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Induced moment antiferromagnet in an external magnetic field—I
I. Phys. Glum. Solids. 1976, Vol. 37, pp. 12!%136. Pergamon Press.
Printed in Gnat Britain
INDUCED MOMENT ANTIFERROMAGNET MAGNETIC FIELD-I
IN AN EXTERNAL
MOLECULAR FIELD THEORY t BAPPANADUN. RAO and YUNG-LI WANG Department of Physics, Florida State University, Tallahassee, FL 32306,U.S.A. (Receioed 1 April 1975;accepted 9 May 1975) Abstract-The behavior of a two-level induced moment antiferromagnet in an external magnetic field is investigated in the molecular field approximation. A significantreduction in the criticalfield and in the sublattice magnetizations is shown. However, the total magnetization rises more rapidly with field and can remain at large value in an external field even at T = 0. The magnetic susceptibility also remains finite at T = 0 in contrast to the case of a permanent moment Ising antiferromagnet. The effects of a ferromagnetic next-nearest neighbor interaction are then examined. It is shown that, in contrast to the usual king antiferromagnets, the ferromagnetic coupling has to exceed a certain value depending on the crvstal field strennth and the antiferromagnetic interaction, to allow for a first order phase transition in a field to occur even at zerotemperature. 1. INTRODUCTION
In rare earth compounds, the crystal field plays a very important role in the nature of the macroscopic magnetic properties and of the collective excitations. This is especially true when the crystal-field-only ground state of the system is a singlet. For such a system, no magnetic ordering can occur even at zero temperature unless the exchange interaction to the crystal field interaction ratio exceeds a certain critical value. In the magnetically ordered state, the moments are induced by the exchange field in a process which leads to the occurrence of the Van Vleck susceptibility in an external magnetic field. Being an induced moment, the moment of each ion varies with the ratio of the effective exchange field to the crystal field. This is in contrast to the usual magnetic system, which has a constant permanent moment on each ion. Induced moment (or singlet-ground-state) systems have been of much interest in recent years and have been found abundant among the rare earth compounds[l]. Most of the discussions have been on the ferromagnetic systems. We present here, in this pair of papers, some theoretical considerations of a simple antiferromagnetic induced moment system, in the hope to motivate some experimental investigations in this direction. As is well known, antiferromagnetic systems have many features not possessed by the ferromagnetic systems, especially in the presence of an external magnetic field. We shall, in this paper, use the molecular field approximation to obtain a gross picture of the behavior of the antiferromagnetic induced moment system in the presence of an external magnetic field. In the following paper, a Green’s function technique in RPA is employed to discuss the collective excitations of the system and their effects on some of the physical quantities calculated in the molecular field approximation. A brief report on part of this work has been given by us[2] in the Sixteenth Annual Conference on Magnetism and Magnetic Materials. Recently Bidaux et al. [3] has also discussed some of the features of the
tSupported by National Science Foundation under Grant No. GH-40174.
antiferromagnetic induced moment system in the molecular field approximation. A two level system with a singlet ground state and a singlet excited state will be assumed in the model calculation. For a more general crystal field scheme, while the molecular field calculation can be generalized straight-forwardly, the Green’s function calculation, which is based on a pseudo-spin formalism[4], becomes much more involved. The qualitative features of our predictions, however, should remain valid in a more complicated system. Perhaps the most studied induced moment antiferromagnet is the TbSb compound[% which has a cubic symmetry. The low lying crystal field levels of Tb ions consist of a singlet ground level and a triplet excited level, for which our calculations can apply only in a qualitative sense. However, compounds with low lying levels consisting of two almost isolated singlets do exist. Some of the cooperative Jahn-Teller rare earth compounds studied recently have also been found to order magnetically. For example, TbAsOa[6] and TbV04[7] order at l+S’K and 0.61”K, respectively, after a Jahn-Teller distortion phase transition at a much higher temperature. For these compounds, in the distorted phase, the energy levels of the Tb ions consist of two closely spaced singlets with the other levels lying at least 60°K above-an ideal singlet-singlet system indeed. The magnetic ordering is of a simple two sublattice antiferromagnetic type. Furthermore, it may be possible to apply a stress to such systems to vary the exchange to the crystal field interaction ratio[8], which can, so far, be changed by diluting the compound as done in TbSb[5]. We shall consider a simple antiferromagnetic induced moment system which consists of two interpenetrating sublattices, and we assume that the nearest neighbors of an ion on one sublattice lie only on the other sublattice. In the next section, the exchange interaction is assumed to exist between an ion and its nearest neighbor ions only, and is antiferromagnetic. We show a significant reduction in the transition temperature, and in the sublattice magnetizations, as the ratio of the crystal field to the exchange interaction increases towards the critical value. The net magnetization in an external field, on the other
129 JPCS VOL. 37 NO. Z-A
130
8. N.
RAO and YUNG-LI WANG
hand, is much Iarger than the corresponding permanent moment system and remains finite even at zero temperature. The magnetic susceptibility also is much higher and remains finite at zero temperature. Specific heat is then calculated. A transition peak is shown on top of the Schottky anomaly associated with the crystal field splitting. The transition peak is lowered as the external magnetic field increases, and disappears at a critical field beyond which the system remains in the paramagnetic phase for all temperatures. The effects of a ferromagnetic Inca-sublatti~e coupling (next nearest neighbor interaction) are examined in the third section. We show that, in contrast to a permanent moment antiferroma~et, which shows a first order phase transition for any finite ferromagnetic intra-sublattice coupling at suthciently low temperatures{9], the phase transition remains second order for an induced moment system even at zero temperature, if the ferromagnetic intra-sublattice coupling is below a certain value depending on the crystal-field to the exchange interaction ratio. 2. A~I~RRO~GNETI~
NEARESTNEIGHBOUR
the two sublattices by A and B with N/2 ions on each sublattice. The angular brackets represent the canonical thermal average. X1 gives the correlations of spin fluctuations and will be ignored in the molecular field approximation. Let IO,) and IL) denote the crystal field ground state and the excited state, and CY= (0, IJ”II, ) be the offdiagonal matrix element of J’ between the two states. A linear combination of these two crystal field states diagonalizes the molecular field Hamiltonian. ~O}~=cos~~JO~)-sin~~~l~)
(2.5a)
fl),=sinBA~Oc)+cosBAJlc)
(2Sb)
IO), =cos &/O,)-sin&jl,)
(2Sc)
I& = sin &lo,) + cos &II,)
(2Sd)
The diagonalizing angles Ba and es are given by tan [email protected] = 2a@YO)(JL.J - g&-I)/A
(2.6)
For convenience of presentation the following reduced quantities are defined.
STETSON
For a two-sublattice ~tife~omagnetic induced moment system with newest-neigh~r-only interaction the HamiItonian is
x = (Ja%
Y = t~~*)~ff, h = g~~ff~~(O)~)* t = kBT/(&O)a2).
x =2 ven +& AJ f I, - gfb@. C X n
n
m
where V,, is the crystal field potential which produces a singlet ground state and a singlet first excited state with an energy gap A; the other eigenstates are assumed to have much higher energies and can be ignored in our discussion. The effective exchange integral j%,*is positive and thereby gives the antiferromagnetic coupling; the summation is over all nearest neighbor pairs. The last term represents the effect of the external magnetic field. To obtain the effective field ~amiltonian we let i;
=J; -(x”)&
(2.7)
(2.1)
(2.2)
We obtain, from eqns (2.5) and (2.6), the equations which determine the reduced moments,
’ = ‘L/(A
&f_
y)*)
‘%lh
t
xhA*+(h-Y)") >’ (2Xaf
h-X VW*+jh - ‘)*)),(2.8b) Y=~fA2i-(h-x)‘)~~~”
where A = A/(Z~(O)~~ is the ratio of the crystal field to the exchange interaction. Phase transition ; critical fields ; sublattice magnetizations From the moment eqns (2.8a, b), it can be shown that, for H = 0 and T = 0, the critical value of A for an infinitesimal moment to exist is (a)
Where i, is a unit vector along the direction of the magnetic ordering. We obtain X=Xllp,sX,
(2.3)
where
A =l,
(2.9)
as in the fe~omagnetic case. For A < 1, the system orders antiferromagnetically with sublattice moments (2.4a)
x=-y
=t/(l-A*),
at zero temperature and zero external field. A second order phase transition to a paramagnetic phase occurs at the Neel temperature,
and
x, =6 &uxl
(2.4b)
&PO is the molecular field Hamiftonian and we have labeled
(h = 0)
(2Sl)
131
Inducedmomentantifenomagnetin an external magnetic field In an external field, the transition temperature is reduced, but the phase transition is always of second order. To obtain the critical temperature in a field, we note that, in the ordered phase, y = x - 2~, where 2r is the difference of the sublattice magnetizations and is small near the transition. If we expand the right hand side of eqn (2.8a) to the first order of E, we obtain the equations determining the critical curve in the h-t plane.
h=
6 +f(i,
(2.13)
f),
where * f(h, r) = V(A!+ h3 tanh
0.2-
I
I
I
I
I
0.2 0.4 0.6 0.8
I
I
I.0 1.2
I
.1 Fig. 1. Phase diagrams in the h-t plane for some representative values of A. A =0 is the permanent moment limit. A = A(2~(0)~2),h = g~~~~~(O)~), t = ~~~/~~O)a~).
(2.14)
The above equations can also be obtained in the more systematic approach of Landau’s expansion of the free energy function as shown in the next section. In Fig. 1, we show the results of some representative calculations. We note that, as A goes to zero, the system approaches the usual permanent moment system, so the curve for A = 0 represents a permanent moment system. The curve for A = 0.8, on the other hand, represents a highly induced moment system and lies much lower. At zero temperature the critical field has a simple expression, h, = (1 f AZ’3)~(1-A*“)
0.4-
v(A*t I?) t .
(t = 0).
(2.15)
h, = 1 for A = 0 (the permanent moment limit) and drops to zero like t/(1 - A) as A approaches the critical value. The bulge appearing in the permanent moment curve is presumably a feature of the molecular field approxjmation, as it disappears in an improved calculation[lO]. The bulge, however, becomes less prominent in the induced moment system, and disappears in the highly induced cases. To examine the more detailed behavior of the system, we solve the moment eqns (2.8) self-consistently to obtain the reduced moments x and y as functions of temperature and field, for various values of A. Figure 2 shows the behavior of the reduced moments as field increases at a fixed temperature Fig. 2(a) represents the permanent moment case, and Fig. 2(b) represents the highly induced moment cases. While a substantiai reduction in the
I.0
A =0.8
0.6-
Fig. 2. Field variation of the reduced sublattice m~ents. (a) A = O-1,representing the permanent moment limit, (b) A = 0.8, ahiiy induced moment system.
132
3. N.
RAO
and YUNG-LIWANG
sublattice moments is observed for the induced moment system (A = 0.8), the net magnetization rises much faster in an external field. Comparing the two figures, we note that the behavior of the induced moment system in a field at zero temperature is more like that of a permanent moment system at a finite temperature; the crystal field in the singlet ground state systems acts in a certain way like the temperature. To show the behavior of the sublattice moments in a fixed field as temperature varies, we again plot, in Fig. 3, the reduced moments as functions of 6 for A = 0.1 and A = 0.8. We choose h = 0.5 in the illustration. The first striking feature of the induced moment system is the high net magnetization at all temperatures in~Iuding zero temperature. This is in contrast to a permanent moment antiferromagnetic system, in which the net magnetization vanishes at zero temperature for all fields less than the critical field. The fact that the magnetization of an induced moment system remains finite at zero temperature is simply due to the fact that, in an induced moment of an ion on one sublattice, and reduces that of and varies with the total field acting on the ion even at T =O. Thus, an external magnetic field increases the moment of an ion of on one sublattice, and reduces that of an ion on the other sublattice. We also note that the transition tem~ra~re and the sublattice moments are substantially reduced in an induced moment system. (b) Magnetic susceptibilities For a two level system, the perpendicular susceptibility is always zero and the parallel susceptibility in units of g’~B’/,$(0) is given by
We calculate x from the moment eqns (2.8) for fixed values of h and A, and plot, in Fig. 4, x as a function of temperature. For h = 0, x shows a maximum at the transition temperature similar to that of a usual antiferromagnet. However, for an induced moment system, x remains finite at T = 0, in contrast to an antiferromagnet with permanent moments, for which x vanishes at T = 0. This phenomenon again is due to the variable moments of ions in an induced moment system. The value of x at T = 0 is given by ,y=A*/(l+A2),
(T=O)
12.17)
which becomes zero in the permanent moment limit (A = 0), and approaching l/2 in the highly induced limit (A + 1). Noting that the value of x at the transition temperature is always l/2 (in the present unit), independent of A, for a highly induced moment system x remains rather constant in temperature in the ordered phase, as shown for the case of A = 0.8. We also note that the ratio of the susceptibility at zero temperature to that the ratio of the susceptibility at zero temperature to that at the N&e1temperature provides a direct measurement of A, (2.18)
~(O)~~(TN)=~A~~(I+A*).
In a magnetic field, there is always a jump in x at the transition temperature. The discontinuity in x increases with the magnetic field. It can be shown that the value of x at the transition temperature as one approaches from the ordered phase is
(2.16)
/a 1.0
h =0.5
I -
A=0.1
-
-
-..._
‘\ 0.8
\
0.2 t
h=O /’ / /
Or2 0.4
I
1
0.6
0.8
I
1.0
1
1.2
d
I.4
0
IB
I.8
t
Fig. 3. Temperature variation of the reduced sublattice moments for A = 0.1 and for A = 0.8.
Fig. 4. Temperature variation of susceptibility (in units of ~‘~B2/,$(0))for A = O-8and A = 0.25 in the absence of external field and in h = 0.6.
Induced moment antifermmagnet in an external magnetic field
where f(h)=
(9) q(A’+ C’) tanh
(2.20)
and f(“’ is the n-th derivative of f(A) with respect to 6, evaluated at 6 given by solving the equatian, 6 +f(B) = I?.
(2.21)
The value of the susceptivity at the transition temperature, approaching from the p~a~eti~ phase, can be easily shown to be l/2, independent of the magnetic field. For h = 0, p’(6) = 0 and x(ti) = (l/2), thus no discontinuous jump in the susceptibility occurs. In the permanent moment limit, A = 0, eqn (2.19) reduces to x =i(4-3&),
(2.22)
as obtained by Bidaux et at.1111. While again the susceptibility of a permanent moment Tsing ~tife~rna~ net always vanishes at T =4 the susce~t~~~~ty of an induced moment system can remain at a large value. (c)
Specific heat The internal energy of the system is given by E = (X~)
The resutts of this calculation are shown in Fig. 5 for an induced moment system of A = 0.8. The broad peak of each curve is the “Schottky anomaly” associated with the crystal field splitting, Because of the existence of the energy gap between the two eigenstates even in the paramagnetic phase, the entropy does not become maximum until at ingnite temperature. As the temperature is lowered, the upper level gets depopulated and the entropy decreases. The rate of change of entropy is greatest for k,T = I%,the energy gap, thus gives rise to a broad peak around b. The sharp peak is associated with the magnetic phase transition. The size of the peak is much smaller than it is usually for the magnetic ordering (in a permanent moment system). This decrease in the size of the peak is clearly due to the lowering of the entropy, as temperature is lowered, before the magnetic ordering sets in. The same behavior has been predicted and induced moment observed in ferromagnetic systemsl4,lZj. In an external magnetic field, while the phase transition of a ferromagnetic system disappears, a phase transition in the ~tiferrom~etic system persists until the zero temperature (or the maximum) critical field is reached. The curve for k =0x5 shows the magnetic transition in a field. The transition peak is further reduced since the level splitting is enhanced by the field. In a large field (k = 0.97, for example), no transition is observed, and the system remains in the paramagnetic phase for all temperatures.
(2.23)
and the specific heat in a constant magnetic field is (2.24)
i-8
Fig. 5. Temperature variation of specific heat for A k = 0.0-5and 0-97.
133
= 0.8 in a B&f,
3.EFFECTS OF NEXT NEAREST-NEIGFIBOR b'TER4CTIONS Itiswelfknown that, in the case of a usual permanent
moment Ising ant~e~om~et, the presence of any &rite amount of ferromaguetic next-newest-nei~~r (intrasublat~ce} inte~ction would cause the transition in a lieId to become first order at suthciently low temperatures. For the induced moment system, on the other hand, as we shall show below, the ratio of the nnn ferromagnetic interaction to the nn antiferromagnetic interaction has to exceed a certain value for the system to undergo a first order phase transition in a field even at T = 0. We shall, in this section, employ the Landau’s theory of second order phase transition113J in conjunction with a full examination of the behavior of the free energy func~on to obtain the phase boundaries in the magnetic ~eId-tem~ra~ plaue, to locate the tricritical pointsIl4f, and to map out, in the nnn to nn exchange interaction ~ti~temperat~e plane, regions in which different types of transition would occur. It should be noted that for a permanent moment antiferromagnet, if the ferromagnetic to the antiferromagnetic interaction ratio and the temperature fall in a certain narrow region, the system would undergo a first order phase transition to an intermediate phase, as the magnetic field increases, before making a second order phase transition to the paramagnetic phase. This type of multiple-transition has been named by Bidaux et aI.ltl] as the @-type transition. Similar phase ~~si~~a occm in the induced moment systems, but on even more strict conditions, and only for systems with small values of A (permanent mom&-liie systems),
134
B. N.
RAO
and lr _ ._ YUNG-LI WANG
(a) Free energy and moment equations In the presence of the nnn ferromagnetic interaction, the only additional term to the Hamiltonian (2.1) are 2”“. = -&a;.$ ,’ 6.4
*X.-&&j, I ’ rB
..?##, (3.1)
where y;,, and $drn, are the effective exchange integrals which couple an ion to its next nearest neighbors (intrasublattice coupling). As in Section 1, the Hamiltonian (2.1) with the additional terms (3.1) can be split into the molecular field term X0 and the spin fluctuation term XI. In the molecular field approximation, the partition function is defined as 2 = Tr(e-@“),
(3.2)
where the trace is taken over the eigenstates lO),lO),, IO).,I&, II), IO), and Il)A(&. The Gibb’s free energy G =-kTlnZ
(3.3)
and the result is G 2= $(2b t 1)m’t ~‘1 2@(O)+dP’(O))a - f In cash - f In cash
t ~(A2t[ht(2btl)m-cl* t
(3.4)
b = -j(O)U(O)
(3.5)
+ $V’)l,
and is defined in accordance with Ref. [ll] in which an Ising antiferromagnetic system was discussed. b = -1 corresponds to the case of vanishing ferromagnetic intra-sublattice coupling, and b = 0 corresponds to the case of a much stronger ferromagnetic intra-sublattice coupling. The reduced quantities h, t, and A are redefined by
G=GotGz~‘tGq~~tG6~‘+...
Idp(O)tdp’(O)] in
eqns
(2.7).
Minimizing the free energy (3.4) with respect to m and l we obtain the equations relating m and E or the sublattice magnetizations x and y. The latter pair of equations are the same as eqns (2.8) with h-y replaced by h t (b t 1)x t by and h-x by h t (b t 1)y t bx. As in Section 2, the pair of equations should be solved simultaneously to obtain the sublattice magnetizations. Here, since the phase transition can be first order, it is essential to obtain the free energy associated with the solution. We therefore look for the minima of the free energy as a function of E after m is eliminated with the help of the moment equations. This is a more convenient procedure, which yields all the information we need in one calculation. As we shall show later in this section, for E 2 0, there may be
(3.6)
For a second order phase transition to occur, the necessary conditions are Gdt, h) = 0, Gd(t, h)>O.
(3.7a) (3.7b)
Equations (3.7) give the critical curve in the (h, t) plane for a second order phase transition. As pointed out by Landau[l3], the boundary separating two phases of different symmetry can not stop at a point. It may, however, change at some point into a first order phase transition line. This happens when Gd(f, h) = 0, with the stability condition G6(t, h) >O. The point at which a second order phase transition line ends and joins to the first order phase transition line is called the tricritical point by Griffiths[l4]. Thus the tricritical point is given by G&r h) = 0, Gn(t, h) > 0.
>
where m = (x + y)/2 is the reduced total magnetization, 2r =x - y, the difference of the reduced sublattice magnetizations as defined before. Vanishing in the paramagnetic phase, E can be taken as an ordering parameter for an antiferromagnet in an external field.
f(0)
(b) Phase boundaries and tricritical points Following Landau[l3] we expand the free energy in powers of the ordering parameter E:
v’(A* t [h t (26 + 1)m t ~1’)
-tln2
by replacing
more than one minimum existing simultaneously in the free energy; the one with the lowest free energy corresponds to the physical state.
(3.8a) (3.8b)
If condition (3.8b) is not satisfied, the presupposition of a continuous phase transition is obviously false; the behavior of the free energy function needs to be examined in detail. Indeed, this happens in a narrow range of values of b and t, in which the free energy shows a development of an additional minimum at a smaller but nonzero value of E as field increases, and the system would first undergo a first order phase transition to an intermediate phase before making a second order transition to the paramagnetic phase. This falls into the category of P-type transition occurring in the permanent moment Ising antiferromagnets and discussed by Kanamori et al. [9] and by Bidaux et al. [ll] In fact even if conditions (3.8) are satisfied, the point may still not be a tricritical point. An example showing that conditions (3.8) are necessary but not sufficient conditions has been given by Rao and Wang[lSl. It is a simple matter to show that eqns (3.7) yield f”’ = 1
(3.9a)
and (3.9b) where f(& r) and f’“‘(6, t) have been defined in Section 2 [eqn (2.14)], and 6 is related to the external field h by 6 = h + (2b t l)f(h)
(3.10)
For t = 0, eqn (3.9a) gives h,. = (A 2’3- (26 t l))d( 1- A 2’3)
(3.11)
Induced moment ~ntjfe~oma~et
The N&l temperature at h = 0 is once again given by eqn (2.31) with the present definition of A. The tricritical conditions (3.8) are given by (3.12a)
in an extem& magnetic field
135
for the tricritical point,
(3.14)
for -$ < b c 0, as shown by Kanamori et al. [91 and by Bidaux a nL f 1II- For 8r1yEnik value of A, the caIculation (3. i2b) can be carried out by first obtairiing 6 from eqns (3_9a)for where a Exed vaIue of t and A and then ev~ua~g the (3.13a) derivatives off(&) at this value of K Equation (3.12a) theri gives the critical value of b at which the nature of the transition changes from second order to first order, if (3.13b) condition (3.12b) holds. If condition (3.12b) fails, the system is found tn undergo the multiple-transition (P-type with transition) as discussed above. A = _f(3)/p Figure 6 shows tho LOCUS of the tricritical points in the (3.13c) b-A plane at T = 0. the tricritical line ends at the tip of a For A = I), the permanent moment limit, eqlrs (3.12) gives, crescent shaped are&, and we find three regions. If the vdues of b and A of %Isystem fall in the F-region7 a single first order phase transitioa will take place in a magnetic field; if they fall in the S-region, a single seco& order phase ~ansi~ou in a fiefd is expected; if they faI1 in the small p-region, the $3 type transition would occur. The &type transition ceases to exist for A >053536_ FOF A ~0.3536, the ranges of values of b and t to allow for a P-type transition to occur is also very narrow so we do not perceive any practical interest in the P-type transition. As is c&r from the figure that a first order phase transition does rrol:occur for any small amount of ferromagnetic intraWsublattice coupling even at T = 0, as does in a permanent moment system. The larger the value of A is, the higher the v&x of b (stronger ferroma~etic any-sublattic~ ~u~li~~) is required for a first order phase transition to occur. For finite temperatures~ a stronger ferroma~~t~c in~~s~bl~t~ce s&lattice coupling is Fig, 6, Regionsin (& A ) piarre in v&i& a sin& first a&r (z;‘), a needed. Figure 7 shows the regions in the b-t plane for singly secondorder(31,and a firstorder foIlowed by a secondorder three representative cases. The first order region is to the i/3) transition in a field may occur at T = 0. B measuresthe nn antiferrormgnetic to the nnn ferromagneticcouplingratio and is Ieft of each curve lab&d by the value of A for the system definedin eqn (3.5). under consideration, and the second order region lies to
136
B. N. Rho and
0.3
-
0.2
-
YUNG-U WANG
b=-0.45
b=-03
-e-----m.__ h
0.1 A-O.8
0.0
Mdewlor
field
theory
1
1
I
0.1
0.2
0.3
I 0.4
0.5
0.6
0.7
0%
t
Fig. 8. Phase diagrams in the h-t pIane for A = 0.8, b = - 0.45 and b = - 0.3.6 measures the relative strength of the nn antife~om~etic to the nnn fe~oma~etic couplings and is defined in eqn (3.5). Solid lines indicate second order phase transitions, and dashed lines, first order phase transitions.
the right of the curve. For A = 0.25, a small P-region exists. The ferromagnetic intra-sublattice coupling needed to bring about a first order phase transition in a field can be found from the figure; it increases with the temperature as expected. It is also clear that the larger the value of A is, the smaller the first order region becomes. Finally, to give an example of the phase diagram in (h, t) plane, we consider a highly induced moment system with A = O-8 for two values of b. Figure 8 shows the phase boundaries. For b = -0.45, which corresponds to a ferromagnetic coupling being I.2 times the antiferromagnetic coupling, the phase transition is always second order. Increasing the strength of ferromagnetic coupling to 2.3 times that of the antiferromagnetic coupling (b = - 0.3), the second order line (solid line) ends at t = 0,484, and a first order line starts (dashed line). This latter feature is similar to a permanent moment system in which, however, the tricritical point always exists, for any size of ferromagnetic intra-sublattice coupling. REFE~NC~ 1.
For a review see Cooper B. R., CRC C~i~calReviewsof Solid State Sciences 3, 83 (1972).
2. Wang Y. L. and Rao B. N., J. Appl. Phys. 42, 1414(1971). 3. Bidaux R., Gavignet-Tillard A. and Hamman J., J. de Phys. 34, 19 (1973). 4. Wang Y. L. and Cooper B. R., Phys. Rev. 172,539(1968);Ibid. 185, 6% (1969). 5. Cooper B. R. and Vogt 0.. Phys. Rev. B 1,1218 (1970).Holden T. M., Svensson E. C. and Buyers W. J. L., Phys. Rev. B 10, 3864 (1974). 6. Wtichner W., Biihm W., Kable H. G., Kasten A. and Laugsch J.. Phvs. Stat. Sal. (bf 54, 273 (1972). 7. KableH. G., Simon‘A~,Wtichner W., Phys. Stat. SOL (b) 61, KS3 (1974). 8. Mcpherson J. W. and Wang Y. L. To be published. 9. Kanamori J., Motizuki K. and Yoshida K., Busseiron Kenkyu 63, 28 (1953) (in Japanese). A summary of the results in English appears as an appendix to Motizuki K., L Pkys. SW. Japan 14, 759 (1959). 10. Kasteleijn P. W., Physica 22, 387 (1956). 11. Bidaux R., Carrara P. and Vivet, J. Phys. Chem. Solids 28, 2453 (1967). 12. Andres K., Bucher E., Darack S. and Maita J. P., Phys. Reu. B 6, 2716 (1972);Lee K. N., Bachmann R., Beballe T. H. and Maita J. P., Phys. Rev. B 2, 4580 (1970). 13. Landau L. D. and Liishitz E. M., Statistical Physics, 2nd Edn. p. 424 Addison-Wesley, Mass. (1969). 14, G~ffiths R. B., P&s. Rev. Let&. 24, 715 (1970). 15. Rao B. N. and Wang Y. L., J. Phys. CRem. S~~~d~36, 699 (1974).
Printed in Gnat Britain
INDUCED MOMENT ANTIFERROMAGNET MAGNETIC FIELD-I
IN AN EXTERNAL
MOLECULAR FIELD THEORY t BAPPANADUN. RAO and YUNG-LI WANG Department of Physics, Florida State University, Tallahassee, FL 32306,U.S.A. (Receioed 1 April 1975;accepted 9 May 1975) Abstract-The behavior of a two-level induced moment antiferromagnet in an external magnetic field is investigated in the molecular field approximation. A significantreduction in the criticalfield and in the sublattice magnetizations is shown. However, the total magnetization rises more rapidly with field and can remain at large value in an external field even at T = 0. The magnetic susceptibility also remains finite at T = 0 in contrast to the case of a permanent moment Ising antiferromagnet. The effects of a ferromagnetic next-nearest neighbor interaction are then examined. It is shown that, in contrast to the usual king antiferromagnets, the ferromagnetic coupling has to exceed a certain value depending on the crvstal field strennth and the antiferromagnetic interaction, to allow for a first order phase transition in a field to occur even at zerotemperature. 1. INTRODUCTION
In rare earth compounds, the crystal field plays a very important role in the nature of the macroscopic magnetic properties and of the collective excitations. This is especially true when the crystal-field-only ground state of the system is a singlet. For such a system, no magnetic ordering can occur even at zero temperature unless the exchange interaction to the crystal field interaction ratio exceeds a certain critical value. In the magnetically ordered state, the moments are induced by the exchange field in a process which leads to the occurrence of the Van Vleck susceptibility in an external magnetic field. Being an induced moment, the moment of each ion varies with the ratio of the effective exchange field to the crystal field. This is in contrast to the usual magnetic system, which has a constant permanent moment on each ion. Induced moment (or singlet-ground-state) systems have been of much interest in recent years and have been found abundant among the rare earth compounds[l]. Most of the discussions have been on the ferromagnetic systems. We present here, in this pair of papers, some theoretical considerations of a simple antiferromagnetic induced moment system, in the hope to motivate some experimental investigations in this direction. As is well known, antiferromagnetic systems have many features not possessed by the ferromagnetic systems, especially in the presence of an external magnetic field. We shall, in this paper, use the molecular field approximation to obtain a gross picture of the behavior of the antiferromagnetic induced moment system in the presence of an external magnetic field. In the following paper, a Green’s function technique in RPA is employed to discuss the collective excitations of the system and their effects on some of the physical quantities calculated in the molecular field approximation. A brief report on part of this work has been given by us[2] in the Sixteenth Annual Conference on Magnetism and Magnetic Materials. Recently Bidaux et al. [3] has also discussed some of the features of the
tSupported by National Science Foundation under Grant No. GH-40174.
antiferromagnetic induced moment system in the molecular field approximation. A two level system with a singlet ground state and a singlet excited state will be assumed in the model calculation. For a more general crystal field scheme, while the molecular field calculation can be generalized straight-forwardly, the Green’s function calculation, which is based on a pseudo-spin formalism[4], becomes much more involved. The qualitative features of our predictions, however, should remain valid in a more complicated system. Perhaps the most studied induced moment antiferromagnet is the TbSb compound[% which has a cubic symmetry. The low lying crystal field levels of Tb ions consist of a singlet ground level and a triplet excited level, for which our calculations can apply only in a qualitative sense. However, compounds with low lying levels consisting of two almost isolated singlets do exist. Some of the cooperative Jahn-Teller rare earth compounds studied recently have also been found to order magnetically. For example, TbAsOa[6] and TbV04[7] order at l+S’K and 0.61”K, respectively, after a Jahn-Teller distortion phase transition at a much higher temperature. For these compounds, in the distorted phase, the energy levels of the Tb ions consist of two closely spaced singlets with the other levels lying at least 60°K above-an ideal singlet-singlet system indeed. The magnetic ordering is of a simple two sublattice antiferromagnetic type. Furthermore, it may be possible to apply a stress to such systems to vary the exchange to the crystal field interaction ratio[8], which can, so far, be changed by diluting the compound as done in TbSb[5]. We shall consider a simple antiferromagnetic induced moment system which consists of two interpenetrating sublattices, and we assume that the nearest neighbors of an ion on one sublattice lie only on the other sublattice. In the next section, the exchange interaction is assumed to exist between an ion and its nearest neighbor ions only, and is antiferromagnetic. We show a significant reduction in the transition temperature, and in the sublattice magnetizations, as the ratio of the crystal field to the exchange interaction increases towards the critical value. The net magnetization in an external field, on the other
129 JPCS VOL. 37 NO. Z-A
130
8. N.
RAO and YUNG-LI WANG
hand, is much Iarger than the corresponding permanent moment system and remains finite even at zero temperature. The magnetic susceptibility also is much higher and remains finite at zero temperature. Specific heat is then calculated. A transition peak is shown on top of the Schottky anomaly associated with the crystal field splitting. The transition peak is lowered as the external magnetic field increases, and disappears at a critical field beyond which the system remains in the paramagnetic phase for all temperatures. The effects of a ferromagnetic Inca-sublatti~e coupling (next nearest neighbor interaction) are examined in the third section. We show that, in contrast to a permanent moment antiferroma~et, which shows a first order phase transition for any finite ferromagnetic intra-sublattice coupling at suthciently low temperatures{9], the phase transition remains second order for an induced moment system even at zero temperature, if the ferromagnetic intra-sublattice coupling is below a certain value depending on the crystal-field to the exchange interaction ratio. 2. A~I~RRO~GNETI~
NEARESTNEIGHBOUR
the two sublattices by A and B with N/2 ions on each sublattice. The angular brackets represent the canonical thermal average. X1 gives the correlations of spin fluctuations and will be ignored in the molecular field approximation. Let IO,) and IL) denote the crystal field ground state and the excited state, and CY= (0, IJ”II, ) be the offdiagonal matrix element of J’ between the two states. A linear combination of these two crystal field states diagonalizes the molecular field Hamiltonian. ~O}~=cos~~JO~)-sin~~~l~)
(2.5a)
fl),=sinBA~Oc)+cosBAJlc)
(2Sb)
IO), =cos &/O,)-sin&jl,)
(2Sc)
I& = sin &lo,) + cos &II,)
(2Sd)
The diagonalizing angles Ba and es are given by tan [email protected] = 2a@YO)(JL.J - g&-I)/A
(2.6)
For convenience of presentation the following reduced quantities are defined.
STETSON
For a two-sublattice ~tife~omagnetic induced moment system with newest-neigh~r-only interaction the HamiItonian is
x = (Ja%
Y = t~~*)~ff, h = g~~ff~~(O)~)* t = kBT/(&O)a2).
x =2 ven +& AJ f I, - gfb@. C X n
n
m
where V,, is the crystal field potential which produces a singlet ground state and a singlet first excited state with an energy gap A; the other eigenstates are assumed to have much higher energies and can be ignored in our discussion. The effective exchange integral j%,*is positive and thereby gives the antiferromagnetic coupling; the summation is over all nearest neighbor pairs. The last term represents the effect of the external magnetic field. To obtain the effective field ~amiltonian we let i;
=J; -(x”)&
(2.7)
(2.1)
(2.2)
We obtain, from eqns (2.5) and (2.6), the equations which determine the reduced moments,
’ = ‘L/(A
&f_
y)*)
‘%lh
t
xhA*+(h-Y)") >’ (2Xaf
h-X VW*+jh - ‘)*)),(2.8b) Y=~fA2i-(h-x)‘)~~~”
where A = A/(Z~(O)~~ is the ratio of the crystal field to the exchange interaction. Phase transition ; critical fields ; sublattice magnetizations From the moment eqns (2.8a, b), it can be shown that, for H = 0 and T = 0, the critical value of A for an infinitesimal moment to exist is (a)
Where i, is a unit vector along the direction of the magnetic ordering. We obtain X=Xllp,sX,
(2.3)
where
A =l,
(2.9)
as in the fe~omagnetic case. For A < 1, the system orders antiferromagnetically with sublattice moments (2.4a)
x=-y
=t/(l-A*),
at zero temperature and zero external field. A second order phase transition to a paramagnetic phase occurs at the Neel temperature,
and
x, =6 &uxl
(2.4b)
&PO is the molecular field Hamiftonian and we have labeled
(h = 0)
(2Sl)
131
Inducedmomentantifenomagnetin an external magnetic field In an external field, the transition temperature is reduced, but the phase transition is always of second order. To obtain the critical temperature in a field, we note that, in the ordered phase, y = x - 2~, where 2r is the difference of the sublattice magnetizations and is small near the transition. If we expand the right hand side of eqn (2.8a) to the first order of E, we obtain the equations determining the critical curve in the h-t plane.
h=
6 +f(i,
(2.13)
f),
where * f(h, r) = V(A!+ h3 tanh
0.2-
I
I
I
I
I
0.2 0.4 0.6 0.8
I
I
I.0 1.2
I
.1 Fig. 1. Phase diagrams in the h-t plane for some representative values of A. A =0 is the permanent moment limit. A = A(2~(0)~2),h = g~~~~~(O)~), t = ~~~/~~O)a~).
(2.14)
The above equations can also be obtained in the more systematic approach of Landau’s expansion of the free energy function as shown in the next section. In Fig. 1, we show the results of some representative calculations. We note that, as A goes to zero, the system approaches the usual permanent moment system, so the curve for A = 0 represents a permanent moment system. The curve for A = 0.8, on the other hand, represents a highly induced moment system and lies much lower. At zero temperature the critical field has a simple expression, h, = (1 f AZ’3)~(1-A*“)
0.4-
v(A*t I?) t .
(t = 0).
(2.15)
h, = 1 for A = 0 (the permanent moment limit) and drops to zero like t/(1 - A) as A approaches the critical value. The bulge appearing in the permanent moment curve is presumably a feature of the molecular field approxjmation, as it disappears in an improved calculation[lO]. The bulge, however, becomes less prominent in the induced moment system, and disappears in the highly induced cases. To examine the more detailed behavior of the system, we solve the moment eqns (2.8) self-consistently to obtain the reduced moments x and y as functions of temperature and field, for various values of A. Figure 2 shows the behavior of the reduced moments as field increases at a fixed temperature Fig. 2(a) represents the permanent moment case, and Fig. 2(b) represents the highly induced moment cases. While a substantiai reduction in the
I.0
A =0.8
0.6-
Fig. 2. Field variation of the reduced sublattice m~ents. (a) A = O-1,representing the permanent moment limit, (b) A = 0.8, ahiiy induced moment system.
132
3. N.
RAO
and YUNG-LIWANG
sublattice moments is observed for the induced moment system (A = 0.8), the net magnetization rises much faster in an external field. Comparing the two figures, we note that the behavior of the induced moment system in a field at zero temperature is more like that of a permanent moment system at a finite temperature; the crystal field in the singlet ground state systems acts in a certain way like the temperature. To show the behavior of the sublattice moments in a fixed field as temperature varies, we again plot, in Fig. 3, the reduced moments as functions of 6 for A = 0.1 and A = 0.8. We choose h = 0.5 in the illustration. The first striking feature of the induced moment system is the high net magnetization at all temperatures in~Iuding zero temperature. This is in contrast to a permanent moment antiferromagnetic system, in which the net magnetization vanishes at zero temperature for all fields less than the critical field. The fact that the magnetization of an induced moment system remains finite at zero temperature is simply due to the fact that, in an induced moment of an ion on one sublattice, and reduces that of and varies with the total field acting on the ion even at T =O. Thus, an external magnetic field increases the moment of an ion of on one sublattice, and reduces that of an ion on the other sublattice. We also note that the transition tem~ra~re and the sublattice moments are substantially reduced in an induced moment system. (b) Magnetic susceptibilities For a two level system, the perpendicular susceptibility is always zero and the parallel susceptibility in units of g’~B’/,$(0) is given by
We calculate x from the moment eqns (2.8) for fixed values of h and A, and plot, in Fig. 4, x as a function of temperature. For h = 0, x shows a maximum at the transition temperature similar to that of a usual antiferromagnet. However, for an induced moment system, x remains finite at T = 0, in contrast to an antiferromagnet with permanent moments, for which x vanishes at T = 0. This phenomenon again is due to the variable moments of ions in an induced moment system. The value of x at T = 0 is given by ,y=A*/(l+A2),
(T=O)
12.17)
which becomes zero in the permanent moment limit (A = 0), and approaching l/2 in the highly induced limit (A + 1). Noting that the value of x at the transition temperature is always l/2 (in the present unit), independent of A, for a highly induced moment system x remains rather constant in temperature in the ordered phase, as shown for the case of A = 0.8. We also note that the ratio of the susceptibility at zero temperature to that the ratio of the susceptibility at zero temperature to that at the N&e1temperature provides a direct measurement of A, (2.18)
~(O)~~(TN)=~A~~(I+A*).
In a magnetic field, there is always a jump in x at the transition temperature. The discontinuity in x increases with the magnetic field. It can be shown that the value of x at the transition temperature as one approaches from the ordered phase is
(2.16)
h =0.5
I -
A=0.1
-
-
-..._
‘\ 0.8
\
0.2 t
h=O /’ / /
Or2 0.4
I
1
0.6
0.8
I
1.0
1
1.2
d
I.4
0
IB
I.8
t
Fig. 3. Temperature variation of the reduced sublattice moments for A = 0.1 and for A = 0.8.
Fig. 4. Temperature variation of susceptibility (in units of ~‘~B2/,$(0))for A = O-8and A = 0.25 in the absence of external field and in h = 0.6.
Induced moment antifermmagnet in an external magnetic field
where f(h)=
(9) q(A’+ C’) tanh
(2.20)
and f(“’ is the n-th derivative of f(A) with respect to 6, evaluated at 6 given by solving the equatian, 6 +f(B) = I?.
(2.21)
The value of the susceptivity at the transition temperature, approaching from the p~a~eti~ phase, can be easily shown to be l/2, independent of the magnetic field. For h = 0, p’(6) = 0 and x(ti) = (l/2), thus no discontinuous jump in the susceptibility occurs. In the permanent moment limit, A = 0, eqn (2.19) reduces to x =i(4-3&),
(2.22)
as obtained by Bidaux et at.1111. While again the susceptibility of a permanent moment Tsing ~tife~rna~ net always vanishes at T =4 the susce~t~~~~ty of an induced moment system can remain at a large value. (c)
Specific heat The internal energy of the system is given by E = (X~)
The resutts of this calculation are shown in Fig. 5 for an induced moment system of A = 0.8. The broad peak of each curve is the “Schottky anomaly” associated with the crystal field splitting, Because of the existence of the energy gap between the two eigenstates even in the paramagnetic phase, the entropy does not become maximum until at ingnite temperature. As the temperature is lowered, the upper level gets depopulated and the entropy decreases. The rate of change of entropy is greatest for k,T = I%,the energy gap, thus gives rise to a broad peak around b. The sharp peak is associated with the magnetic phase transition. The size of the peak is much smaller than it is usually for the magnetic ordering (in a permanent moment system). This decrease in the size of the peak is clearly due to the lowering of the entropy, as temperature is lowered, before the magnetic ordering sets in. The same behavior has been predicted and induced moment observed in ferromagnetic systemsl4,lZj. In an external magnetic field, while the phase transition of a ferromagnetic system disappears, a phase transition in the ~tiferrom~etic system persists until the zero temperature (or the maximum) critical field is reached. The curve for k =0x5 shows the magnetic transition in a field. The transition peak is further reduced since the level splitting is enhanced by the field. In a large field (k = 0.97, for example), no transition is observed, and the system remains in the paramagnetic phase for all temperatures.
(2.23)
and the specific heat in a constant magnetic field is (2.24)
i-8
Fig. 5. Temperature variation of specific heat for A k = 0.0-5and 0-97.
133
= 0.8 in a B&f,
3.EFFECTS OF NEXT NEAREST-NEIGFIBOR b'TER4CTIONS Itiswelfknown that, in the case of a usual permanent
moment Ising ant~e~om~et, the presence of any &rite amount of ferromaguetic next-newest-nei~~r (intrasublat~ce} inte~ction would cause the transition in a lieId to become first order at suthciently low temperatures. For the induced moment system, on the other hand, as we shall show below, the ratio of the nnn ferromagnetic interaction to the nn antiferromagnetic interaction has to exceed a certain value for the system to undergo a first order phase transition in a field even at T = 0. We shall, in this section, employ the Landau’s theory of second order phase transition113J in conjunction with a full examination of the behavior of the free energy func~on to obtain the phase boundaries in the magnetic ~eId-tem~ra~ plaue, to locate the tricritical pointsIl4f, and to map out, in the nnn to nn exchange interaction ~ti~temperat~e plane, regions in which different types of transition would occur. It should be noted that for a permanent moment antiferromagnet, if the ferromagnetic to the antiferromagnetic interaction ratio and the temperature fall in a certain narrow region, the system would undergo a first order phase transition to an intermediate phase, as the magnetic field increases, before making a second order phase transition to the paramagnetic phase. This type of multiple-transition has been named by Bidaux et aI.ltl] as the @-type transition. Similar phase ~~si~~a occm in the induced moment systems, but on even more strict conditions, and only for systems with small values of A (permanent mom&-liie systems),
134
B. N.
RAO
and lr _ ._ YUNG-LI WANG
(a) Free energy and moment equations In the presence of the nnn ferromagnetic interaction, the only additional term to the Hamiltonian (2.1) are 2”“. = -&a;.$ ,’ 6.4
*X.-&&j, I ’ rB
..?##, (3.1)
where y;,, and $drn, are the effective exchange integrals which couple an ion to its next nearest neighbors (intrasublattice coupling). As in Section 1, the Hamiltonian (2.1) with the additional terms (3.1) can be split into the molecular field term X0 and the spin fluctuation term XI. In the molecular field approximation, the partition function is defined as 2 = Tr(e-@“),
(3.2)
where the trace is taken over the eigenstates lO),lO),, IO).,I&, II), IO), and Il)A(&. The Gibb’s free energy G =-kTlnZ
(3.3)
and the result is G 2= $(2b t 1)m’t ~‘1 2@(O)+dP’(O))a - f In cash - f In cash
t ~(A2t[ht(2btl)m-cl* t
(3.4)
b = -j(O)U(O)
(3.5)
+ $V’)l,
and is defined in accordance with Ref. [ll] in which an Ising antiferromagnetic system was discussed. b = -1 corresponds to the case of vanishing ferromagnetic intra-sublattice coupling, and b = 0 corresponds to the case of a much stronger ferromagnetic intra-sublattice coupling. The reduced quantities h, t, and A are redefined by
G=GotGz~‘tGq~~tG6~‘+...
Idp(O)tdp’(O)] in
eqns
(2.7).
Minimizing the free energy (3.4) with respect to m and l we obtain the equations relating m and E or the sublattice magnetizations x and y. The latter pair of equations are the same as eqns (2.8) with h-y replaced by h t (b t 1)x t by and h-x by h t (b t 1)y t bx. As in Section 2, the pair of equations should be solved simultaneously to obtain the sublattice magnetizations. Here, since the phase transition can be first order, it is essential to obtain the free energy associated with the solution. We therefore look for the minima of the free energy as a function of E after m is eliminated with the help of the moment equations. This is a more convenient procedure, which yields all the information we need in one calculation. As we shall show later in this section, for E 2 0, there may be
(3.6)
For a second order phase transition to occur, the necessary conditions are Gdt, h) = 0, Gd(t, h)>O.
(3.7a) (3.7b)
Equations (3.7) give the critical curve in the (h, t) plane for a second order phase transition. As pointed out by Landau[l3], the boundary separating two phases of different symmetry can not stop at a point. It may, however, change at some point into a first order phase transition line. This happens when Gd(f, h) = 0, with the stability condition G6(t, h) >O. The point at which a second order phase transition line ends and joins to the first order phase transition line is called the tricritical point by Griffiths[l4]. Thus the tricritical point is given by G&r h) = 0, Gn(t, h) > 0.
>
where m = (x + y)/2 is the reduced total magnetization, 2r =x - y, the difference of the reduced sublattice magnetizations as defined before. Vanishing in the paramagnetic phase, E can be taken as an ordering parameter for an antiferromagnet in an external field.
f(0)
(b) Phase boundaries and tricritical points Following Landau[l3] we expand the free energy in powers of the ordering parameter E:
v’(A* t [h t (26 + 1)m t ~1’)
-tln2
by replacing
more than one minimum existing simultaneously in the free energy; the one with the lowest free energy corresponds to the physical state.
(3.8a) (3.8b)
If condition (3.8b) is not satisfied, the presupposition of a continuous phase transition is obviously false; the behavior of the free energy function needs to be examined in detail. Indeed, this happens in a narrow range of values of b and t, in which the free energy shows a development of an additional minimum at a smaller but nonzero value of E as field increases, and the system would first undergo a first order phase transition to an intermediate phase before making a second order transition to the paramagnetic phase. This falls into the category of P-type transition occurring in the permanent moment Ising antiferromagnets and discussed by Kanamori et al. [9] and by Bidaux et al. [ll] In fact even if conditions (3.8) are satisfied, the point may still not be a tricritical point. An example showing that conditions (3.8) are necessary but not sufficient conditions has been given by Rao and Wang[lSl. It is a simple matter to show that eqns (3.7) yield f”’ = 1
(3.9a)
and (3.9b) where f(& r) and f’“‘(6, t) have been defined in Section 2 [eqn (2.14)], and 6 is related to the external field h by 6 = h + (2b t l)f(h)
(3.10)
For t = 0, eqn (3.9a) gives h,. = (A 2’3- (26 t l))d( 1- A 2’3)
(3.11)
Induced moment ~ntjfe~oma~et
The N&l temperature at h = 0 is once again given by eqn (2.31) with the present definition of A. The tricritical conditions (3.8) are given by (3.12a)
in an extem& magnetic field
135
for the tricritical point,
(3.14)
for -$ < b c 0, as shown by Kanamori et al. [91 and by Bidaux a nL f 1II- For 8r1yEnik value of A, the caIculation (3. i2b) can be carried out by first obtairiing 6 from eqns (3_9a)for where a Exed vaIue of t and A and then ev~ua~g the (3.13a) derivatives off(&) at this value of K Equation (3.12a) theri gives the critical value of b at which the nature of the transition changes from second order to first order, if (3.13b) condition (3.12b) holds. If condition (3.12b) fails, the system is found tn undergo the multiple-transition (P-type with transition) as discussed above. A = _f(3)/p Figure 6 shows tho LOCUS of the tricritical points in the (3.13c) b-A plane at T = 0. the tricritical line ends at the tip of a For A = I), the permanent moment limit, eqlrs (3.12) gives, crescent shaped are&, and we find three regions. If the vdues of b and A of %Isystem fall in the F-region7 a single first order phase transitioa will take place in a magnetic field; if they fall in the S-region, a single seco& order phase ~ansi~ou in a fiefd is expected; if they faI1 in the small p-region, the $3 type transition would occur. The &type transition ceases to exist for A >053536_ FOF A ~0.3536, the ranges of values of b and t to allow for a P-type transition to occur is also very narrow so we do not perceive any practical interest in the P-type transition. As is c&r from the figure that a first order phase transition does rrol:occur for any small amount of ferromagnetic intraWsublattice coupling even at T = 0, as does in a permanent moment system. The larger the value of A is, the higher the v&x of b (stronger ferroma~etic any-sublattic~ ~u~li~~) is required for a first order phase transition to occur. For finite temperatures~ a stronger ferroma~~t~c in~~s~bl~t~ce s&lattice coupling is Fig, 6, Regionsin (& A ) piarre in v&i& a sin& first a&r (z;‘), a needed. Figure 7 shows the regions in the b-t plane for singly secondorder(31,and a firstorder foIlowed by a secondorder three representative cases. The first order region is to the i/3) transition in a field may occur at T = 0. B measuresthe nn antiferrormgnetic to the nnn ferromagneticcouplingratio and is Ieft of each curve lab&d by the value of A for the system definedin eqn (3.5). under consideration, and the second order region lies to
136
B. N. Rho and
0.3
-
0.2
-
YUNG-U WANG
b=-0.45
b=-03
-e-----m.__ h
0.1 A-O.8
0.0
Mdewlor
field
theory
1
1
I
0.1
0.2
0.3
I 0.4
0.5
0.6
0.7
0%
t
Fig. 8. Phase diagrams in the h-t pIane for A = 0.8, b = - 0.45 and b = - 0.3.6 measures the relative strength of the nn antife~om~etic to the nnn fe~oma~etic couplings and is defined in eqn (3.5). Solid lines indicate second order phase transitions, and dashed lines, first order phase transitions.
the right of the curve. For A = 0.25, a small P-region exists. The ferromagnetic intra-sublattice coupling needed to bring about a first order phase transition in a field can be found from the figure; it increases with the temperature as expected. It is also clear that the larger the value of A is, the smaller the first order region becomes. Finally, to give an example of the phase diagram in (h, t) plane, we consider a highly induced moment system with A = O-8 for two values of b. Figure 8 shows the phase boundaries. For b = -0.45, which corresponds to a ferromagnetic coupling being I.2 times the antiferromagnetic coupling, the phase transition is always second order. Increasing the strength of ferromagnetic coupling to 2.3 times that of the antiferromagnetic coupling (b = - 0.3), the second order line (solid line) ends at t = 0,484, and a first order line starts (dashed line). This latter feature is similar to a permanent moment system in which, however, the tricritical point always exists, for any size of ferromagnetic intra-sublattice coupling. REFE~NC~ 1.
For a review see Cooper B. R., CRC C~i~calReviewsof Solid State Sciences 3, 83 (1972).
2. Wang Y. L. and Rao B. N., J. Appl. Phys. 42, 1414(1971). 3. Bidaux R., Gavignet-Tillard A. and Hamman J., J. de Phys. 34, 19 (1973). 4. Wang Y. L. and Cooper B. R., Phys. Rev. 172,539(1968);Ibid. 185, 6% (1969). 5. Cooper B. R. and Vogt 0.. Phys. Rev. B 1,1218 (1970).Holden T. M., Svensson E. C. and Buyers W. J. L., Phys. Rev. B 10, 3864 (1974). 6. Wtichner W., Biihm W., Kable H. G., Kasten A. and Laugsch J.. Phvs. Stat. Sal. (bf 54, 273 (1972). 7. KableH. G., Simon‘A~,Wtichner W., Phys. Stat. SOL (b) 61, KS3 (1974). 8. Mcpherson J. W. and Wang Y. L. To be published. 9. Kanamori J., Motizuki K. and Yoshida K., Busseiron Kenkyu 63, 28 (1953) (in Japanese). A summary of the results in English appears as an appendix to Motizuki K., L Pkys. SW. Japan 14, 759 (1959). 10. Kasteleijn P. W., Physica 22, 387 (1956). 11. Bidaux R., Carrara P. and Vivet, J. Phys. Chem. Solids 28, 2453 (1967). 12. Andres K., Bucher E., Darack S. and Maita J. P., Phys. Reu. B 6, 2716 (1972);Lee K. N., Bachmann R., Beballe T. H. and Maita J. P., Phys. Rev. B 2, 4580 (1970). 13. Landau L. D. and Liishitz E. M., Statistical Physics, 2nd Edn. p. 424 Addison-Wesley, Mass. (1969). 14, G~ffiths R. B., P&s. Rev. Let&. 24, 715 (1970). 15. Rao B. N. and Wang Y. L., J. Phys. CRem. S~~~d~36, 699 (1974).