a harmonic or involutory homology. Accordingly, second-order curves are transformed into other secondorder curves. The correspondence for some important lines is reported in table 1. Two types of F O M P are distinguished, depending on the fact whether the final state after the transition is the saturation state (type-1 F O M P ) or not (type-2 FOMP). They are represented in fig. 2. From the degree and the symmetry of the equilibrium equations, it turns out that when the easy direction is coincident with a cone (cone C +) type-2 F O M P cannot occur. From the equality of the total energy in the two states and from the equilibrium equations, one obtains the conditions for the occurrence of the two types of FOMP. Figure 3 summarizes the results, giving the complete diagram of F O M P in the planes of reduced anisotropy constants x and y for the two cases K 1 > 0 and K 1 < 0. The various regions are
406
G. ASTI c-axis
'
(a)
.r--. ;
Ha2
H
(b)
(c)
Ib
Fig. ½. Examples of FOMP different types (right-hand side): (a) type 1. (b) type 1 with easy cone. (c) type 2. The diagrams on the left-hand side are cross-sections of the anisotropy-energy surface (Asti and Bolzoni 1980). The arrows on the vertical portion of the M(H) curve (full line) indicate that the transition occurs at the same critical field both with increasing and decreasing magnetic field, when magnetic states having equal free energy are involved. This is actually the most common case for experimentally observed FOMPs, because the hysteresis around the transition is either very narrow or even not detectable. Upward and downward arrows on the two sides indicate upper and lower limits for the critical field, located at the onset of instability. They define the maximum theoretical width of the hysteresis cycle connected with the FOMP. distinguished by the labels A 1 , A 2 , P1, P2, P1C, A 1 C which specify the m a g n e t i c field direction (A, axial; P, planar) and the type o f F O M P (1 and 2), and the easy-cone o f the regions with type-1 F O M P ( A 1 C , P1C). T h e b o u n d a r y lines are those specified in table 1. T h e areas that are c o n j u g a t e d in the K ~ R transformation are s h a d e d in the same way. T h e analysis of the equations also provides the values o f the critical field her and the critical m a g n e t i z a t i o n mcr as functions o f the c o o r d i n a t e s x, y a n d the sign of K 1. T h e critical magnetization is defined as the m a g n e t i z a t i o n at which the transition takes place. Figure 4 gives the same diagrams as in fig. 3, but n o w with the lines of constant h , and incr. T h e s e plots can be used in analyzing experimental observations o f F O M P , and allow precise
FIRST-ORDER MAGNETIC PROCESSES .....
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4
~
-
.;:,(U:,-',-
407
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x
....,......,........... -"-'--',-" .......- , - , ~ , ~ - p ( ~ i ' .
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!iiii! ~
'
"
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Fig. 3. Diagram of FOMP in the plane of reduced anisotropy constants x = K2/K1, y = K 3 / K x. Top and bottom figures refer to the cases Kx > 0 and K~ < 0, respectively. The different regions are distinguished by a letter, i.e., P for the P-case or A for the A-case (see text), followed by the numeral 1 or 2 indicating the type of FOME Type-1 FOMP in regions of easy cone are specified by the addition of the letter C. The boundary lines are those specified in table 1. The areas that are conjugated in the K ~ R transformation (see text) are shaded in the same way (Asti and Bolzoni 1980). e v a l u a t i o n s o f t h e r a t i o s o f t h e a n i s o t r o p y constan~ts. I t is w o r t h n o t i n g t h a t i n t h e case o f t y p e - 2 F O M P t h e s y s t e m o f e q u a t i o n s c a n b e a n a l y t i c a l l y s o l v e d g i v i n g e x p l i c i t e x p r e s s i o n s for her as well as mcr = m 1 a n d m2, t h e r e d u c e d m a g n e t i z a t i o n s for t h e i n i t i a l a n d final s t a t e s , r e s p e c t i v e l y . T h e s e a r e , for t h e P - c a s e , hc~ = ~ A { 2 - [17x 2 + x ( 6 0 y
-
llx2)1/2]/30y}, (2.9)
m x = ½(A-
D),
m 2 = ½(A + D ) ,
where A = {[-3x + (60y - llx2)~/2]/lOy} D = {[-5x - (60y - llx2)l/2]/6y}
~/2 , t/2
408
G. ASTI 6
-2
-4
¥
-6
- 4
-2
.3
Fig. 4. Lines of constant values for the reduced critical magnetization, mcr (dashed lines), and reduced critical magnetic field her (full lines), in the plane of reduced anisotropy constants x = K 2 / K 1 , y = K 3 / K ~. The various regions are denoted as in fig. 3. In the case of type-2 FOMP, mcr is the initial state, i.e., m 1. The spacing between the values relative to contiguous lines of mcr or her, changes only when crossing a line labelled by its value. (a) K1 > 0, x > 0, y < 0. (b) K 1 > 0, x < 0, y > 0. (c) K~ > 0, x<0, y<0.(d) Kl<0, x>0, y>0.(e) Kl<0, x>0, y<0.(f) K~<0, x<0, y>0(AstiandBolzoni 1980).
409
FIRST-ORDER MAGNETIC PROCESSES -~
X
-4
0
-2
(c)
e ___ ~--~-~_
--_
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . -
. . . . . . .
2
~
. . . .
4
(a) Fig. 4.
(Cont.)
. 3 s -
6
410
G. ASTI 2
4
6
~---<.~ ,+.<
-2
I
i
i
~
',
-..<)
-4, Y
(e)
~',
\~
\;
t
I /
,'/
/,,1¥
e""4 ~X" \ \ \
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\\
,
I
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i
., \\
I
,A2
\x
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\\
,
A1
,
/I
//
I 111
'
II
~I
//
[ t /X,,'/ .J.-" 1.'/. ~..--"t=
- 7 \ / " -~
'~..~.'-,.~,~,7 ~7~ j ""-
~_.~_~
~~-~.~
J( -i
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Fig.
(f) 4. (Cont.)
-~
'
0
FIRST-ORDER MAGNETIC PROCESSES
411
and, for the A-case, her = } A ' { 2 - (x + 3y)[17(x + 3y) + G l / [ 3 O y ( 1 + 2 x + 3y)]}, (2.10) m I = (A' - D')/2,
m 2 =
(A' + 0')/2,
where G = (60y - l l x 2 + 81y 2 + 5 4 x y ) 1/2 , D' = {[5(x + 3y) - G]/6y}
A' = ([3(x + 3y) + G l / l O y ) 1/2
1/~ .
If we impose the condition m 2 = 1, we find the boundary with the region of transitions of type 1 [line labelled 'r', see fig. 3 (top)] for the P case. The equation obtained is X 4 -- 5x3y + 61x2y 2 + 255xy 3 + 225y 4 -- 8x2y + 52xy 2 + 105y 3 + 16y 2 = 0 .
(2.11) After applying the R ~ K transformation it becomes the ' r " line [see fig. 3 (bottom) and table 1], which is relative to the A-case. Similar diagrams can also be obtained in terms of the anisotropy coefficients Y{Z,minstead of the constants K i. These coefficients apear in the expansion of the anisotropy energy in spherical harmonics. It is known, in fact, that there are advantages in using this representation when analyzing the t e m p e r a t u r e dependence of magnetocrystalline anisotropy, because of the congruence with the Callen and Callen (1966) theory. The relations between the two sets of p a r a m e ters are, K 1 = al ~'/2, 0 + bl ~'{4, 0 + cl if{6,0 + . . . , K2 =
(2.12)
b2•{4, 0 q- C2Y~'6, 0 "-[- . . . ,
K3 ~_~
c3Y{6,0 ql_ . . .
,
where aI
=
-V~
p ,
c2 = ~ X / ~ p ,
b a = -10p ,
cI = -7X/-~ p
c3 = - ~X/-~ p ,
b2 = ~p ,
,
p = 3/(4x/-~).
The inverse transformation is, ~/~'2,0 = a71K1 - b l ( a l b 2 ) - l K 2 ~/'4,0 = ffff6,0 =
b 21K2
+ ( b l C 2 - b2Cl)(alb2C3)-lK3 , - c 2(b 2c3)- 1K3,
c31K3 ,
(2.13)
412
G. ASTI
The FOMP diagrams in terms of the ratios u = Y{4,0/Y{2,0and v = ~{6,o/Y{2,o are reported in figs. 5 and 6, for the cases Y~2,0> 0 and Y{2,0< 0. The straight line labeled L' corresponds to K 1 = 0. Above this line, K 1 has a sign opposite to Y{2,0, and changes sign when crossing it. The same letters denoting the boundary lines in fig. 1 are used here in capitals to distinguish corresponding lines according to transformation (2.12). By the same transformation the FOMP diagrams of figs. 3 and 4 become those given in fig. 5 and 6, respectively. It is worth noting that the diagrams in the plane ff(2,o > 0 are symmetric to those in the plane ~Lr2,0< 0 in a projective sense, i.e., including the points at infinity and considering continuity through it; a property that is absent in the diagrams in terms of the K's. It is also of interest to consider a typical trajectory of the representative point at low temperatures as predicted by the ll(l + 1) power law according to the Callen and Callen (1966) theory. This yields u x m y and v ~ m is, so the line described by the
':!:!:i: i:!:i:i:!!!:i:i:!:i:i:ii!!!i!ii:!
:::!:!:::::::::: li~:}:ii ::~::i::~i!~:!~ P1C
ci!i!iiiii!iiiiiiiiii! ';'iiii!!ii;iiii iiiii?iii!i'oJ;:ii:iiiiii!!iii:!i:i:!i!ii::ii!i
:::::::::::::::::::::::::::::::::::::::::::::::::
;i;!S;~;{~ttt-~t::~:~':-:':-:-:-:-:':':-:': 'Ic !!ii!!iiiiii!!!iiii!iiii1 ~ ;-:-:-:-:-:-:-:':-:-:':':-:':~ v!ii i!i i!!i!i i i i i !i i ...F,2
i::i~ii!ii:!-ii!i!i~iii~Lo'::::::::::::::::::: | :::::::::::::::::::::::::::::: i.:;::P_t ,
o
~--..."::,.,:
Fig. 5. Diagram of F O M P in the plane of reduced anisotropy coefficients u = Y{n.01Y{z,0, v = Y{6,0/Sff2,0. Top and bottom figures refer to the cases Y{2,0> 0 and ~2,0 < 0, respectively. The different regions are distinguished as in fig. 3. The same letters denoting the boundary lines in fig. 1 are used here in capitals, to distinguish corresponding lines according to the transformation of eq. (2.12). The areas that correspond (according to the same transformation) to the conjugated areas represented in fig. 3 are shaded in the same way (Asti and Bolzoni 1980).
FIRST-ORDER
MAGNETIC
.... 06. . . . . . . .
A2
er-'O
\-,\
' ~
,
i
PROCESSES
I_~~--,
",
, ___2_.
-
413
f,,r.,-" PlC~
~
.
--l-
"-._
-..
~.'~
\
~-.%,
\
•o,,.-.~,~, \
A'iC ~ \ -08. `
~xx
~ / •
,
/
,, / , ,
\ , // ~
II
-
,,' .,~
.4//."" y/
//"" .-
,
,>..oo~,
(,IA2 ~,,oo4~._.-4 3
5
t'ig. 6. Lines of constant values for the reduced critical magnetization mot (dashed lines), and reduced critical magnetic field hot (full lines), in the planes of reduced anisotropy coefficients u = 5(4,0/5(2,0 and v = ~6,0/5(2,0 . T h e different regions and the values relative to the various lines of mcr and her are distinguished as in fig. 4. (a) and (b) figures refer- to the cases 5(2.0 > 0 and 5(2.0 < 0, respectively. T h e inset of (b) shows a part of the plane 5(2,0 < 0, with an expanded scale and with the origin (u = v = 0) in the same point. T h e curve . . . . . . . is an example of the ½1(l + 1) power law for the t e m p e r a t u r e dependence of the anisotropy. T h e equation of the curve is o = c u 1817 with c = 1 (Asti and Bolzoni 1980).
414
G. ASTI
representative point is a curve of the type v = CU18/7 or V = c u 2"57, with c constant. Figure 6a shows a typical curve for c = 1. It is worth noting that an analogy exists between the expression for the anisotropy energy [see eq. (2.1)] and the Landau free-energy expansion in the magnetization M, i.e., F = 1 A M z + ¼BM 4 + 1 C M 6 - H M .
This similarity was used by Khan and Melville (1983) to construct a complete set of phase diagrams for zero field and metamagnetic transitions from paramagnetic to ordered state. In particular, the critical magnetic field and specific magnetizations corresponding to first-order magnetic processes were determined. 2.3. Cases o f trigonal, hexagonal a n d tetragonal s y m m e t r y
Numerous ferrites of the hexagonal system may have either hexagonal or rhombohedral crystal structures, depending on whether there is or there is not a symmetry plane perpendicular to the three-fold axis. In this system of compounds, cobalt ions have been found to have strong effects on magnetic properties, introducing high-order anisotropy constants that favor, in general, easymagnetization directions in or near the basal plane, and spin-reorientation phase transitions (see section 2.5). This behaviour is attributed to the degeneracy, even in a crystal field having trigonal symmetry, of the ground state of the Co ions. Investigations (Bolzoni and Pareti 1984) on single crystals of the series of ferrimagnetic oxides Ba2(Zn1_xCOx)2Fea2022 (Y-type ferrites, abbreviated as (Zn,Co)2-Y) having trigonal symmetry, gave evidence of spin-reorientation phase transitions and FOMP. The transition is observed at 77 K with magneticfield direction along the symmetry c axis in all samples containing cobalt. For instance, in the sample with composition Znl.sCo0.5-Y, the critical field is l l . 4 k O e and the magnetization jumps to saturation. The critical parameters (magnetic field and magnetization) measured as functions of temperature and composition, together with the values of the anisotropy field when FOMPs are absent, allow to obtain complete information on the magnetic anisotropy of this system. For carrying out this analysis, Bolzoni and Pareti (1984) have studied the magnetic phase diagram of a trigonal system and determined the conditions for the observation of FOMP in terms of anisotropy-constant ratios. They use the following expression for the anisotropy energy of a trigonal crystal, E 1 = K 1 sin 2 0 + K z sin 4 0 + K t sin 3 0 cos 0 sin 3q~,
(2.14)
where 0 and q~ are the polar angles of magnetization vector M s. Proceeding in a way similar to that described in section 2.2, they obtained the magnetic phase diagram, reported in fig. 7, in terms of the anisotropy-constant ratios x = K 2 / K 1 and y = Kt/K1, giving easy-magnetization directions and other extrema of aniso-
FIRST-ORDER MAGNETIC PROCESSES
415
Y 3
t
X~ I -,
I
-9
0
I
2
I
4
Fig. 7. Magnetic phase diagram of a trigonal ferromagnet with K1> 0. The coordinates are x = K2/K ~ and y = K ] K 1. A symbol similar to that of fig. 1 has been utilized. It marks every region and gives informations about the easy direction of magnetization vector M~, as well as about the other extrema of the anisotropy energy surface with respect to variable 0 at q~= 90° (0 and ~0are the polar and the azimuthal angles of M~with respect to the tfigonal axis); vertical and oblique stems refer to the ternary axis and to 'cones', respectively. A concave tip denotes a minimum while a convex tip refers to a maximum of the energy. The absolute minimum (easy direction) is indicated by a filled tip. For K1< 0 no boundary line exists; the system is easy cone throughout: its representative symbol is "~ (Bolzoni and Pareti 1984).
tropy energy. The diagram of F O M P is reported in fig. 8 which gives the lines of constant reduced critical field her -- H c r M s / I K l l and critical reduced magnetization mcr = M , / M s. The shaded zone indicates the region where type-2 F O M P is present. Figure 9 instead reports the diagram of lines of constant values of the easy 'cone' angle together with the curves mcr--const. By using this approach, Bolzoni and Pareti (1984) have been able to obtain for the first time precise and consistent data about magnetic anisotropy in these trigonal ferrites. The value they obtained for K t (i.e., 3.5 x 106 erg/cm, extrapolated to 0 K) reconciles the dramatic discrepancies between theory and experiment. In fact, Bickford (1960) used a torque technique to measure anisotropy of C o 2 - Y and (K 1 and K2) data obtained by Casimir et al. (1959) in magnetic fields far below saturation. The experimental value thus obtained by Bickford for K t at llT K (i.e., 6 x 105 e r g / c m 3) was more than one order of magnitude lower than the one he calculated on the basis of atomic and structural considerations, i.e., 1 0 . 7 x 106 erg/cm. For rare earth transition metal compounds having hexagonal symmetry, a two-sublattice model has been developed by Sinnema et al. (1987). It applies to compounds of the type R2(Co,Fe)I 7. The attention was especially devoted to the study of easy-plane ferrimagnetic R2T~7 compounds (T stands for transition metal), which display first-order magnetic transitions at high magnetic fields. The
416
G. ASTI
,
Y I/
.
.
\
...
.
a,.
.
.
.
,
,, e
':iiii:iiii ,.....:.:..: ........ . . . . . Y." , O;,,,v
i::!$ii!:?'
- ::!~:::. . . . . . 5 : ~:!!i' ....... .7 - i!iiiii!:
. . . . . . . . .
3
F
64
0
-2
2 ,
\
t,
,
4
i - /
i
(b)
-3 ~
,,,;):
s
Y • --~
I
'
I I
i /~',,
,/
,~ ~
, ,
/ e°
'
Fig. 8. Lines of constant values for the reduced critical field h , = Hc,Ms/[KI[ (solid lines), and reduced critical magnetization rn, = M c r / M s (dashed lines), for type-1 FOMP, in the planes of reduced anisotropy constants x = K2/K~ and y = KJK~. (a)/(1 > 0. (b) K 1 < 0. The shaded zone indicates the region where type-2 FOMP is present (Bolzoni and Pareti 1984). f r e e - e n e r g y e x p r e s s i o n u s e d is,
E = E Kil sin 20i + Ki2 sin 40i + Ki3 sin 60i + Ki4 sin 60i
COS6 ~i
i
Jr nRTM R ' M T - M
R "H--M
T'H,
(2.15)
w h e r e i = T o r R , a n d 0i a n d ~i a r e t h e p o l a r a n d a z i m u t h a l a n g l e s o f m a g n e t i z a t i o n v e c t o r s M T o r M R w i t h r e s p e c t t o c axis a n d a axis, r e s p e c t i v e l y ; nRT is t h e i n t e r s u b l a t t i c e m o l e c u l a r field c o e f f i c i e n t a n d H is t h e e x t e r n a l m a g n e t i c field. T h e m a g n e t i z a t i o n c u r v e s f o r H a p p l i e d p a r a l l e l t o a, b a n d c a x e s h a v e b e e n o b t a i n e d
FIRST-ORDER MAGNETIC PROCESSES 4
,\
,
,\,
3-
,~
"
',
,
'
\ ",
,
, ;
~,
,
,
417
,
~, , ,
I
(a) --
1
e,
'4
.
""
.
"',
-...
o
64
-2
0
'
x'-L+ 2 X
"
4
4
-3
Y + -4
Fig. 9. Lines of constant values for the easy 'cone' angle 0c (solid lines), and reduced critical magnetization, mcr (dashed lines), in the planes of reduced anisotropy constants x = K J K 1 and y = K t / K ~. (a) Ka >0. (b) K~ <0 (Bolzoni and Pareti 1984).
by numerical methods. In easy-plane ferrimagnetic R2T17 compounds, the magnetization curves, with the field in the plane, show large first-order transitions (termed F O M R , from 'first-order moment reorientation processes') resulting mostly from competition between exchange and magnetostatic energy. The assumption is made that K~ is negligible as evident from investigations on Y2C017 and Y2Fe17 compounds. At zero field the resultant moment is directed along one of the six easy directions in the plane. As the magnetic field rises, successive transition points are reached where different moment configurations are energetically favorable. The critical fields correspond to the points where the energy of the two magnetic phases have the same value. An example of the obtained
418
G. ASTI
experimental and theoretical curves is shown in fig. 10 for Dy2Co17. The measurements have been performed at 4.2 K in a field up to 40 T at the high-field laboratory of the University of Amsterdam. A variety of compounds have been investigated using this approach, obtaining precise and consistent data about anisotropy constants for both sublattices. The obtained values of K i are, in most cases, one order of magnitude larger than those derived from other authors' low-field measurements and assuming rigid exchange bonding between the moments (see section 3.3.3). In tetragonal crystals, due to the presence of four-fold anisotropy in the basal plane, it is even more important to take the tetragonal term into account. An interesting case is that of the ternary intermetallic compound Nd2Fe14B , which represents a very important development in the field of hard magnetic materials [for a comprehensive review on this topic see the work of Buschow (1986)]. Working on this material, Asti et al. (1984) and Gr6ssinger et al. (1984) revealed, at temperatures below 210 K, the presence of FOMPs with critical fields above 14 T. For these experiments, the technique used was that of the singular point detection (SPD, see section 4.2). The existence of FOMP was confirmed by direct observations on single-crystal samples by Pareti et al. (1985a). However, the important role of the tetragonal symmetry appeared immediately (Bolzoni et al. 1987, Kajiwara et al. 1987), as the FOMP was not visible when the applied field was parallel to (110) axes. Subsequent measurements in other high-field laboratories [see references below in addition to Andreyev et al. (1985) and Sinnema (1988)] confirmed the existence of the transition in a variety of compositions in the system R2T~aB. For the understanding of magnetic anisotropy in this
Oy2[o1~ 6C
42
K
+ a axis
,
"--
x ~ X _
X
~
-
~
~
z
,
1
1
f
10 20 Mognefic Fietd B(T)
,
i
i
30 =
Fig. 10. Magnetization curves at 4.2K along different crystallographic directions of hexagonal Dy2Co17. The a axis is the easy magnetization axis within basal plane. The full lines are computed magnetization curves; after Sinnema et al. (1987).
FIRST-ORDER MAGNETIC PROCESSES
419
system a theoretical approach based on tetragonal symmetry is necessary. Theoretical models based on the crystalline electric field (CEF) Hamiltonian ~CZF = ~2 B~O~,
(2.16)
l,m
have been developed and applied to several compounds of this class. It is found that CEF terms up to fourth order are necessary to explain the observed FOMP in Nd2Fe14B and its dependence on crystal orientation (Kido et al. 1987). The case of Pr2FelgB is worth noting because the observed FOMP is of type 2, as the magnetization, after the abrupt increase, is well below saturation (Gr6ssinger et al. 1985, Pareti et al. 1985b, 1988, Huang Ying-Kai et al. 1987, Pareti et al. 1988, Pareti 1988). Moreover, the critical field changes from 13 to 16 T when the magnetic field is moved from [100] to [110] direction. In this case, the theoretical interpretation is complicated because it is found that CEF terms up to sixth order are necessary for reproduction of the observed phenomena (Hiroyoshi et al. 1987). On the other hand, the role of sixth-order terms in the case of type-2 FOMP was already clear from phenomenological arguments, even in uniaxial systems [Asti et al. (1980), see section 2.2]. The necessity of sixth-order terms has, in this context, also been evidenced by Zhao Tiesong and Jin Hanmin (1987), although the various authors adopt different criteria for simplifying the CEF Hamiltonian, because of the great number of adjustable parameters.
2.4. Cubic symmetry First-order magnetic processes of the same nature as those described in the preceding sections (FOMP) have also been observed in magnetic materials with cubic symmetry. Preliminary discussions of the effect were already present in a work by Bozorth (1936) and the theoretical description of the effect has been partially developed by Krause (1964) and Rebouillat (1971) on the basis of the usual phenomenological theory of magnetic anisotropy. Furthermore, Rebouillat (1971) gave some experimental evidence of a first-order transition of this kind in iron single crystals at low temperatures. Asti and Rinaldi (1974b) have examined the conditions for the existence of FOMP with the magnetic field H parallel to one of the crystallographic axes, taking into account only the first two anisotropy constants, K 1 and K2, both in the case of cubic and uniaxial crystals. Two different kinds of FOMP are possible: (1) When the irrreversible rotation of M~ occurs in the plane defined by the magnetic field direction and the easy direction, i.e., the initial direction of M~ (type-I transition): (2) When M~ starts to rotate in one plane and, at the critical field, jumps to another plane where it continues to rotate towards the magnetic field direction (type-II transition). The analysis of the various possible magnetization curves has been made assuming the direction of magnetic field H to be parallel to an extremal direction
420
G. ASTI
of the anisotropy energy surface, i.e., a stationary point but not an absolute minimum. In the presence of only K 1 and K 2 constants these directions coincide with crystallographic symmetry directions. The results are summarized in the graph displayed in fig. 11, which also gives the phase diagram of easy and hard directions in the (K 1, K2) plane. Depending
Fig. 11. Conditions for the existence of FOMP in cubic crystals, which depends on the ratio K 2 / K 1 of the first two anisotropy constants and the sign of K 1 (on the outside part). The letters A, B and C indicate magnetic field orientations parallel to (100), (110) and (111) directions, respectively. The superscript indicates whether the transition is of type I or II. The magnetic phase diagram of a cubic crystal is given by the inner shaded circular areas relative to the three crystallographic directions. The various ranges of K 2 / K ~ are distinguished by different symbols, indicating a minimum for the anisotropy energy (e), a maximum (h), a saddle point having a maximum for rotations of Ms in the {110) planes (sx) or in the {100} planes (sy). For K 2 / K ~ < - 2 (see the ranges indicated by 0 and ~oin the white circular zone) the anisotropy energy surface has saddle points along non-symmetry directions of M~ lying in the (110) planes and having two direction cosines equal to ( - K 1 / K 2 ) ~/2 For rotations of Ms within the (110) planes the anisotropy energy has a maximum or a minimum when K z / K ~ is in the range 0 or ~, respectively (Asti and Rinaldi 1974a,b). A continuous solid line, inside the shaded areas, joins all the ranges on the three crystallographic areas labelled 'e', where t h e crystal is easy axis (absolute minimum).
FIRST-ORDER MAGNETIC PROCESSES
421
on the direction of H, the following intervals for the anisotropy constants ratio k = K2/K 1 are indicated for the two types of FOMP: (1) H[I(100) - T y p e I for k > 4 with g I < 0 , and k < - 9 with K~ > 0 ; type II for-9 0. (3) H I I ( l l l ) - T y p e I for k < - 9 with K 1 < 0 and k > - 9 with K1 > 0 , i.e., the whole range where (111} directions are not easy axes; type II for k > 9 for K 1 > 0. In this range both types of transition coexist. A more detailed discussion of the different magnetic phases and modes of magnetization in the cubic ferromagnet has been given by Birss et al. (1977) including also the eight-order anisotropy constant K3, but again considering only magnetic field directions parallel to the symmetry axes. The results of Birss et al. (1977) are in agreement with those of Asti and Rinaldi (1974b) and differ in several respects from those of Rebouillat (1971). Thus, the behaviour of a cubic ferromagnet in an applied magnetic field is an area which is rich in examples of phase transitions. For this reason, it has been especially considered from the point of view of the study of the critical phenomena. Mukamel et al. (1976) showed that a cubic ferromagnet, in which (111) directions are hard axes, under the action of a magnetic field, is a realization of the three-state Potts model. Cullen and Callen (1984) extended this type of analysis, also including sixth-order terms of magnetic anisotropy, and calculating various aspects of the magnetic-field phase diagram depending on the ratio k = K2/K ~. That is, they display in a diagram the critical field-strengths f o r all field orientations, discussing the topological variations in the phase diagram in relation to the different k values. From the experimental point of view, besides the experiments on iron by Rebouillat (1971) and by Hathaway and Prinz (1981), it is important to mention that low-temperature measurements in cubic RFe 2 compounds (R is a heavy rare earth) reveal values of K 2 and K 1 which lie in all four quadrants of the (Ka, K2) plane (Atzmony et al. 1973). Good examples of anisotropic cubic ferromagnets are the RAI 2 compounds. For instance, DyA12 has been shown to have a large FOMP for fields applied in the (110) plane (Barbara et al. 1978). These works, besides the inherent importance for the study of critical phenomena, provide a reliable guide for the investigation of cubic ferromagnets and ferrimagnets which display FOMP type transitions. FOMPs also have remarkable effects of magnetostriction, giving rise to large discontinuities, as observed by Melville et al. (1981) in CeFe 2 as well as, more recently, in hexagonal systems, namely Er2(Col_xFex)17 (Kudrevatykh et al. 1986).
2.5. Mechanisms responsible for high-order anisotropy constants
From the above discussion, it is clear that the existence of FOMP in a ferromagnet implies high-order anisotropy terms. These circumstances are not easy to be
422
G. ASTI
included in the theory and, in general, require special mechanisms that take into account the nature of the magnetic ions and of their mutual interactions. For instance, in cubic crystals the f electrons of rare earth ions can support no term above sixth order in the CEF Hamiltonian (Cullen and Callen 1984). In mixed crystals, namely pseudobinary compounds, the competition between the individual anisotropies of the constituent rare earths is responsible for the onset of higher-order terms in the anisotropy energy. These terms are generated if we treat the crystalline electric field in higher than first-order perturbation (in terms of the ratio crystalline field/exchange field), either by admixing higher multiplets in the J quantum number, or by allowing a small canting between magnetic moments (Cnllen and Callen 1985). Indeed, section 3 is devoted to the effect of canting on the anisotropic behaviour of multisublattice systems on a phenomenological basis. As regards magnetic oxides of the transition metals, the mechanisms for generating high-order anisotropy constants must be different. In fact, we are in this case in the limit of large exchange and the situation for a 3d ion is very different as for the relative importance of the various terms in the Hamiitonian. As a matter of fact, there are examples of hexagonal ferrites that display large FOMPs within wide ranges of temperature and magnetic fields, both parallel and perpendicular to the symmetry c axis (Asti et al. 1978, Graetsch et al. 1984, Bolzoni and Pareti 1984, Paoluzi et al. 1988). All these examples have in common the presence of Co as a constituent. Lotgering et al. (1980) have given an explanation of the peculiar influence of Co 2+ ions on the anisotropic behaviour of all hexagonal ferrites. Similar effects have also been observed for Fe 2+ ions in L ~ 2+~ 3 + ~ are re11 u19 (Lotgering 1974). The interpretation is based on the Slonczewski theory (1961), which considers the incomplete quenching of the orbital momentum in the local crystal field, present at octahedral sites which have an uniaxial component, in addition to the cubic one. In fact the local uniaxial crystal field is not sufficient to completely remove the degeneracy and in one case we can have an orbital doublet as the lowest state. The orbital momentum is then either +h or - h along the local axis, so that the spin-orbit coupling gives a torque rTI = ALS sin 0 (where L = 1 and 0 is the angle of the spin with respect to the local axis). This fact implies a sign change in the orbital momentum at 0 = 90°, giving rise to a discontinuity in the torque curve and a very high anisotropy. Even if, in real cases, there are substantial deviations from such ideal conditions, due to various perturbations, the model gives an account of the strong distortions that can be induced in the pure sine function, representing the torque of the spin when only second-order anisotropy exists. As a consequence, high-order terms enter the anisotropy energy. If the orbital singlet is lowest, an anisotropy can again arise from admixing of the excited state with an unquenched orbital moment by spin-orbit coupling (Lotgering et al. 1980). The fact that Co 2+ ions in all hexagonal ferrites reside on lattice sites having octahedral coordination, allows to deduce relations between the anisotropy constants in the hexagonal structure. In fact the anisotropy term associated with Co 2+ in octahedral coordination has the form,
F I R S T - O R D E R M A G N E T I C PROCESSES E
2 2 2 2 2 2 = Kc(O/lO~ 2 + o~20t 3 + o t 3 0 t l )
423
(2.17)
where the a's are the direction cosines of the magnetization with respect to the (x, y, z) reference system centered at the cation site and with axes containing the coordination anions. The c axis of the hexagonal structure has direction cosines ( ½X/-3, IV~, )V~). Expressing eq. (2.17) in terms of polar coordinates (0, ~o) with the c axis as the polar axis, and the azimuthal axis having direction cosines (~X/2, ½V2, 0), we obtain, E=Kc( 1-2sin 20+7
sin 4 0 + } s i n 3 0 c o s 0 s i n 3 ~ ) .
(2.18)
The last term on the right-hand side has trigonal symmetry. However, for hexagonal structures (like W- and Z-type compounds) we must take into account that for each octahedron there is another one rotated by 180°. Averaging over the two, the trigonal term vanishes. Equation (2.18) indicates that a positive singleion cubic anisotropy contribution gives rise, in the hexagonal structure, to both a negative K 1 and a positive K 2 (Asti and Rinaldi 1977). There is another kind of transition in hexagonal ferrites that cannot be considered as ordinary FOMP, but is associated with a more complex magnetic behaviour (see section 3.4). These phenomena are characteristic of ferrites having block-angled magnetic structures and are caused by mechanisms which are different from those responsible for the above-mentioned FOMP. 2.6. Domain wall processes at the transition point
As we have seen, a FOMP is due to an irreversible rotation of the magnetization vector M s between two inequivalent minima of the free energy of the crystal under the action of the magnetic field. Normal hysteresis in a ferromagnetic material is caused by irreversible rotation processes between equivalent crystallographic directions and is therefore symmetrical. In the case of FOMP we expect to observe minor hysteresis loops displaced to an unsymmetrical position in the M versus H diagram. In reality, it is difficult to reveal the hysteresis associated with FOMP. The few known examples are experiments at liquid-helium temperatures, as reported by Barbara et al. (1978) and by Ermolenko and Rozhda (1978). The reasons are probably in the inherent low coercivity of the magnetization processes in the critical range of the phase transition. Mitsek et al. (1974) and Melville et al. (1987) have calculated the domain structure existing in this range. The analysis starts from the consideration of a 180° domain wall in a uniaxial ferromagnet with K 1 > 0 and x = K ~ / K 1 in the range ( - 1 , - ~1) , which corresponds to a P1 type FOMP (see section 2.2). The magnetic field H is oriented along the positive direction of the x axis while the plane of the wall is parallel to the xz plane and neighbouring domains are oriented in opposite directions of the z axis. Under the action of the magnetic field, both domain-magnetization vectors rotate by an angle 0 towards the field direction, so that the wall, polarized along the magnetic
424
G. ASTI
field direction, becomes a ( 1 8 0 - 20) degree wall. The free energy is
--de f E= k[ A \(d0) d y / 2 + K l s i n Z O + K 2 s i n g o - M s H s i n O ] dy,
(2.19)
where A is the exchange parameter. The Euler equation has the form, K1
d20
dy z
A sin 0 cos 0
2K2 MsH ~-- sin 3 0 cos 0 + ~ cos 0 = 0 ,
(2.20)
and the boundary condition are, d0 ~yy y=--+~= 0 ,
0 ( - ~ ) = OH,
0(~) = ~ - - OH,
where OH is the equilibrium angle corresponding to field H inside the domains. The solution of eq. (2.20) turns out to be
0
y =
fl
~ dO,
(2.21)
re/2 with U=
Kx K2 -)- (sin 2 0 - sin 2 OH) + ~ - (sin g 0 -- sin g OH)
]1/2 MsH (sin 0 - sin OH)] A
.
The slope dO/dy can be calculated from differentiation of both parts of eq. (2.21), obtaining, dO -- = U. dy
(2.22)
It is found that dO/dy vanishes at the center of the wall, i.e., for 0 = ½~-, when H is equal to the critical field of F O M P H , = ½HAahcr = Klhc]M ~. The reduced field h~ is given by eq. (2.3), where y is taken equal to zero and m is equal to the critical magnetization given by
mcr = Mc,./Ms = ½[-1 + ( - 2 - 3/x)1/2],
(2.23)
which is a solution of eq. (2.4). The fact that the derivative dO/dy vanishes at the center of the wall means that, at the critical field, the original 180 ° wall is split into two symmetrical walls of (90 - 0) degrees, giving rise to a nucleus of the new phase that appears inside the
FIRST-ORDER MAGNETICPROCESSES
425
domain wall. These new walls are in equilibrium in the presence of a magnetic field equal to Her , because the two adjacent domains belong to different phases, but have the same free energy. The same results can be obtained in a very direct way if one utilizes a simple energy criterion that is valid for domain structures in general, and for a single wall in particular. Let us consider a one-dimensional problem as before, so that we consider a system of plane domain walls parallel to the xz plane. Then we can rewrite eq. (2.19) as follows, cr2
KI(x + y2) + K2(x z + y2)2 _ Ms(xH~ + yHy + zH~) + 2~MZy 2 o"1
where x, y and z are the Cartesian coordinates of the unit vector Ms~Ms, Hx, H e and H z are the components of the magnetic field H and tr is an axis parallel to the y axis. In the case of static domains, if Hy = 0, we may exclude the y component because it gives rise to demagnetizing fields that would drive the wall motion. In this form, the integral of eq. (2.24) can be regarded as the action integral of a mechanical system consisting of a body of mass 2A, free to move on a sphere. The mechanical forces acting on it are opposite to the forces acting on the magnetic vector M s. They are a gravitation-like force directed opposite to H, and elastic forces repelling the body away from the z axis (when K 1 > 0) and away from the xy plane. Hence, cr represents the time, and the function in the integral is the Lagrangian function T - V of the mechanical system, where the kinetic energy T corresponds to the exchange term and the potential energy V is represented by the other terms with changed sign. This mechanical analogy enables derivation of an important property of magnetic domain structures directly from the energy conservation principle in the mechanical system. In fact, T + V must be a constant throughout the whole range of or. As a consequence, we can say that everywhere in the magnetic system the increase in exchange-energy density is balanced by an equal increment in the remaining part of the energy density, consisting of the addition of anisotropy and magnetostatic terms. This kind of equipartition principle has easy and immediate applications. For instance for the calculation of domain-wall thickness (see section 4.2) and to the above results concerning domain structure in the presence of FOMP. In fact, eq. (2.22) can be written directly from the balance of the energy densities, and, likewise, the vanishing of dO/dy at the center of the wall, when a nucleus of the new phase appears, is obvious, because the two phases have, by definition, the same energy density in terms of anisotropy plus magnetostatic contributions. As a consequence, there must be no variation in exchange energy density. The mechanical analogy also makes intuitive the effect on domain structure of the magnetization process of a uniaxial crystal with applied magnetic field parallel
426
G. ASTI
to the c axis, which is taken as easy magnetization direction. For instance, it is easy to understand why reverse domains are reduced with respect to those oriented parallel to magnetic field: If we consider the body on the sphere to rotate continuously, it means that we have a periodic system of parallel walls having alternate polarization, towards positive and negative x direction. Then, in the mechanical analogue the moving body spends less 'time' in the opposite domains because it has a higher kinetic energy there. Equilibrium is reached, because there is a repulsion between adjacent walls with opposite polarization. In the limit of high magnetic fields they join to form a 360° wall. Instead, solutions with magnetization vector pointing in the same direction must be unstable, because they present the opposite effect, i.e., the mean magnetization of the crystal will be opposite to H. All these conclusions are in agreement with detailed theoretical studies reported by Forlani and Minnaia (1969), Wasilewski (1973) and Odozynsky and Zieter (1977). The same approach can be used to study domain wall mobility and dynamic domain-wall structure, if one includes in eq. (2.24) the terms in y and effective dynamic forces deducible from the dynamic equation of magnetization, e.g., in the Gilbert form, dMJdt
= 7M x H - (M s x dM/dt)a/M
s.
(2.25)
If we admit that there are solutions characterized by a rigid displacement of the domain wall with a constant velocity v, differentiation with respect to time of a function L becomes d L / d t = v d L / d o - = v L , where we have indicated differentiaton with respect to o- by a point above the function. Then the dynamic equilibrium of the moving body can be written in the form ( 7 / M s ) M s × ~:
= O,
(2.26)
where ~ is the total dynamical force given by = (-grad V - 2 A ~ ' I I M s - v M s G I 7 - v a M l y ) ,
(2.27)
and G = M x ~ is a Lorentz-like force caused by a pseudo-magnetic field perpendicular to the surface of the sphere in outward direction and having a uniform intensity equal to 1/M s. The term containing the damping parameter a is responsible for the energy loss and acts as a typical viscous resistance. The dynamic equations are then obtained by projecting eq. (2.26) on the Cartesian axes, or from the condition that the component of ~r on the sphere must be equal to zero. This condition can be expressed in the form, o% = 0 ,
o~0 -- 0,
(2.28)
where ~ and ~0 are the components of ~: along the directions of the local spherical-coordinate lines ~p = const., and 0 = const., having taken the polar axis
FIRST-ORDER MAGNETIC PROCESSES
427
coincident with the z direction. If we suppose that the magnetic field H is applied parallel to the positive direction of the z axis and that we have a single domain wall with the boundary conditions z = 1 for cr---~ - ~ , z --- - 1 for or--->+ ~ , then we can see that a very simple solution exists, namely the one given by ~0 = const. [the solution given by Walker and reported by Dillon (1963)]. In fact, in this case eqs. (2.28) become,
o%~ = 4~yM~ cos q~ - vi(/lly = 27rM 2 sin 0 sin 2~o - v M / y = O, (2.29)
~o = MsH sin 0 - a v ~ l / y - 2Alf4/M s + 2 K 1 sin 0 cos 0 + 27rM 2 × sin 2 qo sin 20 = O. The solution corresponds to the condition that the viscous resistance is perfectly balanced on every 'instant' tr, by the term due to the applied field, i.e.,
MsH sin 0 = a v i f l / y .
(2.30)
This means that the magnetostatic energy - M . H is entirely dissipated inside the wall during its damped motion. This condition together with the first equation of (2.28) yields, sin 2~ = H / ( 2 7 r a M s ) .
(2.31)
Then h)/can be deduced from the principle of energy conservation, i.e.,
A(i(4/Ms) = K 1 sin 2 0 + 27rM~ sin 2 0 sin 2 q~,
(2.32)
so, for the wall velocity the following expression is obtained,
v = H ( y / a ) (A/K1)l/2[1 + 7rM2(1 - c o s 2~)/K1] - v 2 .
(2.33)
The case ~ = ¼7r corresponds to the maximum value for v, which is known as the Walker limit (Dillon 1963).
3. Processes involving competition between exchange and anisotropy 3.1. Introduction In the previous chapter we have seen that the simple consideration of high-order terms in the magnetic anisotropy energy expansion justifies the existence of F O M P in an 'effective' ferromagnet. By using this expression, we emphasize the fact that the internal magnetic structure of the system has no relevance, because during the magnetization process it was assumed to remain absolutely rigid. Examples are ordinary ferrimagnetic materials, such a s the oxides of transition
428
G. ASTI
metals which, at ordinary field intensities, display a constant spontaneous magnetization as well as perfectly collinear magnetic moments, due to the extremely high exchange interactions. However, this is not true for all magnetically ordered substances. It is well-known indeed that in RE intermetallic compounds the situation can be such that the fundamental interactions have comparable intensities, so that we can have exchange and crystal-field terms in the Hamiltonian which are essentially of the same order of magnitude. The result is obviously that we can no longer consider the system as a rigid collinear magnetic structure, but we must allow for substantial deviations from the equilibrium configuration at zero field. Hence, the role of the exchange interaction in the magnetic process can in certain conditions be so critical, as to produce extraordinary effects. As a matter of fact, it can be responsible for: variations of the anisotropy fields by orders of magnitudes (even leading to change of sign); stabilization of easy-cone directions; or it can give rise to field-induced first-order transitions, even when only the first anisotropy constants of the individual sublattices are present. In treating effects of this kind in the next two sections, we shall use, once more, a phenomenological approach that is based on the classical model of exchange. Although simple in principle, it can be considered as a useful guide to the experimentalist, because it contains the essential features and allows to gain a better understanding of the fundamental processes which form the basis of the observed phenomena. In section 3.2, as a first application of this concept, a detailed analysis is made of the magnetic system in the vicinity of the transition points, so as to obtain exact relations between some fundamental parameters, such as the expressions for the anisotropy fields and the conditions for the existence of FOMP. In section 3.3, the same concept is developed within the approximation of small deviations from collinear magnetic order (small-angle canting). In this limit, the effect of canting can be accounted for, in a sufficiently accurate approximation, by the use of effective anisotropy constants that are slightly field dependent. So, the system can be conveniently treated as an effective ferromagnet having certain anisotropy constants. A typical application of this model is an immediate explanation of the easy-axis to easy-cone transition observed at 110 K in PrC%, thus, indicating in a very simple way the important role of canting (see section 3.3.3). In section 3.4, a brief review is given of other cases and treatments of the problem of canted systems and magnetic phase transitions.
3.2. Role of exchange at transition points 3.2.1. Multisublattice magnetic systems with uniaxial anisotropy We shall focus our attention on an ordered magnetic system with uniaxial magnetic anisotropy, consisting of L interacting sublattices, Ma, M b , . . . , ML, when it is near the transition point, in general, in the presence of an external magnetic field. In what follows, we will consider transitions involving a change in magnetic symmetry, in particular, transitions in which one of the phases is a collinear-type equilibrium configuration, i.e., a transition between a 'collinear' phase and an 'angular' or canted phase.
FIRST-ORDER MAGNETICPROCESSES
429
So, the equilibrium states we are considering in the collinear phase are essentially: the fundamental state in zero field, normally characterized by a collinear-type magnetic structure, and the various other collinear states that are stabilized by an external magnetic field H parallel to a crystallographic symmetry direction. The latter are intermediate saturated states and are stable in principle within certain regions on the temperature-magnetic field plane ( H - T plane). Among these states, the one at the lowest field, when possible, is that having the s a m e magnetic structure as the fundamental state at zero field and occurs at H = HA, the effective anisotropy field, or at H = Her, the first critical field, in case a FOMP is present. Other collinear states can be stabilized by higher fields. They are characterized by different magnetic structures, corresponding, in principle, to all possible combinations of the mutual orientations of vectors M i. Among these 'excited' states the one having the highest magnetization corresponds to the forced ferromagnetic order. The phase transitions associated with these magnetic states are of two types: s p o n t a n e o u s spin reorientation transitions (SRT), a n d f i e l d - i n d u c e d SRT. One can imagine the reverse situation, i.e., to keep the magnetic field H constant and to change the temperature, or even a combination of these two, which corresponds to crossing the transition line in the H - T plane along different directions. In this sense, the first type of SRT reduces to a particular case of the second type. As an example, let us consider again the phase diagram of figs. 1, 3 and 4. The line m represents first-order SRT. If the representative point, with changing temperature, crosses that line at a certain critical value T = Tcr, it means that the system undergoes a first-order SRT in which the easy-magnetization axis changes discontinuously from axis to plane or vice versa. Alternatively, the lines o and o' are those where discontinuous transitions occur of the type easy axis-easy cone or easy plane-easy cone. Obviously, the same lines have the meaning of lines where the value of the critical field of FOMP reduces to zero. Since in the present treatment we are confined to small deviations from collinear order, the orientation angles of the magnetic moments are treated as infinitesimal, in general, all of the same order. So in our considerations we are not limited to the classical 'large exchange' case, but we may consider systems in which the intensity of the anisotropy forces can be comparable to or even higher than, the exchange interactions. The analysis of the behaviour of the system in the neighbourhood of the transition point leads to exact relations between important magnetic parameters. Let us consider a system of L sublattices M a , M b , M ~ , . . . , M L having uniaxial anisotropy constants K l a , K l b , K l c , . . . , K1L, Kza , K2b , Kz~, . . . , K2L up to a certain order N, and intersublattice molecular field coefficients J~t3. The free energy of the system, in the presence of an external magnetic field H parallel to the symmetry axis c has the form, L
F=-
~ a,a >fl
J~M~M~
cos(0~ - 0~) + ~ a=a
N
L
~ Ki, ~ sin2i 0~ - ~ HM,~ cos 0~, i=1
a=a
(3.a)
430
G. ASTI
where Oa, 0b . . . . , 0 L are the orientation angles of the magnetization vectors M, with respect to the c axis. This is the typical situation we have when we magnetize a crystal along a crystallographic axis: when the field approaches a certain critical value, which in the ordinary cases is the anisotropy field H A relative to that direction (hard magnetization direction), all the O's are vanishing and at H = H A we have a metastable equilibrium. Above that field, the magnetic structure is collinear and does not change until H (in the case of ferrimagnetic order) eventually reaches a further critical value which starts another SRT, in general, towards a canted state having higher magnetization. In the limit of 'infinite' exchange, the effective anisotropy of the system is merely given by the sum of the sublattice anisotropies, so that the anisotropy field turns out to be, H A ---
2K13
M,
(3.2)
with M = E~L =, M . Moreover, when the exchange is of comparable magnitude as the anisotropy energy, we can have considerable modifications in that expression because there are important contributions to the total energy from relaxation processes due to substantial deviations from the coUinear order. The particular collinear configuration is specified by the sequence of the signs of Ms, and can be chosen independently from the sequence of the signs of J ~ , because, in general, the stability of a configuration cannot be decided a priori, being dependent on the values of the K's and H. Furthermore, we neglect here possible effects due to the splitting of one or more sublattices, a phenomenon of the Yafet-Kittel type, (Boucher et al. 1970) which is possible, in principle, if we consider competing intrasublattice-exchange interactions.
3.2.2. Field-induced spin reorientation transitions. Critical fields It is easy to write down the equations for the critical field Ht, at which a certain state having collinear order becomes unstable. The equilibrium equations of the system are, L
OF~dOe = ~
N
J~t~M~Mt~ sin(0~ - 0t~) + ~] Ki~2i cos 0~ sin 2i-1 0 a
/~=a
i=1
+ M ~ H sin O~ = O,
(3.3)
with a = a, b , . . . , L. Expanding up to first order in the O's, we obtain a system of homogeneous linear equations, L
/3=a
J.~M,~M~(O,~ - Or3) + (2K1~ + M~H)O~ = 0 ,
(a=a,b,...
,L). (3.4)
It is evident that in these equations the anisotropy constants of an order higher than two do not appear, because we have considered the transition with the field parallel to the c axis.
F I R S T - O R D E R M A G N E T I C PROCESSES
431
The condition at the transition point is that the determinant of the matrix of the coefficients is vanishing, i.e.,
a =o.
(3.5)
This is an equation of degree L, which, in principle, can be solved giving the expression for the critical fields H t as functions of the sublattice magnetizations, the anisotropy constants and the exchange coefficients. Note that, in principle, we have L distinct solutions for each collinear magnetic structure. So, already for a small number of sublattices the situation appears very complicated. Among the real solutions H = H 1, H 2 , . . . , Hp of eq. (3.5), with p ~< L, the solutions having physical meaning are those corresponding to transitions to (or from) a minimum of F. They define ranges of H for which a particular collinear magnetic structure is metastable. The other solutions of eq. (3.5) are, in general, only transitions between saddle points of different kinds or between a saddle point and a maximum, i.e., unstable states. An important case is when all the exchange coefficients are zero except those related to one particular sublattice, say M a. This can be a good approximation for several rare earth-transition metal compounds (RE-T), and especially for solid solutions containing various RE species or RE occupying different lattice sites. In this case, we can indeed neglect all interactions between RE and consider only those between RE and TM. Hence, the system of eq. (3.4) reduces to,
(Z
L J~M.M~
) L "}-2gla "]- M a l l Oa-- Z Jl3MaM,t3Ot3 -~-0,
-
¢~=b
(3.6)
- J y o M , Oa+(JyoM
+2K1 + M H)O =0,
(¢=b,c,... ,L),
where we have taken J~ = Ja.~. Then eq. (3.5) becomes,
-
JoMoM + 2Kla + Mon,
¢l=b
-
"y=b
(LMaM, + 2K,.,, + M ,Ht)
(JvM.M:, + 2K1. -I- M:,Ht) J ~: M a2M ~2/ ( J ~ M a M ~ + 2 K ~
+ M~Ht) = O, which yields, L
2K,,, + Mo, H t + ~, [(2Kit3 + Mt3Ht) -1
+ (J~MaMi3)-l1-1 = 0 .
(3.7)
In the case H t = 0 this equation, as well as eq. (3.5), gives the condition for the spontaneous SRT. All these results are easily extended to the case of a field perpendicular to the c axis, by the use of the K-~-->R transformations (see section
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G. ASTI
2.2). In the most simple case, H t in eq. (3.7) can be identified with the anisotropy field H A. An important consequence of the effect of canting is that we have two different expressions for the anisotropy field, depending on whether we apply the magnetic field parallel or perpendicular to the symmetry axis of the crystal (see section 3.3.2). These directions of the applied magnetic field are typical experimental arrangements for measuring magnetization curves, depending on the fact that the symmetry direction is a hard or an easy direction, respectively. In the absence of canting effects the anisotropy field in both cases is given by eq. (3.2), if only second-order anisotropy constants contribute to the anisotropy energy. However, the anisotropy field given by eq. (3.7) could also have physical meaning when the symmetry direction is an easy direction, when we think of processes involved in phenomena like the nucleation and growth of a reverse domain, that have relevance for coercivity mechanisms. Equation (3.7) reduces to a very simple form in the case of two sublattices, i.e., (2Kla + m a n t ) -~ + (2Kab + MbHt) -~ + ( J m a m b ) -1 = O,
(3.8)
with J = Jb = Jab" Then, the condition for the spontaneous SRT becomes, (2Kaa) -a + (2Kab) -1 + ( J M a M b ) -1 = O,
(3.9)
which merely says that the sum of the compliances due to anisotropy forces is balanced by the exchange compliance. 3.2.3. B o u n d a r y conditions f o r first-order transitions
Another problem concerning these magnetic transitions is connected with their order. In fact, as we have seen in section 2.2, the approach to saturation in certain circumstances can occur with a first-order transition that is a discontinuous rotation of the magnetization vector (FOMP). The present approach, based on the analysis of an infinitesimal region around the collinear state, can be used to study the conditions for the existence of first-order magnetic processes in general, which, in particular cases, can be identified as ordinary FOMP. For instance, the transi:ion to the first intermediate saturation state can be considered to be an ordinary FOMP. Instead, first-order transitions involving higher magnetization states, such as the spin-flop transition, unavoidably bring drastic changes in the magnedc order. When necessary, it may be convenient to distinguish between these two types of FOMP, so, hereafter we will call them 'o.FOMP' and 'h.FOMP' (for 'ordinary' and 'high-magnetization state' FOMP, respectively). Let us consider the system of equilibrium equations (3.3) and expand up to terms of third order in the variables 0. The magnetic field will be expressed in terms of an infinitesimal reduced variable given by ~, = ( n -
H,)/I4 t ,
(3.10)
where H t is a real positive solution of eq. (3.7). The an~les 0 can be written as the
FIRST-ORDER MAGNETIC PROCESSES
433
sum of an unperturbed part, 0, plus an infinitesimal of an higher order, e, i.e., we write 0~ + e~ instead of 0~ for each a. We obtain a system of L linear equations in the variables e having the same coefficient matrix as the homogeneous system of eq. (3.4). The column of terms on the right-hand side are, instead, functions of the 0's and of y, so the system obtained is L
J ~ M Mt~(e~ - e¢) + (2K1,, + M~Ht)e ~ = 0 ( 0 3, TO), t3=a
(a = a, b , . . . ,
L),
(3.11)
Since A is equal to zero, the system has a solution only if the augmented matrix (i.e., the matrix including the column of the right-hand side terms) has the same rank as the matrix of the coefficients. This, in general, allows the writing down of a condition in the form A' = 0, where A' is one of the determinants obtained by replacing one of the columns of A with the column of the right-hand side terms. This existence condition allows one to find the relation between y and the angles 0, which means that we can obtain the slope of the magnetization curve M ( H ) at the transition: a negative slope indicates an unstable equilibrium, which implies that a F O M P is possible. That is, if M ( H ) connects stable states, the transition is of first order, otherwise it means that M ( H ) represents metastable states, and so the transition does not occur at all. In the particular case already considered, in which all the exchange interactions are zero except those with a single sublattice Ma, the coefficient matrix of system (3.11) reduces to that of system (3.6), while the column of the right-hand side terms turns out to be, L
-~ ~
Jt~Mt~Ma(Ot~ Oa)3 + ( ~ M a n t + 4 gla -4Kza)O 3 - MantOa,Y , -
[3=b
~Jt3Mt3M~(Ot3 - 0,) 3 + (~Mt3H t + 4K1~ - 4K2t~)013 - Mt3HtyOt~, (B=b,c,...
(3.12)
,L).
Note that anisotropy constants above fourth order give no contribution, thus indicating that the boundary between first- and second-order magnetic transitions depends, in general, only o n K 1 and K 2. This is consistent with the results obtained in section 2.2, where we have seen that the boundary line n', separating A1 and A I C from normal regions, is vertical, which means that K 3 has no effect. The equation of n' is, in fact, 4 K 2 = K 1.
3.2.4. The two-sublattice system In what follows, we shall limit ourselves to the simplest case of only two sublattices, M a and M b. As already mentioned, the smaller of the two magnetizations, Mb, can be either positive (ferromagnetic order) or negative (ferrimagnetic order), it is convenient to introduce the reduced variables
G. ASTI
434 x = 2Kla/JMM
a ,
y = 2Kab/JMM
v = 4 K 2a / J M M
b = 1 - a = Mb/M, h = H/JM,
a = M / Ma
b ,
a
,
w =4K2b/JMM
(3.13)
b ,
m : MII/M = a cos 0a + b cos 0b ,
h t = Ht/JM,
where M = M, + Mb, J = Jb = Jab and MII = M a cos O, + M b cos 0b is the component of the total magnetization parallel to H. Hence, we can write the existence condition A ' = 0 as follows, -a
[ _ l a(Oa _ 0b)3 + (~ht + 2 y _ w)O~ - htTOb]
A, = b+ht+x
[~b(O
a -
0b) 3 + ( l h t + 2 x - 0 ) 0 3 - htTOa]
=0.
(3.14)
Note that in the case of ferrimagnetic order we have J < 0 so that also both h and h t are negative. We have to take into account that h t is the value of the field at the transition. In the simplest case, it is the anisotropy field h A in the hard direction. In terms of the reduced variables it is related to the other p a r a m e t e r s by the condition that A = 0, i.e., -a
A=
a+ht+Y
b+h t+x
-b
=0,
(3.15)
which is identical to eq. (3.7) for L = b, i.e., to eq. (3.8). The direction angles 0 are determined but for a proportionality factor Oa/Ob = ( a + h t + y ) / a
(3.16)
.
A m o n g the p a r a m e t e r s a, b, x, y and h t only three are independent variables. By solving eq. (3.14) we can express (Y/0bz) in terms of y ' , h t , a, v and w, y/0~ = (h* - h t ) [ ( y ' + a) 4 + a3(1 - a ) ] / [ 2 a 2 h t ( Y '2 + 2 a y ' + a ) ] ,
(3.17)
where, h* = [-by'2(y'
+ 2a)2 _ 2 v ( y ' + a ) 4 - 2 w a 3 b l / [ ( y '
+ a) 4 + a 3 b ] ,
(3.18)
and x' = x + h t ,
y' =y + h t .
(3.19)
In case v = w = 0, our two-sublattice magnetic system is fixed by giving a, x and y. It can be represented by a point in the plane of the sublattice anisotropy fields (x, y). Transformation (3.19) allows reference to overall effective fields x', y ' which
FIRST-ORDER MAGNETIC PROCESSES
435
are a superposition of anisotropy and applied field. Equation (3.17) is related to the approach to saturation, because the component of the measured magnetization parallel to H is MII, or m = MII/M. In fact, by using eqs. (3.13), (3.15), (3.16) and (3.17) we obtain the slope of the magnetization curve at h = ht, S = d m / d ( h / h t ) l h = h t = a ( y '2 + 2 a y ' + a ) 2 h t / { [ ( y ' + a) 4 + a3b](ht - h*)}.
(3.20) The boundary between second- and first-order transitions is determined by the condition ht = h*.
(3.21)
In the limit of infinite exchange (J---~ oo), this condition gives (K2, + K2b )/(Kla + Klb ) = ~, which, as expected, is the equation of the boundary line n' of FOMP of type 1 (see section 2.2). The use of eq. (3.20) allows us to evaluate what is the role of the canting effects in determining FOMPs in different conditions. In particular, one is interested to ascertain whether FOMPs are possible when only K~ constants are present in the individual anisotropies of the sublattices. Hence, we shall examine the various situations under the hypothesis that g2a and K2b are vanishing, so that eq. (3.18) reduces to h* = - b y ' 2 ( y
' + 2 a ) 2 / [ ( y ' + a) 4 q- a3b] .
(3.22)
In the case J > 0 we have ferromagnetic order. It can immediately be seen that in this case S > 0: in fact, 0 < a < 1, h* < 0 and h t = h A > 0. So, in general, we ought not to expect a FOMP when the two sublattices are coupled ferromagnetically and the highest sublattice anisotropy terms are of second order. At least, we can exclude type-1 F O M P with S < 0. In the case of ferrimagnetic order (J < 0), the situation is more complex. In fact, we have three possible transitions with increasing field. The first one at H = Htl(=HA), where the system reaches the state of intermediate saturation with the two vectors antiparallel. The system does not change until another critical field H = Ht2 is reached that triggers the break of the collinear symmetry, and the two vectors start to open an angle. With further increase of H the system reaches the forced ferromagnetic state at a field H = Ht3, with the two vectors parallel. However, depending on the values of the parameters, it is possible that the intermediate saturation is completely absent, so that there is a continuous regime of the magnetization in the canted phase from H = 0 up to H = Ht3. This can happen because the intermediate saturation can be either unstable or metastable. The condition for the existence of FOMP of any kind, o.FOMP or h.FOMP, at each of the transition points H t l , Ht2 and Ht3 is deduced from eq. (3.20), where H t is always a solution of eq. (3.15). So, the negative value of S, relative to a certain transition point, together with the fact that the collinear state involved is
436
G. ASTI
stable, are sufficient conditions to assure that the transition occurs through a FOMP. In the case of ferrimagnetic order, a is larger than 1, the denominator of eq. (3.22) is less than zero for y' below a certain value Y0. So that for y' < y ~ we have S < 0 if h t < h*, i.e., Ihtl > Ih*l. In the case of the transition to forced ferromagnetic order (at H = H~3), we have 0 < a < 1 so that always h * < 0 and S < 0 if h , > h * , i.e., Ihtl < Ih*l. We have to remember that, in this case, the reduced variables (3.13) are referred to a different unit field, because for ferrimagnetic order it is J M = J ( M a - I M b l ) , while for ferromagnetic order it is J M = J(Ma + IMbl), which implies a factor (a - b) between the reduced expressions of the same quantities [see eq. (3.32)]. We can obtain the equation in parametric form for the boundary line, 'h*' between first- and second-order transitions. The definition equations are x0=x'-h*,
Y0=Y'-h*"
(3.23)
We must take into account that the effective fields x' and y', defined by eq. (3.19) obey eq. (3.15), that represents a hyperbola having an upper branch that crosses the origin, i.e., ax' + by' + x ' y ' = 0,
(3.24)
so that eq. (3.23) becomes, x o = -by'/(y'
+ a)- h*,
Yo = Y ' -
h*,
(3.25)
in which y' is the variable parameter. Its graph for the case M a = 21Mbl will be shown in fig. 17. For the aforementioned value y' = y;, the denominator of eq. (3.22) vanishes and one has h*---~ ~. In the graph the line denoted by 'Y0" is then the asymptote of line h*. From the above analysis, we can conclude that the canting effects can be responsible for FOMP transitions, even in the absence of anisotropy constants of higher than second order. However, this possibility is limited to ferrimagnetic order, certainly for h.FOMP, while it is doubtful that o.FOMP could exist. Obviously, by admitting K2a a n d / o r K2b # 0, there are a broad range of possibilities for a variety of FOMPs, and the equations obtained and critical parameters could be used for the analysis of the experimental curves. In the limit of J---~ 0o we find y' ~ 0, h* ~ 0 and S ~ 1, as expected. It may be useful to write down the explicit expressions for h t of the various critical fields. They can be deduced from eq. (3.15), which in explicit form is, ht2 + (1 + x + y ) h t + ax + by + x y = 0.
(3.26)
We take into account definitions (3.13) of the reduced variables, which are valid for all cases. As already mentioned, this implies that for the same set of values of the physical parameters, Ma, IMbl, gl,,, glb and IJI we obtain different sets of
FIRST-ORDER MAGNETIC PROCESSES
437
values for a, b, x and y in the various cases. For the case of ferromagnetic order (J > O) we have only one transition at (3.27)
htf = h A = ½(C + D ) , where, C= -1-x-y
,
D = [(1 + x - y ) 2 _ 4 a ( x - y)]1/2.
If htf is negative, it means that the c axis is an easy direction. The solution with negative sign has no physical meaning in this case, because it merely gives the value of the field at which the c axis changes from a maximum to a saddle point for the free energy F. A geometrical interpretation of htf is given in fig. 12. Given a representative point P(x, y) in the plane (x, y) of the reduced sublattice anisotropy fields, we draw the line g from P having angular coefficient equal to 1
~Y
X
-1
¢/:
Fig. 12. Magnetic phase diagram for the case of a two-sublattice system having ferromagnetic order ( J > 0 , M b > 0 ) with a=Ma/M= 3, in the plane of reduced sublattice anisotropy fields x = 2Kx,,/(JMM,~), y=2K1J(JMMb). Line o,~ is the upper branch of the hyperbola with equation ax + by + xy = 0. The region above o-[ corresponds to collinear magnetic states with the sublattice moments parallel to the c axis. When the representative point P(x, y) drops below ¢r[, the magnetic system again has the same magnetic order provided the magnetic field is stronger than the anisotropy field H A. A geometrical interpretation of this critical-field value is given by the components of vector PQ, oriented at 45° from the x axis, that turn out to be equal to the reduced anisotropy field h A = htf = HA/JM.
438
G. ASTI
and crossing the upper branch o-I of the hyperbola ax + by + x y = 0 ,
(3.28)
in a point Q: the components of the vector PQ are equal to htf. This procedure stems from the fact that the equation for the critical fields [(3.26)] is obtained from eq. (3.28) by the substitution x---~x + h, y---~y + h. The condition for the c axis to be a hard direction turns into the requirement that point P must not be above o-I . For the case of ferrimagnetic order (J < 0) we have, htl = h A = ½(C + D ) ,
(3.29)
htz= ½(C- D) .
Referring to fig. 13, we have that htl and ht2 are represented by the components of vectors PQ1 and P Q z , respectively, which, indeed, are both negative in the given example. Q1 and Q2 are the intersections with the upper branch o-a+ of the hyperbola given by eq. (3.28), which is different from o-f because now a > 1 (and
t,
(3.30)
In fact, in this case, a and b are both positive, as in the ordinary ferromagnet, but J < 0 so that F is a minimum below the lower branch o-~- of hyperbola (3.28) (see fig. 14). It is evident that when P drops below the straight line 'r', of equation y-x-p = 0 and tangent to O'a+, then the intermediate saturation state is unstable. The value of p comes from the condition D = 0 and turns out to be p = 1 - 2a + 2(a 2 - a) 1/2 .
(3.31)
If we want to compare ht3 with the other two critical fields related to the ferrimagnetic order, it may be convenient to express ht3 in the same units as used for htl and ht2. For this we have to refer to the same unit field J M - = J(M,, -[Mb]) instead o f J M + = J ( M a + IMbl), where we have denoted by M - and M + the modulus of the resultant magnetization in the cases of ferri- and ferromagnetic order, respectively. Then, assuming for all the reduced variables the meaning they have in the case of ferrimagnetic order, we must multiply ht3 by a factor, ~1 = M + / M - = (M,~ + [ M b l ) / ( M a - IMb[) = a -
b,
(3.32)
in order to obtain its value in the new units. Or we can give directly in terms of a, b, x and y (as defined in the ferrimagnetic order) the new expression h~3 , by using the substitution, x---~ x / ( a - b) ,
y---~ - y / ( a - b) ,
ht3--~ h~/rl .
(3.33)
F I R S T - O R D E R M A G N E T I C PROCESSES
439
::Y
-3
-2
)
~
X
u
+
.........................................
i:i ......
Fig. 13. Magnetic phase diagrams for the case of a two-sublattice system having ferrimagnetic order (J < 0, M b < 0 ) , with a = Ma/M = 2, in the plane of reduced sublattice anisotropy fields x = 2K1,,/ (JMMa), y = 2K1J(JMMb). Line o-~+ is the upper branch of the hyperbola with equation ax + by + xy = 0. The region above tra+ corresponds to collinear magnetic states with the sublattice moments parallel to the c axis but having opposite orientations. When the representative point P(x, y) drops between line O'a+ and the straight line r, which is the upper part of the tangent to (r~ in its vertex, the magnetic system again has the same magnetic order provided the magnetic field is stronger than the anisotropy field H A = Htx but lower than the second critical field, H,2, at which the collinear state becomes unstable (see section 3.2.2). A geometrical interpretation of these critical-field values is given by the components of vectors PQ1 and PQ2, oriented at 225 ° from the x axis, that turn out to be equal to the reduced critical fields, h A = ht~ = H~/JM and ht2 = H~2/JM , respectively. Line h* defines the boundary between second-order and first-order transitions [see eq. (3.22) and fig. 17]. In using this diagram one has to take i n t o a c c o u n t that, because of the fact that J < 0 and M b < 0, the reduced critical fields are negative, x has a sign opposite to K~a and y has the same sign as K~b.
So, we have, (3.34)
h~ = ½(a - b ) ( C - - D - ) ,
with, C- = -1-
x/(a-
b) + y / ( a -
b) ,
D - = {[1 + x / ( a -
b) + y / ( a -
b)] 2 - 4a(x + y ) / ( a - b)} 1/2 ,
(3.35)
which yields, h(3=X{b-a-x+y-[(a-b+x+y)2-4a(a-b)(x+y)]l/z}.
(3.36)
440
G. ASTI
Y
-2
-1 I
I
X
h* g
iP
-2
Fig. 14. Magnetic phase diagram for the case of a two-sublattice system having forced ferromagnetic order ( J < 0 , Mb>0), with a=Ma/M= z3 in the plane of reduced sublattice anisotropy fields x = 2Kxa/(JMMa), y = 2Kxb/(JMMb). Line try- is the lower branch of the hyperbola with equation ax + by + xy = 0. The region below trf corresponds to collinear magnetic states with the sublattice moments parallel to the c axis. When the representative point P(x, y) drops above line trf the magnetic system again has the same magnetic order provided the magnetic field is stronger than the third critical field Ht3. A geometrical interpretation of this critical field value is given by the components of vector PQ, oriented at 225° from the x axis, that turn out to be equal to the reduced critical field, hi3 Ht3/JM. Note that, in this case, h, x and y have signs opposite to H, Kla and K~b, respectively. Line h* defines the boundary between second-order and first-order transitions [see eq. (3.22) and fig. 17)]. =
A s a l r e a d y n o t i c e d , t h e a b o v e given e x p r e s s i o n s a n d r e p r e s e n t a t i o n s for t h e critical fields h t a r e g e n e r a l l y v a l i d , b e c a u s e t h e y d e p e n d o n l y On t h e s e c o n d o r d e r anisotropy constants. 3.2.5. L i n e a r regime a n d magnetic transitions in the canted p h a s e I t is k n o w n t h a t t h e m o l e c u l a r field t h e o r y a p p l i e d to a N r e l m u l t i s u b l a t t i c e s y s t e m , in t h e a b s e n c e o f m a g n e t i c a n i s o t r o p y , p r e d i c t s t h e p o s s i b i l i t y o f a
FIRST-ORDER MAGNETIC PROCESSES
441
non-collinear magnetic order that can be achieved via spontaneous SRT with changing temperature, or via field-induced SRT which are normally of second order. A peculiar feature of the magnetic behaviour of a canted system of this kind is that the resultant molecular field, acting on each one of the canted sublattice magnetization vectors, has the form,
lij = xj tj,
(3.37)
where Aj is a function of the set {J~K} of the molecular field coefficients only (Acquarone and Asti 1975). That is, the canted sublattices are effectively decoupled and evolve with temperature as an assembly of non-interacting ferromagnets, each with a different 'effective Curie constant' (°quasi-ferromagnetic' behaviour). In this respect, also the applied field H can be formally treated as one of the canted vectors to which a 'molecular field' x H = r. M i corresponds, i.e., the resultant magnetization. The consequence is that X, the susceptibility of the system, is also a constant, being only a function of the set {Jir}; this phenomenon was observed by Clark and Callen (1968) in rare earth-iron garnets. Indeed, they have observed temperature ranges over which the magnetization, at fixed field strength, is temperature independent. For a two sublattice N~el ferrimagnet having molecular fields, H a -= JxlMa + J 1 2 M b ,
l i b = JzlMa + J22Mb,
(3.38)
we have in the canted phase, (3.39)
X = - 1/.112.
It is worth noting that the two-sublattice ferrimagnet discussed in section 3.2.4, having only second-order anisotropy constants, does show a similar behaviour within certain conditions. In fact, it is found that if the representative point P ( x , y ) belongs to the upper branch '1' of the hyperbola (see fig. 17), b x + ay - x y
-= 0,
(3.40)
then it turns out that: (i) Either htl or ht2 is equal to --1; (ii) The same point P(x, y) in the reference frame of the forced ferromagnetic order (J < 0, M b > 0) belongs to the lower branch 1 of hyperbola (3.40) so that also ht3 = - 1 [or h~3 = - a + b, if expressed in ferrimagnetic reduced variables as in eq. (3.36)]; (iii) The magnetization curve is linear and takes the simple form m = - h , where h and m a r e the reduced field and magnetization defined by relations (3.13). This means that the magnetic susceptibility is a constant up to h = h~3 and is equal to, X = M H/H = m M / h J M
= - 1/J,
(3.39')
442
G. A S T I
an expression that coincides with eq. (3.39), which is related to a N6el ferrimagnet (without anisotropy). In order to demonstrate this property, we start from the equilibrium equations, b sin(0a - 0b) + h sin 0a + x sin 0~ cos 0, = 0 , (3.41) - a sin(0~ - 0b) + h sin 0b + y sin 0b cos 0b = 0. These equations can be obtained by means of relations (3.3) using the same definitions for x , y , a and b as given in connection with relations (3.13). Denoting s~ = sin 0~, c I = cos 0a, s 2 = sin 0~ and c 2 = cos 0b, and multiplying the first equation by c 2 and the second by Cl, we obtain after subtraction, ( m + h ) ( s l c 2 - seca)
+ (xs 1 -
YS2)CaC 2 = O.
(3.42)
Now, elimination of h from eqs. (3.41) yields, (b - Y)C2S~/S 2 - (a
2 -
X)ClS1/S
2
2 '[- a C 2 S 1 / S 2 - - b c 1 = O .
(3.43)
Taking into account condition (3.40), we can verify that XS 1 -- ys 2 = O,
(3.44)
is a solution of eq. (3.43), so that eq. (3.42) becomes, (m + h) sin(0a - 0b) = 0.
(3.45)
Since in the canted phase 0a ~ Ob, we conclude that m = - h . Note that hyperbola (3.40) is obtained by a translation of hyperbola (3.28) parallel to a vector v, i.e., (1, 1), having unitary components. In fact eq. (3.40) can be obtained from eq. (3.26) by taking h t = - 1 . The results are also consistent with eq. (3.20), where S is equal to the differential susceptibility at the transition point. In fact, substitution of h t = - 1 in eq. (3.20), where h* is given by eq. (3.22), yields, S = (y,2 + 2 a y ' + a ) 2 / ( y '2 + 2 a y ' + a) 2 = 1.
(3.46)
Moreover, we can show that if htl = - 1 (or ht2 = - 1 ) , then also ht3 = - 1 and vice versa. In fact, let x + and y+ be the coordinates of the representative point P(x +, y+) in the flame of forced ferromagnetic o r d e r (i.e., x + and y+ are expressed in units of J M + = J ( M , + [M b [). Then by the substitution (3.33) for x + and y+ in eq. (3.26), we obtain, ht23 + [1 + (x - y ) / r l ] h t 3 + (ax + b y - x y ) / r l 2 = O,
(3.47)
F I R S T - O R D E R M A G N E T I C PROCESSES
443
with 7/= a - b. If b x + a y - x y = 0, eq. (3.47) becomes ht2 + [1 + ( x - y ) / ' o ] h t 3 + ( x - y ) / r l = 0 ,
(3.48)
for which ht3 -- - 1 is a solution. Or, if we put ht3 = - 1, we obtain b x + a y - x y = 0, which implies that either htl or ht2 is equal to - 1 . Now we can observe that the variation of 01 and 02, according to the law of eq. (3.44), is such that, in general, the m i n i m u m value m 0 of the magnetization in the linear regime is above m = 0, and occurs at a certain critical field h 0 (when h t l = - 1 ) . Obviously, depending on the values of x and y , such a m i n i m u m can coincide with intermediate saturation (when ht2 = - 1 ) , or it can coincide with the origin (h = 0, rn = 0) when x = y [see the paragraph below eq. (3.49)]. That is, the range of the minimum m 0 is 0 ~< m 0 <~ 1. The transition to the linear regime is
Ill¸ 3
1 ...........
0
Ta,
T_'I~===.T 2
-I1.1
-ht2
I
2
;
-h
J,
Fig. 15. Magnetization curve for the case of a two-sublattice ferrimagnet with a = 2, x = 0.75 and y = 0.6. This is an example of the linear regime that occurs when the system has only second-order sublattice anisotropy constants and the representative point P(x, y) belongs to the upper branch T of the hyperbola b x + a y - x y = O, shown in fig. 17. The equation of the linear part, in terms of reduced variables, is m = - h , which implies that susceptibility is a constant equal to X = - 1 / J . The transition to the linear regime occurs at a certain magnetization m 0 and is of second order. When m 0 < 1, as in the example shown, there are actually two superimposed magnetization curves in the range m 0 < rn < 1, corresponding to the same rn and the same energy but to different magnetic structures.
444
G. ASTI
a s e c o n d - o r d e r transition. In the case m 0 < 1, which occurs w h e n htl = - 1 , the m i n i m u m in the m a g n e t i z a t i o n implies that we have a range of h (Ih0l < Ihl < 1) w h e r e two different m a g n e t i c structures c o r r e s p o n d to the same m. Actually, there are in this range two b r a n c h e s of the m a g n e t i z a t i o n curve that c o r r e s p o n d to different angled-magnetic structures. Figure 15 shows an e x a m p l e for the case w h e n a = 2, x = 0.75 and y = 0.6. A w a y to elucidate the shape of the magnetization curve m o r e clear, is to consider it as the limit of a t w o - b r a n c h curve of the type given in fig. 16, c o r r e s p o n d i n g to a point just a b o v e line 1 [see eq. (3.40) and fig. 17]. Besides the s e c o n d - o r d e r transition, at the lower limit of the linear curve, w h e r e h = h 0 and m = m o , there is a first-order transition in the angles 01 and 02 that is not visible in the parallel c o m p o n e n t m , as defined by the last of the relations in eq. (3.13), but in the transverse c o m p o n e n t m I = a sin 0a + b sin 0b. A s a m a t t e r of fact, in the whole interval Ih0l < fhl < 1 the two magnetic states, besides having the same m, have equal free-energy density. In fact, it is easy to show that the free e n e r g y of the system in the linear regime d e p e n d s on 0a and 0b only t h r o u g h the variable m = a cos 0a + b cos 0b. F o r this p u r p o s e , it is sufficient to utilize relations (3.40) and (3.44) in the expression of F given by eq. (3.1) for the case of two sublattices, after substitution o f the r e d u c e d variables of eq.
1.05
.95
d
I
,9'5
. . . .
1
,
,
,
,
I
%05'
'-h ~
Fig. 16. Magnetization curve for the case of a two-sublattice ferrimagnet with a = 2, x = 0.745 and y = 0.595 corresponding to a point close to line '1' [see eq. (3.40) and fig. 17]. The case of fig. 15 can be considered as the limit of a two-branches curve of the type shown here. A FOMP transition is shown that brings the system from the intermediate saturated state into the spin-flop phase.
FIRST-ORDER MAGNETIC PROCESSES
445
(3.13). H e n c e , the system is d e g e n e r a t e a n d the two m a g n e t i c phases m a y coexist in the w h o l e r a n g e of h u n t i l h = h t l , w h e r e the p h a s e c o n n e c t e d with the i n t e r m e d i a t e s a t u r a t i o n state disappears. This t r a n s i t i o n occurs with n o discont i n u i t y i n the v a r i a b l e m a n d c a n b e c o n s i d e r e d as the limit of a F O M P of the type s h o w n in fig. 16. A n o t h e r possible l i n e a r s o l u t i o n of eq. (3.43) is 0a = 0 b = 0 w h e n the r e d u c e d a n i s o t r o p y fields of the two sublattices are e q u a l , i.e., w h e n the r e p r e s e n t a t i v e p o i n t P ( x , y ) b e l o n g s to the l i n e ' s ' (see fig. 17) which has the e q u a t i o n x = y. T h e n , eqs. (3.41) b e c o m e , (h + x cos 0) sin 0 = 0 , i.e., m = - h / x
=-h/y.
(3.49)
I n the case of f e r r o m a g n e t i c order, we m u s t have
.8 'O.
.6
.4
.4
.6
.8
X
1
Fig. 17. The phase diagram, shown in fig. 13 for the ferrimagnetic case, is shown here in greater detail in the area of the first quadrant which is crossed by various critical lines: h*, O'a+, 1, r, s and y~. For explanation of h*, ~r,+ and r see fig. 13. Hyperbola 1is mentioned in fig. 15 and, together with straight line s, is discussed in section 3.2.5. Line y~ is the asymptote of line h*. Line h* crosses line 1 and is tangent to it in a critical point G with coordinates x = a - ~aa2 - a, y = 1 - x. In G they are both tangent to straight line r.
446
G. A S T I
x = y < 0, while in the ferrimagnetic case x = y > 0. Obviously, the linear relationship b e t w e e n magnetization and field holds only up to htt. It is worth noting that we have no canting, even in the presence of 'competing' anisotropies (Kaa < 0 and Klb > 0 in the ferrimagnet). The various critical lines h*, o-+, 1, r, s and Y0, deduced in the analysis of t h e ferrimagnetic case, are shown in fig. 17. Line h* crosses line 1 and is tangent to it in a critical point G with coordinates x = a - ~/a 2 - a and y = 1 - x. In G, they are both tangent to the straight line r. Line 1 crosses line s in a point u, i.e., (1, 1). B e l o w line r the intermediate saturation state does not exist. A b o v e this line various magnetization curves are possible that, in general, are c o m p o s e d of two branches. These connect with continuity two couples of the four critical points: O (h = 0, m = 0), the origin; T1 (h = hta , m = 1) and T2 (h = ht2 , m = 1), the extremes of the intermediate saturation state; and T3 (h = ht3 m = a - b). In addition to these two branches, there is another equilibrium state represented by the line which connects T~ and T e (see, e.g., figs. 15, 16, 18 and 19) which covers
m
3
T3
2
T1 ~
1
o
1
2
2
-h
3
Fig. 18. The magnetization curve for the case of a two-sublattice ferrimagnet is, in general, composed of two branches connecting with continuity two couples of the four critical points: O (h = 0, m = 0), the origin; T1 (h = h,1, m = 1) and T2 (h = ht2, m = 1), the extremes of the intermediate-saturation state; and T3 (h = ht3 , m a - b ) the transition point to the forced ferromagnetic state. In addition to these two branches, there is a n o t h e r equilibrium state represented by line T 1 - T 2 which covers the range of intermediate-saturation state. The present graph, obtained for a = 2, x = 1.1 and y = 1.2, i s an example of a magnetization curve consisting of branch O - T 2 and branch T 1 - T 3 . T h e collinear state for h = 0 (0a = 0b = 17r) belongs to branch O - T 2 and is not the equilibrium state. The state of m i n i m u m free energy at h = 0 is, instead, on the other branch where we have 0a ~ 0b ~ ½~" and m ~ 0. This branch and its continuity is better evaluated w h e n considering it as a connection between T1 and the symmetric point of T3 with respect to the origin (i.e., for h = - h t 3 , m = - a + b ) . Note that in the present case, at h = 0 we have a noncollinear state with the resultant magnetization pointing out of the basal plane, but very close to it. =
F I R S T - O R D E R M A G N E T I C PROCESSES
447
m
T3
3
2
1
T1
~--._._... T2
1
Fig. 19. As in fig. 18 but for a = 2, x = 0.9700 and y = 0.9047. The two branches are of the type O - T 3 and T1-T2.
the range of intermediate saturation state. Besides figs. 15 and 16, other examples are reported in figs. 18 and 19 showing different connections. In the case of fig. 16 it is easy to predict a FOMP, because, with increasing field, there must necessarily be a jump from one branch to the other. Depending on the characteristics of the branches, we understand that we can have in the various cases a variety of transitions also involving the intermediate saturation state. A condition that can be favourable for observing these phenomena may be that of ferrimagnets in the neighbourhood of compensation points, because the field JM can be made small, and the various critical conditions are achieved at relatively low fields. We must remember that the examples given (figs. 15, 16, 18 and 19) are obtained in the limit of only second-order anisotropy terms (i.e., v = w = 0) and accordingly also lines h* and 1 in fig. 17 refer to this situation. Instead, the graphs and the expressions relative to critical fields h t ' s remain the same, even in the presence of higher-order anisotropy terms because they are independent of v and W.
3.3. The small-angle canting model 3.3.1. General remarks The magnetic behaviour of a multisublattice magnetic system, under conditions of small deviations from the collinear magnetic order, can be conveniently treated by the use of an approximate theory, the small-angle canting model (SAC) (Rinaldi and Pareti 1979). The SAC model appears as a simple extension of the classical phenomenological theory of magnetic anisotropy. By the admission of deviations from the collinear order, the system acquires a further degree of freedom that
448
G. ASTI
results in a relaxation of the free energy, which appears as a strong distortion of the anisotropy-energy surface. The opening of the angles between the interacting moments occurs at the expense of exchange energy, with a subsequent gain in anisotropy energy. The important point is that such a distortion can be readily accounted for, by appropriate corrections of the anisotropy constants. It means that the system can be described as a simple ferromagnet characterized by 'effective anisotropy constants' K1, K2, 1 £ 3 , . . . , that are functions of sublattice anisotropy constants and of the molecular-field coefficients. The effective anisotropy constants are not rigorously constant because there remains a magnetic field dependence to first order in the parameter h = H / J M , where H is the magnetic field and J M is a quantity of the order of the exchange field. High-order anisotropy constants appear, even if we attribute to the individual sublattices only second-order constants. For instance, /£1 can be very different from the simple algebraic sum of the individual Ka's. It can even have opposite sign. The small-angle canting model has general validity, as it has been shown (Asti 1981) that it applies to both ferromagnetic and ferrimagnetic order, as well as to any combination of anisotropy constants. The conditions for the application of the SAC model may refer: (i) To the orientation of the magnetic vectors within a narrow range close to the crystallographic symmetry directions (as in the case treated in section 3.2); (ii) To the whole range of orientations, when exchange interactions are much stronger than magnetic anisotropy forces in the system. The region well inside the spin-flop phase of ferrimagnets is obviously excluded (see section 3.2). In the case of magnetization processes close to the symmetry directions, the model provides exact expressions for important physical parameters, such as initial susceptibility and anisotropy fields, or relations concerning important phenomena such as the spin-reorientation transitions. The concept can be applied, in principle, to complex structures such as the case of a three-sublattice system (Asti 1987). In section 3.3.2, the case of a twosublattice system will be described and examples of applications will be given in section 3.3.3. 3.3.2.
The two-sublattice m o d e l
Let us consider a uniaxial system of two magnetic sublattices a and b having magnetizations M a and M b and anisotropy constants Kaa, 1£2o, K 3 a , . . . , and Klb, K2b, K 3 b , . . . . We refer the orientation angles of all vectors with respect to the symmetry axis, c. So, 0~, 0b, 0 and q~ are the angles of M a, M b , M = M a + M b and magnetic field H respectively. The free energy of the system is, E = - JMan b
-
-
M" H + ~ i=1
K ~ sin 2i 0a +
k Kjb sin2~0b ,
(3.50)
j=l
where - J M a • M b is the exchange energy and J the intersublattice molecular-field coefficient. The interaction is ferromagnetic if J > 0 and antiferromagnetic if J < 0. If 'a' denotes the major sublattice, one may have M b > 0 or M b < 0. The
FIRST-ORDER MAGNETIC PROCESSES
449
assumption that the minor magnetization M b c a n take negative values, allows to utilize the same definition for the angles, so as to extend immediately all the results obtained for ferromagnetic interaction to the ferrimagnetic case, simple by changing the sign of both J and M b (see section 3.2). We will consider only situations for which M~ and M b a r e nearly parallel, so that we can write e a = 0 , - 0 and 8 b -~-0b --O, where G and e b are infinitesimal deflection angles of M~ and M b with respect to M. T h e n eq. (3.50) becomes E = - J M a M o cOS(G - % ) - M a l l cos(q) - 0 - G ) - M b H cos(~ - 0 - %) 4- ~
giasin2i(o-} - Ca)'+ ~
i=t
Kjb sin2J(0 + % ) .
(3.51)
j=l
By putting p = M b / M ~ , M = M¢ + M b and e = eb, and taking into account that M~ e a + M b e b = 0, we obtain by expansion up to second order in e, E = E o + A e + B e 2,
with E o = - J M , , M b - M H cos(q~ - 0) + ~
(3.52)
(K~ + K~b).sin 2~ 0 ,
i=1
A = k
- 2 i ( P K i ~ - Kib) COS 0 sin 2/-x 0 ,
i=1
B = IjpM2
+ lpMH
i(Kiap 2 +
cos(q~ - 0) + ~
gib ) sin 2i-2 0
i=1
x [ ( 2 i - 1) cos 2 0 - sin 2 0] . H e r e E 0 is the ordinary expression for the total energy of the collinear system. The value of e is determined by the condition 3 E / O e = 0 , which gives e = -A/2B. Substitution in eq. (3.52) yields, E = E o - A2/4B ,
(3.53)
i,e.
E = Eo -
- i g i c s 2i-1 -
)7{
e +
ifde~-2[(2i i=1
}
1) - 2is 2] ,
(3.54)
with gi = PKia - Kib ,
c=cos0
and
~ = p2Kia q- Kib '
s=sin0.
e = ½[ p J M 2 + p M H cos(q~ - 0 ) ] ,
450
G. ASTI
If we suppose that gia and Kib are zero above a certain order N, eq. (3.54) can be expressed as,
)2/{
E = Eo- c
_
igis2i 1
e + fl
-
2N2fN
S2N
N--1 + ~
} [ ( j + 1)(2]
+ 1)f]+ 1 --
2]2fjls 2j .
(3.55)
j=l
This expression for the energy can be transformed in such a way that it assumes the usual form of a power series in sin 2 0. Hence, the coefficients K~ are the effective anisotropy.constants of the two-sublattice canted system. In the simplest case, i.e., for N = 17 it gives, c~
E = - M H cos(q~ - 0) + J M ~ M b + ~.
K i
sin2i 0 ,
i=1
with,
K1 = Kla + Klb -- g21/(e + fl) ,
K 2 = g21(e - f ~ ) / ( e
+ L)
2 ,
(3.56)
gn = gn-12fl/(e
-{-]el) =
2 , - n - 2 ~t e - f l ) / ( e + f ~ ) gaY1
n.
These results are the same as those reported in the work by Asti (1981) in which t h e p a r a m e t e r s a, b and h are related to the p a r a m e t e r s used here, by the following relations, a = 2g2/JpM 2
b
=
2fl/JPM 2 ,
h + 1 = 2e/JpM 2 •
Let us now consider the case N = 2, which means that we have sublattices with anisotropy constants of second and fourth order. Then from eq. (3.55) we obtain, 4-[4-k1 E= Eo-[g~/(e+
fl)] ~ k=l
r[(k+r),2]
r~=l - Pr-l[
j~--O __ ( k
_
r - , ] Jcelk-r-2j a 2j]12k JJa ,
j
(3.57) where [w] indicates the largest integer in w, with the assumption that [w] = 0 for w ~< 0, and Pa = - 1 + 4g2/g 1 ,
Po = 1 , 2
2
P3 = - 4 g z / g a ,
P2 = ( - 1 + g2/gl)4g2/ga ,
% = -(6f2 - 2fl)/(e + fl),
a2 = 8fz/(e + f l ) .
One can easily verify that for g2a = g2b = O, the coefficient K k of S 2k a r e coincident with expressions (3.56). It can be convenient to write down the first
FIRST-ORDER MAGNETIC PROCESSES
451
few terms of eq. (3.57) explicitly, if we limit the series to k = 4, it becomes, E = E o - [g~l(e + f~)]{s 2 + (Pl + 0/1)$4 -~- (0/21 -~ 0/2 q- P10/1 + P 2 ) $6
+ [0/31 + 20/10/2 + P,(°~21 + 0/z) + P z a l + P31ss + ' " "}.
(3.58)
The coefficients of s 2~ in eq. (3.57), including the terms contained in E 0, give analytical expressions for the effective anisotropy constants K k as functions of the parameters of the two sublattices, i.e., K~a, Kza, Klb , K2o , m a , m b and J. The dependence of K k on the magnetic field is through the parameter e. It is negligible in the transverse configuration, i.e., when the initial susceptibility is measured with H perpendicular to the c axis, both because H is small compared with J M and because q~ = ½7r and 0 is small, so that cos(q~ - 0) ~-0. Instead, when H is parallel to the c axis and this is a hard direction, the parameter e plays an important role. As a consequence, the anisotropy field has two different expressions depending on whether the axis under consideration is a hard or an easy direction. The same expressions given in sections 3.2.2 and 3.2.4 for the anisotropy field H A in a hard direction, can be obtained here from the definition, H A = 2Ka/M,
(3.59)
where K 1 is given by eq. (3.57) or (3.58), taking into account expression (3.52) for E 0. More easily, K 1 is directly taken from the first part of eq. (3.56) because H A depends only on second-order anisotropy constants. So, the condition for the spontaneous SRT has the usual form K 1 -= 0. The expression for the anisotropy field in easy direction is again deducible from eq. (3.59), here we have to put the term p M H cos(q~ - 0), which appears in the parameter e, equal to zero. All these considerations are also valid if we consider a planar direction instead of the c axis. In fact, if we make a K ~ R transformation (see section 2.2), we can exchange axial with planar directions, which means that all the expressions obtained can be interchanged once we replace all K constants by the corresponding R constants. Models including K 2 constants have been utilized successfully by Ermolenko (1979) and by Sinnema et al. (1987), for computing magnetization curves of R C % and R2Co17. This means that higher-order terms in the crystal-field interaction must be taken into account. The advantage of the present treatment is that it gives analytical expressions that are exact for magnetic processes that involve small angles between the sublattice moments, e.g., concerning magnetic transitions involving symmetry directions (see section 3.2). In the analysis of the magnetization curves, it is convenient to make reference to the ordinary K anisotropy constants when the applied magnetic field is along or near the c axis. On the other hand, when we are magnetizing along a direction close to the basal plane, we have a more accurate description of the magnetization curve in terms of the R constants, which are obtained from the above mentioned K<--->R transformation.
452
G. ASTI
3.3.3. Some applications The first application of the small-angle canting model was by Rinaldi and Pareti (1979) on PrCos, giving a straightforward explanation of the SRT easy-axis to cone transition observed at 110 K by Tatsumoto et al. (1971), Ermolenko (1976) and Asti et al. (1980), despite the fact that the s u m gla "~ Klb of the Co and Pr sublattice anisotropy constants is positive. In fact, as explained in sections 3.2.2 and 3.3.2, according to this model, the condition for the existence of a SRT is merely the vanishing of the effective/£1 constant, i.e., K 1 -- 0. By the use of the formulas given in section 3.3.2 it is possible to analyze, in a very simple way, the magnetization curves of both ferro- and ferrimagnetic materials. The approximation of the small-angle canting model is valid in a large number of cases and a variety of experimental conditions. As an example, we will report here the analysis carried out by Asti and Deriu (1982) of the initial susceptibility data, in the hard direction, obtained by Ermolenko (1979) on the series Yl_zGdzCos. In fact, the effective anisotropy constant K~ given by eq. (3.56) can be written in the form,
K1 = ~7(K~. + Klb),
(3.60)
where ~ is a correction parameter and K~a and Klb are the anisotropy constants relative to the Co and Gd sublattice, respectively. A convenient expression for 77 in terms of reduced exchange and anisotropy fields, as defined by eqs. (3.13) is r/= [1 + xy/(ax + by)]/(1 + bx + ay).
(3.61)
The case of the Gd sublattice is a special one, since Klb 0. It means that there is no competition between the anisotropies of the two sublattices. In spite of this, the canting effects exist because the competiton is with the applied magnetic field H. Hence, eq. (3.61) reduces to =
= 1/(1 + bx).
(3.62)
From the data on the initial susceptibility X as a function of the composition z, taken from the work of Ermolenko (1979), we can calculate the effective anisotropy constant K 1 = M2/(2X) as a function of z, and plot the experimental values of ~7= KI(z)/KI(O). Figure 20 shows the theoretical curve together with the experimental points. The agreement between the point and the curve supports the validity of the model, and shows how much the effective anisotropy is reduced due to the onset of a small deviation from collinearity. Indeed, in the absence of canting effects, the parmeter 77 ought to be independent of z. Another example is also taken from the work by Asti and Deriu (1982) and shows how general the validity of the Rinaldi and Pareti (1979) model is. As explained in section 3.3.2, it applies to any case of magnetic order (ferro- and ferrimagnetic) and sign of sublattice anisotropy constants. Besides that, for magnetic processes close to the symmetry directions, it is rigorously valid, even
FIRST-ORDER MAGNETIC PROCESSES
453
8
6
4
o
a
z'
,
Fig. 20. Plot of the values of the correction parameter ~7 [see eq. 3.60] calculated from the initial susceptibility data versus composition, taken from the work of Ermolenko (1979) for the series Yl_zGdzCos. The solid line is the theoretical curve deduced from eq. (3.62).
when anisotropy and exchange interactions are of the same order. Indeed, the case of ferrimagnetic order is that for which canting can produce dramatic effects. Peculiar examples are those of HozFe17 and Ho2Co17 , for which enormous discrepancies were obtained between anisotropy constants, as deduced from spin-wave spectra, and macroscopic magnetic measurements (Clausen and Lebech 1980). The apparent paradox is easily removed by the admission of canting, and the model presented above gives a complete account of the effect. Sarkis and Callen (1982), in order to study this problem, have developed an ad hoe calculation that led them to the following expression for the effective anisotropy constant, in terms of the constants (K R, KT) and the magnetizations (MR, MT) of the individual sublattices, Keff = (K 1 + 2K2)eff = (K R + K T + 2KRKT/L)/[1 + 2(KR M2 + KTMZ)/LM2],
(3.63)
where L = JMTM R and J is the intersublattice molecular-field coefficient. This result can be derived directly from eqs. (3.60) and (3.61). To do this, we have to use the K ~ R transformations described in section 2.2, which allow the immediate transfer of all the equations valid for the case HIIc axis to that of H_l_c axis and vice versa. So we can write the conjugate equation of (3.60), R 1 = 'rl(Rla + R~b),
(3.64)
where R~, Rla and Rlb a r e the anisotropy constants conjugate to K~, Kla and Klb , i.e., R I = - K 1 - 2 K 2 - 3 K 3 + ' " , and similarly for R1, and Rib. Taking into account expression (3.61) for ,/, eq. (3.64) becomes
R 1 = (Rla + R~b + 2RI,RIb/J')/[1 + 2(RaaM ~ + RlbMZa)/J'M2],
(3.65)
with J ' = JMaM b. This expression is coincident with eq. (3.63), once R1, Rla ,
454
G. ASTI
J', M a and M b are substituted by the corresponding quantities Keff, Ka-, KR, L, M r and M R. An approach based on the hypothesis of the small canting-angle has also been utilized by Cullen (1981) for the study of the problem of easy-magnetization directions lying along non-major axes in ternary cubic compounds of composition R~R~_xFe 2. This work also confirms that a classical model of canted magnetic moments can give rise to high-order anisotropy energy and phase instabilities. This type of approach is not as accurate as the direct numerical diagonalization of the total Hamiltonian containing crystal-field terms and the molecular field Hex. However, the classical treatment provides a transparent picture of the physical processes responsible for the observed phenomena.
Rib ,
3.4. Hexagonal ferrites having complex block-angled and spiral structures. Role of antisymmetric exchange During the investigation of magnetic properties of hexagonal ferrites, Enz (1961) encountered unusual properties in the system Bal_xSrxZn2Fe12022 (abbreviated as Bal_xSr~Zn 2 - Y ) . The compound has no spontaneous magnetization, but a field of the order of 1 kOe induces a first-order transition to a magnetization of about 85% of saturation. At a higher field, after a rather fiat plateau of the magnetization, another transition starts a further increase at a constant susceptibility up to saturation. Enz interpreted this behaviour as being due to a helical spin configuration having the axis parallel to the c axis and the magnetization vector rotating in the basal plane. The same behaviour has been observed in other hexagonal ferrites, both of the same Y structure [Ba2Mg 2 - Y (Albanese et al. 1975)] and of another structure, namely Ba3_~Sr~Zn2Fe24041 [Z-type, usually shortened as Ba3_~SrxZn2-Z (Namtalishvili et al. 1972)]. Like all hexagonal ferrites, also Y- and Z-type crystal structures are formed by the stacking of chemical blocks (usually labelled S, T and R) along the c axis (Smit and Wijn 1959). The starting point of the theory is the fact that no anomaly is observed in Zn2-Y and Zn2-Z containing Ba, and that the substituting Sr ions are localized in well-defined layers inside the T blocks for both structures. Hence, the suggestion of Enz was that substitution of Sr results in a weakening of the superexchange interaction patterns across the planes of the crystal. Then any further counteracting interaction across the same planes may result in a deviation from collinearity of the spin. In fact, neutron-diffraction investigations on a variety of hexagonal ferrites have shown the existence of a peculiar type of magnetic order that is known as 'block' structure (Aleshko Ozhewsky et al. 1969, Namtalishvily et al. 1972, Sizov and Zaitser 1974). The structure of these ferrites is such'that in it one can identify symmetric magnetic blocks, possibly comprising several chemical blocks: inside each magnetic block the localized spins stay mutually parallel, but the total magnetic moment of the block, taken as a whole, may assume a canted orientation with respect to contiguous blocks. The resulting configuration may be either a flat or conical spiral ('block conical spiral', or 'helix'), or an alternating structure
FIRST-ORDER MAGNETIC PROCESSES
455
when the rotation angle from block to block is 180°, thus producing an 'antiphase block' structure (also termed 'block angular' structure). As for the case of compounds containing Sr, in the case of Ba2Mg2-Y, the occupation by a nonmagnetic ion (Mg 2+ ) in the inner octahedral sites of the T block is expected to give rise to a similar weakening of the superexchange coupling between adjacent magnetic blocks. Besides the transition from spiral tO fan structure, these compounds having block-magnetic structures display another transition at a higher field. As a result, the magnetization curve appears as a two-step curve with a sharp knee at the end of the first fiat part. An example is given in fig. 10 of Chapter 6 of Vol. 3 of this handbook, p. 408 (Sugimoto 1982), for the case of Sr2Zna-Y. The figure has been taken from the publication by Enz (1961). Various theoretical models have been proposed for two-sublattice systems, indicating that different canted states and second-order spin-flop transitions are possible (Yamashita 1972, Morrison 1973). Mita and Momozava (1975) give an interpretation of the transition from helix to fan state on the basis of Nagamyia theory (1967). Just as in the interpretation given by Enz (1961), however, no explanation is given for the transition observed at higher fields. The existence of a critical field at the end of the first plateau in the magnetization curve has been interpreted by Sannikov and Perekalina (1969) in terms of a two-sublattice model, but only assuming different values for both the magnetic moments and the anisotropy constants of the sublattices. However, according to this model no spiral structure is possible, and moreover, the hypothesis is contradictory to the inherent symmetry of the two sublattices to be associated with the angled blocks. A step forward in the theory comes from an important qualitative consideration on the effects of the substitution. Indeed, the breaking of the symmetry centers inside the T block opens the way to the action of antisymmetric exchange as proposed by Koroleva and Mitina (1971). A complete theoretical study based on this line of thought has been given by Acquarone (1981). It provides the equilibrium configurations and the phase transitions of a magnetic model system, consisting of two sets of moments of equal magnitude, coupled by isotropic, anisotropic, and antisymmetric exchange, in the presence of uniaxial anisotropy and a transverse magnetic field. It applies both to two-sublattice and to spiral magnetic structures, and gives exhaustive explanations of a variety of two-step magnetization curves of the type described above. The model does not assume any restriction upon the relative strength of the different interactions. 4. Considerations on the experimental methods
4.1. Measurements in high magnetic fields First-order magnetization processes are normally observed at low temperatures, i.e., well below the Curie point, due to the rapid decrease of the anisotropy energy with temperature. It is clear from the graph shown in fig. 6, where an example of the ~l(l + 1) power law for the temperature dependence of anisotropy
456
G. ASTI
is given, that the effect of this behaviour is both to reduce the critical field and to move the state of the system towards the boundary with the normal state. In the majority of cases, at the higher limit of the temperature range, the phenomenon vanishes when the critical field is still relatively high. With only a few exceptions, as is the case with ferrites (Asti et al. 1978) and with Pra(Fe,Co)x 7 (Shanley and Harmer 1973, Melville et al. 1976), this means that magnetic fields above the limit of ordinary laboratory electromagnets are necessary to investigate this type of magnetic phenomena. On the other hand, the measurement of the critical parameters (field and magnetizations) provides useful data that allow deduction of important and precise information on high-order terms of the anisotropy energy. The use of single-crystal samples is obviously the most favourable condition for detecting the transition and a unique way to achieve precise evaluations of the critical magnetization mcr (or m s and m2, in the case of type-2 FOMP). However, the orientation of the crystal in the magnet appears, in most cases, to be quite critical for performing precise measurements, because of the strong sensitivity of Her to deviations from perfect alignment: e.g., a deflection by only two degrees of the applied-field direction from the basal plane in PrCo 5 gives rise to a change of Her by 2 T. Another source of error, which is important only for transitions occurring at low fields, is the demagnetizing field of the sample. If the parallel component M of the magnetization of the crystal is plotted versus the applied magnetic field intensity He, one gets a straight line at the transition that is not vertical, but has a slope equal to d M / d H e = 1 / N , where N is the demagnetizing factor of the specimen. So, in general, a better procedure is to plot directly M as a function of the internal magnetic field H = H e - N M . The use of spherical or ellipsoidal specimens offers obvious advantages: (i) accurate values of N can be determined, and (ii) one can avoid the broadening of the transition due to the non-uniformity of H. This non-uniformity produces distortions in the magnetization curve M ( H ) , particularly at the edges of the transition where the values of m 1 and m: must be identified. The use of continuous magnetic fields allows accurate and reliable determinations, if a large number of measurements are performed in the neighbourhood of the transition point. However, with the exception of a few magnets in the world giving constant fields above 20-25 T the practical limit is in the range around 15 T as is well-known, for both superconducting and water-cooled solenoids. Besides that, one has to take into account the cost of the equipment, which changes by orders of magnitude going through a threshold at about 8-10 T. As a matter of fact, rare earth intermetallic compounds having H~r above 15 T are fairly common. The only way to carry out magnetic measurements at very high magnetic fields is by using pulsed methods. The magnet laboratory at the Amsterdam University is unique, in that it provides the possibility of achieving 40 T, keeping the field intensity constant for a time interval of the order of one tenth of a second. Other installations in the world allow to go even above that limit in a reproducible way by using various pulsed-field techniques (Gr6ssinger 1982), which, in general, give sinusoidal pulse shapes. The main problems arising with the use of pulsed-field apparatus, in performing magnetic measurements, are
FIRST-ORDER MAGNETIC PROCESSES
457
connected with the inherent difficulty in overcoming the various noise sources. For instance, in using the induction method, the flux caused by the specimen has to be detected in the huge background signal coming from the applied field. Despite this, there are several reasons why a pulsed field could be used for magnetic measurements: (i) in a single shot it is possible to measure the whole magnetization curve from zero up to the maximum field; (ii) the continuous and regular variation of the field is a favorable condition for detecting sudden changes such as those associated with a transition; (iii) the experimental setup is relatively simple and economical.
4.2. The singular-point detection technique The inherent high sensitivity of the pulsed-field method for detecting transitions is more and more enhanced if one observes the successive derivatives of the magnetization M with respect to the field H, or to time. This is indeed the principle at the basis of the singular-point detection technique (SPD) (Asti and Rinaldi 1974a), an experimental method that allows to reveal the singularities associated with the field-induced magnetic transitions, utilizing polycrystalline specimens. The SPD theory was originally developed for measurements of the hard-direction anisotropy fields, and has been subsequently extended to the case of FOMP (Asti 1981, Asti and Bolzoni 1985). The magnetization curve of a ferromagnetic crystal shows, in general, a singuarity when it reaches saturation at the anisotropy field H A. The transition is, in principle, of second order and occurs only if the magnetic field is perfectly aligned with the hard direction. For field orientations other than hard and easy directions, the reversible magnetization curve is, in general, a regular function for H = HA, never achieving complete saturation for finite values of H. When we apply a magnetic field to a polycrystalline material, we obtain, in general, a smooth curve M(H) which, at first sight, seems to give no indication of the singularity at H = H A. However, in reality the averaging over all the crystal orientations does not completely cancel the singularity, which is due to the contribution of the crystallites oriented in such a way that their hard directions are nearly parallel to H. The singular point can be detected by observing the successive derivatives d~M/dH n. The shape of the singularity and the order of differentiation n at which it becomes apparent depends on the symmetry of the hard axis and on the ratios of the anisotropy constants. Asti and Rinaldi (1974a), besides giving explicit expressions for the singularity in the most important cases of both cubic and uniaxial symmetry, have determined general rules for obtaining the order of differentiation at which a discontinuity appears at the singular point, both for the cases of longitudinal and transverse susceptibilities. The principle allows precise and reliable measurements of the anisotropy fields using polycrystalline specimens. The most important case is that of uniaxial materials having hard directions in the basal plane; the singularity is observed in the derivative of the differential susceptibility, d2M/dH 2, and has the shape of a cusp on the left-hand side (H < HA) of the peak at H A. Usually, SPD measurements are
458
G. A S T I
easily performed in pulsed fields by direct differentiation of the signal from the pick-up coil. However, also continuous-field techniques can be used by computing the derivatives of the measured M(H) curve (Obradors et al. 1984), or by applying single or double modulation, or triangular waves. The grain size has no effect, while the grain orientation distribution only affects the amplitude of the observed peak, in a way similar to X-ray diffraction in powders. Indeed, this inherent sampling characteristic of the technique, to select the crystallites oriented with their hard axes nearly parallel to H, allows studying distributions of anisotropy fields, and even to determine the angular distribution function of the crystallites (Asti 1987), an application that could be of interest for characterization of permanent magnets and for the study of coercivity mechanisms. The detection of the singularity in the transverse susceptibility configuration is difficult because domain-wall displacement is superimposed to the reversible rotation process for H < H A. However, this high sensitivity to domain walls offers a convenient method for the study and characterization of single-domain particles, as shown recently by Pareti and Turilli (1987). SPD measurements have been performed on a wide variety of hard magnetic materials, such as ferrites, various types of alloys such as MnA1, PtCo, CoCr, and rare earth intermetallic compounds. In all cases, the SPD technique proved to be particularly convenient for extensive studies on solid solutions, on the effects of additives and various chemical modifications, as well as on materials obtained via melt spinning and similar methods. An interesting case is that of the system Zr(Fel_xAlx) 2 which is hexagonal over a wide range of x values (Hilscher and Gr6ssinger 1980). Among the most recent applications are measurements of anisotropy fields made on new classes of rare earth intermetallics, i.e., on Nd2Fe14B with various substitutions (Asti et al. 1987, Pareti et al. 1988, Pareti 1988) and on SmFe,,Ti and similar compounds (Li et al. 1988). Also worth noting is the case of NdCos, for which SPD measurements have been performed of anisotropy fields relative to the c axis as well as to in-plane anisotropy, in the whole range of temperatures where the compound changes its easy direction, from axis to cone and from cone to plane (Marusi 1988). When applied to materials that exhibit FOMP transitions, the SPD technique provides exact determinations of the critical field Her. In fact, it was demonstrated, in the case of uniaxial materials (Asti 1981, Asti and Bolzoni 1985), that Her increases continuously if the crystal is rotated out of the exact orientation, e.g., the c axis parallel (A-case) or perpendicular (P-case) to the magnetic field. As a consequence, the magnetization curve of a polycrystalline aggregate shows a singularity exactly located at Her (see fig. 21). For a random orientation, in the case of a P-type FOMP, there is a discontinuity in the differential susceptibility, dM/dH, equal to, A X = (Ms/Hcr)(m
2
ml)2/(~/-]
- m 1
(4.1)
where m I and m e are the critical magnetizations as defined in section 2.2 [(eq. (2.9)]. In the case of A-type FOMP the discontinuity is in the slope of the
F I R S T - O R D E R M A G N E T I C PROCESSES
459
M/Ms
Her
H
Fig. 21. (Top) Magnetization curve M(H) of an easy axis uniaxial single crystal at various orientations close to ~p = 90 °, where q~ is the angle of magnetic-field direction with respect to the c axis. The F O M P transition is progressively displaced to higher fields with decreasing ~. From left to right the various curves in the figure refer to ~p = 90 °, 86 °, 82 °, 78 °, 74 ° and 70 °. (Bottom) M(H) and first derivative dM/dH for a polycrystalline aggregate having crystallites oriented at random.
differential susceptibility, d 2 M / d H 2, and turns out to be, A X ' = ( M J H c2~ ) ( m 2 - m 1)3/(W~- _ m 2I - wry-_ m22)2 .
(4.2)
For the P-type case Asti and Bolzoni (1985) calculated the slope of AX just above the critical field, i.e., d A x / d H , a quantity that is related to the peak width appearing in d M / d H at H = Her. Perfect agreement was found with computed curves based on an extended Stoner-Wohlfarth model that includes anisotropy terms up to sixth order and arbitrary distribution functions for crystallite orientation. Figure 22 shows an example of a computed curve M ( H ) simulating the case of an oriented polycrystalline sample of PrCo s (Asti et al. 1980). The computed curve turns out to be in perfect agreement with the experimental curve obtained by a pulsed-field experiment at a temperature of 78 K. For comparison the theoretical curve for a single crystal is also shown. The accurate measurement of Her by the SPD technique, together with the value estimated for M~r, allowed precise determinations of the ratios x = K 2 / K 1 and y = K3/K1, through c o m parison with the phase diagram of FOMP (see fig. 4). Above 155 K, the FOMP vanishes and the same technique gave for the same sample the anisotropy field, H A = 2(K, + 2K 2 + 3 K 3 ) / M ~. As regards the determination of the anisotropy constants K 1, K 2 and K 3 from the magnetization curve M ( H ) in oriented polycrystalline materials, there is another remarkable phenomenon, described below, that is important to point out, because it gives strong support for a suitable best-fit procedure. In fact, it is found (Asti et al. 1980) by computer simulations with the above-mentioned extended
460
G. ASTI
M
s
..:-=---.-----"
~
me r Hcr
;
0
;
;
I
6
I
I
I
I
12
I
I
I
I
18
I
;
;
"-
-"
H (Tesla)
Fig. 22. Computed magnetization curve M(H) based on an extended Stoner-Wohlfarth model that includes anisotropy terms up to sixth order and arbitrary distribution function for crystallite orientation. The case shown in the figure is a simulation of an oriented polycrystalline specimen of PrCo 5, after Asti et al. (1980). Solid line: computed curve, dashed line: single-crystal curve with magnetic field in the basal plane, (a) experimental points.
Stoner-Wohlfarth model, that the magnetization curve of an oriented polycrystalline material is reproduced very well by the single-crystal curve oriented at an angle p, which represents the average angle of misalignment of the crystallites. The coincidence of the two curves is surprisingly good up to values of M near saturation, and for ~p of the order of 10-20 °, which means that it is insensitive to changes both of the type and of the width of the angular distribution function of the crystallites. As a consequence, it is possible to use a very simple and rapid best-fit program based on the analytical expression of M(H) for a tilted single crystal, in which the angle enters as a further adjustable parameter just as K1, K 2 and K 3. Even when there is a FOMP, this procedure can be conveniently used in the part of the curve below Her. Besides PrCos, there are other cases where SPD techniques proved to be very effective in giving evidence of the existence of FOMP. Figure 23 shows the progressive change with decreasing temperature of the SPD signal, d2M/dH 2, from a magnet of Nd2Fe14B. As is evident, the downward cusp located at H A is modified until, at 200 K, a sharp positive peak appears that grows more and more with further decrease of the temperature. Clearly, the positive peak is an approximation to the delta function representing the derivative of the discontinuity in the differential susceptibility given by eq. (4.1). Other examples are intermetallic compounds of ThMn12 structure (Li et al. 1988, Deriu et al. 1989, Solzi et al. 1988). Finally, it is worth noting that there is a surprising resemblance between the shape of the SPD peak in d2M/dH 2 for a random polycrystal of a uniaxial easy-axis ferromagnet, and the thickness, 6(H), of a domain wall in a magnetic field perpendicular to the c axis and opposite to its magnetic moment. Expressions for 8 have been obtained by Drring (1966) and Wasilewski (1973), and fig. 24 shows the graphical representation of 8. By using the energy density criterion explained in section 2.6, it is very easy to deduce the expression for 8. In fact, if we assume that all anisotropy constants of higher than second order are vanishing,
FIRST-ORDER MAGNETIC PROCESSES
I
d=M/dt2
f
' T - 290
461
K
l
f
.~
184
0
I
50
100
H(KOe)-~-
Fig. 23. Progressive change with decreasing temperature of the SPD signal,
L
dEM/dH 2, for a magnet of
Nd2Fe14B. Evidently, the downward cusp located at H = H A is modified until, at 200K, a sharp positive peak appears that grows continuously.
E i
i
i
i
I
i
I
~1.4 1.2 1 i
0
i
4
i
6
H(Koe)
I
12
Fig. 24. The thickness, 3(H), of a domain wall of a uniaxial easy-axis ferromagnet, under the action of a magnetic field perpendicular to the c axis and opposite to its magnetic moment [see eqs. (4.4) and (4.5)], after Wasilewski (1973).
we have,
A(dO/do')~ = K , ( 1 - sin 2 OH) + HMs(1 + sin OH),
(4.3)
w h e r e OH is the equilibrium angle within the domains. R e m b e r i n g that h = HMs/2K ~ and sin 0H = h, f r o m the usual definition of the wall thickness, we obtain, 3 = (7r + 2 0 H ) / ( d 0 / d r r ) = (A/K~)I/2(~r + 20H)/(1 + h ) ,
for h < 1 ,
(4.4)
462
G. ASTI
and
= (A/K1)~/2~'/V~,
for h > 1
(4.5)
Expressions (4.4) and (4.5) are in agreement with the expressions given by Wasilewski (1973). References Acquarone, M., 1981, Phys. Rev. B 24, 3847. Acquarone, M., and G. Asti, 1975, J. Magn. & Magn. Mater. 1, 48. Albanese, G., G. Asti, M. Carbucicchio, A. Deriu and S. Rinaldi, 1975, Appl. Phys. 7, 227. Aleshko-Ozhewskii, O.P., R.A. Sizov, I.I. Yamzin and V.A. Lubimtsev, 1969, Sov. Phys.-JETP 28, 425. Andreyev, A.V., A.V. Deryagin, N.V. Kudrevatykh, V.A. Reimer and S.V. Terentiev, 1985, JETP 90, 1042. Asti, G., 1981, IEEE Trans. Magn. MAG-17, 2630. Asti, G., 1987, in: Proc. 5th Int. Symp. on Magnetic Anisotropy and Coercivity in RETM Alloys, Bad Soden 1987, eds C. Herget, H. Kronmuller and R. Poerschke (DPGGMBH, D-5340 Bad Honnef 1, FRG) p. 1. Asti, G., and F. Bolzoni, 1980, J. Magn. & Magn. Mater. 20, 29. Asti, G., and F. Bolzoni, 1985, J. Appl. Phys. 58, 1924. Asti, G., and A. Deriu, 1982, in: Proc. 3rd Symp. on Magnetic Anisotropy and Coercivity in RE-TM Alloys, Vienna 1982, ed. J. Fidler (Technical University of Vienna, Vienna) p. 525. Asti, G., and S. Rinaldi, 1974a, J. Appl. Phys. 45, 3600. Asti, G., and S. Rinaldi, 1974b, in: Proc. 3rd Eur. Conf. on Hard Magnetic Materials, Amsterdam 1974, ed. H. Zijlstra (Bond voor Materialenkennis, Den Haag, The Netherlands) p. 302. Asti, G., and S. Rinaldi, 1977, AIP Conf. Proc. 34, 214. Asti, G., F. Bolzoni, F. Licci and M. Canali, 1978, IEEE Trans. Magn. MAG-14, 883. Asti, G., F. Bolzoni, F. Leccabue, R. Panizzieri, L. Pareti and S. Rinaldi, 1980, J. Magn. & Magn. Mater. 15-18, 561. Asti, G., F. Bolzoni, F. Leccabue, L. Pareti
and R. Panizzieri, 1984, Nd-Fe Permanent Magnets - Their Present and Future Applications, Rep. and Proc. of a Workshop Meeting, Brussels, October 25, 1984, ed. I.V. Mitchel (Commission of the European Community) p. 161. Asti, G., F. Bolzoni and L. Pareti, 1987, IEEE Trans. Magn. MAG-23, 2521. Atzimony, U., M. Dariel, E. Bauminger, D. Lebenbaum, I. Novik and S. Ofer, 1973, Phys. Rev. B 7, 4220. Barbara, B., M.F. Rossignol, H.G. Purwins and E. Walker, 1978, J. Phys. (USA) C 77, L183. Bickford Jr, L.R., 1960, Phys. Rev., Suppl. llS, 1000. Birss, R.R., G.R. Evans and D.J. Martin, 1977, Physica B 86-88, 1371. Bolzoni, F., and L. Pareti, 1984, J. Magn. & Magn. Mater. 42, 44. Bolzoni, F., O. Moze and L. Pareti, 1987, J. Appl. Phys. 62, 615. Boucher, B., R. Buhl and M. Perrin, 1970, J. Phys. & Chem. Solids 31, 2251. Bozorth, R.M., 1936, Phys. Rev. 50, 1076. Buschow, K.H.J., 1980, Rare earth compounds, ch. 4, in: Ferromagnetic Materials, Vol. 1, ed. E.P. Wohlfarth (North Holland, Amsterdam) p. 299. Buschow, K.H.J., 1986, Mat. Sci. Rep. 1, 1. Callen, H.B., and E. Callen, 1966, J. Phys. & Chem. Solids 27, 1271. Casimir, H.B., J. Smit, U. Enz, J.F. Fast, H.P.J. Wijn, E.W. Gorter, A.J.W. Duyvesteyn, J.D. Fast and J.J. de Jong, 1959, J. Phys. Radium 20, 360. Clark, A.E., and E. Callen, 1968, J. Appl. Phys. 39, 5972. Clausen, K.N., and B. Lebech, 1980, J. Magn. & Magn. Mater. 15-18, 347. Cullen, J.R., 1981, J. Appl. Phys. 52, 2038. Cullen, J.R., and E. Callen, 1984, J. AppI. Phys. 55, 2426.
FIRST-ORDER MAGNETIC PROCESSES Cullen, J.R., and E. Callen, 1985, Physica B 130, 289. Deriu, A., G. Leo, O. Moze, L. Pareti, M. Solzi and R.H. Xue, 1989, Hyperfine Interactions, 45, 241. Dillon Jr, J.F., 1963, Domains and domain walls, ch. 9, in: Magnetism, Vol. 3, eds G.T. Rado and H. Suhl (Academic Press, New York) p. 415. D6ring, W., 1966, Mikromagnetismus, in: Encyclopedia of Physics, ed. S. Flfigge, Vol. XVIII/2, Ferromagnetismus, ed. H.P.J. Wijn (Springer, Berlin) p. 405. Eibler, R., R. Gr6ssinger, G. Hilscher, H.R. Kirchmayr, O. Mayerhofer, H. Sassik, X.K. Sun and G. Wiesinger, 1984, Nd-Fe Permanent Magnets - Their Present and Future Applications, Rep. and Proc. of a Workshop Meeting, Brussels, October 25, 1984, ed. I.V. Mitchell (Commission of the European Community) p. 167. Enz, U., 1961, J. Appl. Phys., Suppl. 32, 22S. Ermolenko, A.S., 1976, IEEE Trans. Magn. MAG-12, 992. Ermolenko, A.S., 1979, IEEE Trans. Magn. MAG-15, 1765. Ermolenko, A.S., 1982, in: Proc. 3rd Symp. on Magnetic Anisotropy and Coercivity in RETM Alloys, Vienna 1982, ed. J. Fidler (Technical University of Vienna, Vienna) p. 771. Ermolenko, A.S., and A.F. Rozhda, 1978, IEEE Trans. Magn. MAG-14, 676. Forlani, F., and N. Minnaja, 1969, J. Appl. Phys. 40, 1092. Graetsch, H., F. Haberey, R. Leckebusch, M.S. Rosenberg and K. Sahl, 1984, IEEE Trans. Magn. MAG-20, 495. Gr6ssinger, R., 1982, J. Phys. (USA) D 15, 1545. Gr6ssinger, R., and J. Liedl, 1981, IEEE Trans. Magn. MAG-17, 3005. Gr6ssinger, R., P. Obitsch, X.K. Sun, R. Eibler, H.R. Kirchmayr, F. Rothwarf and H. Sassik, 1984, Mater. Lett. 2, 539. Gr6ssinger, R., X.K. Sun, R. Eibler, K.H.J. Buschow and H.R. Kirchmayr, 1985, J. Phys. (France), Colloq. 46(6), C6-221. Hathaway, K.B., and G.A. Prinz, 1981, Phys. Rev. Lett. 47, 1761. Hilscher, G., and R. Gr6ssinger, 1980, J. Magn. & Magn. Mater. 15-18, 1189. Hiroyoshi, H., H. Kato, M. Yamada, N. Saito, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, Solid State Commun. 62, 475.
463
Huang, Ying-Kai, C.H. Wu, Y.C. Chuang, FuMing Yang and F.R. de Boer, 1987, J. Less-Common Met. 132, 317. Kajiwara, S., G. Kido, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, J. Phys. Soc. Jpn. 58, 829. Kazakov, A.A., and N.A. Litvinenko, 1978, Fiz. Met. & Metallloved. 45, 940. Khan, W.I., and D. Melville, 1983, J. Magn. & Magn. Mater. 36, 265. Kido, G., S. Kajiwara, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, IEEE Trans. Magn. MAG-23, 3107. Koroleva, L.I., and L.P. Mitina, 1971, Phys. Status Solidi A 5, K55. Krause, D., 1964, Phys. Status Solidi 6, 125. Kudrevatykh, N.V., E.W. Lee and D. Melville, 1986, Fiz. Met. & Metalloved. 61, 898. Li, H.S., B.P. Hu, J.P. Gavigan, J.M.D. Coey, L. Pareti and O. Moze, 1988, J. Phys. (France) 49, 541. Lotgering, F., 1974, J. Phys. & Chem. Solids 35, 1633. Lotgering, F., P.R. Locher and R.P. van Stapele, 1980, J. Phys. & Chem. Solids 41, 481. Marusi, G., 1988, Thesis (University of Parma). Melville, D., W.I. Khan and S. Rinaldi, 1976, IEEE Trans, Magn. MAG-12, 1012. Melville, D., K.M. A1 Rawi and W.I. Khan, 1981, Phys. Status Solidi A 66, 133. Melville, D., J.M. Machado da Silva and J.F.D. Montenegro, 1987, Port. Phys. 18, 49. Meyer, A.J., 1964, C.R. Acad. Sci. 258, 4935. Mita, M., and N. Momozava, 1975, AIP Conf. Proc. 24, 406. Mitsek, A.I., N.P. Kolmakova and D.I. Sirota, 1974, Fiz. Met. & Metalloved. 38, 35. Morrison, B.R., 1973, Phys. Status Solidi B 59, 551. Mukamel, D., E.M. Fisher and E. Domany, 1976, Phys. Rev. Lett. 37, 565. Nagamiya, T., 1967, Solid State Phys. 20, 305. Namtalishvili, M.I., O.P. Aleshko-Ozhewskii and I.I. Yamzin, 1972, Sov. Phys.-JETP 35, 370. Obradors, X., A. Collomb, M. Pernet, J.C. Joubert and A. Isalgue, 1984, J. Magn. & Magn. Mater. 44, 118. Odozynski, R., and W.J. Zietek, 1977, Physica B 86-88, 1373. Paoluzi, A., F. Licci, O. Moze and G. Turilli, 1988, J. Appl. Phys. 63, 5074.
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Pareti, L., 1988, in: Proc. Int. Conf. on Magnetics, Paris 1988. Pareti, L., and G. Turilli, 1987, J. Appl. Phys. 61, 5098. Pareti, L., F. Bolzoni and O. Moze, 1985a, Phys. Rev. 32, 7604. Pareti, L., H. Szymczak and H.K. Lachowicz, 1985b, Phys. Status Solidi a 92, K65. Pareti, L., O. Moze, M. Solzi, F. Bolzoni, G. Asti and G. Marusi, 1988, Final C.E.A.M. Meeting, Madrid, April 12-16, 1988. Radwanski, R.J., 1986, Physica B 142, 57. Radwanski, R.J., J.J.M. Franse and S. Sinnema, 1985, J. Phys. (USA) F 15, 969. Rebouillat, J.E, 1971, J. Phys. (France) 32, C1-547. Rinaldi, S., and L. Pareti, 1979, J. Appl. Phys. 50, 7719. Rozenfeld, Y.V., 1978, Fiz. Met. & Metalloved. 45, 911. Sannikov, D.G., and T.M. Perekalina, 1969, Sov. Phys. JETP 29, 396. Sarkis, A., and E. Callen, 1982, Phys. Rev. 26, 3870. Shanley, C.W., and S. Harmer, 1973, AIP Conf. Proc. 18, 1217. Sinnema, S., 1988, Doctoral Thesis (Amsterdam University, Amsterdam). Sinnema, S., J.J.M. Franse, A. Menovsky, F.R. de Boer and R,J. Radwanski, 1987,
3rd Conf. on Phys. of Magnetic Materials, Szczyrk-Billa 1986 (World Scientific, Singapore) p. 324. Sizov, R.A., and K.N. Zaitsev, 1974, Sov. Phys.-JETP 39, 175. Slonczewski, J.C., 1961, J. Appl. Phys., Suppl. 32, 253S. Smit, J., and H.EJ. Wijn, 1959, Ferrites (Philips Technical Laboratory, Eindhoven). Solzi, M., L. Pareti, O. Moze and W.I.F. David, 1988, J.Appl. Phys. 64, 5084. Sugimoto, M., 1982, Properties of ferroxplanatype hexagonal ferrites, ch. 6, in: Ferromagnetic Materials, Vol. 3, ed. E.P. Wohlfarth (North Holland, Amsterdam) p. 408. Tatsumoto, E., T. Okamoto, H. Fujii and C. Inoue, 1971, J. Phys. (France), Colloq. 32, C1-550. Tiesong Zhao, and Jin Hanmin, 1987, Solid State Commun. 64, 103. Wasilewski, W., 1973, Acta Phys. Pol. A 43, 729. Wohlfarth, E.P., 1980, J. Magn. & Magn. Mater. 20, 77. Wohlfarth, E.E, 1983, First order magnetic transitions in some metallic materials, in: High Field Magnetism, ed. M. Date (North Holland, Amsterdam) p. 69. Yamashita, N., 1972, J. Phys. Soc. Jpn. 32, 610.
G. ASTI Physics Department University of Parma 43100 Parma Italy
Ferromagnetic Materials, Vol. 5 Edited by K.H.J. Buschow and E.P. Wohlfartht © Elsevier Science Publishers B.V., 1990
CONTENTS 1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Phenomenology of first-order magnetization processes in terms of anisotropy constants. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Uniaxial symmetry . . . . . . . . . . . . . . . . . . . . . . . 2.3. Cases of trigonal, hexagonal and tetragonal symmetry . . . . . . . . . . . 2.4. Cubic symmetry . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Mechanisms responsible for high-order anisotropy constants . . . . . . . . . 2.6. Domain wall processes at the transition point . . . . . . . . . . . . . . 3. Processes involving competition between exchange and anisotropy . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Role of exchange at transition points . . . . . . . . . . . . . . . . . 3.2.1. Multisublattice magnetic systems with uniaxial anisotropy . . . . . . . . 3.2.2. Field-induced spin reorientation transitions. Critical fields . . . . . . . . 3.2.3. Boundary conditions for first-order transitions . . . . . . . . . . . 3.2.4. The two-sublattice system . . . . . . . . . . . . . . . . . . 3.2.5. Linear regime and magnetic transitions in the canted phase . . . . . . . 3.3. The small-angle canting model . . . . . . . . . . . . . . . . . . . 3.3.1. General remarks . . . . . . . . . . . . . . . . . . . . . 3.3.2. The two-sublattice model . . . . . . . . . . . . . . . . . . 3.3.3. Some applications . . . . . . . . . . . . . . . . . . . . . 3.4. Hexagonal ferrites having block-angled and spiral structures. Role of antisymmetric exchange . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Considerations on the experimental methods . . . . . . . . . . . . . . . . 4.1. Measurements in high magnetic fields . . . . . . . . . . . . . . . . 4.2. The singular-point detection technique . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
398
399 402 402 402 414 419 421 423 427 427 428 428 430 432 433 440 447 447 448 453 454 455 455 457 462
1. General
The magnetization curve of a magnetically ordered crystal, ferro- or ferrimagnetic, can, under special circumstances, show anomalous discontinuities that appear as sudden increases of the magnetization with increasing magnetic field. In these conditions, the crystal actually undergoes a first-order magnetic transition, with characteristics that are, in general, strongly dependent on the relative orientation of the applied magnetic field direction and the crystal axes. The observed phenomenon is an irreversible process of a different nature than the ordinary magnetization reversal, which always occurs when the applied magnetic field intensity is equal to the coercive field; a fact that is reflected in the well-known symmetry of the hysteresis loop of all ferromagnets (by the general term 'ferromagnet', we shall refer hereafter to both types of material, properly called ferromagnetic or ferrimagnetic, when the magnetic order is considered to be irrelevant for the phenomenon under study). These anomalous magnetization jumps, being first-order transitions, also ought to be accompanied by hysteresis loops, which would essentially be localized in the first quadrant of the magnetization curve, but, in general, the hysteresis around the transition happens to be very small and can only be revealed with difficulty. The observed jump in the magnetization curve can be due to a rotation of the magnetization vector, to a change of the modulus of the magnetization vectors, or, in complex systems, to a change of the magnetic order. In the first case, we deal with changes in the magnetic state of the system that can effectively be ascribed to its magnetic anisotropy, while in the second case, we are, in most cases, faced with direct manifestations of microscopic effects, such as metamagnetic transitions that can be the consequences of particular crystal-field effects, and/or the effect of the magnetic field on the energy-band structure. The third case involves effects of the type of spin-flop transitions, with drastic changes in magnetic structures. In the present chapter we will not consider the second type of transitions, which has been treated in the reviews by Wohlfarth (1980, 1983), Buschow (1980) and Khan and Melville (1983). Some aspects of the third type of phenomena, i.e., the case of anisotropic ferrimagnets, will be examined in section 3.2. In section 2 we deal with the first type of field-induced first-order transition, hereafter referred to as FOMP [first-order magnetization process (Asti and Bolzoni 1980)]. In this case, 399
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G. ASTI
the magnetic anisotropy of the system by itself allows the existence of two inequivalent minima of the free energy that correspond to two distinct directions of the magnetization vector. The effect of the external field is to produce a relative shift between the energies of the two states involved in the transition. When the two states reach the same energy the transition can start, provided a mechanism exists that allows the nucleation and growth of the new phase. It is likely that this nucleation occurs inside the domain wall (see section 2.6). When the changing parameter is temperature, the system can make the transition in zero external field, making a sudden rotation of the easy-axis direction. This type of transition is usually referred to as a spin reorientation transition (SRT), and is essentially the same phenomenon as FOMP. In fact, in two-dimensional phase diagrams, energy versus H or T, the transition is represented by lines that in general can be crossed by changing either T or H. FOMPs in cubic crystals have been considered in the frame of the problem of phase transitions and critical phenomena (Mukamel et al. 1976, Cullen and Callen 1984) as examples of the Potts model (see section 2.4). However, the widest application of the concepts developed around the FOMP transition is indeed in studies of the anisotropic properties of multisublattice magnetic systems, especially rare earth (R) intermetallic compounds. In practice, it is very common to find this type of behaviour in these materials. The study of FOMP and the measurement of critical parameters, on one hand, provide a great deal of information about the magnetic anisotropy and, on the other hand, the interpretation of observed FOMPs represents a formidable test for a theory of magnetic anisotropy in these systems. Moreover, one has to consider the inherent precision and proper reliability of data about critical parameters, as compared to best fit approach to saturation and similar procedures. A phenomenological description of FOMP in terms of the anisotropy constants or the anisotropy coefficients shows that these transitions require high-order terms in the expansion of the free energy. Crystal-field theory applied to single ions does not in many cases justify the existence of such high-order terms. This fact by itself indicates how important it is to take the competing interactions typical of multisublattice systems into account, as, e.g., the competition between anisotropy of different sublattices (Cullen and Callen 1985). In this respect, the exchange interaction also plays an important role, especially for the case of ferrimagnetic order (Sarkis and Callen 1982). Indeed, the presence in R intermetallic compounds of exchange of the same order of magnitude as the anisotropy energy implies deviations from the collinear magnetic order. Even for small canting angles, the free energy of the system is strongly modified, and the macroscopic anisotropy properties are influenced in various ways. Hence, the classical phenomenological description is inadequate, sometimes leading to wrong conclusions. For instance, non-monotonic dependences of the anisotropy constant in pseudobinaries, such as (Y, Nd)Cos, (Y, Gd)Cos, and (Sm,Nd)Cos, have erroneously been considered as evidence of the non single-ion origin of the anisotropy energy (Ermolenko 1979). A convenient approach to the problem, which applies to the majority of cases, is to treat the
FIRST-ORDER MAGNETICPROCESSES
401
small-angle canting (SAC) between sublattice moments as a first-order perturbation (Rinaldi and Pareti 1979). The advantage of this model (hereafter referred as the SAC model) is that the system can again in many respects be treated as an ordinary ferromagnet having certain effective anisotropy constants K1, g2,.... It means that the effect of canting of the sublattice moments can be merely accounted for by using proper expressions for the effective anisotropy constants; these include the exchange-interaction parameters and reduce to the ordinary algebraic addition of the anisotropy constants of the individual sublattices when the anisotropy is negligible with respect to the exchange-energy density, a condition that corresponds to normal collinear magnetic order. Actually, the K's obtained are not rigorously constant because they depend on the intensity of the magnetic field, but the model in many cases allows to get a simple physical picture of the mechanisms responsible for the observed phenomena. As a matter of fact, it gives an explanation of the existence in these intermetallic compounds of effective anisotropy constants of high order. Certainly, canting effects play a role in the presence of FOMP and non-linear effects in the magnetization curves. As mentioned above, at high magnetic fields, pronounced transitions of the spin-flop type are possible for ferrimagnetically ordered RE intermetaUic compounds. They most frequently involve direct competition between magnetostatic energy and exchange. With increasing fields, magnetic configurations with large angles between the magnetic moments can be energetically favorable. A particular type of first-order transition of this category, named first-order moment reorientations (FOMR), was observed by Radwanski et al. (1985) and extensively studied in easy-plane ferrimagnetic compounds of the R2T17 type (here T is a transition metal). In this case, the applied field indeed lies in the basal plane and the six-fold symmetry plays a role in allowing non-collinear moment configurations to become energetically favorable. Computed curves for ferrimagnetic systems with magnetic anisotropy, presenting transitions between different magnetic configurations have also been described by other authors (Sannikov and Perekalina 1969, Kazakov and Litvinenko 1978, Rozenfeld 1978, Gr6ssinger and Liedl 1981, Ermolenko 1982, Radwanski 1986). A certain analogy exists with the phase diagrams of antiferromagnets that show spin-flop transitions. Classic treatments on angled spin structures but without anisotropy (Clark and Callen 1968), show continuous transitions between various structures. Among the peculiar effects found, are a temperature-independent susceptibility (Clark and Callen 1968) and an effective decoupling between the sublattices in the canted phase (Meyer 1964, Boucher et al. 1970, Acquarone and Asti 1975). A theoretical model (Acquarone 1981) of an antiferromagnetic chain including anisotropic and antisymmetric exchange, as well as anisotropy terms up to sixth order, explains the observed phenomena in the so-called 'angled block structures' in complex ferrimagnetic oxides of hexagonal and trigonal structure. Certain compositions show field-induced transitions and unusual effects, because the system behaves like an effective two-sublattice canted antiferromagnet, so as to give rise to various possible configurations: angled, conical helix or flat spiral.
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G. ASTI
2. Phenomenology of first-order magnetization processes in terms of anisotropy constants 2.1. Introduction
The anisotropy energy of a ferromagnetic crystal can be expressed as a power series of the direction cosines of the magnetic moment with respect to the crystal axes. So the expansion has different forms depending on the symmetry of the crystal. The coefficients of the various terms are the anisotropy constants and, in general, a satisfactory description of the magnetization curve is obtained by limiting the expansion to the first few terms. Under normal conditions, this curve is continuous in the region preceding saturation, where the magnetization process takes place by reversible rotation of the magnetization vector M s . However, when the anisotropy constants fall within certain ranges corresponding to some specific conditions, irreversible rotations of M S are possible, implying first-order transitions between inequivalent magnetization states. In fact, depending on the values of the anisotropy constants, the system may possess two or more inequivalent minima of free energy E for different magnetization directions; so it is evident that an applied magnetic field H of suitable intensity and orientation can induce transitions between these minima. Actually, the addition of a magnetostatic term modifies the total energy surface in such a way that the two energy levels become equal. The existence of two minima of E in zero field is not a necessary condition: the effect of the magnetostatic energy component is in certain circumstances enough to produce an additional minimum besides the absolute one (easy direction). In principle, in the case of F O M P we expect to observe minor hysteresis loops around the critical field, for which the two states have equal energy. However, it is difficult to observe hysteresis in F O M P even at low temperatures (Melville et al. 1976, Ermolenko and Rozhda 1978). The reasons probably lie in the very low coercivity inherent in this type of magnetization process (see section 2.6). 2.2. Uniaxial s y m m e t r y
A very special but important case is that of uniaxial symmetry. In this case the free energy of the system can be expressed as E -- K 1 sin e 0 + K 2 sin 4 0 + K 3 sin 6 0 + . . . .
H M s cos(0 - q0,
(2.1)
where K1, Kz,/(3,... , are the anisotropy constants and 0 and ~p are the orientation angles of the magnetization vector M s and the magnetic field H with respect to the symmetry axis c, respectively, As a first step, it is important to know the phase diagram of easy and hard directions. For reasons of symmetry, the c axis (0 = 0) and the basal plane (0 = ½It) are always points of extremum. With the anisotropy energy expanded up to the sixth order, we can have, at most,
FIRST-ORDER MAGNETIC PROCESSES
403
two additional extrema along conical directions at angles given by sin 0c = ( [ - K 2 ++-( K 2 - 3 K1K3) 1/2]/3K 3 }1/2.
(2.2)
Let C ÷ and C - be the cones associated with the + and - sign in eq. (2.2). It turns out that C ÷ is always a minimum while C - is always a maximum, so that only C ÷ can be an easy direction. A convenient representation of the diagram of the easy directions and the other extrema is representation in terms of reduced anisotropy constants x = K 2 / K 1 and y -- K 3 / K 1. These two variables are sufficient to distinguish all the possible cases. In fig. 1 the phase diagram for the two cases K i > 0 (upper) and K 1 < 0 (lower) is shown. All the information concerning the easy directions and the other extrema are contained in a special symbol that marks every different region. It simulates a polar type of energy representation indicating existing extrema by concave (minimum) and convex tips (maxima). Vertical and horizontal stems refer to the symmetry axis and the basal plane, respectively. The left-hand and right-hand oblique stems indicate the C - and C ÷ cones, respectively. The absolute minimum (easy direction) is indicated by filling
I
4
¥
L, ×
-2 Fig. 1. Magnetic phase diagram of the uniaxial ferromagnet: (Top) K~ > 0; (Bottom) K 1 < 0. T h e coordinates are x = Kz/K ~ and y = K3/K~. For the explanation of the symbols see text (Asti and Bolzoni 1980).
404
G. ASTI
of the tip. The boundary lines of the various regions in the phase diagram of fig. 1 are specified by the equations labelled by the same letters in table 1. The determination of the conditions for the existence of FOMP requires the analysis of the magnetization curve and its dependence of the values of the anisotropy constants, for different directions of the magnetic field. For most purposes it is enough to consider the cases of H parallel or perpendicular to the c axis, hereafter indicated as the A-case and P-case, respectively, where A denotes axial while P stands for planar. The general case is relevant when considering the effects of FOMP in polycrystalline materials (see section 4). Here, it suffices to mention that a peculiar phenomenon is only observed for H lying not in a symmetry direction; under certain conditions of anisotropy constants and field orientations, an additional transition is possible, so that one observes a double FOMP, consisting of an ordinary and extraordinary FOMP (Asti and Bolzoni 1985). The equilibrium equation is OE/O0 = 0, where E is given by eq. (2.1). For the P-case (~ = ½) it turns out to be, h = 2m(1 + 2 x m a + 3 y m 4 ) K 1 / l K l [ ,
(2.3)
where m = M / M s is the reduced magnetization, h = 2H/IHAII the reduced field and HA1 = 2K1/Ms, the c axis anisotropy field. Using eq. (2.3) and the condition that the total energy is equal in the two phases, we obtain the equation for the critical magnetization, (3m 2 + 2m + 1)x + (5m 4 + 4m 3 + 3 m 2 + 2m + 1)y + 1 = 0.
(2.4)
For the A-case, the equilibrium equations corresponding to eqs. (2.3) and (2.4) are h = - 2 m [ 1 + 2x(1 - m 2) + 3y(1 -
m2)2]KJ[KI(1+ 2x + 3y)[,
(1 + m)(1 - 3 m ) x + (1 + m)(1 - m2)(1 - 5 m ) y + 1 = 0,
and
(2.3') (2.4')
TABLE 1 Equations of the boundary lines of the various regions in the phase diagram (fig. 1) and in the F O M P diagram (figs. 3 and 4). The lines are labelled by a letter that appears in figs. 1 and 3. The couples of curves conjugated in the K ~ R transformation [eqs. (2.6)-(2.8)] appear on the same line in the table. Curve
Label
Conjugated curve
Label
3y+2x+l=0 x+y+l=0 15y + 6 x + 1 = 0
1 m n
K I = 0 , i.e., x+y+l=0 4x- 1 =0
1'
4y - x 2=0
o
3 y 2 + 2xy - x z + 4y =0
3y-
p
3y-x
q
3x z + 8xy + 12y z - 5y = 0
r
x4+
x 2 =0
5y - 3x z = 0
x,y---~
n' o'
2 =0 q'
Eq. (2.11) in the text
x3y - 8xay - 3 6 x y a -
27y 3 +
16y 2 = 0
r'
FIRST-ORDER MAGNETIC PROCESSES
405
where h = 2 H / I H A 2 ] with HA2 = -2(K~ + 2 K 2 + 3 K 3 ) / M ~ , the anisotropy field along the basal plane. There is a transformation, called the K*+ R transformation, of the anisotropy constants into conjugate quantities that allows to transfer immediately all the results obtained for the P-case to the A-case and vice versa, according to the following symmetrical dual correspondence,
basal plane
c axis
K1 K2 K3
(2.5)
R1 R2 R3
The K ~ R transformation is obtained by imposing on the energy expressions the same formal dependence on the reduced magnetization m, for the A and P cases. So one obtains the following linear transformation, RI=-Ka-2K2-3K
3,
R2=K2+3K
3,
3.
(2.6)
/£3 = - R 3 .
(2.7)
R3=-K
The inverse transformation obviously has the same form, K 1 = - R 1 - 2R 2 - 3R 3 ,
K2 = R 2 + 3R 3 ,
In terms of reduced variables it becomes, 2= (-x-
3y)/(1 + 2 x + 3 y ) ,
x = (-2-
337)/(1 + 22 + 337),
fi = y / ( 1 + 2 x +
3y), (2.8)
y = 37/(1 + 2£ + 337),
where 2 = R 2 / R 1 and 37= R 3 / R 1. One can easily recognize that the obtained correspondence is a projective transformation, i.e., a homology having the center in the point C ( - 1 , 0) and the axis being the line with equation 2x + 3y = 0, with an invariant equal to - 1 , i.e., the transformation is a harmonic or involutory homology. Accordingly, second-order curves are transformed into other secondorder curves. The correspondence for some important lines is reported in table 1. Two types of F O M P are distinguished, depending on the fact whether the final state after the transition is the saturation state (type-1 F O M P ) or not (type-2 FOMP). They are represented in fig. 2. From the degree and the symmetry of the equilibrium equations, it turns out that when the easy direction is coincident with a cone (cone C +) type-2 F O M P cannot occur. From the equality of the total energy in the two states and from the equilibrium equations, one obtains the conditions for the occurrence of the two types of FOMP. Figure 3 summarizes the results, giving the complete diagram of F O M P in the planes of reduced anisotropy constants x and y for the two cases K 1 > 0 and K 1 < 0. The various regions are
406
G. ASTI c-axis
'
(a)
.r--. ;
Ha2
H
(b)
(c)
Ib
Fig. ½. Examples of FOMP different types (right-hand side): (a) type 1. (b) type 1 with easy cone. (c) type 2. The diagrams on the left-hand side are cross-sections of the anisotropy-energy surface (Asti and Bolzoni 1980). The arrows on the vertical portion of the M(H) curve (full line) indicate that the transition occurs at the same critical field both with increasing and decreasing magnetic field, when magnetic states having equal free energy are involved. This is actually the most common case for experimentally observed FOMPs, because the hysteresis around the transition is either very narrow or even not detectable. Upward and downward arrows on the two sides indicate upper and lower limits for the critical field, located at the onset of instability. They define the maximum theoretical width of the hysteresis cycle connected with the FOMP. distinguished by the labels A 1 , A 2 , P1, P2, P1C, A 1 C which specify the m a g n e t i c field direction (A, axial; P, planar) and the type o f F O M P (1 and 2), and the easy-cone o f the regions with type-1 F O M P ( A 1 C , P1C). T h e b o u n d a r y lines are those specified in table 1. T h e areas that are c o n j u g a t e d in the K ~ R transformation are s h a d e d in the same way. T h e analysis of the equations also provides the values o f the critical field her and the critical m a g n e t i z a t i o n mcr as functions o f the c o o r d i n a t e s x, y a n d the sign of K 1. T h e critical magnetization is defined as the m a g n e t i z a t i o n at which the transition takes place. Figure 4 gives the same diagrams as in fig. 3, but n o w with the lines of constant h , and incr. T h e s e plots can be used in analyzing experimental observations o f F O M P , and allow precise
FIRST-ORDER MAGNETIC PROCESSES .....
~, :,.-':-,
4
~
-
.;:,(U:,-',-
407
"Y
-:','-'.- 2 . , ' - ' , , . " ~ - - ' ~
x
....,......,........... -"-'--',-" .......- , - , ~ , ~ - p ( ~ i ' .
.....
!iiii! ~
'
"
!~:!i!i!~:##::!!?!ii:~!ii?i!!!!: !!':::il;i!il}i ii!},
.
x
.......! :::::::::::::::::::::::
Fig. 3. Diagram of FOMP in the plane of reduced anisotropy constants x = K2/K1, y = K 3 / K x. Top and bottom figures refer to the cases Kx > 0 and K~ < 0, respectively. The different regions are distinguished by a letter, i.e., P for the P-case or A for the A-case (see text), followed by the numeral 1 or 2 indicating the type of FOME Type-1 FOMP in regions of easy cone are specified by the addition of the letter C. The boundary lines are those specified in table 1. The areas that are conjugated in the K ~ R transformation (see text) are shaded in the same way (Asti and Bolzoni 1980). e v a l u a t i o n s o f t h e r a t i o s o f t h e a n i s o t r o p y constan~ts. I t is w o r t h n o t i n g t h a t i n t h e case o f t y p e - 2 F O M P t h e s y s t e m o f e q u a t i o n s c a n b e a n a l y t i c a l l y s o l v e d g i v i n g e x p l i c i t e x p r e s s i o n s for her as well as mcr = m 1 a n d m2, t h e r e d u c e d m a g n e t i z a t i o n s for t h e i n i t i a l a n d final s t a t e s , r e s p e c t i v e l y . T h e s e a r e , for t h e P - c a s e , hc~ = ~ A { 2 - [17x 2 + x ( 6 0 y
-
llx2)1/2]/30y}, (2.9)
m x = ½(A-
D),
m 2 = ½(A + D ) ,
where A = {[-3x + (60y - llx2)~/2]/lOy} D = {[-5x - (60y - llx2)l/2]/6y}
~/2 , t/2
408
G. ASTI 6
-2
-4
¥
-6
- 4
-2
.3
Fig. 4. Lines of constant values for the reduced critical magnetization, mcr (dashed lines), and reduced critical magnetic field her (full lines), in the plane of reduced anisotropy constants x = K 2 / K 1 , y = K 3 / K ~. The various regions are denoted as in fig. 3. In the case of type-2 FOMP, mcr is the initial state, i.e., m 1. The spacing between the values relative to contiguous lines of mcr or her, changes only when crossing a line labelled by its value. (a) K1 > 0, x > 0, y < 0. (b) K 1 > 0, x < 0, y > 0. (c) K~ > 0, x<0, y<0.(d) Kl<0, x>0, y>0.(e) Kl<0, x>0, y<0.(f) K~<0, x<0, y>0(AstiandBolzoni 1980).
409
FIRST-ORDER MAGNETIC PROCESSES -~
X
-4
0
-2
(c)
e ___ ~--~-~_
--_
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . -
. . . . . . .
2
~
. . . .
4
(a) Fig. 4.
(Cont.)
. 3 s -
6
410
G. ASTI 2
4
6
~---<.~ ,+.<
-2
I
i
i
~
',
-..<)
-4, Y
(e)
~',
\~
\;
t
I /
,'/
/,,1¥
e""4 ~X" \ \ \
\\
\\
,
I
\\
i
., \\
I
,A2
\x
~\~~ \! \ '/'.
\\
,
A1
,
/I
//
I 111
'
II
~I
//
[ t /X,,'/ .J.-" 1.'/. ~..--"t=
- 7 \ / " -~
'~..~.'-,.~,~,7 ~7~ j ""-
~_.~_~
~~-~.~
J( -i
'
Fig.
(f) 4. (Cont.)
-~
'
0
FIRST-ORDER MAGNETIC PROCESSES
411
and, for the A-case, her = } A ' { 2 - (x + 3y)[17(x + 3y) + G l / [ 3 O y ( 1 + 2 x + 3y)]}, (2.10) m I = (A' - D')/2,
m 2 =
(A' + 0')/2,
where G = (60y - l l x 2 + 81y 2 + 5 4 x y ) 1/2 , D' = {[5(x + 3y) - G]/6y}
A' = ([3(x + 3y) + G l / l O y ) 1/2
1/~ .
If we impose the condition m 2 = 1, we find the boundary with the region of transitions of type 1 [line labelled 'r', see fig. 3 (top)] for the P case. The equation obtained is X 4 -- 5x3y + 61x2y 2 + 255xy 3 + 225y 4 -- 8x2y + 52xy 2 + 105y 3 + 16y 2 = 0 .
(2.11) After applying the R ~ K transformation it becomes the ' r " line [see fig. 3 (bottom) and table 1], which is relative to the A-case. Similar diagrams can also be obtained in terms of the anisotropy coefficients Y{Z,minstead of the constants K i. These coefficients apear in the expansion of the anisotropy energy in spherical harmonics. It is known, in fact, that there are advantages in using this representation when analyzing the t e m p e r a t u r e dependence of magnetocrystalline anisotropy, because of the congruence with the Callen and Callen (1966) theory. The relations between the two sets of p a r a m e ters are, K 1 = al ~'/2, 0 + bl ~'{4, 0 + cl if{6,0 + . . . , K2 =
(2.12)
b2•{4, 0 q- C2Y~'6, 0 "-[- . . . ,
K3 ~_~
c3Y{6,0 ql_ . . .
,
where aI
=
-V~
p ,
c2 = ~ X / ~ p ,
b a = -10p ,
cI = -7X/-~ p
c3 = - ~X/-~ p ,
b2 = ~p ,
,
p = 3/(4x/-~).
The inverse transformation is, ~/~'2,0 = a71K1 - b l ( a l b 2 ) - l K 2 ~/'4,0 = ffff6,0 =
b 21K2
+ ( b l C 2 - b2Cl)(alb2C3)-lK3 , - c 2(b 2c3)- 1K3,
c31K3 ,
(2.13)
412
G. ASTI
The FOMP diagrams in terms of the ratios u = Y{4,0/Y{2,0and v = ~{6,o/Y{2,o are reported in figs. 5 and 6, for the cases Y~2,0> 0 and Y{2,0< 0. The straight line labeled L' corresponds to K 1 = 0. Above this line, K 1 has a sign opposite to Y{2,0, and changes sign when crossing it. The same letters denoting the boundary lines in fig. 1 are used here in capitals to distinguish corresponding lines according to transformation (2.12). By the same transformation the FOMP diagrams of figs. 3 and 4 become those given in fig. 5 and 6, respectively. It is worth noting that the diagrams in the plane ff(2,o > 0 are symmetric to those in the plane ~Lr2,0< 0 in a projective sense, i.e., including the points at infinity and considering continuity through it; a property that is absent in the diagrams in terms of the K's. It is also of interest to consider a typical trajectory of the representative point at low temperatures as predicted by the ll(l + 1) power law according to the Callen and Callen (1966) theory. This yields u x m y and v ~ m is, so the line described by the
':!:!:i: i:!:i:i:!!!:i:i:!:i:i:ii!!!i!ii:!
:::!:!:::::::::: li~:}:ii ::~::i::~i!~:!~ P1C
ci!i!iiiii!iiiiiiiiii! ';'iiii!!ii;iiii iiiii?iii!i'oJ;:ii:iiiiii!!iii:!i:i:!i!ii::ii!i
:::::::::::::::::::::::::::::::::::::::::::::::::
;i;!S;~;{~ttt-~t::~:~':-:':-:-:-:-:':':-:': 'Ic !!ii!!iiiiii!!!iiii!iiii1 ~ ;-:-:-:-:-:-:-:':-:-:':':-:':~ v!ii i!i i!!i!i i i i i !i i ...F,2
i::i~ii!ii:!-ii!i!i~iii~Lo'::::::::::::::::::: | :::::::::::::::::::::::::::::: i.:;::P_t ,
o
~--..."::,.,:
Fig. 5. Diagram of F O M P in the plane of reduced anisotropy coefficients u = Y{n.01Y{z,0, v = Y{6,0/Sff2,0. Top and bottom figures refer to the cases Y{2,0> 0 and ~2,0 < 0, respectively. The different regions are distinguished as in fig. 3. The same letters denoting the boundary lines in fig. 1 are used here in capitals, to distinguish corresponding lines according to the transformation of eq. (2.12). The areas that correspond (according to the same transformation) to the conjugated areas represented in fig. 3 are shaded in the same way (Asti and Bolzoni 1980).
FIRST-ORDER
MAGNETIC
.... 06. . . . . . . .
A2
er-'O
\-,\
' ~
,
i
PROCESSES
I_~~--,
",
, ___2_.
-
413
f,,r.,-" PlC~
~
.
--l-
"-._
-..
~.'~
\
~-.%,
\
•o,,.-.~,~, \
A'iC ~ \ -08. `
~xx
~ / •
,
/
,, / , ,
\ , // ~
II
-
,,' .,~
.4//."" y/
//"" .-
,
,>..oo~,
(,IA2 ~,,oo4~._.-4 3
5
t'ig. 6. Lines of constant values for the reduced critical magnetization mot (dashed lines), and reduced critical magnetic field hot (full lines), in the planes of reduced anisotropy coefficients u = 5(4,0/5(2,0 and v = ~6,0/5(2,0 . T h e different regions and the values relative to the various lines of mcr and her are distinguished as in fig. 4. (a) and (b) figures refer- to the cases 5(2.0 > 0 and 5(2.0 < 0, respectively. T h e inset of (b) shows a part of the plane 5(2,0 < 0, with an expanded scale and with the origin (u = v = 0) in the same point. T h e curve . . . . . . . is an example of the ½1(l + 1) power law for the t e m p e r a t u r e dependence of the anisotropy. T h e equation of the curve is o = c u 1817 with c = 1 (Asti and Bolzoni 1980).
414
G. ASTI
representative point is a curve of the type v = CU18/7 or V = c u 2"57, with c constant. Figure 6a shows a typical curve for c = 1. It is worth noting that an analogy exists between the expression for the anisotropy energy [see eq. (2.1)] and the Landau free-energy expansion in the magnetization M, i.e., F = 1 A M z + ¼BM 4 + 1 C M 6 - H M .
This similarity was used by Khan and Melville (1983) to construct a complete set of phase diagrams for zero field and metamagnetic transitions from paramagnetic to ordered state. In particular, the critical magnetic field and specific magnetizations corresponding to first-order magnetic processes were determined. 2.3. Cases o f trigonal, hexagonal a n d tetragonal s y m m e t r y
Numerous ferrites of the hexagonal system may have either hexagonal or rhombohedral crystal structures, depending on whether there is or there is not a symmetry plane perpendicular to the three-fold axis. In this system of compounds, cobalt ions have been found to have strong effects on magnetic properties, introducing high-order anisotropy constants that favor, in general, easymagnetization directions in or near the basal plane, and spin-reorientation phase transitions (see section 2.5). This behaviour is attributed to the degeneracy, even in a crystal field having trigonal symmetry, of the ground state of the Co ions. Investigations (Bolzoni and Pareti 1984) on single crystals of the series of ferrimagnetic oxides Ba2(Zn1_xCOx)2Fea2022 (Y-type ferrites, abbreviated as (Zn,Co)2-Y) having trigonal symmetry, gave evidence of spin-reorientation phase transitions and FOMP. The transition is observed at 77 K with magneticfield direction along the symmetry c axis in all samples containing cobalt. For instance, in the sample with composition Znl.sCo0.5-Y, the critical field is l l . 4 k O e and the magnetization jumps to saturation. The critical parameters (magnetic field and magnetization) measured as functions of temperature and composition, together with the values of the anisotropy field when FOMPs are absent, allow to obtain complete information on the magnetic anisotropy of this system. For carrying out this analysis, Bolzoni and Pareti (1984) have studied the magnetic phase diagram of a trigonal system and determined the conditions for the observation of FOMP in terms of anisotropy-constant ratios. They use the following expression for the anisotropy energy of a trigonal crystal, E 1 = K 1 sin 2 0 + K z sin 4 0 + K t sin 3 0 cos 0 sin 3q~,
(2.14)
where 0 and q~ are the polar angles of magnetization vector M s. Proceeding in a way similar to that described in section 2.2, they obtained the magnetic phase diagram, reported in fig. 7, in terms of the anisotropy-constant ratios x = K 2 / K 1 and y = Kt/K1, giving easy-magnetization directions and other extrema of aniso-
FIRST-ORDER MAGNETIC PROCESSES
415
Y 3
t
X~ I -,
I
-9
0
I
2
I
4
Fig. 7. Magnetic phase diagram of a trigonal ferromagnet with K1> 0. The coordinates are x = K2/K ~ and y = K ] K 1. A symbol similar to that of fig. 1 has been utilized. It marks every region and gives informations about the easy direction of magnetization vector M~, as well as about the other extrema of the anisotropy energy surface with respect to variable 0 at q~= 90° (0 and ~0are the polar and the azimuthal angles of M~with respect to the tfigonal axis); vertical and oblique stems refer to the ternary axis and to 'cones', respectively. A concave tip denotes a minimum while a convex tip refers to a maximum of the energy. The absolute minimum (easy direction) is indicated by a filled tip. For K1< 0 no boundary line exists; the system is easy cone throughout: its representative symbol is "~ (Bolzoni and Pareti 1984).
tropy energy. The diagram of F O M P is reported in fig. 8 which gives the lines of constant reduced critical field her -- H c r M s / I K l l and critical reduced magnetization mcr = M , / M s. The shaded zone indicates the region where type-2 F O M P is present. Figure 9 instead reports the diagram of lines of constant values of the easy 'cone' angle together with the curves mcr--const. By using this approach, Bolzoni and Pareti (1984) have been able to obtain for the first time precise and consistent data about magnetic anisotropy in these trigonal ferrites. The value they obtained for K t (i.e., 3.5 x 106 erg/cm, extrapolated to 0 K) reconciles the dramatic discrepancies between theory and experiment. In fact, Bickford (1960) used a torque technique to measure anisotropy of C o 2 - Y and (K 1 and K2) data obtained by Casimir et al. (1959) in magnetic fields far below saturation. The experimental value thus obtained by Bickford for K t at llT K (i.e., 6 x 105 e r g / c m 3) was more than one order of magnitude lower than the one he calculated on the basis of atomic and structural considerations, i.e., 1 0 . 7 x 106 erg/cm. For rare earth transition metal compounds having hexagonal symmetry, a two-sublattice model has been developed by Sinnema et al. (1987). It applies to compounds of the type R2(Co,Fe)I 7. The attention was especially devoted to the study of easy-plane ferrimagnetic R2T~7 compounds (T stands for transition metal), which display first-order magnetic transitions at high magnetic fields. The
416
G. ASTI
,
Y I/
.
.
\
...
.
a,.
.
.
.
,
,, e
':iiii:iiii ,.....:.:..: ........ . . . . . Y." , O;,,,v
i::!$ii!:?'
- ::!~:::. . . . . . 5 : ~:!!i' ....... .7 - i!iiiii!:
. . . . . . . . .
3
F
64
0
-2
2 ,
\
t,
,
4
i - /
i
(b)
-3 ~
,,,;):
s
Y • --~
I
'
I I
i /~',,
,/
,~ ~
, ,
/ e°
'
Fig. 8. Lines of constant values for the reduced critical field h , = Hc,Ms/[KI[ (solid lines), and reduced critical magnetization rn, = M c r / M s (dashed lines), for type-1 FOMP, in the planes of reduced anisotropy constants x = K2/K~ and y = KJK~. (a)/(1 > 0. (b) K 1 < 0. The shaded zone indicates the region where type-2 FOMP is present (Bolzoni and Pareti 1984). f r e e - e n e r g y e x p r e s s i o n u s e d is,
E = E Kil sin 20i + Ki2 sin 40i + Ki3 sin 60i + Ki4 sin 60i
COS6 ~i
i
Jr nRTM R ' M T - M
R "H--M
T'H,
(2.15)
w h e r e i = T o r R , a n d 0i a n d ~i a r e t h e p o l a r a n d a z i m u t h a l a n g l e s o f m a g n e t i z a t i o n v e c t o r s M T o r M R w i t h r e s p e c t t o c axis a n d a axis, r e s p e c t i v e l y ; nRT is t h e i n t e r s u b l a t t i c e m o l e c u l a r field c o e f f i c i e n t a n d H is t h e e x t e r n a l m a g n e t i c field. T h e m a g n e t i z a t i o n c u r v e s f o r H a p p l i e d p a r a l l e l t o a, b a n d c a x e s h a v e b e e n o b t a i n e d
FIRST-ORDER MAGNETIC PROCESSES 4
,\
,
,\,
3-
,~
"
',
,
'
\ ",
,
, ;
~,
,
,
417
,
~, , ,
I
(a) --
1
e,
'4
.
""
.
"',
-...
o
64
-2
0
'
x'-L+ 2 X
"
4
4
-3
Y + -4
Fig. 9. Lines of constant values for the easy 'cone' angle 0c (solid lines), and reduced critical magnetization, mcr (dashed lines), in the planes of reduced anisotropy constants x = K J K 1 and y = K t / K ~. (a) Ka >0. (b) K~ <0 (Bolzoni and Pareti 1984).
by numerical methods. In easy-plane ferrimagnetic R2T17 compounds, the magnetization curves, with the field in the plane, show large first-order transitions (termed F O M R , from 'first-order moment reorientation processes') resulting mostly from competition between exchange and magnetostatic energy. The assumption is made that K~ is negligible as evident from investigations on Y2C017 and Y2Fe17 compounds. At zero field the resultant moment is directed along one of the six easy directions in the plane. As the magnetic field rises, successive transition points are reached where different moment configurations are energetically favorable. The critical fields correspond to the points where the energy of the two magnetic phases have the same value. An example of the obtained
418
G. ASTI
experimental and theoretical curves is shown in fig. 10 for Dy2Co17. The measurements have been performed at 4.2 K in a field up to 40 T at the high-field laboratory of the University of Amsterdam. A variety of compounds have been investigated using this approach, obtaining precise and consistent data about anisotropy constants for both sublattices. The obtained values of K i are, in most cases, one order of magnitude larger than those derived from other authors' low-field measurements and assuming rigid exchange bonding between the moments (see section 3.3.3). In tetragonal crystals, due to the presence of four-fold anisotropy in the basal plane, it is even more important to take the tetragonal term into account. An interesting case is that of the ternary intermetallic compound Nd2Fe14B , which represents a very important development in the field of hard magnetic materials [for a comprehensive review on this topic see the work of Buschow (1986)]. Working on this material, Asti et al. (1984) and Gr6ssinger et al. (1984) revealed, at temperatures below 210 K, the presence of FOMPs with critical fields above 14 T. For these experiments, the technique used was that of the singular point detection (SPD, see section 4.2). The existence of FOMP was confirmed by direct observations on single-crystal samples by Pareti et al. (1985a). However, the important role of the tetragonal symmetry appeared immediately (Bolzoni et al. 1987, Kajiwara et al. 1987), as the FOMP was not visible when the applied field was parallel to (110) axes. Subsequent measurements in other high-field laboratories [see references below in addition to Andreyev et al. (1985) and Sinnema (1988)] confirmed the existence of the transition in a variety of compositions in the system R2T~aB. For the understanding of magnetic anisotropy in this
Oy2[o1~ 6C
42
K
+ a axis
,
"--
x ~ X _
X
~
-
~
~
z
,
1
1
f
10 20 Mognefic Fietd B(T)
,
i
i
30 =
Fig. 10. Magnetization curves at 4.2K along different crystallographic directions of hexagonal Dy2Co17. The a axis is the easy magnetization axis within basal plane. The full lines are computed magnetization curves; after Sinnema et al. (1987).
FIRST-ORDER MAGNETIC PROCESSES
419
system a theoretical approach based on tetragonal symmetry is necessary. Theoretical models based on the crystalline electric field (CEF) Hamiltonian ~CZF = ~2 B~O~,
(2.16)
l,m
have been developed and applied to several compounds of this class. It is found that CEF terms up to fourth order are necessary to explain the observed FOMP in Nd2Fe14B and its dependence on crystal orientation (Kido et al. 1987). The case of Pr2FelgB is worth noting because the observed FOMP is of type 2, as the magnetization, after the abrupt increase, is well below saturation (Gr6ssinger et al. 1985, Pareti et al. 1985b, 1988, Huang Ying-Kai et al. 1987, Pareti et al. 1988, Pareti 1988). Moreover, the critical field changes from 13 to 16 T when the magnetic field is moved from [100] to [110] direction. In this case, the theoretical interpretation is complicated because it is found that CEF terms up to sixth order are necessary for reproduction of the observed phenomena (Hiroyoshi et al. 1987). On the other hand, the role of sixth-order terms in the case of type-2 FOMP was already clear from phenomenological arguments, even in uniaxial systems [Asti et al. (1980), see section 2.2]. The necessity of sixth-order terms has, in this context, also been evidenced by Zhao Tiesong and Jin Hanmin (1987), although the various authors adopt different criteria for simplifying the CEF Hamiltonian, because of the great number of adjustable parameters.
2.4. Cubic symmetry First-order magnetic processes of the same nature as those described in the preceding sections (FOMP) have also been observed in magnetic materials with cubic symmetry. Preliminary discussions of the effect were already present in a work by Bozorth (1936) and the theoretical description of the effect has been partially developed by Krause (1964) and Rebouillat (1971) on the basis of the usual phenomenological theory of magnetic anisotropy. Furthermore, Rebouillat (1971) gave some experimental evidence of a first-order transition of this kind in iron single crystals at low temperatures. Asti and Rinaldi (1974b) have examined the conditions for the existence of FOMP with the magnetic field H parallel to one of the crystallographic axes, taking into account only the first two anisotropy constants, K 1 and K2, both in the case of cubic and uniaxial crystals. Two different kinds of FOMP are possible: (1) When the irrreversible rotation of M~ occurs in the plane defined by the magnetic field direction and the easy direction, i.e., the initial direction of M~ (type-I transition): (2) When M~ starts to rotate in one plane and, at the critical field, jumps to another plane where it continues to rotate towards the magnetic field direction (type-II transition). The analysis of the various possible magnetization curves has been made assuming the direction of magnetic field H to be parallel to an extremal direction
420
G. ASTI
of the anisotropy energy surface, i.e., a stationary point but not an absolute minimum. In the presence of only K 1 and K 2 constants these directions coincide with crystallographic symmetry directions. The results are summarized in the graph displayed in fig. 11, which also gives the phase diagram of easy and hard directions in the (K 1, K2) plane. Depending
Fig. 11. Conditions for the existence of FOMP in cubic crystals, which depends on the ratio K 2 / K 1 of the first two anisotropy constants and the sign of K 1 (on the outside part). The letters A, B and C indicate magnetic field orientations parallel to (100), (110) and (111) directions, respectively. The superscript indicates whether the transition is of type I or II. The magnetic phase diagram of a cubic crystal is given by the inner shaded circular areas relative to the three crystallographic directions. The various ranges of K 2 / K ~ are distinguished by different symbols, indicating a minimum for the anisotropy energy (e), a maximum (h), a saddle point having a maximum for rotations of Ms in the {110) planes (sx) or in the {100} planes (sy). For K 2 / K ~ < - 2 (see the ranges indicated by 0 and ~oin the white circular zone) the anisotropy energy surface has saddle points along non-symmetry directions of M~ lying in the (110) planes and having two direction cosines equal to ( - K 1 / K 2 ) ~/2 For rotations of Ms within the (110) planes the anisotropy energy has a maximum or a minimum when K z / K ~ is in the range 0 or ~, respectively (Asti and Rinaldi 1974a,b). A continuous solid line, inside the shaded areas, joins all the ranges on the three crystallographic areas labelled 'e', where t h e crystal is easy axis (absolute minimum).
FIRST-ORDER MAGNETIC PROCESSES
421
on the direction of H, the following intervals for the anisotropy constants ratio k = K2/K 1 are indicated for the two types of FOMP: (1) H[I(100) - T y p e I for k > 4 with g I < 0 , and k < - 9 with K~ > 0 ; type II for-9
2.5. Mechanisms responsible for high-order anisotropy constants
From the above discussion, it is clear that the existence of FOMP in a ferromagnet implies high-order anisotropy terms. These circumstances are not easy to be
422
G. ASTI
included in the theory and, in general, require special mechanisms that take into account the nature of the magnetic ions and of their mutual interactions. For instance, in cubic crystals the f electrons of rare earth ions can support no term above sixth order in the CEF Hamiltonian (Cullen and Callen 1984). In mixed crystals, namely pseudobinary compounds, the competition between the individual anisotropies of the constituent rare earths is responsible for the onset of higher-order terms in the anisotropy energy. These terms are generated if we treat the crystalline electric field in higher than first-order perturbation (in terms of the ratio crystalline field/exchange field), either by admixing higher multiplets in the J quantum number, or by allowing a small canting between magnetic moments (Cnllen and Callen 1985). Indeed, section 3 is devoted to the effect of canting on the anisotropic behaviour of multisublattice systems on a phenomenological basis. As regards magnetic oxides of the transition metals, the mechanisms for generating high-order anisotropy constants must be different. In fact, we are in this case in the limit of large exchange and the situation for a 3d ion is very different as for the relative importance of the various terms in the Hamiitonian. As a matter of fact, there are examples of hexagonal ferrites that display large FOMPs within wide ranges of temperature and magnetic fields, both parallel and perpendicular to the symmetry c axis (Asti et al. 1978, Graetsch et al. 1984, Bolzoni and Pareti 1984, Paoluzi et al. 1988). All these examples have in common the presence of Co as a constituent. Lotgering et al. (1980) have given an explanation of the peculiar influence of Co 2+ ions on the anisotropic behaviour of all hexagonal ferrites. Similar effects have also been observed for Fe 2+ ions in L ~ 2+~ 3 + ~ are re11 u19 (Lotgering 1974). The interpretation is based on the Slonczewski theory (1961), which considers the incomplete quenching of the orbital momentum in the local crystal field, present at octahedral sites which have an uniaxial component, in addition to the cubic one. In fact the local uniaxial crystal field is not sufficient to completely remove the degeneracy and in one case we can have an orbital doublet as the lowest state. The orbital momentum is then either +h or - h along the local axis, so that the spin-orbit coupling gives a torque rTI = ALS sin 0 (where L = 1 and 0 is the angle of the spin with respect to the local axis). This fact implies a sign change in the orbital momentum at 0 = 90°, giving rise to a discontinuity in the torque curve and a very high anisotropy. Even if, in real cases, there are substantial deviations from such ideal conditions, due to various perturbations, the model gives an account of the strong distortions that can be induced in the pure sine function, representing the torque of the spin when only second-order anisotropy exists. As a consequence, high-order terms enter the anisotropy energy. If the orbital singlet is lowest, an anisotropy can again arise from admixing of the excited state with an unquenched orbital moment by spin-orbit coupling (Lotgering et al. 1980). The fact that Co 2+ ions in all hexagonal ferrites reside on lattice sites having octahedral coordination, allows to deduce relations between the anisotropy constants in the hexagonal structure. In fact the anisotropy term associated with Co 2+ in octahedral coordination has the form,
F I R S T - O R D E R M A G N E T I C PROCESSES E
2 2 2 2 2 2 = Kc(O/lO~ 2 + o~20t 3 + o t 3 0 t l )
423
(2.17)
where the a's are the direction cosines of the magnetization with respect to the (x, y, z) reference system centered at the cation site and with axes containing the coordination anions. The c axis of the hexagonal structure has direction cosines ( ½X/-3, IV~, )V~). Expressing eq. (2.17) in terms of polar coordinates (0, ~o) with the c axis as the polar axis, and the azimuthal axis having direction cosines (~X/2, ½V2, 0), we obtain, E=Kc( 1-2sin 20+7
sin 4 0 + } s i n 3 0 c o s 0 s i n 3 ~ ) .
(2.18)
The last term on the right-hand side has trigonal symmetry. However, for hexagonal structures (like W- and Z-type compounds) we must take into account that for each octahedron there is another one rotated by 180°. Averaging over the two, the trigonal term vanishes. Equation (2.18) indicates that a positive singleion cubic anisotropy contribution gives rise, in the hexagonal structure, to both a negative K 1 and a positive K 2 (Asti and Rinaldi 1977). There is another kind of transition in hexagonal ferrites that cannot be considered as ordinary FOMP, but is associated with a more complex magnetic behaviour (see section 3.4). These phenomena are characteristic of ferrites having block-angled magnetic structures and are caused by mechanisms which are different from those responsible for the above-mentioned FOMP. 2.6. Domain wall processes at the transition point
As we have seen, a FOMP is due to an irreversible rotation of the magnetization vector M s between two inequivalent minima of the free energy of the crystal under the action of the magnetic field. Normal hysteresis in a ferromagnetic material is caused by irreversible rotation processes between equivalent crystallographic directions and is therefore symmetrical. In the case of FOMP we expect to observe minor hysteresis loops displaced to an unsymmetrical position in the M versus H diagram. In reality, it is difficult to reveal the hysteresis associated with FOMP. The few known examples are experiments at liquid-helium temperatures, as reported by Barbara et al. (1978) and by Ermolenko and Rozhda (1978). The reasons are probably in the inherent low coercivity of the magnetization processes in the critical range of the phase transition. Mitsek et al. (1974) and Melville et al. (1987) have calculated the domain structure existing in this range. The analysis starts from the consideration of a 180° domain wall in a uniaxial ferromagnet with K 1 > 0 and x = K ~ / K 1 in the range ( - 1 , - ~1) , which corresponds to a P1 type FOMP (see section 2.2). The magnetic field H is oriented along the positive direction of the x axis while the plane of the wall is parallel to the xz plane and neighbouring domains are oriented in opposite directions of the z axis. Under the action of the magnetic field, both domain-magnetization vectors rotate by an angle 0 towards the field direction, so that the wall, polarized along the magnetic
424
G. ASTI
field direction, becomes a ( 1 8 0 - 20) degree wall. The free energy is
--de f E= k[ A \(d0) d y / 2 + K l s i n Z O + K 2 s i n g o - M s H s i n O ] dy,
(2.19)
where A is the exchange parameter. The Euler equation has the form, K1
d20
dy z
A sin 0 cos 0
2K2 MsH ~-- sin 3 0 cos 0 + ~ cos 0 = 0 ,
(2.20)
and the boundary condition are, d0 ~yy y=--+~= 0 ,
0 ( - ~ ) = OH,
0(~) = ~ - - OH,
where OH is the equilibrium angle corresponding to field H inside the domains. The solution of eq. (2.20) turns out to be
0
y =
fl
~ dO,
(2.21)
re/2 with U=
Kx K2 -)- (sin 2 0 - sin 2 OH) + ~ - (sin g 0 -- sin g OH)
]1/2 MsH (sin 0 - sin OH)] A
.
The slope dO/dy can be calculated from differentiation of both parts of eq. (2.21), obtaining, dO -- = U. dy
(2.22)
It is found that dO/dy vanishes at the center of the wall, i.e., for 0 = ½~-, when H is equal to the critical field of F O M P H , = ½HAahcr = Klhc]M ~. The reduced field h~ is given by eq. (2.3), where y is taken equal to zero and m is equal to the critical magnetization given by
mcr = Mc,./Ms = ½[-1 + ( - 2 - 3/x)1/2],
(2.23)
which is a solution of eq. (2.4). The fact that the derivative dO/dy vanishes at the center of the wall means that, at the critical field, the original 180 ° wall is split into two symmetrical walls of (90 - 0) degrees, giving rise to a nucleus of the new phase that appears inside the
FIRST-ORDER MAGNETICPROCESSES
425
domain wall. These new walls are in equilibrium in the presence of a magnetic field equal to Her , because the two adjacent domains belong to different phases, but have the same free energy. The same results can be obtained in a very direct way if one utilizes a simple energy criterion that is valid for domain structures in general, and for a single wall in particular. Let us consider a one-dimensional problem as before, so that we consider a system of plane domain walls parallel to the xz plane. Then we can rewrite eq. (2.19) as follows, cr2
KI(x + y2) + K2(x z + y2)2 _ Ms(xH~ + yHy + zH~) + 2~MZy 2 o"1
where x, y and z are the Cartesian coordinates of the unit vector Ms~Ms, Hx, H e and H z are the components of the magnetic field H and tr is an axis parallel to the y axis. In the case of static domains, if Hy = 0, we may exclude the y component because it gives rise to demagnetizing fields that would drive the wall motion. In this form, the integral of eq. (2.24) can be regarded as the action integral of a mechanical system consisting of a body of mass 2A, free to move on a sphere. The mechanical forces acting on it are opposite to the forces acting on the magnetic vector M s. They are a gravitation-like force directed opposite to H, and elastic forces repelling the body away from the z axis (when K 1 > 0) and away from the xy plane. Hence, cr represents the time, and the function in the integral is the Lagrangian function T - V of the mechanical system, where the kinetic energy T corresponds to the exchange term and the potential energy V is represented by the other terms with changed sign. This mechanical analogy enables derivation of an important property of magnetic domain structures directly from the energy conservation principle in the mechanical system. In fact, T + V must be a constant throughout the whole range of or. As a consequence, we can say that everywhere in the magnetic system the increase in exchange-energy density is balanced by an equal increment in the remaining part of the energy density, consisting of the addition of anisotropy and magnetostatic terms. This kind of equipartition principle has easy and immediate applications. For instance for the calculation of domain-wall thickness (see section 4.2) and to the above results concerning domain structure in the presence of FOMP. In fact, eq. (2.22) can be written directly from the balance of the energy densities, and, likewise, the vanishing of dO/dy at the center of the wall, when a nucleus of the new phase appears, is obvious, because the two phases have, by definition, the same energy density in terms of anisotropy plus magnetostatic contributions. As a consequence, there must be no variation in exchange energy density. The mechanical analogy also makes intuitive the effect on domain structure of the magnetization process of a uniaxial crystal with applied magnetic field parallel
426
G. ASTI
to the c axis, which is taken as easy magnetization direction. For instance, it is easy to understand why reverse domains are reduced with respect to those oriented parallel to magnetic field: If we consider the body on the sphere to rotate continuously, it means that we have a periodic system of parallel walls having alternate polarization, towards positive and negative x direction. Then, in the mechanical analogue the moving body spends less 'time' in the opposite domains because it has a higher kinetic energy there. Equilibrium is reached, because there is a repulsion between adjacent walls with opposite polarization. In the limit of high magnetic fields they join to form a 360° wall. Instead, solutions with magnetization vector pointing in the same direction must be unstable, because they present the opposite effect, i.e., the mean magnetization of the crystal will be opposite to H. All these conclusions are in agreement with detailed theoretical studies reported by Forlani and Minnaia (1969), Wasilewski (1973) and Odozynsky and Zieter (1977). The same approach can be used to study domain wall mobility and dynamic domain-wall structure, if one includes in eq. (2.24) the terms in y and effective dynamic forces deducible from the dynamic equation of magnetization, e.g., in the Gilbert form, dMJdt
= 7M x H - (M s x dM/dt)a/M
s.
(2.25)
If we admit that there are solutions characterized by a rigid displacement of the domain wall with a constant velocity v, differentiation with respect to time of a function L becomes d L / d t = v d L / d o - = v L , where we have indicated differentiaton with respect to o- by a point above the function. Then the dynamic equilibrium of the moving body can be written in the form ( 7 / M s ) M s × ~:
= O,
(2.26)
where ~ is the total dynamical force given by = (-grad V - 2 A ~ ' I I M s - v M s G I 7 - v a M l y ) ,
(2.27)
and G = M x ~ is a Lorentz-like force caused by a pseudo-magnetic field perpendicular to the surface of the sphere in outward direction and having a uniform intensity equal to 1/M s. The term containing the damping parameter a is responsible for the energy loss and acts as a typical viscous resistance. The dynamic equations are then obtained by projecting eq. (2.26) on the Cartesian axes, or from the condition that the component of ~r on the sphere must be equal to zero. This condition can be expressed in the form, o% = 0 ,
o~0 -- 0,
(2.28)
where ~ and ~0 are the components of ~: along the directions of the local spherical-coordinate lines ~p = const., and 0 = const., having taken the polar axis
FIRST-ORDER MAGNETIC PROCESSES
427
coincident with the z direction. If we suppose that the magnetic field H is applied parallel to the positive direction of the z axis and that we have a single domain wall with the boundary conditions z = 1 for cr---~ - ~ , z --- - 1 for or--->+ ~ , then we can see that a very simple solution exists, namely the one given by ~0 = const. [the solution given by Walker and reported by Dillon (1963)]. In fact, in this case eqs. (2.28) become,
o%~ = 4~yM~ cos q~ - vi(/lly = 27rM 2 sin 0 sin 2~o - v M / y = O, (2.29)
~o = MsH sin 0 - a v ~ l / y - 2Alf4/M s + 2 K 1 sin 0 cos 0 + 27rM 2 × sin 2 qo sin 20 = O. The solution corresponds to the condition that the viscous resistance is perfectly balanced on every 'instant' tr, by the term due to the applied field, i.e.,
MsH sin 0 = a v i f l / y .
(2.30)
This means that the magnetostatic energy - M . H is entirely dissipated inside the wall during its damped motion. This condition together with the first equation of (2.28) yields, sin 2~ = H / ( 2 7 r a M s ) .
(2.31)
Then h)/can be deduced from the principle of energy conservation, i.e.,
A(i(4/Ms) = K 1 sin 2 0 + 27rM~ sin 2 0 sin 2 q~,
(2.32)
so, for the wall velocity the following expression is obtained,
v = H ( y / a ) (A/K1)l/2[1 + 7rM2(1 - c o s 2~)/K1] - v 2 .
(2.33)
The case ~ = ¼7r corresponds to the maximum value for v, which is known as the Walker limit (Dillon 1963).
3. Processes involving competition between exchange and anisotropy 3.1. Introduction In the previous chapter we have seen that the simple consideration of high-order terms in the magnetic anisotropy energy expansion justifies the existence of F O M P in an 'effective' ferromagnet. By using this expression, we emphasize the fact that the internal magnetic structure of the system has no relevance, because during the magnetization process it was assumed to remain absolutely rigid. Examples are ordinary ferrimagnetic materials, such a s the oxides of transition
428
G. ASTI
metals which, at ordinary field intensities, display a constant spontaneous magnetization as well as perfectly collinear magnetic moments, due to the extremely high exchange interactions. However, this is not true for all magnetically ordered substances. It is well-known indeed that in RE intermetallic compounds the situation can be such that the fundamental interactions have comparable intensities, so that we can have exchange and crystal-field terms in the Hamiltonian which are essentially of the same order of magnitude. The result is obviously that we can no longer consider the system as a rigid collinear magnetic structure, but we must allow for substantial deviations from the equilibrium configuration at zero field. Hence, the role of the exchange interaction in the magnetic process can in certain conditions be so critical, as to produce extraordinary effects. As a matter of fact, it can be responsible for: variations of the anisotropy fields by orders of magnitudes (even leading to change of sign); stabilization of easy-cone directions; or it can give rise to field-induced first-order transitions, even when only the first anisotropy constants of the individual sublattices are present. In treating effects of this kind in the next two sections, we shall use, once more, a phenomenological approach that is based on the classical model of exchange. Although simple in principle, it can be considered as a useful guide to the experimentalist, because it contains the essential features and allows to gain a better understanding of the fundamental processes which form the basis of the observed phenomena. In section 3.2, as a first application of this concept, a detailed analysis is made of the magnetic system in the vicinity of the transition points, so as to obtain exact relations between some fundamental parameters, such as the expressions for the anisotropy fields and the conditions for the existence of FOMP. In section 3.3, the same concept is developed within the approximation of small deviations from collinear magnetic order (small-angle canting). In this limit, the effect of canting can be accounted for, in a sufficiently accurate approximation, by the use of effective anisotropy constants that are slightly field dependent. So, the system can be conveniently treated as an effective ferromagnet having certain anisotropy constants. A typical application of this model is an immediate explanation of the easy-axis to easy-cone transition observed at 110 K in PrC%, thus, indicating in a very simple way the important role of canting (see section 3.3.3). In section 3.4, a brief review is given of other cases and treatments of the problem of canted systems and magnetic phase transitions.
3.2. Role of exchange at transition points 3.2.1. Multisublattice magnetic systems with uniaxial anisotropy We shall focus our attention on an ordered magnetic system with uniaxial magnetic anisotropy, consisting of L interacting sublattices, Ma, M b , . . . , ML, when it is near the transition point, in general, in the presence of an external magnetic field. In what follows, we will consider transitions involving a change in magnetic symmetry, in particular, transitions in which one of the phases is a collinear-type equilibrium configuration, i.e., a transition between a 'collinear' phase and an 'angular' or canted phase.
FIRST-ORDER MAGNETICPROCESSES
429
So, the equilibrium states we are considering in the collinear phase are essentially: the fundamental state in zero field, normally characterized by a collinear-type magnetic structure, and the various other collinear states that are stabilized by an external magnetic field H parallel to a crystallographic symmetry direction. The latter are intermediate saturated states and are stable in principle within certain regions on the temperature-magnetic field plane ( H - T plane). Among these states, the one at the lowest field, when possible, is that having the s a m e magnetic structure as the fundamental state at zero field and occurs at H = HA, the effective anisotropy field, or at H = Her, the first critical field, in case a FOMP is present. Other collinear states can be stabilized by higher fields. They are characterized by different magnetic structures, corresponding, in principle, to all possible combinations of the mutual orientations of vectors M i. Among these 'excited' states the one having the highest magnetization corresponds to the forced ferromagnetic order. The phase transitions associated with these magnetic states are of two types: s p o n t a n e o u s spin reorientation transitions (SRT), a n d f i e l d - i n d u c e d SRT. One can imagine the reverse situation, i.e., to keep the magnetic field H constant and to change the temperature, or even a combination of these two, which corresponds to crossing the transition line in the H - T plane along different directions. In this sense, the first type of SRT reduces to a particular case of the second type. As an example, let us consider again the phase diagram of figs. 1, 3 and 4. The line m represents first-order SRT. If the representative point, with changing temperature, crosses that line at a certain critical value T = Tcr, it means that the system undergoes a first-order SRT in which the easy-magnetization axis changes discontinuously from axis to plane or vice versa. Alternatively, the lines o and o' are those where discontinuous transitions occur of the type easy axis-easy cone or easy plane-easy cone. Obviously, the same lines have the meaning of lines where the value of the critical field of FOMP reduces to zero. Since in the present treatment we are confined to small deviations from collinear order, the orientation angles of the magnetic moments are treated as infinitesimal, in general, all of the same order. So in our considerations we are not limited to the classical 'large exchange' case, but we may consider systems in which the intensity of the anisotropy forces can be comparable to or even higher than, the exchange interactions. The analysis of the behaviour of the system in the neighbourhood of the transition point leads to exact relations between important magnetic parameters. Let us consider a system of L sublattices M a , M b , M ~ , . . . , M L having uniaxial anisotropy constants K l a , K l b , K l c , . . . , K1L, Kza , K2b , Kz~, . . . , K2L up to a certain order N, and intersublattice molecular field coefficients J~t3. The free energy of the system, in the presence of an external magnetic field H parallel to the symmetry axis c has the form, L
F=-
~ a,a >fl
J~M~M~
cos(0~ - 0~) + ~ a=a
N
L
~ Ki, ~ sin2i 0~ - ~ HM,~ cos 0~, i=1
a=a
(3.a)
430
G. ASTI
where Oa, 0b . . . . , 0 L are the orientation angles of the magnetization vectors M, with respect to the c axis. This is the typical situation we have when we magnetize a crystal along a crystallographic axis: when the field approaches a certain critical value, which in the ordinary cases is the anisotropy field H A relative to that direction (hard magnetization direction), all the O's are vanishing and at H = H A we have a metastable equilibrium. Above that field, the magnetic structure is collinear and does not change until H (in the case of ferrimagnetic order) eventually reaches a further critical value which starts another SRT, in general, towards a canted state having higher magnetization. In the limit of 'infinite' exchange, the effective anisotropy of the system is merely given by the sum of the sublattice anisotropies, so that the anisotropy field turns out to be, H A ---
2K13
M,
(3.2)
with M = E~L =, M . Moreover, when the exchange is of comparable magnitude as the anisotropy energy, we can have considerable modifications in that expression because there are important contributions to the total energy from relaxation processes due to substantial deviations from the coUinear order. The particular collinear configuration is specified by the sequence of the signs of Ms, and can be chosen independently from the sequence of the signs of J ~ , because, in general, the stability of a configuration cannot be decided a priori, being dependent on the values of the K's and H. Furthermore, we neglect here possible effects due to the splitting of one or more sublattices, a phenomenon of the Yafet-Kittel type, (Boucher et al. 1970) which is possible, in principle, if we consider competing intrasublattice-exchange interactions.
3.2.2. Field-induced spin reorientation transitions. Critical fields It is easy to write down the equations for the critical field Ht, at which a certain state having collinear order becomes unstable. The equilibrium equations of the system are, L
OF~dOe = ~
N
J~t~M~Mt~ sin(0~ - 0t~) + ~] Ki~2i cos 0~ sin 2i-1 0 a
/~=a
i=1
+ M ~ H sin O~ = O,
(3.3)
with a = a, b , . . . , L. Expanding up to first order in the O's, we obtain a system of homogeneous linear equations, L
/3=a
J.~M,~M~(O,~ - Or3) + (2K1~ + M~H)O~ = 0 ,
(a=a,b,...
,L). (3.4)
It is evident that in these equations the anisotropy constants of an order higher than two do not appear, because we have considered the transition with the field parallel to the c axis.
F I R S T - O R D E R M A G N E T I C PROCESSES
431
The condition at the transition point is that the determinant of the matrix of the coefficients is vanishing, i.e.,
a =o.
(3.5)
This is an equation of degree L, which, in principle, can be solved giving the expression for the critical fields H t as functions of the sublattice magnetizations, the anisotropy constants and the exchange coefficients. Note that, in principle, we have L distinct solutions for each collinear magnetic structure. So, already for a small number of sublattices the situation appears very complicated. Among the real solutions H = H 1, H 2 , . . . , Hp of eq. (3.5), with p ~< L, the solutions having physical meaning are those corresponding to transitions to (or from) a minimum of F. They define ranges of H for which a particular collinear magnetic structure is metastable. The other solutions of eq. (3.5) are, in general, only transitions between saddle points of different kinds or between a saddle point and a maximum, i.e., unstable states. An important case is when all the exchange coefficients are zero except those related to one particular sublattice, say M a. This can be a good approximation for several rare earth-transition metal compounds (RE-T), and especially for solid solutions containing various RE species or RE occupying different lattice sites. In this case, we can indeed neglect all interactions between RE and consider only those between RE and TM. Hence, the system of eq. (3.4) reduces to,
(Z
L J~M.M~
) L "}-2gla "]- M a l l Oa-- Z Jl3MaM,t3Ot3 -~-0,
-
¢~=b
(3.6)
- J y o M , Oa+(JyoM
+2K1 + M H)O =0,
(¢=b,c,... ,L),
where we have taken J~ = Ja.~. Then eq. (3.5) becomes,
-
JoMoM + 2Kla + Mon,
¢l=b
-
"y=b
(LMaM, + 2K,.,, + M ,Ht)
(JvM.M:, + 2K1. -I- M:,Ht) J ~: M a2M ~2/ ( J ~ M a M ~ + 2 K ~
+ M~Ht) = O, which yields, L
2K,,, + Mo, H t + ~, [(2Kit3 + Mt3Ht) -1
+ (J~MaMi3)-l1-1 = 0 .
(3.7)
In the case H t = 0 this equation, as well as eq. (3.5), gives the condition for the spontaneous SRT. All these results are easily extended to the case of a field perpendicular to the c axis, by the use of the K-~-->R transformations (see section
432
G. ASTI
2.2). In the most simple case, H t in eq. (3.7) can be identified with the anisotropy field H A. An important consequence of the effect of canting is that we have two different expressions for the anisotropy field, depending on whether we apply the magnetic field parallel or perpendicular to the symmetry axis of the crystal (see section 3.3.2). These directions of the applied magnetic field are typical experimental arrangements for measuring magnetization curves, depending on the fact that the symmetry direction is a hard or an easy direction, respectively. In the absence of canting effects the anisotropy field in both cases is given by eq. (3.2), if only second-order anisotropy constants contribute to the anisotropy energy. However, the anisotropy field given by eq. (3.7) could also have physical meaning when the symmetry direction is an easy direction, when we think of processes involved in phenomena like the nucleation and growth of a reverse domain, that have relevance for coercivity mechanisms. Equation (3.7) reduces to a very simple form in the case of two sublattices, i.e., (2Kla + m a n t ) -~ + (2Kab + MbHt) -~ + ( J m a m b ) -1 = O,
(3.8)
with J = Jb = Jab" Then, the condition for the spontaneous SRT becomes, (2Kaa) -a + (2Kab) -1 + ( J M a M b ) -1 = O,
(3.9)
which merely says that the sum of the compliances due to anisotropy forces is balanced by the exchange compliance. 3.2.3. B o u n d a r y conditions f o r first-order transitions
Another problem concerning these magnetic transitions is connected with their order. In fact, as we have seen in section 2.2, the approach to saturation in certain circumstances can occur with a first-order transition that is a discontinuous rotation of the magnetization vector (FOMP). The present approach, based on the analysis of an infinitesimal region around the collinear state, can be used to study the conditions for the existence of first-order magnetic processes in general, which, in particular cases, can be identified as ordinary FOMP. For instance, the transi:ion to the first intermediate saturation state can be considered to be an ordinary FOMP. Instead, first-order transitions involving higher magnetization states, such as the spin-flop transition, unavoidably bring drastic changes in the magnedc order. When necessary, it may be convenient to distinguish between these two types of FOMP, so, hereafter we will call them 'o.FOMP' and 'h.FOMP' (for 'ordinary' and 'high-magnetization state' FOMP, respectively). Let us consider the system of equilibrium equations (3.3) and expand up to terms of third order in the variables 0. The magnetic field will be expressed in terms of an infinitesimal reduced variable given by ~, = ( n -
H,)/I4 t ,
(3.10)
where H t is a real positive solution of eq. (3.7). The an~les 0 can be written as the
FIRST-ORDER MAGNETIC PROCESSES
433
sum of an unperturbed part, 0, plus an infinitesimal of an higher order, e, i.e., we write 0~ + e~ instead of 0~ for each a. We obtain a system of L linear equations in the variables e having the same coefficient matrix as the homogeneous system of eq. (3.4). The column of terms on the right-hand side are, instead, functions of the 0's and of y, so the system obtained is L
J ~ M Mt~(e~ - e¢) + (2K1,, + M~Ht)e ~ = 0 ( 0 3, TO), t3=a
(a = a, b , . . . ,
L),
(3.11)
Since A is equal to zero, the system has a solution only if the augmented matrix (i.e., the matrix including the column of the right-hand side terms) has the same rank as the matrix of the coefficients. This, in general, allows the writing down of a condition in the form A' = 0, where A' is one of the determinants obtained by replacing one of the columns of A with the column of the right-hand side terms. This existence condition allows one to find the relation between y and the angles 0, which means that we can obtain the slope of the magnetization curve M ( H ) at the transition: a negative slope indicates an unstable equilibrium, which implies that a F O M P is possible. That is, if M ( H ) connects stable states, the transition is of first order, otherwise it means that M ( H ) represents metastable states, and so the transition does not occur at all. In the particular case already considered, in which all the exchange interactions are zero except those with a single sublattice Ma, the coefficient matrix of system (3.11) reduces to that of system (3.6), while the column of the right-hand side terms turns out to be, L
-~ ~
Jt~Mt~Ma(Ot~ Oa)3 + ( ~ M a n t + 4 gla -4Kza)O 3 - MantOa,Y , -
[3=b
~Jt3Mt3M~(Ot3 - 0,) 3 + (~Mt3H t + 4K1~ - 4K2t~)013 - Mt3HtyOt~, (B=b,c,...
(3.12)
,L).
Note that anisotropy constants above fourth order give no contribution, thus indicating that the boundary between first- and second-order magnetic transitions depends, in general, only o n K 1 and K 2. This is consistent with the results obtained in section 2.2, where we have seen that the boundary line n', separating A1 and A I C from normal regions, is vertical, which means that K 3 has no effect. The equation of n' is, in fact, 4 K 2 = K 1.
3.2.4. The two-sublattice system In what follows, we shall limit ourselves to the simplest case of only two sublattices, M a and M b. As already mentioned, the smaller of the two magnetizations, Mb, can be either positive (ferromagnetic order) or negative (ferrimagnetic order), it is convenient to introduce the reduced variables
G. ASTI
434 x = 2Kla/JMM
a ,
y = 2Kab/JMM
v = 4 K 2a / J M M
b = 1 - a = Mb/M, h = H/JM,
a = M / Ma
b ,
a
,
w =4K2b/JMM
(3.13)
b ,
m : MII/M = a cos 0a + b cos 0b ,
h t = Ht/JM,
where M = M, + Mb, J = Jb = Jab and MII = M a cos O, + M b cos 0b is the component of the total magnetization parallel to H. Hence, we can write the existence condition A ' = 0 as follows, -a
[ _ l a(Oa _ 0b)3 + (~ht + 2 y _ w)O~ - htTOb]
A, = b+ht+x
[~b(O
a -
0b) 3 + ( l h t + 2 x - 0 ) 0 3 - htTOa]
=0.
(3.14)
Note that in the case of ferrimagnetic order we have J < 0 so that also both h and h t are negative. We have to take into account that h t is the value of the field at the transition. In the simplest case, it is the anisotropy field h A in the hard direction. In terms of the reduced variables it is related to the other p a r a m e t e r s by the condition that A = 0, i.e., -a
A=
a+ht+Y
b+h t+x
-b
=0,
(3.15)
which is identical to eq. (3.7) for L = b, i.e., to eq. (3.8). The direction angles 0 are determined but for a proportionality factor Oa/Ob = ( a + h t + y ) / a
(3.16)
.
A m o n g the p a r a m e t e r s a, b, x, y and h t only three are independent variables. By solving eq. (3.14) we can express (Y/0bz) in terms of y ' , h t , a, v and w, y/0~ = (h* - h t ) [ ( y ' + a) 4 + a3(1 - a ) ] / [ 2 a 2 h t ( Y '2 + 2 a y ' + a ) ] ,
(3.17)
where, h* = [-by'2(y'
+ 2a)2 _ 2 v ( y ' + a ) 4 - 2 w a 3 b l / [ ( y '
+ a) 4 + a 3 b ] ,
(3.18)
and x' = x + h t ,
y' =y + h t .
(3.19)
In case v = w = 0, our two-sublattice magnetic system is fixed by giving a, x and y. It can be represented by a point in the plane of the sublattice anisotropy fields (x, y). Transformation (3.19) allows reference to overall effective fields x', y ' which
FIRST-ORDER MAGNETIC PROCESSES
435
are a superposition of anisotropy and applied field. Equation (3.17) is related to the approach to saturation, because the component of the measured magnetization parallel to H is MII, or m = MII/M. In fact, by using eqs. (3.13), (3.15), (3.16) and (3.17) we obtain the slope of the magnetization curve at h = ht, S = d m / d ( h / h t ) l h = h t = a ( y '2 + 2 a y ' + a ) 2 h t / { [ ( y ' + a) 4 + a3b](ht - h*)}.
(3.20) The boundary between second- and first-order transitions is determined by the condition ht = h*.
(3.21)
In the limit of infinite exchange (J---~ oo), this condition gives (K2, + K2b )/(Kla + Klb ) = ~, which, as expected, is the equation of the boundary line n' of FOMP of type 1 (see section 2.2). The use of eq. (3.20) allows us to evaluate what is the role of the canting effects in determining FOMPs in different conditions. In particular, one is interested to ascertain whether FOMPs are possible when only K~ constants are present in the individual anisotropies of the sublattices. Hence, we shall examine the various situations under the hypothesis that g2a and K2b are vanishing, so that eq. (3.18) reduces to h* = - b y ' 2 ( y
' + 2 a ) 2 / [ ( y ' + a) 4 q- a3b] .
(3.22)
In the case J > 0 we have ferromagnetic order. It can immediately be seen that in this case S > 0: in fact, 0 < a < 1, h* < 0 and h t = h A > 0. So, in general, we ought not to expect a FOMP when the two sublattices are coupled ferromagnetically and the highest sublattice anisotropy terms are of second order. At least, we can exclude type-1 F O M P with S < 0. In the case of ferrimagnetic order (J < 0), the situation is more complex. In fact, we have three possible transitions with increasing field. The first one at H = Htl(=HA), where the system reaches the state of intermediate saturation with the two vectors antiparallel. The system does not change until another critical field H = Ht2 is reached that triggers the break of the collinear symmetry, and the two vectors start to open an angle. With further increase of H the system reaches the forced ferromagnetic state at a field H = Ht3, with the two vectors parallel. However, depending on the values of the parameters, it is possible that the intermediate saturation is completely absent, so that there is a continuous regime of the magnetization in the canted phase from H = 0 up to H = Ht3. This can happen because the intermediate saturation can be either unstable or metastable. The condition for the existence of FOMP of any kind, o.FOMP or h.FOMP, at each of the transition points H t l , Ht2 and Ht3 is deduced from eq. (3.20), where H t is always a solution of eq. (3.15). So, the negative value of S, relative to a certain transition point, together with the fact that the collinear state involved is
436
G. ASTI
stable, are sufficient conditions to assure that the transition occurs through a FOMP. In the case of ferrimagnetic order, a is larger than 1, the denominator of eq. (3.22) is less than zero for y' below a certain value Y0. So that for y' < y ~ we have S < 0 if h t < h*, i.e., Ihtl > Ih*l. In the case of the transition to forced ferromagnetic order (at H = H~3), we have 0 < a < 1 so that always h * < 0 and S < 0 if h , > h * , i.e., Ihtl < Ih*l. We have to remember that, in this case, the reduced variables (3.13) are referred to a different unit field, because for ferrimagnetic order it is J M = J ( M a - I M b l ) , while for ferromagnetic order it is J M = J(Ma + IMbl), which implies a factor (a - b) between the reduced expressions of the same quantities [see eq. (3.32)]. We can obtain the equation in parametric form for the boundary line, 'h*' between first- and second-order transitions. The definition equations are x0=x'-h*,
Y0=Y'-h*"
(3.23)
We must take into account that the effective fields x' and y', defined by eq. (3.19) obey eq. (3.15), that represents a hyperbola having an upper branch that crosses the origin, i.e., ax' + by' + x ' y ' = 0,
(3.24)
so that eq. (3.23) becomes, x o = -by'/(y'
+ a)- h*,
Yo = Y ' -
h*,
(3.25)
in which y' is the variable parameter. Its graph for the case M a = 21Mbl will be shown in fig. 17. For the aforementioned value y' = y;, the denominator of eq. (3.22) vanishes and one has h*---~ ~. In the graph the line denoted by 'Y0" is then the asymptote of line h*. From the above analysis, we can conclude that the canting effects can be responsible for FOMP transitions, even in the absence of anisotropy constants of higher than second order. However, this possibility is limited to ferrimagnetic order, certainly for h.FOMP, while it is doubtful that o.FOMP could exist. Obviously, by admitting K2a a n d / o r K2b # 0, there are a broad range of possibilities for a variety of FOMPs, and the equations obtained and critical parameters could be used for the analysis of the experimental curves. In the limit of J---~ 0o we find y' ~ 0, h* ~ 0 and S ~ 1, as expected. It may be useful to write down the explicit expressions for h t of the various critical fields. They can be deduced from eq. (3.15), which in explicit form is, ht2 + (1 + x + y ) h t + ax + by + x y = 0.
(3.26)
We take into account definitions (3.13) of the reduced variables, which are valid for all cases. As already mentioned, this implies that for the same set of values of the physical parameters, Ma, IMbl, gl,,, glb and IJI we obtain different sets of
FIRST-ORDER MAGNETIC PROCESSES
437
values for a, b, x and y in the various cases. For the case of ferromagnetic order (J > O) we have only one transition at (3.27)
htf = h A = ½(C + D ) , where, C= -1-x-y
,
D = [(1 + x - y ) 2 _ 4 a ( x - y)]1/2.
If htf is negative, it means that the c axis is an easy direction. The solution with negative sign has no physical meaning in this case, because it merely gives the value of the field at which the c axis changes from a maximum to a saddle point for the free energy F. A geometrical interpretation of htf is given in fig. 12. Given a representative point P(x, y) in the plane (x, y) of the reduced sublattice anisotropy fields, we draw the line g from P having angular coefficient equal to 1
~Y
X
-1
¢/:
Fig. 12. Magnetic phase diagram for the case of a two-sublattice system having ferromagnetic order ( J > 0 , M b > 0 ) with a=Ma/M= 3, in the plane of reduced sublattice anisotropy fields x = 2Kx,,/(JMM,~), y=2K1J(JMMb). Line o,~ is the upper branch of the hyperbola with equation ax + by + xy = 0. The region above o-[ corresponds to collinear magnetic states with the sublattice moments parallel to the c axis. When the representative point P(x, y) drops below ¢r[, the magnetic system again has the same magnetic order provided the magnetic field is stronger than the anisotropy field H A. A geometrical interpretation of this critical-field value is given by the components of vector PQ, oriented at 45° from the x axis, that turn out to be equal to the reduced anisotropy field h A = htf = HA/JM.
438
G. ASTI
and crossing the upper branch o-I of the hyperbola ax + by + x y = 0 ,
(3.28)
in a point Q: the components of the vector PQ are equal to htf. This procedure stems from the fact that the equation for the critical fields [(3.26)] is obtained from eq. (3.28) by the substitution x---~x + h, y---~y + h. The condition for the c axis to be a hard direction turns into the requirement that point P must not be above o-I . For the case of ferrimagnetic order (J < 0) we have, htl = h A = ½(C + D ) ,
(3.29)
htz= ½(C- D) .
Referring to fig. 13, we have that htl and ht2 are represented by the components of vectors PQ1 and P Q z , respectively, which, indeed, are both negative in the given example. Q1 and Q2 are the intersections with the upper branch o-a+ of the hyperbola given by eq. (3.28), which is different from o-f because now a > 1 (and
t,
(3.30)
In fact, in this case, a and b are both positive, as in the ordinary ferromagnet, but J < 0 so that F is a minimum below the lower branch o-~- of hyperbola (3.28) (see fig. 14). It is evident that when P drops below the straight line 'r', of equation y-x-p = 0 and tangent to O'a+, then the intermediate saturation state is unstable. The value of p comes from the condition D = 0 and turns out to be p = 1 - 2a + 2(a 2 - a) 1/2 .
(3.31)
If we want to compare ht3 with the other two critical fields related to the ferrimagnetic order, it may be convenient to express ht3 in the same units as used for htl and ht2. For this we have to refer to the same unit field J M - = J(M,, -[Mb]) instead o f J M + = J ( M a + IMbl), where we have denoted by M - and M + the modulus of the resultant magnetization in the cases of ferri- and ferromagnetic order, respectively. Then, assuming for all the reduced variables the meaning they have in the case of ferrimagnetic order, we must multiply ht3 by a factor, ~1 = M + / M - = (M,~ + [ M b l ) / ( M a - IMb[) = a -
b,
(3.32)
in order to obtain its value in the new units. Or we can give directly in terms of a, b, x and y (as defined in the ferrimagnetic order) the new expression h~3 , by using the substitution, x---~ x / ( a - b) ,
y---~ - y / ( a - b) ,
ht3--~ h~/rl .
(3.33)
F I R S T - O R D E R M A G N E T I C PROCESSES
439
::Y
-3
-2
)
~
X
u
+
.........................................
i:i ......
Fig. 13. Magnetic phase diagrams for the case of a two-sublattice system having ferrimagnetic order (J < 0, M b < 0 ) , with a = Ma/M = 2, in the plane of reduced sublattice anisotropy fields x = 2K1,,/ (JMMa), y = 2K1J(JMMb). Line o-~+ is the upper branch of the hyperbola with equation ax + by + xy = 0. The region above tra+ corresponds to collinear magnetic states with the sublattice moments parallel to the c axis but having opposite orientations. When the representative point P(x, y) drops between line O'a+ and the straight line r, which is the upper part of the tangent to (r~ in its vertex, the magnetic system again has the same magnetic order provided the magnetic field is stronger than the anisotropy field H A = Htx but lower than the second critical field, H,2, at which the collinear state becomes unstable (see section 3.2.2). A geometrical interpretation of these critical-field values is given by the components of vectors PQ1 and PQ2, oriented at 225 ° from the x axis, that turn out to be equal to the reduced critical fields, h A = ht~ = H~/JM and ht2 = H~2/JM , respectively. Line h* defines the boundary between second-order and first-order transitions [see eq. (3.22) and fig. 17]. In using this diagram one has to take i n t o a c c o u n t that, because of the fact that J < 0 and M b < 0, the reduced critical fields are negative, x has a sign opposite to K~a and y has the same sign as K~b.
So, we have, (3.34)
h~ = ½(a - b ) ( C - - D - ) ,
with, C- = -1-
x/(a-
b) + y / ( a -
b) ,
D - = {[1 + x / ( a -
b) + y / ( a -
b)] 2 - 4a(x + y ) / ( a - b)} 1/2 ,
(3.35)
which yields, h(3=X{b-a-x+y-[(a-b+x+y)2-4a(a-b)(x+y)]l/z}.
(3.36)
440
G. ASTI
Y
-2
-1 I
I
X
h* g
iP
-2
Fig. 14. Magnetic phase diagram for the case of a two-sublattice system having forced ferromagnetic order ( J < 0 , Mb>0), with a=Ma/M= z3 in the plane of reduced sublattice anisotropy fields x = 2Kxa/(JMMa), y = 2Kxb/(JMMb). Line try- is the lower branch of the hyperbola with equation ax + by + xy = 0. The region below trf corresponds to collinear magnetic states with the sublattice moments parallel to the c axis. When the representative point P(x, y) drops above line trf the magnetic system again has the same magnetic order provided the magnetic field is stronger than the third critical field Ht3. A geometrical interpretation of this critical field value is given by the components of vector PQ, oriented at 225° from the x axis, that turn out to be equal to the reduced critical field, hi3 Ht3/JM. Note that, in this case, h, x and y have signs opposite to H, Kla and K~b, respectively. Line h* defines the boundary between second-order and first-order transitions [see eq. (3.22) and fig. 17)]. =
A s a l r e a d y n o t i c e d , t h e a b o v e given e x p r e s s i o n s a n d r e p r e s e n t a t i o n s for t h e critical fields h t a r e g e n e r a l l y v a l i d , b e c a u s e t h e y d e p e n d o n l y On t h e s e c o n d o r d e r anisotropy constants. 3.2.5. L i n e a r regime a n d magnetic transitions in the canted p h a s e I t is k n o w n t h a t t h e m o l e c u l a r field t h e o r y a p p l i e d to a N r e l m u l t i s u b l a t t i c e s y s t e m , in t h e a b s e n c e o f m a g n e t i c a n i s o t r o p y , p r e d i c t s t h e p o s s i b i l i t y o f a
FIRST-ORDER MAGNETIC PROCESSES
441
non-collinear magnetic order that can be achieved via spontaneous SRT with changing temperature, or via field-induced SRT which are normally of second order. A peculiar feature of the magnetic behaviour of a canted system of this kind is that the resultant molecular field, acting on each one of the canted sublattice magnetization vectors, has the form,
lij = xj tj,
(3.37)
where Aj is a function of the set {J~K} of the molecular field coefficients only (Acquarone and Asti 1975). That is, the canted sublattices are effectively decoupled and evolve with temperature as an assembly of non-interacting ferromagnets, each with a different 'effective Curie constant' (°quasi-ferromagnetic' behaviour). In this respect, also the applied field H can be formally treated as one of the canted vectors to which a 'molecular field' x H = r. M i corresponds, i.e., the resultant magnetization. The consequence is that X, the susceptibility of the system, is also a constant, being only a function of the set {Jir}; this phenomenon was observed by Clark and Callen (1968) in rare earth-iron garnets. Indeed, they have observed temperature ranges over which the magnetization, at fixed field strength, is temperature independent. For a two sublattice N~el ferrimagnet having molecular fields, H a -= JxlMa + J 1 2 M b ,
l i b = JzlMa + J22Mb,
(3.38)
we have in the canted phase, (3.39)
X = - 1/.112.
It is worth noting that the two-sublattice ferrimagnet discussed in section 3.2.4, having only second-order anisotropy constants, does show a similar behaviour within certain conditions. In fact, it is found that if the representative point P ( x , y ) belongs to the upper branch '1' of the hyperbola (see fig. 17), b x + ay - x y
-= 0,
(3.40)
then it turns out that: (i) Either htl or ht2 is equal to --1; (ii) The same point P(x, y) in the reference frame of the forced ferromagnetic order (J < 0, M b > 0) belongs to the lower branch 1 of hyperbola (3.40) so that also ht3 = - 1 [or h~3 = - a + b, if expressed in ferrimagnetic reduced variables as in eq. (3.36)]; (iii) The magnetization curve is linear and takes the simple form m = - h , where h and m a r e the reduced field and magnetization defined by relations (3.13). This means that the magnetic susceptibility is a constant up to h = h~3 and is equal to, X = M H/H = m M / h J M
= - 1/J,
(3.39')
442
G. A S T I
an expression that coincides with eq. (3.39), which is related to a N6el ferrimagnet (without anisotropy). In order to demonstrate this property, we start from the equilibrium equations, b sin(0a - 0b) + h sin 0a + x sin 0~ cos 0, = 0 , (3.41) - a sin(0~ - 0b) + h sin 0b + y sin 0b cos 0b = 0. These equations can be obtained by means of relations (3.3) using the same definitions for x , y , a and b as given in connection with relations (3.13). Denoting s~ = sin 0~, c I = cos 0a, s 2 = sin 0~ and c 2 = cos 0b, and multiplying the first equation by c 2 and the second by Cl, we obtain after subtraction, ( m + h ) ( s l c 2 - seca)
+ (xs 1 -
YS2)CaC 2 = O.
(3.42)
Now, elimination of h from eqs. (3.41) yields, (b - Y)C2S~/S 2 - (a
2 -
X)ClS1/S
2
2 '[- a C 2 S 1 / S 2 - - b c 1 = O .
(3.43)
Taking into account condition (3.40), we can verify that XS 1 -- ys 2 = O,
(3.44)
is a solution of eq. (3.43), so that eq. (3.42) becomes, (m + h) sin(0a - 0b) = 0.
(3.45)
Since in the canted phase 0a ~ Ob, we conclude that m = - h . Note that hyperbola (3.40) is obtained by a translation of hyperbola (3.28) parallel to a vector v, i.e., (1, 1), having unitary components. In fact eq. (3.40) can be obtained from eq. (3.26) by taking h t = - 1 . The results are also consistent with eq. (3.20), where S is equal to the differential susceptibility at the transition point. In fact, substitution of h t = - 1 in eq. (3.20), where h* is given by eq. (3.22), yields, S = (y,2 + 2 a y ' + a ) 2 / ( y '2 + 2 a y ' + a) 2 = 1.
(3.46)
Moreover, we can show that if htl = - 1 (or ht2 = - 1 ) , then also ht3 = - 1 and vice versa. In fact, let x + and y+ be the coordinates of the representative point P(x +, y+) in the flame of forced ferromagnetic o r d e r (i.e., x + and y+ are expressed in units of J M + = J ( M , + [M b [). Then by the substitution (3.33) for x + and y+ in eq. (3.26), we obtain, ht23 + [1 + (x - y ) / r l ] h t 3 + (ax + b y - x y ) / r l 2 = O,
(3.47)
F I R S T - O R D E R M A G N E T I C PROCESSES
443
with 7/= a - b. If b x + a y - x y = 0, eq. (3.47) becomes ht2 + [1 + ( x - y ) / ' o ] h t 3 + ( x - y ) / r l = 0 ,
(3.48)
for which ht3 -- - 1 is a solution. Or, if we put ht3 = - 1, we obtain b x + a y - x y = 0, which implies that either htl or ht2 is equal to - 1 . Now we can observe that the variation of 01 and 02, according to the law of eq. (3.44), is such that, in general, the m i n i m u m value m 0 of the magnetization in the linear regime is above m = 0, and occurs at a certain critical field h 0 (when h t l = - 1 ) . Obviously, depending on the values of x and y , such a m i n i m u m can coincide with intermediate saturation (when ht2 = - 1 ) , or it can coincide with the origin (h = 0, rn = 0) when x = y [see the paragraph below eq. (3.49)]. That is, the range of the minimum m 0 is 0 ~< m 0 <~ 1. The transition to the linear regime is
Ill¸ 3
1 ...........
0
Ta,
T_'I~===.T 2
-I1.1
-ht2
I
2
;
-h
J,
Fig. 15. Magnetization curve for the case of a two-sublattice ferrimagnet with a = 2, x = 0.75 and y = 0.6. This is an example of the linear regime that occurs when the system has only second-order sublattice anisotropy constants and the representative point P(x, y) belongs to the upper branch T of the hyperbola b x + a y - x y = O, shown in fig. 17. The equation of the linear part, in terms of reduced variables, is m = - h , which implies that susceptibility is a constant equal to X = - 1 / J . The transition to the linear regime occurs at a certain magnetization m 0 and is of second order. When m 0 < 1, as in the example shown, there are actually two superimposed magnetization curves in the range m 0 < rn < 1, corresponding to the same rn and the same energy but to different magnetic structures.
444
G. ASTI
a s e c o n d - o r d e r transition. In the case m 0 < 1, which occurs w h e n htl = - 1 , the m i n i m u m in the m a g n e t i z a t i o n implies that we have a range of h (Ih0l < Ihl < 1) w h e r e two different m a g n e t i c structures c o r r e s p o n d to the same m. Actually, there are in this range two b r a n c h e s of the m a g n e t i z a t i o n curve that c o r r e s p o n d to different angled-magnetic structures. Figure 15 shows an e x a m p l e for the case w h e n a = 2, x = 0.75 and y = 0.6. A w a y to elucidate the shape of the magnetization curve m o r e clear, is to consider it as the limit of a t w o - b r a n c h curve of the type given in fig. 16, c o r r e s p o n d i n g to a point just a b o v e line 1 [see eq. (3.40) and fig. 17]. Besides the s e c o n d - o r d e r transition, at the lower limit of the linear curve, w h e r e h = h 0 and m = m o , there is a first-order transition in the angles 01 and 02 that is not visible in the parallel c o m p o n e n t m , as defined by the last of the relations in eq. (3.13), but in the transverse c o m p o n e n t m I = a sin 0a + b sin 0b. A s a m a t t e r of fact, in the whole interval Ih0l < fhl < 1 the two magnetic states, besides having the same m, have equal free-energy density. In fact, it is easy to show that the free e n e r g y of the system in the linear regime d e p e n d s on 0a and 0b only t h r o u g h the variable m = a cos 0a + b cos 0b. F o r this p u r p o s e , it is sufficient to utilize relations (3.40) and (3.44) in the expression of F given by eq. (3.1) for the case of two sublattices, after substitution o f the r e d u c e d variables of eq.
1.05
.95
d
I
,9'5
. . . .
1
,
,
,
,
I
%05'
'-h ~
Fig. 16. Magnetization curve for the case of a two-sublattice ferrimagnet with a = 2, x = 0.745 and y = 0.595 corresponding to a point close to line '1' [see eq. (3.40) and fig. 17]. The case of fig. 15 can be considered as the limit of a two-branches curve of the type shown here. A FOMP transition is shown that brings the system from the intermediate saturated state into the spin-flop phase.
FIRST-ORDER MAGNETIC PROCESSES
445
(3.13). H e n c e , the system is d e g e n e r a t e a n d the two m a g n e t i c phases m a y coexist in the w h o l e r a n g e of h u n t i l h = h t l , w h e r e the p h a s e c o n n e c t e d with the i n t e r m e d i a t e s a t u r a t i o n state disappears. This t r a n s i t i o n occurs with n o discont i n u i t y i n the v a r i a b l e m a n d c a n b e c o n s i d e r e d as the limit of a F O M P of the type s h o w n in fig. 16. A n o t h e r possible l i n e a r s o l u t i o n of eq. (3.43) is 0a = 0 b = 0 w h e n the r e d u c e d a n i s o t r o p y fields of the two sublattices are e q u a l , i.e., w h e n the r e p r e s e n t a t i v e p o i n t P ( x , y ) b e l o n g s to the l i n e ' s ' (see fig. 17) which has the e q u a t i o n x = y. T h e n , eqs. (3.41) b e c o m e , (h + x cos 0) sin 0 = 0 , i.e., m = - h / x
=-h/y.
(3.49)
I n the case of f e r r o m a g n e t i c order, we m u s t have
.8 'O.
.6
.4
.4
.6
.8
X
1
Fig. 17. The phase diagram, shown in fig. 13 for the ferrimagnetic case, is shown here in greater detail in the area of the first quadrant which is crossed by various critical lines: h*, O'a+, 1, r, s and y~. For explanation of h*, ~r,+ and r see fig. 13. Hyperbola 1is mentioned in fig. 15 and, together with straight line s, is discussed in section 3.2.5. Line y~ is the asymptote of line h*. Line h* crosses line 1 and is tangent to it in a critical point G with coordinates x = a - ~aa2 - a, y = 1 - x. In G they are both tangent to straight line r.
446
G. A S T I
x = y < 0, while in the ferrimagnetic case x = y > 0. Obviously, the linear relationship b e t w e e n magnetization and field holds only up to htt. It is worth noting that we have no canting, even in the presence of 'competing' anisotropies (Kaa < 0 and Klb > 0 in the ferrimagnet). The various critical lines h*, o-+, 1, r, s and Y0, deduced in the analysis of t h e ferrimagnetic case, are shown in fig. 17. Line h* crosses line 1 and is tangent to it in a critical point G with coordinates x = a - ~/a 2 - a and y = 1 - x. In G, they are both tangent to the straight line r. Line 1 crosses line s in a point u, i.e., (1, 1). B e l o w line r the intermediate saturation state does not exist. A b o v e this line various magnetization curves are possible that, in general, are c o m p o s e d of two branches. These connect with continuity two couples of the four critical points: O (h = 0, m = 0), the origin; T1 (h = hta , m = 1) and T2 (h = ht2 , m = 1), the extremes of the intermediate saturation state; and T3 (h = ht3 m = a - b). In addition to these two branches, there is another equilibrium state represented by the line which connects T~ and T e (see, e.g., figs. 15, 16, 18 and 19) which covers
m
3
T3
2
T1 ~
1
o
1
2
2
-h
3
Fig. 18. The magnetization curve for the case of a two-sublattice ferrimagnet is, in general, composed of two branches connecting with continuity two couples of the four critical points: O (h = 0, m = 0), the origin; T1 (h = h,1, m = 1) and T2 (h = ht2, m = 1), the extremes of the intermediate-saturation state; and T3 (h = ht3 , m a - b ) the transition point to the forced ferromagnetic state. In addition to these two branches, there is a n o t h e r equilibrium state represented by line T 1 - T 2 which covers the range of intermediate-saturation state. The present graph, obtained for a = 2, x = 1.1 and y = 1.2, i s an example of a magnetization curve consisting of branch O - T 2 and branch T 1 - T 3 . T h e collinear state for h = 0 (0a = 0b = 17r) belongs to branch O - T 2 and is not the equilibrium state. The state of m i n i m u m free energy at h = 0 is, instead, on the other branch where we have 0a ~ 0b ~ ½~" and m ~ 0. This branch and its continuity is better evaluated w h e n considering it as a connection between T1 and the symmetric point of T3 with respect to the origin (i.e., for h = - h t 3 , m = - a + b ) . Note that in the present case, at h = 0 we have a noncollinear state with the resultant magnetization pointing out of the basal plane, but very close to it. =
F I R S T - O R D E R M A G N E T I C PROCESSES
447
m
T3
3
2
1
T1
~--._._... T2
1
Fig. 19. As in fig. 18 but for a = 2, x = 0.9700 and y = 0.9047. The two branches are of the type O - T 3 and T1-T2.
the range of intermediate saturation state. Besides figs. 15 and 16, other examples are reported in figs. 18 and 19 showing different connections. In the case of fig. 16 it is easy to predict a FOMP, because, with increasing field, there must necessarily be a jump from one branch to the other. Depending on the characteristics of the branches, we understand that we can have in the various cases a variety of transitions also involving the intermediate saturation state. A condition that can be favourable for observing these phenomena may be that of ferrimagnets in the neighbourhood of compensation points, because the field JM can be made small, and the various critical conditions are achieved at relatively low fields. We must remember that the examples given (figs. 15, 16, 18 and 19) are obtained in the limit of only second-order anisotropy terms (i.e., v = w = 0) and accordingly also lines h* and 1 in fig. 17 refer to this situation. Instead, the graphs and the expressions relative to critical fields h t ' s remain the same, even in the presence of higher-order anisotropy terms because they are independent of v and W.
3.3. The small-angle canting model 3.3.1. General remarks The magnetic behaviour of a multisublattice magnetic system, under conditions of small deviations from the collinear magnetic order, can be conveniently treated by the use of an approximate theory, the small-angle canting model (SAC) (Rinaldi and Pareti 1979). The SAC model appears as a simple extension of the classical phenomenological theory of magnetic anisotropy. By the admission of deviations from the collinear order, the system acquires a further degree of freedom that
448
G. ASTI
results in a relaxation of the free energy, which appears as a strong distortion of the anisotropy-energy surface. The opening of the angles between the interacting moments occurs at the expense of exchange energy, with a subsequent gain in anisotropy energy. The important point is that such a distortion can be readily accounted for, by appropriate corrections of the anisotropy constants. It means that the system can be described as a simple ferromagnet characterized by 'effective anisotropy constants' K1, K2, 1 £ 3 , . . . , that are functions of sublattice anisotropy constants and of the molecular-field coefficients. The effective anisotropy constants are not rigorously constant because there remains a magnetic field dependence to first order in the parameter h = H / J M , where H is the magnetic field and J M is a quantity of the order of the exchange field. High-order anisotropy constants appear, even if we attribute to the individual sublattices only second-order constants. For instance, /£1 can be very different from the simple algebraic sum of the individual Ka's. It can even have opposite sign. The small-angle canting model has general validity, as it has been shown (Asti 1981) that it applies to both ferromagnetic and ferrimagnetic order, as well as to any combination of anisotropy constants. The conditions for the application of the SAC model may refer: (i) To the orientation of the magnetic vectors within a narrow range close to the crystallographic symmetry directions (as in the case treated in section 3.2); (ii) To the whole range of orientations, when exchange interactions are much stronger than magnetic anisotropy forces in the system. The region well inside the spin-flop phase of ferrimagnets is obviously excluded (see section 3.2). In the case of magnetization processes close to the symmetry directions, the model provides exact expressions for important physical parameters, such as initial susceptibility and anisotropy fields, or relations concerning important phenomena such as the spin-reorientation transitions. The concept can be applied, in principle, to complex structures such as the case of a three-sublattice system (Asti 1987). In section 3.3.2, the case of a twosublattice system will be described and examples of applications will be given in section 3.3.3. 3.3.2.
The two-sublattice m o d e l
Let us consider a uniaxial system of two magnetic sublattices a and b having magnetizations M a and M b and anisotropy constants Kaa, 1£2o, K 3 a , . . . , and Klb, K2b, K 3 b , . . . . We refer the orientation angles of all vectors with respect to the symmetry axis, c. So, 0~, 0b, 0 and q~ are the angles of M a, M b , M = M a + M b and magnetic field H respectively. The free energy of the system is, E = - JMan b
-
-
M" H + ~ i=1
K ~ sin 2i 0a +
k Kjb sin2~0b ,
(3.50)
j=l
where - J M a • M b is the exchange energy and J the intersublattice molecular-field coefficient. The interaction is ferromagnetic if J > 0 and antiferromagnetic if J < 0. If 'a' denotes the major sublattice, one may have M b > 0 or M b < 0. The
FIRST-ORDER MAGNETIC PROCESSES
449
assumption that the minor magnetization M b c a n take negative values, allows to utilize the same definition for the angles, so as to extend immediately all the results obtained for ferromagnetic interaction to the ferrimagnetic case, simple by changing the sign of both J and M b (see section 3.2). We will consider only situations for which M~ and M b a r e nearly parallel, so that we can write e a = 0 , - 0 and 8 b -~-0b --O, where G and e b are infinitesimal deflection angles of M~ and M b with respect to M. T h e n eq. (3.50) becomes E = - J M a M o cOS(G - % ) - M a l l cos(q) - 0 - G ) - M b H cos(~ - 0 - %) 4- ~
giasin2i(o-} - Ca)'+ ~
i=t
Kjb sin2J(0 + % ) .
(3.51)
j=l
By putting p = M b / M ~ , M = M¢ + M b and e = eb, and taking into account that M~ e a + M b e b = 0, we obtain by expansion up to second order in e, E = E o + A e + B e 2,
with E o = - J M , , M b - M H cos(q~ - 0) + ~
(3.52)
(K~ + K~b).sin 2~ 0 ,
i=1
A = k
- 2 i ( P K i ~ - Kib) COS 0 sin 2/-x 0 ,
i=1
B = IjpM2
+ lpMH
i(Kiap 2 +
cos(q~ - 0) + ~
gib ) sin 2i-2 0
i=1
x [ ( 2 i - 1) cos 2 0 - sin 2 0] . H e r e E 0 is the ordinary expression for the total energy of the collinear system. The value of e is determined by the condition 3 E / O e = 0 , which gives e = -A/2B. Substitution in eq. (3.52) yields, E = E o - A2/4B ,
(3.53)
i,e.
E = Eo -
- i g i c s 2i-1 -
)7{
e +
ifde~-2[(2i i=1
}
1) - 2is 2] ,
(3.54)
with gi = PKia - Kib ,
c=cos0
and
~ = p2Kia q- Kib '
s=sin0.
e = ½[ p J M 2 + p M H cos(q~ - 0 ) ] ,
450
G. ASTI
If we suppose that gia and Kib are zero above a certain order N, eq. (3.54) can be expressed as,
)2/{
E = Eo- c
_
igis2i 1
e + fl
-
2N2fN
S2N
N--1 + ~
} [ ( j + 1)(2]
+ 1)f]+ 1 --
2]2fjls 2j .
(3.55)
j=l
This expression for the energy can be transformed in such a way that it assumes the usual form of a power series in sin 2 0. Hence, the coefficients K~ are the effective anisotropy.constants of the two-sublattice canted system. In the simplest case, i.e., for N = 17 it gives, c~
E = - M H cos(q~ - 0) + J M ~ M b + ~.
K i
sin2i 0 ,
i=1
with,
K1 = Kla + Klb -- g21/(e + fl) ,
K 2 = g21(e - f ~ ) / ( e
+ L)
2 ,
(3.56)
gn = gn-12fl/(e
-{-]el) =
2 , - n - 2 ~t e - f l ) / ( e + f ~ ) gaY1
n.
These results are the same as those reported in the work by Asti (1981) in which t h e p a r a m e t e r s a, b and h are related to the p a r a m e t e r s used here, by the following relations, a = 2g2/JpM 2
b
=
2fl/JPM 2 ,
h + 1 = 2e/JpM 2 •
Let us now consider the case N = 2, which means that we have sublattices with anisotropy constants of second and fourth order. Then from eq. (3.55) we obtain, 4-[4-k1 E= Eo-[g~/(e+
fl)] ~ k=l
r[(k+r),2]
r~=l - Pr-l[
j~--O __ ( k
_
r - , ] Jcelk-r-2j a 2j]12k JJa ,
j
(3.57) where [w] indicates the largest integer in w, with the assumption that [w] = 0 for w ~< 0, and Pa = - 1 + 4g2/g 1 ,
Po = 1 , 2
2
P3 = - 4 g z / g a ,
P2 = ( - 1 + g2/gl)4g2/ga ,
% = -(6f2 - 2fl)/(e + fl),
a2 = 8fz/(e + f l ) .
One can easily verify that for g2a = g2b = O, the coefficient K k of S 2k a r e coincident with expressions (3.56). It can be convenient to write down the first
FIRST-ORDER MAGNETIC PROCESSES
451
few terms of eq. (3.57) explicitly, if we limit the series to k = 4, it becomes, E = E o - [g~l(e + f~)]{s 2 + (Pl + 0/1)$4 -~- (0/21 -~ 0/2 q- P10/1 + P 2 ) $6
+ [0/31 + 20/10/2 + P,(°~21 + 0/z) + P z a l + P31ss + ' " "}.
(3.58)
The coefficients of s 2~ in eq. (3.57), including the terms contained in E 0, give analytical expressions for the effective anisotropy constants K k as functions of the parameters of the two sublattices, i.e., K~a, Kza, Klb , K2o , m a , m b and J. The dependence of K k on the magnetic field is through the parameter e. It is negligible in the transverse configuration, i.e., when the initial susceptibility is measured with H perpendicular to the c axis, both because H is small compared with J M and because q~ = ½7r and 0 is small, so that cos(q~ - 0) ~-0. Instead, when H is parallel to the c axis and this is a hard direction, the parameter e plays an important role. As a consequence, the anisotropy field has two different expressions depending on whether the axis under consideration is a hard or an easy direction. The same expressions given in sections 3.2.2 and 3.2.4 for the anisotropy field H A in a hard direction, can be obtained here from the definition, H A = 2Ka/M,
(3.59)
where K 1 is given by eq. (3.57) or (3.58), taking into account expression (3.52) for E 0. More easily, K 1 is directly taken from the first part of eq. (3.56) because H A depends only on second-order anisotropy constants. So, the condition for the spontaneous SRT has the usual form K 1 -= 0. The expression for the anisotropy field in easy direction is again deducible from eq. (3.59), here we have to put the term p M H cos(q~ - 0), which appears in the parameter e, equal to zero. All these considerations are also valid if we consider a planar direction instead of the c axis. In fact, if we make a K ~ R transformation (see section 2.2), we can exchange axial with planar directions, which means that all the expressions obtained can be interchanged once we replace all K constants by the corresponding R constants. Models including K 2 constants have been utilized successfully by Ermolenko (1979) and by Sinnema et al. (1987), for computing magnetization curves of R C % and R2Co17. This means that higher-order terms in the crystal-field interaction must be taken into account. The advantage of the present treatment is that it gives analytical expressions that are exact for magnetic processes that involve small angles between the sublattice moments, e.g., concerning magnetic transitions involving symmetry directions (see section 3.2). In the analysis of the magnetization curves, it is convenient to make reference to the ordinary K anisotropy constants when the applied magnetic field is along or near the c axis. On the other hand, when we are magnetizing along a direction close to the basal plane, we have a more accurate description of the magnetization curve in terms of the R constants, which are obtained from the above mentioned K<--->R transformation.
452
G. ASTI
3.3.3. Some applications The first application of the small-angle canting model was by Rinaldi and Pareti (1979) on PrCos, giving a straightforward explanation of the SRT easy-axis to cone transition observed at 110 K by Tatsumoto et al. (1971), Ermolenko (1976) and Asti et al. (1980), despite the fact that the s u m gla "~ Klb of the Co and Pr sublattice anisotropy constants is positive. In fact, as explained in sections 3.2.2 and 3.3.2, according to this model, the condition for the existence of a SRT is merely the vanishing of the effective/£1 constant, i.e., K 1 -- 0. By the use of the formulas given in section 3.3.2 it is possible to analyze, in a very simple way, the magnetization curves of both ferro- and ferrimagnetic materials. The approximation of the small-angle canting model is valid in a large number of cases and a variety of experimental conditions. As an example, we will report here the analysis carried out by Asti and Deriu (1982) of the initial susceptibility data, in the hard direction, obtained by Ermolenko (1979) on the series Yl_zGdzCos. In fact, the effective anisotropy constant K~ given by eq. (3.56) can be written in the form,
K1 = ~7(K~. + Klb),
(3.60)
where ~ is a correction parameter and K~a and Klb are the anisotropy constants relative to the Co and Gd sublattice, respectively. A convenient expression for 77 in terms of reduced exchange and anisotropy fields, as defined by eqs. (3.13) is r/= [1 + xy/(ax + by)]/(1 + bx + ay).
(3.61)
The case of the Gd sublattice is a special one, since Klb 0. It means that there is no competition between the anisotropies of the two sublattices. In spite of this, the canting effects exist because the competiton is with the applied magnetic field H. Hence, eq. (3.61) reduces to =
= 1/(1 + bx).
(3.62)
From the data on the initial susceptibility X as a function of the composition z, taken from the work of Ermolenko (1979), we can calculate the effective anisotropy constant K 1 = M2/(2X) as a function of z, and plot the experimental values of ~7= KI(z)/KI(O). Figure 20 shows the theoretical curve together with the experimental points. The agreement between the point and the curve supports the validity of the model, and shows how much the effective anisotropy is reduced due to the onset of a small deviation from collinearity. Indeed, in the absence of canting effects, the parmeter 77 ought to be independent of z. Another example is also taken from the work by Asti and Deriu (1982) and shows how general the validity of the Rinaldi and Pareti (1979) model is. As explained in section 3.3.2, it applies to any case of magnetic order (ferro- and ferrimagnetic) and sign of sublattice anisotropy constants. Besides that, for magnetic processes close to the symmetry directions, it is rigorously valid, even
FIRST-ORDER MAGNETIC PROCESSES
453
8
6
4
o
a
z'
,
Fig. 20. Plot of the values of the correction parameter ~7 [see eq. 3.60] calculated from the initial susceptibility data versus composition, taken from the work of Ermolenko (1979) for the series Yl_zGdzCos. The solid line is the theoretical curve deduced from eq. (3.62).
when anisotropy and exchange interactions are of the same order. Indeed, the case of ferrimagnetic order is that for which canting can produce dramatic effects. Peculiar examples are those of HozFe17 and Ho2Co17 , for which enormous discrepancies were obtained between anisotropy constants, as deduced from spin-wave spectra, and macroscopic magnetic measurements (Clausen and Lebech 1980). The apparent paradox is easily removed by the admission of canting, and the model presented above gives a complete account of the effect. Sarkis and Callen (1982), in order to study this problem, have developed an ad hoe calculation that led them to the following expression for the effective anisotropy constant, in terms of the constants (K R, KT) and the magnetizations (MR, MT) of the individual sublattices, Keff = (K 1 + 2K2)eff = (K R + K T + 2KRKT/L)/[1 + 2(KR M2 + KTMZ)/LM2],
(3.63)
where L = JMTM R and J is the intersublattice molecular-field coefficient. This result can be derived directly from eqs. (3.60) and (3.61). To do this, we have to use the K ~ R transformations described in section 2.2, which allow the immediate transfer of all the equations valid for the case HIIc axis to that of H_l_c axis and vice versa. So we can write the conjugate equation of (3.60), R 1 = 'rl(Rla + R~b),
(3.64)
where R~, Rla and Rlb a r e the anisotropy constants conjugate to K~, Kla and Klb , i.e., R I = - K 1 - 2 K 2 - 3 K 3 + ' " , and similarly for R1, and Rib. Taking into account expression (3.61) for ,/, eq. (3.64) becomes
R 1 = (Rla + R~b + 2RI,RIb/J')/[1 + 2(RaaM ~ + RlbMZa)/J'M2],
(3.65)
with J ' = JMaM b. This expression is coincident with eq. (3.63), once R1, Rla ,
454
G. ASTI
J', M a and M b are substituted by the corresponding quantities Keff, Ka-, KR, L, M r and M R. An approach based on the hypothesis of the small canting-angle has also been utilized by Cullen (1981) for the study of the problem of easy-magnetization directions lying along non-major axes in ternary cubic compounds of composition R~R~_xFe 2. This work also confirms that a classical model of canted magnetic moments can give rise to high-order anisotropy energy and phase instabilities. This type of approach is not as accurate as the direct numerical diagonalization of the total Hamiltonian containing crystal-field terms and the molecular field Hex. However, the classical treatment provides a transparent picture of the physical processes responsible for the observed phenomena.
Rib ,
3.4. Hexagonal ferrites having complex block-angled and spiral structures. Role of antisymmetric exchange During the investigation of magnetic properties of hexagonal ferrites, Enz (1961) encountered unusual properties in the system Bal_xSrxZn2Fe12022 (abbreviated as Bal_xSr~Zn 2 - Y ) . The compound has no spontaneous magnetization, but a field of the order of 1 kOe induces a first-order transition to a magnetization of about 85% of saturation. At a higher field, after a rather fiat plateau of the magnetization, another transition starts a further increase at a constant susceptibility up to saturation. Enz interpreted this behaviour as being due to a helical spin configuration having the axis parallel to the c axis and the magnetization vector rotating in the basal plane. The same behaviour has been observed in other hexagonal ferrites, both of the same Y structure [Ba2Mg 2 - Y (Albanese et al. 1975)] and of another structure, namely Ba3_~Sr~Zn2Fe24041 [Z-type, usually shortened as Ba3_~SrxZn2-Z (Namtalishvili et al. 1972)]. Like all hexagonal ferrites, also Y- and Z-type crystal structures are formed by the stacking of chemical blocks (usually labelled S, T and R) along the c axis (Smit and Wijn 1959). The starting point of the theory is the fact that no anomaly is observed in Zn2-Y and Zn2-Z containing Ba, and that the substituting Sr ions are localized in well-defined layers inside the T blocks for both structures. Hence, the suggestion of Enz was that substitution of Sr results in a weakening of the superexchange interaction patterns across the planes of the crystal. Then any further counteracting interaction across the same planes may result in a deviation from collinearity of the spin. In fact, neutron-diffraction investigations on a variety of hexagonal ferrites have shown the existence of a peculiar type of magnetic order that is known as 'block' structure (Aleshko Ozhewsky et al. 1969, Namtalishvily et al. 1972, Sizov and Zaitser 1974). The structure of these ferrites is such'that in it one can identify symmetric magnetic blocks, possibly comprising several chemical blocks: inside each magnetic block the localized spins stay mutually parallel, but the total magnetic moment of the block, taken as a whole, may assume a canted orientation with respect to contiguous blocks. The resulting configuration may be either a flat or conical spiral ('block conical spiral', or 'helix'), or an alternating structure
FIRST-ORDER MAGNETIC PROCESSES
455
when the rotation angle from block to block is 180°, thus producing an 'antiphase block' structure (also termed 'block angular' structure). As for the case of compounds containing Sr, in the case of Ba2Mg2-Y, the occupation by a nonmagnetic ion (Mg 2+ ) in the inner octahedral sites of the T block is expected to give rise to a similar weakening of the superexchange coupling between adjacent magnetic blocks. Besides the transition from spiral tO fan structure, these compounds having block-magnetic structures display another transition at a higher field. As a result, the magnetization curve appears as a two-step curve with a sharp knee at the end of the first fiat part. An example is given in fig. 10 of Chapter 6 of Vol. 3 of this handbook, p. 408 (Sugimoto 1982), for the case of Sr2Zna-Y. The figure has been taken from the publication by Enz (1961). Various theoretical models have been proposed for two-sublattice systems, indicating that different canted states and second-order spin-flop transitions are possible (Yamashita 1972, Morrison 1973). Mita and Momozava (1975) give an interpretation of the transition from helix to fan state on the basis of Nagamyia theory (1967). Just as in the interpretation given by Enz (1961), however, no explanation is given for the transition observed at higher fields. The existence of a critical field at the end of the first plateau in the magnetization curve has been interpreted by Sannikov and Perekalina (1969) in terms of a two-sublattice model, but only assuming different values for both the magnetic moments and the anisotropy constants of the sublattices. However, according to this model no spiral structure is possible, and moreover, the hypothesis is contradictory to the inherent symmetry of the two sublattices to be associated with the angled blocks. A step forward in the theory comes from an important qualitative consideration on the effects of the substitution. Indeed, the breaking of the symmetry centers inside the T block opens the way to the action of antisymmetric exchange as proposed by Koroleva and Mitina (1971). A complete theoretical study based on this line of thought has been given by Acquarone (1981). It provides the equilibrium configurations and the phase transitions of a magnetic model system, consisting of two sets of moments of equal magnitude, coupled by isotropic, anisotropic, and antisymmetric exchange, in the presence of uniaxial anisotropy and a transverse magnetic field. It applies both to two-sublattice and to spiral magnetic structures, and gives exhaustive explanations of a variety of two-step magnetization curves of the type described above. The model does not assume any restriction upon the relative strength of the different interactions. 4. Considerations on the experimental methods
4.1. Measurements in high magnetic fields First-order magnetization processes are normally observed at low temperatures, i.e., well below the Curie point, due to the rapid decrease of the anisotropy energy with temperature. It is clear from the graph shown in fig. 6, where an example of the ~l(l + 1) power law for the temperature dependence of anisotropy
456
G. ASTI
is given, that the effect of this behaviour is both to reduce the critical field and to move the state of the system towards the boundary with the normal state. In the majority of cases, at the higher limit of the temperature range, the phenomenon vanishes when the critical field is still relatively high. With only a few exceptions, as is the case with ferrites (Asti et al. 1978) and with Pra(Fe,Co)x 7 (Shanley and Harmer 1973, Melville et al. 1976), this means that magnetic fields above the limit of ordinary laboratory electromagnets are necessary to investigate this type of magnetic phenomena. On the other hand, the measurement of the critical parameters (field and magnetizations) provides useful data that allow deduction of important and precise information on high-order terms of the anisotropy energy. The use of single-crystal samples is obviously the most favourable condition for detecting the transition and a unique way to achieve precise evaluations of the critical magnetization mcr (or m s and m2, in the case of type-2 FOMP). However, the orientation of the crystal in the magnet appears, in most cases, to be quite critical for performing precise measurements, because of the strong sensitivity of Her to deviations from perfect alignment: e.g., a deflection by only two degrees of the applied-field direction from the basal plane in PrCo 5 gives rise to a change of Her by 2 T. Another source of error, which is important only for transitions occurring at low fields, is the demagnetizing field of the sample. If the parallel component M of the magnetization of the crystal is plotted versus the applied magnetic field intensity He, one gets a straight line at the transition that is not vertical, but has a slope equal to d M / d H e = 1 / N , where N is the demagnetizing factor of the specimen. So, in general, a better procedure is to plot directly M as a function of the internal magnetic field H = H e - N M . The use of spherical or ellipsoidal specimens offers obvious advantages: (i) accurate values of N can be determined, and (ii) one can avoid the broadening of the transition due to the non-uniformity of H. This non-uniformity produces distortions in the magnetization curve M ( H ) , particularly at the edges of the transition where the values of m 1 and m: must be identified. The use of continuous magnetic fields allows accurate and reliable determinations, if a large number of measurements are performed in the neighbourhood of the transition point. However, with the exception of a few magnets in the world giving constant fields above 20-25 T the practical limit is in the range around 15 T as is well-known, for both superconducting and water-cooled solenoids. Besides that, one has to take into account the cost of the equipment, which changes by orders of magnitude going through a threshold at about 8-10 T. As a matter of fact, rare earth intermetallic compounds having H~r above 15 T are fairly common. The only way to carry out magnetic measurements at very high magnetic fields is by using pulsed methods. The magnet laboratory at the Amsterdam University is unique, in that it provides the possibility of achieving 40 T, keeping the field intensity constant for a time interval of the order of one tenth of a second. Other installations in the world allow to go even above that limit in a reproducible way by using various pulsed-field techniques (Gr6ssinger 1982), which, in general, give sinusoidal pulse shapes. The main problems arising with the use of pulsed-field apparatus, in performing magnetic measurements, are
FIRST-ORDER MAGNETIC PROCESSES
457
connected with the inherent difficulty in overcoming the various noise sources. For instance, in using the induction method, the flux caused by the specimen has to be detected in the huge background signal coming from the applied field. Despite this, there are several reasons why a pulsed field could be used for magnetic measurements: (i) in a single shot it is possible to measure the whole magnetization curve from zero up to the maximum field; (ii) the continuous and regular variation of the field is a favorable condition for detecting sudden changes such as those associated with a transition; (iii) the experimental setup is relatively simple and economical.
4.2. The singular-point detection technique The inherent high sensitivity of the pulsed-field method for detecting transitions is more and more enhanced if one observes the successive derivatives of the magnetization M with respect to the field H, or to time. This is indeed the principle at the basis of the singular-point detection technique (SPD) (Asti and Rinaldi 1974a), an experimental method that allows to reveal the singularities associated with the field-induced magnetic transitions, utilizing polycrystalline specimens. The SPD theory was originally developed for measurements of the hard-direction anisotropy fields, and has been subsequently extended to the case of FOMP (Asti 1981, Asti and Bolzoni 1985). The magnetization curve of a ferromagnetic crystal shows, in general, a singuarity when it reaches saturation at the anisotropy field H A. The transition is, in principle, of second order and occurs only if the magnetic field is perfectly aligned with the hard direction. For field orientations other than hard and easy directions, the reversible magnetization curve is, in general, a regular function for H = HA, never achieving complete saturation for finite values of H. When we apply a magnetic field to a polycrystalline material, we obtain, in general, a smooth curve M(H) which, at first sight, seems to give no indication of the singularity at H = H A. However, in reality the averaging over all the crystal orientations does not completely cancel the singularity, which is due to the contribution of the crystallites oriented in such a way that their hard directions are nearly parallel to H. The singular point can be detected by observing the successive derivatives d~M/dH n. The shape of the singularity and the order of differentiation n at which it becomes apparent depends on the symmetry of the hard axis and on the ratios of the anisotropy constants. Asti and Rinaldi (1974a), besides giving explicit expressions for the singularity in the most important cases of both cubic and uniaxial symmetry, have determined general rules for obtaining the order of differentiation at which a discontinuity appears at the singular point, both for the cases of longitudinal and transverse susceptibilities. The principle allows precise and reliable measurements of the anisotropy fields using polycrystalline specimens. The most important case is that of uniaxial materials having hard directions in the basal plane; the singularity is observed in the derivative of the differential susceptibility, d2M/dH 2, and has the shape of a cusp on the left-hand side (H < HA) of the peak at H A. Usually, SPD measurements are
458
G. A S T I
easily performed in pulsed fields by direct differentiation of the signal from the pick-up coil. However, also continuous-field techniques can be used by computing the derivatives of the measured M(H) curve (Obradors et al. 1984), or by applying single or double modulation, or triangular waves. The grain size has no effect, while the grain orientation distribution only affects the amplitude of the observed peak, in a way similar to X-ray diffraction in powders. Indeed, this inherent sampling characteristic of the technique, to select the crystallites oriented with their hard axes nearly parallel to H, allows studying distributions of anisotropy fields, and even to determine the angular distribution function of the crystallites (Asti 1987), an application that could be of interest for characterization of permanent magnets and for the study of coercivity mechanisms. The detection of the singularity in the transverse susceptibility configuration is difficult because domain-wall displacement is superimposed to the reversible rotation process for H < H A. However, this high sensitivity to domain walls offers a convenient method for the study and characterization of single-domain particles, as shown recently by Pareti and Turilli (1987). SPD measurements have been performed on a wide variety of hard magnetic materials, such as ferrites, various types of alloys such as MnA1, PtCo, CoCr, and rare earth intermetallic compounds. In all cases, the SPD technique proved to be particularly convenient for extensive studies on solid solutions, on the effects of additives and various chemical modifications, as well as on materials obtained via melt spinning and similar methods. An interesting case is that of the system Zr(Fel_xAlx) 2 which is hexagonal over a wide range of x values (Hilscher and Gr6ssinger 1980). Among the most recent applications are measurements of anisotropy fields made on new classes of rare earth intermetallics, i.e., on Nd2Fe14B with various substitutions (Asti et al. 1987, Pareti et al. 1988, Pareti 1988) and on SmFe,,Ti and similar compounds (Li et al. 1988). Also worth noting is the case of NdCos, for which SPD measurements have been performed of anisotropy fields relative to the c axis as well as to in-plane anisotropy, in the whole range of temperatures where the compound changes its easy direction, from axis to cone and from cone to plane (Marusi 1988). When applied to materials that exhibit FOMP transitions, the SPD technique provides exact determinations of the critical field Her. In fact, it was demonstrated, in the case of uniaxial materials (Asti 1981, Asti and Bolzoni 1985), that Her increases continuously if the crystal is rotated out of the exact orientation, e.g., the c axis parallel (A-case) or perpendicular (P-case) to the magnetic field. As a consequence, the magnetization curve of a polycrystalline aggregate shows a singularity exactly located at Her (see fig. 21). For a random orientation, in the case of a P-type FOMP, there is a discontinuity in the differential susceptibility, dM/dH, equal to, A X = (Ms/Hcr)(m
2
ml)2/(~/-]
- m 1
(4.1)
where m I and m e are the critical magnetizations as defined in section 2.2 [(eq. (2.9)]. In the case of A-type FOMP the discontinuity is in the slope of the
F I R S T - O R D E R M A G N E T I C PROCESSES
459
M/Ms
Her
H
Fig. 21. (Top) Magnetization curve M(H) of an easy axis uniaxial single crystal at various orientations close to ~p = 90 °, where q~ is the angle of magnetic-field direction with respect to the c axis. The F O M P transition is progressively displaced to higher fields with decreasing ~. From left to right the various curves in the figure refer to ~p = 90 °, 86 °, 82 °, 78 °, 74 ° and 70 °. (Bottom) M(H) and first derivative dM/dH for a polycrystalline aggregate having crystallites oriented at random.
differential susceptibility, d 2 M / d H 2, and turns out to be, A X ' = ( M J H c2~ ) ( m 2 - m 1)3/(W~- _ m 2I - wry-_ m22)2 .
(4.2)
For the P-type case Asti and Bolzoni (1985) calculated the slope of AX just above the critical field, i.e., d A x / d H , a quantity that is related to the peak width appearing in d M / d H at H = Her. Perfect agreement was found with computed curves based on an extended Stoner-Wohlfarth model that includes anisotropy terms up to sixth order and arbitrary distribution functions for crystallite orientation. Figure 22 shows an example of a computed curve M ( H ) simulating the case of an oriented polycrystalline sample of PrCo s (Asti et al. 1980). The computed curve turns out to be in perfect agreement with the experimental curve obtained by a pulsed-field experiment at a temperature of 78 K. For comparison the theoretical curve for a single crystal is also shown. The accurate measurement of Her by the SPD technique, together with the value estimated for M~r, allowed precise determinations of the ratios x = K 2 / K 1 and y = K3/K1, through c o m parison with the phase diagram of FOMP (see fig. 4). Above 155 K, the FOMP vanishes and the same technique gave for the same sample the anisotropy field, H A = 2(K, + 2K 2 + 3 K 3 ) / M ~. As regards the determination of the anisotropy constants K 1, K 2 and K 3 from the magnetization curve M ( H ) in oriented polycrystalline materials, there is another remarkable phenomenon, described below, that is important to point out, because it gives strong support for a suitable best-fit procedure. In fact, it is found (Asti et al. 1980) by computer simulations with the above-mentioned extended
460
G. ASTI
M
s
..:-=---.-----"
~
me r Hcr
;
0
;
;
I
6
I
I
I
I
12
I
I
I
I
18
I
;
;
"-
-"
H (Tesla)
Fig. 22. Computed magnetization curve M(H) based on an extended Stoner-Wohlfarth model that includes anisotropy terms up to sixth order and arbitrary distribution function for crystallite orientation. The case shown in the figure is a simulation of an oriented polycrystalline specimen of PrCo 5, after Asti et al. (1980). Solid line: computed curve, dashed line: single-crystal curve with magnetic field in the basal plane, (a) experimental points.
Stoner-Wohlfarth model, that the magnetization curve of an oriented polycrystalline material is reproduced very well by the single-crystal curve oriented at an angle p, which represents the average angle of misalignment of the crystallites. The coincidence of the two curves is surprisingly good up to values of M near saturation, and for ~p of the order of 10-20 °, which means that it is insensitive to changes both of the type and of the width of the angular distribution function of the crystallites. As a consequence, it is possible to use a very simple and rapid best-fit program based on the analytical expression of M(H) for a tilted single crystal, in which the angle enters as a further adjustable parameter just as K1, K 2 and K 3. Even when there is a FOMP, this procedure can be conveniently used in the part of the curve below Her. Besides PrCos, there are other cases where SPD techniques proved to be very effective in giving evidence of the existence of FOMP. Figure 23 shows the progressive change with decreasing temperature of the SPD signal, d2M/dH 2, from a magnet of Nd2Fe14B. As is evident, the downward cusp located at H A is modified until, at 200 K, a sharp positive peak appears that grows more and more with further decrease of the temperature. Clearly, the positive peak is an approximation to the delta function representing the derivative of the discontinuity in the differential susceptibility given by eq. (4.1). Other examples are intermetallic compounds of ThMn12 structure (Li et al. 1988, Deriu et al. 1989, Solzi et al. 1988). Finally, it is worth noting that there is a surprising resemblance between the shape of the SPD peak in d2M/dH 2 for a random polycrystal of a uniaxial easy-axis ferromagnet, and the thickness, 6(H), of a domain wall in a magnetic field perpendicular to the c axis and opposite to its magnetic moment. Expressions for 8 have been obtained by Drring (1966) and Wasilewski (1973), and fig. 24 shows the graphical representation of 8. By using the energy density criterion explained in section 2.6, it is very easy to deduce the expression for 8. In fact, if we assume that all anisotropy constants of higher than second order are vanishing,
FIRST-ORDER MAGNETIC PROCESSES
I
d=M/dt2
f
' T - 290
461
K
l
f
.~
184
0
I
50
100
H(KOe)-~-
Fig. 23. Progressive change with decreasing temperature of the SPD signal,
L
dEM/dH 2, for a magnet of
Nd2Fe14B. Evidently, the downward cusp located at H = H A is modified until, at 200K, a sharp positive peak appears that grows continuously.
E i
i
i
i
I
i
I
~1.4 1.2 1 i
0
i
4
i
6
H(Koe)
I
12
Fig. 24. The thickness, 3(H), of a domain wall of a uniaxial easy-axis ferromagnet, under the action of a magnetic field perpendicular to the c axis and opposite to its magnetic moment [see eqs. (4.4) and (4.5)], after Wasilewski (1973).
we have,
A(dO/do')~ = K , ( 1 - sin 2 OH) + HMs(1 + sin OH),
(4.3)
w h e r e OH is the equilibrium angle within the domains. R e m b e r i n g that h = HMs/2K ~ and sin 0H = h, f r o m the usual definition of the wall thickness, we obtain, 3 = (7r + 2 0 H ) / ( d 0 / d r r ) = (A/K~)I/2(~r + 20H)/(1 + h ) ,
for h < 1 ,
(4.4)
462
G. ASTI
and
= (A/K1)~/2~'/V~,
for h > 1
(4.5)
Expressions (4.4) and (4.5) are in agreement with the expressions given by Wasilewski (1973). References Acquarone, M., 1981, Phys. Rev. B 24, 3847. Acquarone, M., and G. Asti, 1975, J. Magn. & Magn. Mater. 1, 48. Albanese, G., G. Asti, M. Carbucicchio, A. Deriu and S. Rinaldi, 1975, Appl. Phys. 7, 227. Aleshko-Ozhewskii, O.P., R.A. Sizov, I.I. Yamzin and V.A. Lubimtsev, 1969, Sov. Phys.-JETP 28, 425. Andreyev, A.V., A.V. Deryagin, N.V. Kudrevatykh, V.A. Reimer and S.V. Terentiev, 1985, JETP 90, 1042. Asti, G., 1981, IEEE Trans. Magn. MAG-17, 2630. Asti, G., 1987, in: Proc. 5th Int. Symp. on Magnetic Anisotropy and Coercivity in RETM Alloys, Bad Soden 1987, eds C. Herget, H. Kronmuller and R. Poerschke (DPGGMBH, D-5340 Bad Honnef 1, FRG) p. 1. Asti, G., and F. Bolzoni, 1980, J. Magn. & Magn. Mater. 20, 29. Asti, G., and F. Bolzoni, 1985, J. Appl. Phys. 58, 1924. Asti, G., and A. Deriu, 1982, in: Proc. 3rd Symp. on Magnetic Anisotropy and Coercivity in RE-TM Alloys, Vienna 1982, ed. J. Fidler (Technical University of Vienna, Vienna) p. 525. Asti, G., and S. Rinaldi, 1974a, J. Appl. Phys. 45, 3600. Asti, G., and S. Rinaldi, 1974b, in: Proc. 3rd Eur. Conf. on Hard Magnetic Materials, Amsterdam 1974, ed. H. Zijlstra (Bond voor Materialenkennis, Den Haag, The Netherlands) p. 302. Asti, G., and S. Rinaldi, 1977, AIP Conf. Proc. 34, 214. Asti, G., F. Bolzoni, F. Licci and M. Canali, 1978, IEEE Trans. Magn. MAG-14, 883. Asti, G., F. Bolzoni, F. Leccabue, R. Panizzieri, L. Pareti and S. Rinaldi, 1980, J. Magn. & Magn. Mater. 15-18, 561. Asti, G., F. Bolzoni, F. Leccabue, L. Pareti
and R. Panizzieri, 1984, Nd-Fe Permanent Magnets - Their Present and Future Applications, Rep. and Proc. of a Workshop Meeting, Brussels, October 25, 1984, ed. I.V. Mitchel (Commission of the European Community) p. 161. Asti, G., F. Bolzoni and L. Pareti, 1987, IEEE Trans. Magn. MAG-23, 2521. Atzimony, U., M. Dariel, E. Bauminger, D. Lebenbaum, I. Novik and S. Ofer, 1973, Phys. Rev. B 7, 4220. Barbara, B., M.F. Rossignol, H.G. Purwins and E. Walker, 1978, J. Phys. (USA) C 77, L183. Bickford Jr, L.R., 1960, Phys. Rev., Suppl. llS, 1000. Birss, R.R., G.R. Evans and D.J. Martin, 1977, Physica B 86-88, 1371. Bolzoni, F., and L. Pareti, 1984, J. Magn. & Magn. Mater. 42, 44. Bolzoni, F., O. Moze and L. Pareti, 1987, J. Appl. Phys. 62, 615. Boucher, B., R. Buhl and M. Perrin, 1970, J. Phys. & Chem. Solids 31, 2251. Bozorth, R.M., 1936, Phys. Rev. 50, 1076. Buschow, K.H.J., 1980, Rare earth compounds, ch. 4, in: Ferromagnetic Materials, Vol. 1, ed. E.P. Wohlfarth (North Holland, Amsterdam) p. 299. Buschow, K.H.J., 1986, Mat. Sci. Rep. 1, 1. Callen, H.B., and E. Callen, 1966, J. Phys. & Chem. Solids 27, 1271. Casimir, H.B., J. Smit, U. Enz, J.F. Fast, H.P.J. Wijn, E.W. Gorter, A.J.W. Duyvesteyn, J.D. Fast and J.J. de Jong, 1959, J. Phys. Radium 20, 360. Clark, A.E., and E. Callen, 1968, J. Appl. Phys. 39, 5972. Clausen, K.N., and B. Lebech, 1980, J. Magn. & Magn. Mater. 15-18, 347. Cullen, J.R., 1981, J. Appl. Phys. 52, 2038. Cullen, J.R., and E. Callen, 1984, J. AppI. Phys. 55, 2426.
FIRST-ORDER MAGNETIC PROCESSES Cullen, J.R., and E. Callen, 1985, Physica B 130, 289. Deriu, A., G. Leo, O. Moze, L. Pareti, M. Solzi and R.H. Xue, 1989, Hyperfine Interactions, 45, 241. Dillon Jr, J.F., 1963, Domains and domain walls, ch. 9, in: Magnetism, Vol. 3, eds G.T. Rado and H. Suhl (Academic Press, New York) p. 415. D6ring, W., 1966, Mikromagnetismus, in: Encyclopedia of Physics, ed. S. Flfigge, Vol. XVIII/2, Ferromagnetismus, ed. H.P.J. Wijn (Springer, Berlin) p. 405. Eibler, R., R. Gr6ssinger, G. Hilscher, H.R. Kirchmayr, O. Mayerhofer, H. Sassik, X.K. Sun and G. Wiesinger, 1984, Nd-Fe Permanent Magnets - Their Present and Future Applications, Rep. and Proc. of a Workshop Meeting, Brussels, October 25, 1984, ed. I.V. Mitchell (Commission of the European Community) p. 167. Enz, U., 1961, J. Appl. Phys., Suppl. 32, 22S. Ermolenko, A.S., 1976, IEEE Trans. Magn. MAG-12, 992. Ermolenko, A.S., 1979, IEEE Trans. Magn. MAG-15, 1765. Ermolenko, A.S., 1982, in: Proc. 3rd Symp. on Magnetic Anisotropy and Coercivity in RETM Alloys, Vienna 1982, ed. J. Fidler (Technical University of Vienna, Vienna) p. 771. Ermolenko, A.S., and A.F. Rozhda, 1978, IEEE Trans. Magn. MAG-14, 676. Forlani, F., and N. Minnaja, 1969, J. Appl. Phys. 40, 1092. Graetsch, H., F. Haberey, R. Leckebusch, M.S. Rosenberg and K. Sahl, 1984, IEEE Trans. Magn. MAG-20, 495. Gr6ssinger, R., 1982, J. Phys. (USA) D 15, 1545. Gr6ssinger, R., and J. Liedl, 1981, IEEE Trans. Magn. MAG-17, 3005. Gr6ssinger, R., P. Obitsch, X.K. Sun, R. Eibler, H.R. Kirchmayr, F. Rothwarf and H. Sassik, 1984, Mater. Lett. 2, 539. Gr6ssinger, R., X.K. Sun, R. Eibler, K.H.J. Buschow and H.R. Kirchmayr, 1985, J. Phys. (France), Colloq. 46(6), C6-221. Hathaway, K.B., and G.A. Prinz, 1981, Phys. Rev. Lett. 47, 1761. Hilscher, G., and R. Gr6ssinger, 1980, J. Magn. & Magn. Mater. 15-18, 1189. Hiroyoshi, H., H. Kato, M. Yamada, N. Saito, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, Solid State Commun. 62, 475.
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Huang, Ying-Kai, C.H. Wu, Y.C. Chuang, FuMing Yang and F.R. de Boer, 1987, J. Less-Common Met. 132, 317. Kajiwara, S., G. Kido, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, J. Phys. Soc. Jpn. 58, 829. Kazakov, A.A., and N.A. Litvinenko, 1978, Fiz. Met. & Metallloved. 45, 940. Khan, W.I., and D. Melville, 1983, J. Magn. & Magn. Mater. 36, 265. Kido, G., S. Kajiwara, Y. Nakagawa, S. Hirosawa and M. Sagawa, 1987, IEEE Trans. Magn. MAG-23, 3107. Koroleva, L.I., and L.P. Mitina, 1971, Phys. Status Solidi A 5, K55. Krause, D., 1964, Phys. Status Solidi 6, 125. Kudrevatykh, N.V., E.W. Lee and D. Melville, 1986, Fiz. Met. & Metalloved. 61, 898. Li, H.S., B.P. Hu, J.P. Gavigan, J.M.D. Coey, L. Pareti and O. Moze, 1988, J. Phys. (France) 49, 541. Lotgering, F., 1974, J. Phys. & Chem. Solids 35, 1633. Lotgering, F., P.R. Locher and R.P. van Stapele, 1980, J. Phys. & Chem. Solids 41, 481. Marusi, G., 1988, Thesis (University of Parma). Melville, D., W.I. Khan and S. Rinaldi, 1976, IEEE Trans, Magn. MAG-12, 1012. Melville, D., K.M. A1 Rawi and W.I. Khan, 1981, Phys. Status Solidi A 66, 133. Melville, D., J.M. Machado da Silva and J.F.D. Montenegro, 1987, Port. Phys. 18, 49. Meyer, A.J., 1964, C.R. Acad. Sci. 258, 4935. Mita, M., and N. Momozava, 1975, AIP Conf. Proc. 24, 406. Mitsek, A.I., N.P. Kolmakova and D.I. Sirota, 1974, Fiz. Met. & Metalloved. 38, 35. Morrison, B.R., 1973, Phys. Status Solidi B 59, 551. Mukamel, D., E.M. Fisher and E. Domany, 1976, Phys. Rev. Lett. 37, 565. Nagamiya, T., 1967, Solid State Phys. 20, 305. Namtalishvili, M.I., O.P. Aleshko-Ozhewskii and I.I. Yamzin, 1972, Sov. Phys.-JETP 35, 370. Obradors, X., A. Collomb, M. Pernet, J.C. Joubert and A. Isalgue, 1984, J. Magn. & Magn. Mater. 44, 118. Odozynski, R., and W.J. Zietek, 1977, Physica B 86-88, 1373. Paoluzi, A., F. Licci, O. Moze and G. Turilli, 1988, J. Appl. Phys. 63, 5074.
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