- Email: [email protected]
Phase transition of an antiferromagnetic system of triplet ions with uniaxial anisotropy in a parallel magnetic field
J. Phys. Chem. Solids, 1974, Wol.35, pp. 261-271. Pergamon Press.
Printed in Great Britain
PHASE TRANSITION OF AN ANTIFERROMAGNETIC SYSTEM OF TRIPLET IONS WITH UNIAXIAL ANISOTROPY IN A PARALLEL MAGNETIC FIELD A. R. FERT,J. GELARD and P. CAIL~ARA Laboratoire de Physique des Solides, associ~ au C.N.R.S.
Institut National des Sciences Appliqu~es, Avenue de Rangneil, 31077 Toulouse, Cedex, France (Received 14 February 1973; in revisedform 20July 1973) Abstract- Using a molecular field approximation we study the metamagnetic behavior in a magnetic
field parallel to the anisotropy axis, of an assembly of pseudo spins (S = 1) with a single ion uniaxial anisotropy (D) of the same order of magnitude as the magnetic exchange couplings. The ions are divided in two sublattices with an intrasublattice ferromagnetic coupling (Jl) and an antiferromagnetic coupling (J2) between distinct sublattices. We have taken a particular interest in the evolution of the phase diagram with parameters D, Jt and J2 and we have developed the study of the nature of the point separating first and second order transition lines. A comparison with experimental situations concerning FeCI2 and FeBr2 is given.
I. INTRODUCTION
The ferrous chloride and ferrous bromide antiferromagnetic order corresponds to a parallel alignment along the ternary rhomboedral z-axis of the F e z+ spins within a layer perpendicular to this axis, with opposite orientations of the magnetizations of two adjacent layers. A n experimental study of the FeCI2 and FeBr2 magnetic phase diagrams in a magnetic field along the easy axis of magnetization shows that the field induced transition from the antiferromagnetic phase to the paramagnetic one is of first order below a critical temperature Tc (metamagnetic behavior) and of second order between Tc and the N~el temperature TN[1-5]. In the case of FeBr2 an accurate experimental study [5] shows a singular behavior. F o r T > Tc - 4-7 K the transition field which is, at first, an increasing function of temperature, presents a maximum for T - 7 K, and cancels to zero at TN = 14.2 K. Thus, in some range of temperature and magnetic field, the ordered phase is stable on the high temperature side of the transition line. Various authors have investigated aspects of the metamagnetic behavior of two sublattices antiferromagnets with exchange interaction between different sublattices and within each sublattice. Motizuki[6], G o r t e r et al.[7] and Bidaux et al.[8] use an Ising model (S = ½) in the molecular field approximation. They show that the anti261 I'PCS Vol. 35. No. 2--I
ferromagnetic-paramagnetic transition for a field applied along the axis of the spins is of second order for T near TN and changes to first order with decreasing temperature. Motizuki and Bidaux show that there could exist for some values of the exchange parameters J1 and ,/2 a range of temperature where two successive transitions are observed: the first transition fi-om an antiferromagnetic state to another antfferromagnetic state is of first order type with a discontinuity of the magnetization, the second transition from the antiferromagnetic state to the paramagnetic state occurs at a higher field and is of second order. Bidaux et al. show that the transition field may be, in some temperature and field range, an increasing function of temperature. This particular result was not accepted by the authors who considered that it may be an accident of the model. Earlier authors[9, 10] who considered the Ising case with Ji = 0 and Jz < 0 (a = - 1) have also remarked in this particular case, where there only exists second order phase transitions, that the critical field would increase with temperature. Ziman thinks that this phenomenon does not seem to be physically significant and is introduced by the theoretical model. F o r the same system of Ising spins H e a p [ l l ] and Yomosa[12] have improved the statistical treatment using a Bethe Peieds model. The use of numerical methods does not allow them to
262
A. R. FERTet al.
2. THEORETICALBACKGROUND show the evolution of the phase diagram as a function of the exchange interaction parameters J1 and We shall consider an assembly of N magnetic J2. They give the results in the particular case of ions with a pseudo-spin S = I. The magnetic J1 = -- 10J2 which is a good description of FeClz. Hamiltonian of the system contains single-ion uniaxial anisotropy terms, Zeeman energies in a The principal failure of the Ising spin model in a field parallel to the anisotropy z-axis and Heisenstudy of the magnetic behavior of FeC12 and FeBr2 is to overestimate the importance of the cristalline berg exchange interactions: anisotropy; it seems necessary to consider a system of fictituous spins S = 1 with a single ion uniaxial g f = - D ~ ( SiZ~ - I ) - gl~nH ~ Si z anisotropy of the same order of magnitude as the exchange couplings. Thus Kenan et al.[13] develop a theoretical calculus of the metamagnetic transi(i.j) tion in the particular case of ferrous chloride. A The symbol ( i , j ) indicates that the summation complete basis operator theory[14] allows them to give a detailed study of the critical behavior of is taken over all distinct pairs of ions. The anisoFeCI2 near the tricritical point. tropy parameter D is supposed to be positive so The purpose of the present paper is essentially that z is the easy axis of magnetization. Furtherto investigate the influence of the single ion uni- more, we consider only uniaxial antiferromagnetic axial anisotropy on the metamagnetic behavior of order with two equivalent magnetic sublattices a that type of antiferromagnet, Various kinds of and/3, the mean value of every spin being constant phase diagram are found according to the relative within each sublattice: values of the intralayer ferromagnetic exchange (S/z)=(S~ z)=x, foralli ~ a parameter J , , the interlayer antiferromagnetic exchange parameter J2 and the single ion anisotropy (S~ z) = (Sa z ) = y , for a l l i E /3. term D. We shall take a particular interest in the cases where, for some range of temperature and field, the transition field is an increasing function So, x and 3' are the mean reduced magnetizaof temperature, this fact being confirmed by the tions of the two magnetic sublattices, which are experimental study of FeBr._,[5]. We also analyse in FeCI2 and FeBr., alternate layers of F e -~+ ions the nature of the point separating the first and perpendicular to the z-axis. second order transition lines. We confirm by an In order to investigate the thermodynamical analogous treatment the result of Motizuki in magnetic properties of the system we shall use a the Ising model which appears as the limit case molecular field approximation assuming that of our system of spins (S = l) when D becomes different ions are statistically independent. The infinite. free energy of the system is then a function of the We shall not take into account the possibility temperature T, the field H and the mean reduced of a spin-flop phase, though this phase may be parameters x, y, tt and v observed in the case of a very low anisotropy. We must emphasize that the phase diagrams of x = (S~ z) = mean reduced magnetization of a ions FeClz and FeBrz are excellent experimental illusy = (S~ z) = mean reduced magnetization of/3 ions tration of our results (for FeBr2 the ordered phase is stdble at the high temperature side of the transi- u = mean occupation number of the state Sz = 0 tion point). However the theoretical corresponding of a ions values of the characteristic parameter J~, J.., and D v = mean occupation number of the state Sz = 0 are not in total agreement with those of FeBr2. of/3 ions. The presence of an important magnetoelastic coupling in this compound may be responsible for Neglecting the fluctuations about these mean this quantitative disagreement. Many experiments values we can express the energy E and the entropy show the evidence of such coupling in FeBr._, S of the system in a state (x, y, u, v) at the temperaand FeCl2 [15-21] and a theoretical paper of ture T in a field H : Tsallis[22] proves that magnetoelastic coupling can give the same type of diagram. Only an evaluaE = ½ N D (u + v) - ½ N g t ~ R H ( x + y ) tion of the magnetoelastic parameters can provide a quantitative fit of theory and experiment. - ¼ N ( J i x 2 +Jr Y"- + 2J2xy )
Phase transition of an antiferromagnetic system
S = --Nk[½u Log u -½(1 - u ) Log 2 + ¼ ( 1 - - u + x ) Log ( l - u +
Minimizing the thermodynamical potential g with respect to the variational parameters u, v, m and "0 we obtain a system of four equations (1) giving the equilibrium values of these parameters as implicit functions of t and h:
x)
+ ¼ ( 1 - u - x ) Log ( 1 - u - x ) +½v Log v - ½ (1 - o ) L o g 2 +¼(1 - v + y )
L o g ( 1 - v + y)
+¼(1-v-y)
Log ( 1 - v - y ) ] .
h + a m = t Log ( l - u + m + 4
The effective exchange parameter J~ within each sublattice is positive or ferromagnetic, while the effective exchange parameter Jz between distinct sublattices is negative or antfferromagnetic:
[
jEa
(1.1) .7 =_t Log (l--u+m-t-*7)(l--v--m+*7) 4 (1--u--m--*7)(l--v+m--*7) d =-~Log
]~ 2Jo < 0. iE/3
(1-u+m+*7)
d = _ t Log
It is useful to introduce the antiferromagnetic order parameter *7, the reduced magnetization m and the conjugated exchange parameters A and B: *7 = ½(x--y) m = ½(x+y) A ----J1+J~. B = Jl --J,.
Furthermore, it is convenient to use reduced quantities for the temperature, the field, the anisotropy term and the relative importance of the exchange parameters:
*7)(l-v+m-*7)
(l-u-m-*7)(1-v-m+*7)
t
J~ = ]~' 2Jo > 0 ie~
LJz
263
2
(1.2)
4uZ (1--u-m-*7) 4/)2
(1.3) . (1.4)
(1--v+m--*7)(1--v-m+*7)
We can distinguish two types of solutions of these equations, the first one representing the antiferromagnetic phase (AF)*7 ~ 0 and the second one the paramagnetic phase (P)*7 = 0. The stable phase corresponds to the lowest value of the potential g. We shall concentrate our interest on the transition line in the (h, t) plane between the stability regions of the paramagnetic phase and the antiferromagnetic phase, and particularly on the point separating the first order transition line and the second order transition line. 3. SECOND ORDER PHASE TRANSITION
kT t=--,h=
glznH
B
B
a
A B
,d=--
D B'
J1 + J, Jl-J,"
Then we obtain the expression of the reduced Gibbs thermodynamical potential g per ion in the molecular field approximation:
1 E-TS N B g = ½d. ( u + v) - h m -½am' _~t*Tz
In a second order phase transition, the magnetization m and the order parameter .7 are continuous but their first derivatives (Om[ah)t and (O*7/Oh)tshow a discontinuity. To obtain the second order transition line we first solve the system of the four molecular field equations (1). Then, if we consider the equilibrium free energy g*(h, t, *7)expressed as a function of t, h, .7 the second order transition line is given by the condition (Boccara [23]):
g(h,t,*7, m,u,v) =
+ t [ ½u Log u + ½v Log v - ( l - - ~
02g*l = 0 0.7z ) Log 2
o2g____* I _a2g I
+ ¼ ( 1 - u + r n + * 7 ) Log (1 -u+rn+*7) +¼(1-
o'g ' o,g-'
a n ' Io - a n ' Io - 20--~u o "~U2o = 0
u - m - * 7 ) Log ( 1 - u - m ' - * 7 )
+¼(1-v+m-*7)
where the index zero corresponds to the transition line (*70= 0). This condition:
gives the equation:
Log ( 1 - v + m - - * 7 ) -i
+ ¼(4-- v-- m +.7) Log (1 - v - m + . 7 ) . / .
to = 1 - - Uo - - mo z.
A . R . FERT et al.
264
The final system which defines the second order transition line is:
magnetic field at the transition is given by: t t+m2(t)+m(t) h(t) = ~ Log t + mS(t) -- m ( t )
"r/o = 0, /,to --.=Vo to
1 -
ho + amo = ~- Log I
d= to
=
=fa(t) -ama(t)
uo +
mo --Uo--mo
where fd(t) and rod(t) depend only on the anisotropy parameter d, whereas the magnetic interaction parameter a is present only in the a . m(t) term. Figure 1 shows some curves h(t) for a particular value (a = - 0 . 5 ) of the magnetic interaction parameter a.
4Uo2
to
~ - 2 L°g (1 - Uo+ mo) (1 - U o - mo) 1 - - Uo - - mo 2.
A n equivalent method is to expand the molecular field equations of the antiferromagnetic phase near a transition point (no = O, Uo = Vo, too, ho, to). The compatibility of the system gives again the equation: to =
0'8
h
1 - - Uo - - m o 2.
06
It is not particularly difficult to solve to obtain the second order transition line ho(to). In the following, to simplify writing, we will omit the indices zero. In zero field the transition takes place at "0o - 0, m0 = 0 and t = ts where the N6el temperature tN is solution of the equation:
0,4
02
d=O~-0.5 d-'l I
0'12
1+½exp (-d)
ts =
D J1 - . / 2
kTN Jt -- Jz
0
014
0"6
/,A, I 0'8
d=~
I
= I ts "
We mention that for an Ising spin system ts (d = oo) = 1 and for a Heisenberg (S = 1) system t~ (d = O) = ~. d=
am(t)
0.2
0.5
1
2
oo
~ 0-724
0.78
0.86
0.94
1
Fig. 1. Second order transition lines in the diagram (h, t) (reduced magnetic field h = gv~BH/J~--J2, reduced temperature t = kT/J~--J2) for a particular value of the param e t e r a = J r +JJJ~-J2 characteristic o f the magnetic interactions. W e h a v e c h o s e n the four values of d(d = D/J~ - J 2 ) d = 0; 0.5; 1 and oo.
F o r the analysis of the phase diagram we shall use the reduced temperature: t
Irt a magnetic field the mean magnetization re(t) at the transition is solution of the biquadratic equation:
.--S,m' + (1-t)2-St 2=0 where
This equation has a solution if 0 <~ t ~< ts. The
t~ '
Thus we remove the influence of parameter d on the N~el temperature and center our attention on the evolution with d of the other aspects of the phase diagram, In Fig. 2(a) and (b) the functions fa and ma are represented as functions of the reduced temperature ~- in the extremal cases d = 0 and d = oo. This figure shows that fa(T) and ma('c) are weakly dependent on d. The knowledge of these functions allow us to draw the second order transition line: h(r) = f a ( z ) --ama(z).
Phase transition of an antiferromagnetic system 1.2
°~= .
d=O -.
265
"
d=oa d=O
d=¢o
O'2-=
(o) I 0'25
0
i 0'50
~t I 0.75
h°'8 I= I
r=~--%
0.6
0.4
0.2
o
o
o.~
0.4
0.'6
o'.8
, "
-c= P %
0"25
0"50
r=.f.--
0"?5
I
Fig. 2(a) and (b). f and m are functions of the reduced temperature ~ t/t:: their linear combination f--am gives the magnetic field at the transition. The dotted curves correspond to the limit case d ffi 0, the dashed curves to the limit case d = =. =
Figure 3 shows the results obtained for several values of the parameter a in the cases d = 0 and d - - = . Any value of the parameter d leads to a curve situated between the curves d -- 0 and d -- oo. The relative fluctuation of magnetic transition field with d for a given value of parameter a remains lower than 5 per cent. In some cases, the low temperature part of the transition line obtained by the mathematical solution described above, does not correspond to true physical second order transitions as we shall see Section 6. 4. FIRST ORDER PHASE TRANSITION
In a first order phase transition, magnetization m and order parameter -~ show a discontinuity. We call "0, m, u, u (v/', m', u' = v' ) the values of the characteristic variables in the antiferromagnetic (paramagnetic) phase at the transition.
Fig. 3. Second order transition line in the diagram (h, ~-) (reduced magnetic field h = gl~aH/Jl-J2, reduced temperature ~-= t/tv(d)) for some values of a =J~ + J=/Jl -J=. The dashed lines correspond to the limiting case d = = characteristic of a system of Ising spins; the broken lines correspond to the limiting case d = 0 which is characteristic of a system of triplet ions. The first order phase transition is characterized by equations (1.1)-(1.4) to which we must add equations (1.5)-(1.7):
Og~ = n Ore' v, Og~ _ n
h + am'
t
1 - u' + m'
=~LOgl_u,_m
-t
Ou'--v,
,
(1.5)
4 u '2
d=-2L°g
(1-u'+m')(1-u'-m') (1.6)
g,w = gv,
.u-by g~F = c l - - - ~ - - - - h m - - ½ a m 2 - ½ B ~ 2 +t[½uLogu+~vLogv-(1--~-~)Log2 +¼(1-u+m+~)
Log ( l - u + m + ~ )
+¼(1 - u - m - ~ )
Log (1 - u - m - - ~ )
+¼(1 - v + m - - ~ )
Log ( 1 - - v + m - ' o )
+¼(1 --v--m+~q) Log ( 1 - - v - - m + ~ ) [
(1.7)
266
A. R. FERTet al. g~ = du' - h m ' - ½ a m 'z + t [u' l~og u'
- (l-u')
Log2
+½(l--u'+m')
Log ( l - u '
+m')
+½(l-u'--m')
Log ( l - u ' - m ' ) ] .
This sy.stem of seven equations will give the first order transition line h(t) by tedious numerical calculation§. W e shall consider only the low temperature part of the first order transition line where an expansion can be used, and seek an analytic solution for the crossing point of the first order and second order transition lines. In the low temperature region, we obtain: h = l -- aF - - ~' [ e x p ( - - ~ - ~ - ) + e x p ( -
l + a~ + 2 d ~ ]/ j
This development is a good approximation for a wide range of temperature. To obtain a better fit at higher temperatures we have replaced the factor ( - t / 2 ) by the factor ( - t / 2 + v t z) where the parameter z, is calculated so that the curve h(t) will go through the critical point. Figure 4 shows v as a function of the magnetic interaction parameter a.
5. CRITICAL POINT*
We call the critical point (he, to) the limit point of the second order transition line where the differential isothermal susceptibility (Om/Oh)t becomes infinite. Let mo, Uo=Vo, 7 = 0 be the values of the characteristic variables of the system at a particular point (to,/lo) of the second order transition line. In the vicinity of this point in the A F phase, the first term of the (h -- ho)to and (m -- too)to expansions goes as 7 2. A t the critical point, the differential isothermal susceptibility (arn/ah)to being infinite, the term in ~2 of the ( h - - ho)to expansion cancels. To obtain the critical point, we seek the temperature for which the coefficient of the term in 7 2 of the (h - ho)toexpansion is zero. We choose the following notations: m = m o + / ~ , u = u 0 + e , v = vo+O for t = to and h near ho. First of all, we expand u and v in powers of 7 and ~. Equation (1.3) allows us to write u = u(m, 7) in the form: u= (1--S)-'{--S+ (S)m[1-
( l - - S ) ( m + 7 ) 2 ] m}
where
I
We expand u in powers of 7/and/~: e = U--Uo = P . ( / ~ + 7 ) + Q .
(72+2/~7) + R . 7 3
W-S u°=
I--S
with W = 51/211 - - (1 - - S ) m 0 2 ]
112
I
-0.5
-025
0
p =
O
Fig. 4. Variations with a of parameter v which appears as a coefficient in the expansion of the first order transition line equation h(t). *In recent works this point is often called tricritical point. Our purpose is not to study the "critical behavior" of physical quantities near this point (as is done in [11 and 12]) but to discuss the existence and the position of this point.
R=
Smo.
-w,Q-
S2
2-~,
SS(l--S)mo 2W s
In the same manner, we deduce from the equation (1.4), the v expansion: O = v - v o = P . (/~+7)+ Q • (7 z - 2/~7) - R~ 3 where vo = uo = W - S~ 1 - S. T h e expansion of equation (1.2) where we have introduced the preceding u - Uo and v - v o develop-
Phase transition of an antiferromagnetic system
ments leads to: 2
P+I
7o=--f
P-I -
Z
there is an analytic solution for the critical temperature only i f - - 0 . 5 < a < 1 for every value o l d . It is necessary to point out that critical" points obtained above do not always correspond to true critical points. F o r every value of d, computer calculation should be done to determine the stability region of each magnetic phase. H o w e v e r let us consider the expansion of (h - hO)toin powers of-0. A t the critical point, the tdrm in .~2 cancels and the first term is A ~4. The negative values of the coefficient A correspond to true critical points. F o r t > t~ the transition is really of second order while for t < t~ it becomes of first order. F o r positive values of A, the critical point t = tc has no reality (Fig. 5). In the case of an Ising spins system, (t~ ----TN/B = 1) the critical temperature is:
2
x
where X=l--vo+mo
1
and
1
Y=l--uo--mo
1
1) s
1) a ]
X3
y3
1m
•
Finally, cancellation of ~2 coefficient in the expansion of equation (1.1) leads to the following expression:
a= 1
teC2(te) 4Z(t~)
where a is a function of to. C and Z are functions of W which can be expressed as functions of tc from the relation 2 te
P+I Y
P--I X "
Figure 5 shows the curves a =f(tc/tN). They give the critical temperature of our spin system for every value of a and for several values of d. The variations of t¢ with d remain weak. We see that
o.
o.z
0.4
o.8
"
\go
267
.
s
Fig. 5. Variation with a = J, +J2/J, - J 2 of the temperature ¢c = te/ts of the critical point. T h e solid line curves correspond to true critical points.
Tc te=T~
2 2a-I- 1 3 a+l'
a result analogous to Motizuki's [6]. It is easy to prove with the preceding method that true critical points (,4 < 0) are obtained for - 0 . 2 5 < a < 1, (Motizuki [6]) corresponding to the following values of critical temperatures: 0.444 < tc < 1. The limiting values of a and tc do not change very much with d. N o w we shall study in detail the case where there is not a true critical point. 6. ISOTHERMAL BEHAVIOR OF THE ORDER PARAMETER 11 WITH MAGNETIC FIELD. CROSSING POINT. VANISHING POINT.
In the preceding Sections, we have shown that the behavior of our spin system was not greatly changed by the anisotropy D, if we sketch his behavior in function of • -- T/TM and not in function of t = T/B. In order to study the particular case where there is not a true critical point, we have chosen to work in Ising model (D = oo and TN = B) and we have developed an analysis very similar to that of Motizuki [6]. (a) When there is a critical point (--0.25 < a < 1 in the case of Ising spins)we have drawn (Fig. 6(a)) the characteristic curves ~t(h) for three temperatures tl, to, ts (h < tc < h). We have added the corresponding phase diagram. (b) When the critical point which appears analytically has no physical reality, the curves ~t(h) and the equilibrium diagrams are then represented Fig. 6(b). This situation appears for - 0 . 5 < a < --0-25 in the Ising case. The first and second order transition lines meet at a particular
I
~ ~ ~
268
/-= -/
C
77
"r/ "
h0 hI h-ho= ,4,r/Z
,4>0
h5 h ;
h-h(,=#02 ,4< 0
h
tI tC P3
B>O
h0 h 2
ha
t
(o)
h0 h 3
h 0 hk h-ho=,4'r/2
h_ho=B,r/30u~4 h-h~#r/z B:>O ,4<0
"
FERT et aL
h
ho hl h ~C h h h-h(~,4"tl2 h-ho=B'r}3ou'r/4 h-ho=A'r/S
,4>0"
A. R.
ho
,4<0
J ~._.___ o,2o
ho
h-t~=C(r/-~,,)3
ferromagnetic phases of same symmetry with a discontinuity of the order parameter. The transition is followed by the second order transition previously studied from the antiferromagnetic phase to the paramagnetic one. The transition line of first order inside the antiferromagnetic phase stays in a small range of temperatures (tK < t < ta) between the crossing point K and the vanishing point A (ha, ta). (c) When the analytical resolution gives no critical point (-- 1 < a < - 0 . 5 ) , the phase diagram is the same as the preceding one, but the curves of types tl and t2 no longer appear. We give, in the Figs. 7 and 8, the results obtained with the computer for particular cases which illustrate the behavior of the system described above.
L/'=0.225
C >0
] ..................
...........
/'=-0"254~
;! /', IiI
¢~!~: fs; fr
t-
tk
', I
•
~5
PA
O.5
Fig. 6. Isothermal variation of the order parameter "0 as a function of the magnetic field h for several temperatures. (a) A critical point exists. For t = t~ < tc the transition is of first order and for t = t3 > tc the transition is of
point K which is the crossing point (hK, tK). It is not a critical point; the first and second order transition lines are not tangent. F o r t < tx, we have a first order transition from the antiferromagnetic phase to the paramagnetic one. For t > tK, the analytical transition line corresponding to a first order transition from the antiferromagnetic phase to the paramagnetic one presents a turning point A. For a temperature between tK and ta, when field increases, we observe a first order transition between two anti-
.
.
"/'-'0'20 ~ ~ ~ ; ~
¢
second order. (b) There is no true physical criticalpoint and a crossing point appears. For t = t~, t~, t3 the transition is of first order. For t = t# the transitions of firstand second coincide at h = h#, we have a crossing point of the firstand second order transition lines (point K in phase diagram). For t = ta the transition of firstorder disappears and we have only the transition of second order in h = ho. The point A at h = ha, t = ta is called a vanishing point. The development ~(h) near h = ha is of the form (h-hA)= C('0 -'0,4)3' (C < 0). W e give in the two cases (a) and (b) the corresponding phase diagram h(t)
~.
"''''''''....
0.745
225 0 750
0.755
0.760
h
Fig. 7. Curves "o(h)t for a = -0.5; d = = and for several temperatures. These temperatures are chosen to give the crossing point and vanishing point (ta = 0.254). We have calculated the coordinates (hA, tA) of the vanishing point A for every value of a. Our method consists of specifying that the curve "or(h) presents at A an inflexion point and that h - - h a goes as (-0--0A)z. We have also calculated the coordinates (hK, tK) of the crossing point K. The results are shown in Fig. 9 where we have drawn re, rK and za as functions of the parameter a in the limiting case of the Ising system. The results enable us to present on Fig. 10 the typical transition lines h(z) corresponding to several values of the magnetic interaction parameter a = Jl+J=/Jl-J2. All curves are drawn for the two extremal cases d = 0 and d = oo. This figure shows the essential types of phase diagram.
Phase transition of an antiferromagnetic system
269
I
075 05
~,., d:OO
~~2~!~
0.5
"r/
0"25
~-=0.20
0.25 0745 0750 0'755
O
- 0 355: g-056C
-0.553, d:QO
d=oo
025
!
0,650 0.700 0750 0"800 h '
o%
i I
I
m
if/
0.25
,.-~0.75
,=0.24 a]~p,,
~
"t "r/=O
0745 0750 OE",b-~ h
05
5
h
0350ig4~,.f,l~
075
7
0650 0-700 07'50 0800
h -
1ii
075 /:
I "%,
°._o.5
il
l°s°
,%%0
Fig. 9. Variation with parameter a = J1 + J21J~ --Jz of the reduced temperatures characteristic of the critical point "re = tdtN, of the crossing point Tr----tr/ts and of the vanishing point zA = t a t s . The curves have been obtained in the limiting case d = =.
OllBB al i/3
0745 0.750 0.755 h
;"
0
1.2
0=-[
i.c
o=-o.76
0.6500.700 0,750h
Fig. 8. For the two temperatures t = 0.20 and t = 0.24 w e give the curves ~(h)t, g ( h ) t , m ( h ) t . T h e y permit us to understand the behavior of the different functions 71, g and m at the phase transition. For temperature t -- 0.20 w e have a first order transition for h = 0.748. For t = 0.24
we have a first order phase transition for h--0.749 followed by a second order phase transition for h = 0.758. Vectors sketched under the curves m ( h ) t represent the
magnetization of each sublattices. For instance the case a = 0.5 s h o w s a situation where a critical point exists and the ordered phase is on the l o w temperature side of the transition point. A n experimental e x a m p l e of this case is given by FeCI2. T h e case a = - 0 . 2 5 is an e x a m p l e where a critical point exists, but it appears that for certain field and temperature ranges, the ordered phase is stable on the high temperature side of the transition point. This is true not only for a second order transition but also for a first order one. FeBr2 is an experimental confirmation of this case. T h e case a = - 0 . 5 is an illustration of the existence of a crossing point K and a vanishing point A. 7. CONCLUSION
T h e molecular field approximation has permitted
\ \
p d= m
0=-0.5
o.a h
o=-0.25
0.6 o=0
o4
02
o lL
I 0.2
I 0.4
I 0.6
I 0.8
1 I
T Fig. 10. Phase diagrams in coordinates: reduced magnetic field h = g BH / J , --J2 and reduced temperature ~ = t / t s for different values of a = J, + JdJ~ - J z in the two limiting
cases d -- 0 and d = =. For every value of d(0 < d < =) the corresponding first and second order transition fines are located between the two limit lines d = 0 and d = =. The critical points are marked by a circle.
A. R. FERT et al.
270 J
a complete study of the behavior of a simple antiferromagnet with two exchang~ parameters and an uniaxial anisotropy term, in a magnetic field parallel to the easy axis of magnetization. In this text we have developed in particular, research of the different kinds of phase diagrams, the study of the meeting point of first and second order transition lines and the behavior of the order parameter 71. Other developments are possible such as initial susceptibility, isothermal magnetization curves (we give in Fig. 1 1 an example of such sketch of curves for a = 0.5, d = oo) and critical behavior near T~ (more accurate methods than molecular field approximation are desirable). /'=-0
h Fig. 11. Solid line curves are magnetization isotherms for a = 0.5 and d = ~. Dashed curve II represents the locus
of the second order transition point in the diagramm of reduced magnetic field h vs. magnetization m. Dashed curve I gives the magnetization discontinuity at a first order transition point. The magnetization discontinuity is practically a linear function of H -- He. (a) Our results showed at first the influence of the uniaxial anisotropy characterized by the parameter D. We have pointed out that the incidence of the anisotropy is essential to the magnetic ordering with temperature. F o r instance the N r e l temperature greatly depends on D. But when we study the behavior of our system with a reduced temperature variable r = tits = T/T:~ (0 < r < I) the influence of D is very weak. The phase diagram
in a parallel magnetic field depends above all on the relative importance of the two parameters J~ and J2 representing the ferromagnetic and antiferromagnetic interactions. (b) Different phase diagrams have been obtained. In particular we have shown the existence of a diagram where, in a certain range of temperature, the ordered phase is stable on the high temperature side of the transition point. When there exists a critical point the transition is of first order for t < tc and of second order for t > to, the first and second order transition lines (h, t) being tangent at this point. We can also have a crossing point (hk, tk) of the first and second order transition lines and the transition is of first order for t < t,, whereas, for a temperature just above tk, with increasing field, we observe a first order phase transition between phases of the same symetry, followed by a second order transition to a paramagnetic phase. (c) The phase diagrams of FeCI2 and FeBr2 are experimental illustrations of these results. FeC12, characterized by J~ -> - J 2 , has a classical metamagnetic behavior and corresponds to the situation obtained for a > 0. The theoretical results obtained for a = - 0 - 2 5 describe very well the experimental behavior of FeBr2: the first and second order transition field is an increasing function of temperature near the critical point and the experimental ratio TdTN is close to the theoretical one. This confirms our conviction, based on our recent experimental results (to be published), that FeBr2 is characterised by a great antiferromagnetic interaction - - J 2 ~ J~ in spite of the values previously proposed by us [5]. H o w e v e r this work does not take into account the possibility of a magneto-elastic coupling and we must recall that it may be present in the study of the behavior of compounds as FeCI2 and FeBr~. We have also neglected the possibility of a spinflop phase though this phase may be observed in the extreme case of a very low anisotropy or of a great antiferromagnetic interaction. Preliminary experimental results obtained for FeI2 show that this compound may be an illustration of this case.
REFERENCES I. Start C., Bitter F. and Kaufmann A. R., Phys. Rev. 58, 977 (1940). 2. Jacobs I. S. and Lawrence P. E., Phys. Rev. 164, 866 (1967). 3. Carrara P., Th~se, Orsay (1968). 4. Jacobs I. S. and Lawrence, P. E., J. appl. Phys. 35, 996 (1964).
Phase transition of an antiferromagnetic system 5. Fert A. R., Carrara P., Lanusse M. C., Mischler G. and Redoules J. P., J. Phys. Chem. Solids 34, 223 (1973). 6. Motizuki K., J. Phys. Soc. Jap. 14 (6), 759 (1959); Kanamori J., Motizuki K. and Yosida K., BuseironKenkyu 63, 28 (1953). (In Japanese). 7. Goter C. J. and Van Peski-Tinbergen Tineke, Physical 22, 273 (1956). 8. Bidaux R., Carrara P. and Vivet B., J. Phys. Chem. Solids 28, 2453 (1967). 9. Garrett C. G. B.,J. chem. Phys. 19, 1154(1951). 10. Ziman J. M., Proc. Phys. Soc. 64A, 1108 (1951). I I. Heap B. R., Proc. Phys. Soc. 80, 248 (1962). i 2. Yomosa S., J. phys. Soc. Japan 15, 1068 (1960). 13. Kenan R. P., Mills R. E. and Campbell C. E.,J. appl. Phys. 40 (3), 1027 (1969). 14. Kenah R. P. and Mills R. E. (To be published). 15. Lanusse M. C., Carrara P., Fert A. R., Mischler G.
271
and Redoules J. P., J. Phys. 33, 429 (1972). 16. Schnatterly S. E. and Fontana M., J. Phys. 33, 691 (1972). 17. Laurence G., Phys. Lett. 34A, 308 (1971). 18. Laurence G., Petitgrand D., Phys. Rev. (To be published). 19. Nasser J. A., Hammann J., Phys. Status Solidi 56 (1973). 20. Beille J., Alberts H. L., Bartholin M., Bloch D. and Vettier C., C.R. Acad. Sci. Paris t. 275, B, 713-722 (1972). 21. Vettier C., AIberts H. L., Beille J., and Bioch D., C.R. Acad. Sci. Paris t. 275, B, 915-918 (1972). 22. Tsallis C., J. Phys. 32, 903 (I 971). 23. Boccal'a N., Lea Principes de la Thermodynamique Classique, Voi. 1, p. 83. Prosses Univgrsitaires de France, Pads (1968).
Printed in Great Britain
PHASE TRANSITION OF AN ANTIFERROMAGNETIC SYSTEM OF TRIPLET IONS WITH UNIAXIAL ANISOTROPY IN A PARALLEL MAGNETIC FIELD A. R. FERT,J. GELARD and P. CAIL~ARA Laboratoire de Physique des Solides, associ~ au C.N.R.S.
Institut National des Sciences Appliqu~es, Avenue de Rangneil, 31077 Toulouse, Cedex, France (Received 14 February 1973; in revisedform 20July 1973) Abstract- Using a molecular field approximation we study the metamagnetic behavior in a magnetic
field parallel to the anisotropy axis, of an assembly of pseudo spins (S = 1) with a single ion uniaxial anisotropy (D) of the same order of magnitude as the magnetic exchange couplings. The ions are divided in two sublattices with an intrasublattice ferromagnetic coupling (Jl) and an antiferromagnetic coupling (J2) between distinct sublattices. We have taken a particular interest in the evolution of the phase diagram with parameters D, Jt and J2 and we have developed the study of the nature of the point separating first and second order transition lines. A comparison with experimental situations concerning FeCI2 and FeBr2 is given.
I. INTRODUCTION
The ferrous chloride and ferrous bromide antiferromagnetic order corresponds to a parallel alignment along the ternary rhomboedral z-axis of the F e z+ spins within a layer perpendicular to this axis, with opposite orientations of the magnetizations of two adjacent layers. A n experimental study of the FeCI2 and FeBr2 magnetic phase diagrams in a magnetic field along the easy axis of magnetization shows that the field induced transition from the antiferromagnetic phase to the paramagnetic one is of first order below a critical temperature Tc (metamagnetic behavior) and of second order between Tc and the N~el temperature TN[1-5]. In the case of FeBr2 an accurate experimental study [5] shows a singular behavior. F o r T > Tc - 4-7 K the transition field which is, at first, an increasing function of temperature, presents a maximum for T - 7 K, and cancels to zero at TN = 14.2 K. Thus, in some range of temperature and magnetic field, the ordered phase is stable on the high temperature side of the transition line. Various authors have investigated aspects of the metamagnetic behavior of two sublattices antiferromagnets with exchange interaction between different sublattices and within each sublattice. Motizuki[6], G o r t e r et al.[7] and Bidaux et al.[8] use an Ising model (S = ½) in the molecular field approximation. They show that the anti261 I'PCS Vol. 35. No. 2--I
ferromagnetic-paramagnetic transition for a field applied along the axis of the spins is of second order for T near TN and changes to first order with decreasing temperature. Motizuki and Bidaux show that there could exist for some values of the exchange parameters J1 and ,/2 a range of temperature where two successive transitions are observed: the first transition fi-om an antiferromagnetic state to another antfferromagnetic state is of first order type with a discontinuity of the magnetization, the second transition from the antiferromagnetic state to the paramagnetic state occurs at a higher field and is of second order. Bidaux et al. show that the transition field may be, in some temperature and field range, an increasing function of temperature. This particular result was not accepted by the authors who considered that it may be an accident of the model. Earlier authors[9, 10] who considered the Ising case with Ji = 0 and Jz < 0 (a = - 1) have also remarked in this particular case, where there only exists second order phase transitions, that the critical field would increase with temperature. Ziman thinks that this phenomenon does not seem to be physically significant and is introduced by the theoretical model. F o r the same system of Ising spins H e a p [ l l ] and Yomosa[12] have improved the statistical treatment using a Bethe Peieds model. The use of numerical methods does not allow them to
262
A. R. FERTet al.
2. THEORETICALBACKGROUND show the evolution of the phase diagram as a function of the exchange interaction parameters J1 and We shall consider an assembly of N magnetic J2. They give the results in the particular case of ions with a pseudo-spin S = I. The magnetic J1 = -- 10J2 which is a good description of FeClz. Hamiltonian of the system contains single-ion uniaxial anisotropy terms, Zeeman energies in a The principal failure of the Ising spin model in a field parallel to the anisotropy z-axis and Heisenstudy of the magnetic behavior of FeC12 and FeBr2 is to overestimate the importance of the cristalline berg exchange interactions: anisotropy; it seems necessary to consider a system of fictituous spins S = 1 with a single ion uniaxial g f = - D ~ ( SiZ~ - I ) - gl~nH ~ Si z anisotropy of the same order of magnitude as the exchange couplings. Thus Kenan et al.[13] develop a theoretical calculus of the metamagnetic transi(i.j) tion in the particular case of ferrous chloride. A The symbol ( i , j ) indicates that the summation complete basis operator theory[14] allows them to give a detailed study of the critical behavior of is taken over all distinct pairs of ions. The anisoFeCI2 near the tricritical point. tropy parameter D is supposed to be positive so The purpose of the present paper is essentially that z is the easy axis of magnetization. Furtherto investigate the influence of the single ion uni- more, we consider only uniaxial antiferromagnetic axial anisotropy on the metamagnetic behavior of order with two equivalent magnetic sublattices a that type of antiferromagnet, Various kinds of and/3, the mean value of every spin being constant phase diagram are found according to the relative within each sublattice: values of the intralayer ferromagnetic exchange (S/z)=(S~ z)=x, foralli ~ a parameter J , , the interlayer antiferromagnetic exchange parameter J2 and the single ion anisotropy (S~ z) = (Sa z ) = y , for a l l i E /3. term D. We shall take a particular interest in the cases where, for some range of temperature and field, the transition field is an increasing function So, x and 3' are the mean reduced magnetizaof temperature, this fact being confirmed by the tions of the two magnetic sublattices, which are experimental study of FeBr._,[5]. We also analyse in FeCI2 and FeBr., alternate layers of F e -~+ ions the nature of the point separating the first and perpendicular to the z-axis. second order transition lines. We confirm by an In order to investigate the thermodynamical analogous treatment the result of Motizuki in magnetic properties of the system we shall use a the Ising model which appears as the limit case molecular field approximation assuming that of our system of spins (S = l) when D becomes different ions are statistically independent. The infinite. free energy of the system is then a function of the We shall not take into account the possibility temperature T, the field H and the mean reduced of a spin-flop phase, though this phase may be parameters x, y, tt and v observed in the case of a very low anisotropy. We must emphasize that the phase diagrams of x = (S~ z) = mean reduced magnetization of a ions FeClz and FeBrz are excellent experimental illusy = (S~ z) = mean reduced magnetization of/3 ions tration of our results (for FeBr2 the ordered phase is stdble at the high temperature side of the transi- u = mean occupation number of the state Sz = 0 tion point). However the theoretical corresponding of a ions values of the characteristic parameter J~, J.., and D v = mean occupation number of the state Sz = 0 are not in total agreement with those of FeBr2. of/3 ions. The presence of an important magnetoelastic coupling in this compound may be responsible for Neglecting the fluctuations about these mean this quantitative disagreement. Many experiments values we can express the energy E and the entropy show the evidence of such coupling in FeBr._, S of the system in a state (x, y, u, v) at the temperaand FeCl2 [15-21] and a theoretical paper of ture T in a field H : Tsallis[22] proves that magnetoelastic coupling can give the same type of diagram. Only an evaluaE = ½ N D (u + v) - ½ N g t ~ R H ( x + y ) tion of the magnetoelastic parameters can provide a quantitative fit of theory and experiment. - ¼ N ( J i x 2 +Jr Y"- + 2J2xy )
Phase transition of an antiferromagnetic system
S = --Nk[½u Log u -½(1 - u ) Log 2 + ¼ ( 1 - - u + x ) Log ( l - u +
Minimizing the thermodynamical potential g with respect to the variational parameters u, v, m and "0 we obtain a system of four equations (1) giving the equilibrium values of these parameters as implicit functions of t and h:
x)
+ ¼ ( 1 - u - x ) Log ( 1 - u - x ) +½v Log v - ½ (1 - o ) L o g 2 +¼(1 - v + y )
L o g ( 1 - v + y)
+¼(1-v-y)
Log ( 1 - v - y ) ] .
h + a m = t Log ( l - u + m + 4
The effective exchange parameter J~ within each sublattice is positive or ferromagnetic, while the effective exchange parameter Jz between distinct sublattices is negative or antfferromagnetic:
[
jEa
(1.1) .7 =_t Log (l--u+m-t-*7)(l--v--m+*7) 4 (1--u--m--*7)(l--v+m--*7) d =-~Log
]~ 2Jo < 0. iE/3
(1-u+m+*7)
d = _ t Log
It is useful to introduce the antiferromagnetic order parameter *7, the reduced magnetization m and the conjugated exchange parameters A and B: *7 = ½(x--y) m = ½(x+y) A ----J1+J~. B = Jl --J,.
Furthermore, it is convenient to use reduced quantities for the temperature, the field, the anisotropy term and the relative importance of the exchange parameters:
*7)(l-v+m-*7)
(l-u-m-*7)(1-v-m+*7)
t
J~ = ]~' 2Jo > 0 ie~
LJz
263
2
(1.2)
4uZ (1--u-m-*7) 4/)2
(1.3) . (1.4)
(1--v+m--*7)(1--v-m+*7)
We can distinguish two types of solutions of these equations, the first one representing the antiferromagnetic phase (AF)*7 ~ 0 and the second one the paramagnetic phase (P)*7 = 0. The stable phase corresponds to the lowest value of the potential g. We shall concentrate our interest on the transition line in the (h, t) plane between the stability regions of the paramagnetic phase and the antiferromagnetic phase, and particularly on the point separating the first order transition line and the second order transition line. 3. SECOND ORDER PHASE TRANSITION
kT t=--,h=
glznH
B
B
a
A B
,d=--
D B'
J1 + J, Jl-J,"
Then we obtain the expression of the reduced Gibbs thermodynamical potential g per ion in the molecular field approximation:
1 E-TS N B g = ½d. ( u + v) - h m -½am' _~t*Tz
In a second order phase transition, the magnetization m and the order parameter .7 are continuous but their first derivatives (Om[ah)t and (O*7/Oh)tshow a discontinuity. To obtain the second order transition line we first solve the system of the four molecular field equations (1). Then, if we consider the equilibrium free energy g*(h, t, *7)expressed as a function of t, h, .7 the second order transition line is given by the condition (Boccara [23]):
g(h,t,*7, m,u,v) =
+ t [ ½u Log u + ½v Log v - ( l - - ~
02g*l = 0 0.7z ) Log 2
o2g____* I _a2g I
+ ¼ ( 1 - u + r n + * 7 ) Log (1 -u+rn+*7) +¼(1-
o'g ' o,g-'
a n ' Io - a n ' Io - 20--~u o "~U2o = 0
u - m - * 7 ) Log ( 1 - u - m ' - * 7 )
+¼(1-v+m-*7)
where the index zero corresponds to the transition line (*70= 0). This condition:
gives the equation:
Log ( 1 - v + m - - * 7 ) -i
+ ¼(4-- v-- m +.7) Log (1 - v - m + . 7 ) . / .
to = 1 - - Uo - - mo z.
A . R . FERT et al.
264
The final system which defines the second order transition line is:
magnetic field at the transition is given by: t t+m2(t)+m(t) h(t) = ~ Log t + mS(t) -- m ( t )
"r/o = 0, /,to --.=Vo to
1 -
ho + amo = ~- Log I
d= to
=
=fa(t) -ama(t)
uo +
mo --Uo--mo
where fd(t) and rod(t) depend only on the anisotropy parameter d, whereas the magnetic interaction parameter a is present only in the a . m(t) term. Figure 1 shows some curves h(t) for a particular value (a = - 0 . 5 ) of the magnetic interaction parameter a.
4Uo2
to
~ - 2 L°g (1 - Uo+ mo) (1 - U o - mo) 1 - - Uo - - mo 2.
A n equivalent method is to expand the molecular field equations of the antiferromagnetic phase near a transition point (no = O, Uo = Vo, too, ho, to). The compatibility of the system gives again the equation: to =
0'8
h
1 - - Uo - - m o 2.
06
It is not particularly difficult to solve to obtain the second order transition line ho(to). In the following, to simplify writing, we will omit the indices zero. In zero field the transition takes place at "0o - 0, m0 = 0 and t = ts where the N6el temperature tN is solution of the equation:
0,4
02
d=O~-0.5 d-'l I
0'12
1+½exp (-d)
ts =
D J1 - . / 2
kTN Jt -- Jz
0
014
0"6
/,A, I 0'8
d=~
I
= I ts "
We mention that for an Ising spin system ts (d = oo) = 1 and for a Heisenberg (S = 1) system t~ (d = O) = ~. d=
am(t)
0.2
0.5
1
2
oo
~ 0-724
0.78
0.86
0.94
1
Fig. 1. Second order transition lines in the diagram (h, t) (reduced magnetic field h = gv~BH/J~--J2, reduced temperature t = kT/J~--J2) for a particular value of the param e t e r a = J r +JJJ~-J2 characteristic o f the magnetic interactions. W e h a v e c h o s e n the four values of d(d = D/J~ - J 2 ) d = 0; 0.5; 1 and oo.
F o r the analysis of the phase diagram we shall use the reduced temperature: t
Irt a magnetic field the mean magnetization re(t) at the transition is solution of the biquadratic equation:
.--S,m' + (1-t)2-St 2=0 where
This equation has a solution if 0 <~ t ~< ts. The
t~ '
Thus we remove the influence of parameter d on the N~el temperature and center our attention on the evolution with d of the other aspects of the phase diagram, In Fig. 2(a) and (b) the functions fa and ma are represented as functions of the reduced temperature ~- in the extremal cases d = 0 and d = oo. This figure shows that fa(T) and ma('c) are weakly dependent on d. The knowledge of these functions allow us to draw the second order transition line: h(r) = f a ( z ) --ama(z).
Phase transition of an antiferromagnetic system 1.2
°~= .
d=O -.
265
"
d=oa d=O
d=¢o
O'2-=
(o) I 0'25
0
i 0'50
~t I 0.75
h°'8 I= I
r=~--%
0.6
0.4
0.2
o
o
o.~
0.4
0.'6
o'.8
, "
-c= P %
0"25
0"50
r=.f.--
0"?5
I
Fig. 2(a) and (b). f and m are functions of the reduced temperature ~ t/t:: their linear combination f--am gives the magnetic field at the transition. The dotted curves correspond to the limit case d ffi 0, the dashed curves to the limit case d = =. =
Figure 3 shows the results obtained for several values of the parameter a in the cases d = 0 and d - - = . Any value of the parameter d leads to a curve situated between the curves d -- 0 and d -- oo. The relative fluctuation of magnetic transition field with d for a given value of parameter a remains lower than 5 per cent. In some cases, the low temperature part of the transition line obtained by the mathematical solution described above, does not correspond to true physical second order transitions as we shall see Section 6. 4. FIRST ORDER PHASE TRANSITION
In a first order phase transition, magnetization m and order parameter -~ show a discontinuity. We call "0, m, u, u (v/', m', u' = v' ) the values of the characteristic variables in the antiferromagnetic (paramagnetic) phase at the transition.
Fig. 3. Second order transition line in the diagram (h, ~-) (reduced magnetic field h = gl~aH/Jl-J2, reduced temperature ~-= t/tv(d)) for some values of a =J~ + J=/Jl -J=. The dashed lines correspond to the limiting case d = = characteristic of a system of Ising spins; the broken lines correspond to the limiting case d = 0 which is characteristic of a system of triplet ions. The first order phase transition is characterized by equations (1.1)-(1.4) to which we must add equations (1.5)-(1.7):
Og~ = n Ore' v, Og~ _ n
h + am'
t
1 - u' + m'
=~LOgl_u,_m
-t
Ou'--v,
,
(1.5)
4 u '2
d=-2L°g
(1-u'+m')(1-u'-m') (1.6)
g,w = gv,
.u-by g~F = c l - - - ~ - - - - h m - - ½ a m 2 - ½ B ~ 2 +t[½uLogu+~vLogv-(1--~-~)Log2 +¼(1-u+m+~)
Log ( l - u + m + ~ )
+¼(1 - u - m - ~ )
Log (1 - u - m - - ~ )
+¼(1 - v + m - - ~ )
Log ( 1 - - v + m - ' o )
+¼(1 --v--m+~q) Log ( 1 - - v - - m + ~ ) [
(1.7)
266
A. R. FERTet al. g~ = du' - h m ' - ½ a m 'z + t [u' l~og u'
- (l-u')
Log2
+½(l--u'+m')
Log ( l - u '
+m')
+½(l-u'--m')
Log ( l - u ' - m ' ) ] .
This sy.stem of seven equations will give the first order transition line h(t) by tedious numerical calculation§. W e shall consider only the low temperature part of the first order transition line where an expansion can be used, and seek an analytic solution for the crossing point of the first order and second order transition lines. In the low temperature region, we obtain: h = l -- aF - - ~' [ e x p ( - - ~ - ~ - ) + e x p ( -
l + a~ + 2 d ~ ]/ j
This development is a good approximation for a wide range of temperature. To obtain a better fit at higher temperatures we have replaced the factor ( - t / 2 ) by the factor ( - t / 2 + v t z) where the parameter z, is calculated so that the curve h(t) will go through the critical point. Figure 4 shows v as a function of the magnetic interaction parameter a.
5. CRITICAL POINT*
We call the critical point (he, to) the limit point of the second order transition line where the differential isothermal susceptibility (Om/Oh)t becomes infinite. Let mo, Uo=Vo, 7 = 0 be the values of the characteristic variables of the system at a particular point (to,/lo) of the second order transition line. In the vicinity of this point in the A F phase, the first term of the (h -- ho)to and (m -- too)to expansions goes as 7 2. A t the critical point, the differential isothermal susceptibility (arn/ah)to being infinite, the term in ~2 of the ( h - - ho)to expansion cancels. To obtain the critical point, we seek the temperature for which the coefficient of the term in 7 2 of the (h - ho)toexpansion is zero. We choose the following notations: m = m o + / ~ , u = u 0 + e , v = vo+O for t = to and h near ho. First of all, we expand u and v in powers of 7 and ~. Equation (1.3) allows us to write u = u(m, 7) in the form: u= (1--S)-'{--S+ (S)m[1-
( l - - S ) ( m + 7 ) 2 ] m}
where
I
We expand u in powers of 7/and/~: e = U--Uo = P . ( / ~ + 7 ) + Q .
(72+2/~7) + R . 7 3
W-S u°=
I--S
with W = 51/211 - - (1 - - S ) m 0 2 ]
112
I
-0.5
-025
0
p =
O
Fig. 4. Variations with a of parameter v which appears as a coefficient in the expansion of the first order transition line equation h(t). *In recent works this point is often called tricritical point. Our purpose is not to study the "critical behavior" of physical quantities near this point (as is done in [11 and 12]) but to discuss the existence and the position of this point.
R=
Smo.
-w,Q-
S2
2-~,
SS(l--S)mo 2W s
In the same manner, we deduce from the equation (1.4), the v expansion: O = v - v o = P . (/~+7)+ Q • (7 z - 2/~7) - R~ 3 where vo = uo = W - S~ 1 - S. T h e expansion of equation (1.2) where we have introduced the preceding u - Uo and v - v o develop-
Phase transition of an antiferromagnetic system
ments leads to: 2
P+I
7o=--f
P-I -
Z
there is an analytic solution for the critical temperature only i f - - 0 . 5 < a < 1 for every value o l d . It is necessary to point out that critical" points obtained above do not always correspond to true critical points. F o r every value of d, computer calculation should be done to determine the stability region of each magnetic phase. H o w e v e r let us consider the expansion of (h - hO)toin powers of-0. A t the critical point, the tdrm in .~2 cancels and the first term is A ~4. The negative values of the coefficient A correspond to true critical points. F o r t > t~ the transition is really of second order while for t < t~ it becomes of first order. F o r positive values of A, the critical point t = tc has no reality (Fig. 5). In the case of an Ising spins system, (t~ ----TN/B = 1) the critical temperature is:
2
x
where X=l--vo+mo
1
and
1
Y=l--uo--mo
1
1) s
1) a ]
X3
y3
1m
•
Finally, cancellation of ~2 coefficient in the expansion of equation (1.1) leads to the following expression:
a= 1
teC2(te) 4Z(t~)
where a is a function of to. C and Z are functions of W which can be expressed as functions of tc from the relation 2 te
P+I Y
P--I X "
Figure 5 shows the curves a =f(tc/tN). They give the critical temperature of our spin system for every value of a and for several values of d. The variations of t¢ with d remain weak. We see that
o.
o.z
0.4
o.8
"
\go
267
.
s
Fig. 5. Variation with a = J, +J2/J, - J 2 of the temperature ¢c = te/ts of the critical point. T h e solid line curves correspond to true critical points.
Tc te=T~
2 2a-I- 1 3 a+l'
a result analogous to Motizuki's [6]. It is easy to prove with the preceding method that true critical points (,4 < 0) are obtained for - 0 . 2 5 < a < 1, (Motizuki [6]) corresponding to the following values of critical temperatures: 0.444 < tc < 1. The limiting values of a and tc do not change very much with d. N o w we shall study in detail the case where there is not a true critical point. 6. ISOTHERMAL BEHAVIOR OF THE ORDER PARAMETER 11 WITH MAGNETIC FIELD. CROSSING POINT. VANISHING POINT.
In the preceding Sections, we have shown that the behavior of our spin system was not greatly changed by the anisotropy D, if we sketch his behavior in function of • -- T/TM and not in function of t = T/B. In order to study the particular case where there is not a true critical point, we have chosen to work in Ising model (D = oo and TN = B) and we have developed an analysis very similar to that of Motizuki [6]. (a) When there is a critical point (--0.25 < a < 1 in the case of Ising spins)we have drawn (Fig. 6(a)) the characteristic curves ~t(h) for three temperatures tl, to, ts (h < tc < h). We have added the corresponding phase diagram. (b) When the critical point which appears analytically has no physical reality, the curves ~t(h) and the equilibrium diagrams are then represented Fig. 6(b). This situation appears for - 0 . 5 < a < --0-25 in the Ising case. The first and second order transition lines meet at a particular
I
~ ~ ~
268
/-= -/
C
77
"r/ "
h0 hI h-ho= ,4,r/Z
,4>0
h5 h ;
h-h(,=#02 ,4< 0
h
tI tC P3
B>O
h0 h 2
ha
t
(o)
h0 h 3
h 0 hk h-ho=,4'r/2
h_ho=B,r/30u~4 h-h~#r/z B:>O ,4<0
"
FERT et aL
h
ho hl h ~C h h h-h(~,4"tl2 h-ho=B'r}3ou'r/4 h-ho=A'r/S
,4>0"
A. R.
ho
,4<0
J ~._.___ o,2o
ho
h-t~=C(r/-~,,)3
ferromagnetic phases of same symmetry with a discontinuity of the order parameter. The transition is followed by the second order transition previously studied from the antiferromagnetic phase to the paramagnetic one. The transition line of first order inside the antiferromagnetic phase stays in a small range of temperatures (tK < t < ta) between the crossing point K and the vanishing point A (ha, ta). (c) When the analytical resolution gives no critical point (-- 1 < a < - 0 . 5 ) , the phase diagram is the same as the preceding one, but the curves of types tl and t2 no longer appear. We give, in the Figs. 7 and 8, the results obtained with the computer for particular cases which illustrate the behavior of the system described above.
L/'=0.225
C >0
] ..................
...........
/'=-0"254~
;! /', IiI
¢~!~: fs; fr
t-
tk
', I
•
~5
PA
O.5
Fig. 6. Isothermal variation of the order parameter "0 as a function of the magnetic field h for several temperatures. (a) A critical point exists. For t = t~ < tc the transition is of first order and for t = t3 > tc the transition is of
point K which is the crossing point (hK, tK). It is not a critical point; the first and second order transition lines are not tangent. F o r t < tx, we have a first order transition from the antiferromagnetic phase to the paramagnetic one. For t > tK, the analytical transition line corresponding to a first order transition from the antiferromagnetic phase to the paramagnetic one presents a turning point A. For a temperature between tK and ta, when field increases, we observe a first order transition between two anti-
.
.
"/'-'0'20 ~ ~ ~ ; ~
¢
second order. (b) There is no true physical criticalpoint and a crossing point appears. For t = t~, t~, t3 the transition is of first order. For t = t# the transitions of firstand second coincide at h = h#, we have a crossing point of the firstand second order transition lines (point K in phase diagram). For t = ta the transition of firstorder disappears and we have only the transition of second order in h = ho. The point A at h = ha, t = ta is called a vanishing point. The development ~(h) near h = ha is of the form (h-hA)= C('0 -'0,4)3' (C < 0). W e give in the two cases (a) and (b) the corresponding phase diagram h(t)
~.
"''''''''....
0.745
225 0 750
0.755
0.760
h
Fig. 7. Curves "o(h)t for a = -0.5; d = = and for several temperatures. These temperatures are chosen to give the crossing point and vanishing point (ta = 0.254). We have calculated the coordinates (hA, tA) of the vanishing point A for every value of a. Our method consists of specifying that the curve "or(h) presents at A an inflexion point and that h - - h a goes as (-0--0A)z. We have also calculated the coordinates (hK, tK) of the crossing point K. The results are shown in Fig. 9 where we have drawn re, rK and za as functions of the parameter a in the limiting case of the Ising system. The results enable us to present on Fig. 10 the typical transition lines h(z) corresponding to several values of the magnetic interaction parameter a = Jl+J=/Jl-J2. All curves are drawn for the two extremal cases d = 0 and d = oo. This figure shows the essential types of phase diagram.
Phase transition of an antiferromagnetic system
269
I
075 05
~,., d:OO
~~2~!~
0.5
"r/
0"25
~-=0.20
0.25 0745 0750 0'755
O
- 0 355: g-056C
-0.553, d:QO
d=oo
025
!
0,650 0.700 0750 0"800 h '
o%
i I
I
m
if/
0.25
,.-~0.75
,=0.24 a]~p,,
~
"t "r/=O
0745 0750 OE",b-~ h
05
5
h
0350ig4~,.f,l~
075
7
0650 0-700 07'50 0800
h -
1ii
075 /:
I "%,
°._o.5
il
l°s°
,%%0
Fig. 9. Variation with parameter a = J1 + J21J~ --Jz of the reduced temperatures characteristic of the critical point "re = tdtN, of the crossing point Tr----tr/ts and of the vanishing point zA = t a t s . The curves have been obtained in the limiting case d = =.
OllBB al i/3
0745 0.750 0.755 h
;"
0
1.2
0=-[
i.c
o=-o.76
0.6500.700 0,750h
Fig. 8. For the two temperatures t = 0.20 and t = 0.24 w e give the curves ~(h)t, g ( h ) t , m ( h ) t . T h e y permit us to understand the behavior of the different functions 71, g and m at the phase transition. For temperature t -- 0.20 w e have a first order transition for h = 0.748. For t = 0.24
we have a first order phase transition for h--0.749 followed by a second order phase transition for h = 0.758. Vectors sketched under the curves m ( h ) t represent the
magnetization of each sublattices. For instance the case a = 0.5 s h o w s a situation where a critical point exists and the ordered phase is on the l o w temperature side of the transition point. A n experimental e x a m p l e of this case is given by FeCI2. T h e case a = - 0 . 2 5 is an e x a m p l e where a critical point exists, but it appears that for certain field and temperature ranges, the ordered phase is stable on the high temperature side of the transition point. This is true not only for a second order transition but also for a first order one. FeBr2 is an experimental confirmation of this case. T h e case a = - 0 . 5 is an illustration of the existence of a crossing point K and a vanishing point A. 7. CONCLUSION
T h e molecular field approximation has permitted
\ \
p d= m
0=-0.5
o.a h
o=-0.25
0.6 o=0
o4
02
o lL
I 0.2
I 0.4
I 0.6
I 0.8
1 I
T Fig. 10. Phase diagrams in coordinates: reduced magnetic field h = g BH / J , --J2 and reduced temperature ~ = t / t s for different values of a = J, + JdJ~ - J z in the two limiting
cases d -- 0 and d = =. For every value of d(0 < d < =) the corresponding first and second order transition fines are located between the two limit lines d = 0 and d = =. The critical points are marked by a circle.
A. R. FERT et al.
270 J
a complete study of the behavior of a simple antiferromagnet with two exchang~ parameters and an uniaxial anisotropy term, in a magnetic field parallel to the easy axis of magnetization. In this text we have developed in particular, research of the different kinds of phase diagrams, the study of the meeting point of first and second order transition lines and the behavior of the order parameter 71. Other developments are possible such as initial susceptibility, isothermal magnetization curves (we give in Fig. 1 1 an example of such sketch of curves for a = 0.5, d = oo) and critical behavior near T~ (more accurate methods than molecular field approximation are desirable). /'=-0
h Fig. 11. Solid line curves are magnetization isotherms for a = 0.5 and d = ~. Dashed curve II represents the locus
of the second order transition point in the diagramm of reduced magnetic field h vs. magnetization m. Dashed curve I gives the magnetization discontinuity at a first order transition point. The magnetization discontinuity is practically a linear function of H -- He. (a) Our results showed at first the influence of the uniaxial anisotropy characterized by the parameter D. We have pointed out that the incidence of the anisotropy is essential to the magnetic ordering with temperature. F o r instance the N r e l temperature greatly depends on D. But when we study the behavior of our system with a reduced temperature variable r = tits = T/T:~ (0 < r < I) the influence of D is very weak. The phase diagram
in a parallel magnetic field depends above all on the relative importance of the two parameters J~ and J2 representing the ferromagnetic and antiferromagnetic interactions. (b) Different phase diagrams have been obtained. In particular we have shown the existence of a diagram where, in a certain range of temperature, the ordered phase is stable on the high temperature side of the transition point. When there exists a critical point the transition is of first order for t < tc and of second order for t > to, the first and second order transition lines (h, t) being tangent at this point. We can also have a crossing point (hk, tk) of the first and second order transition lines and the transition is of first order for t < t,, whereas, for a temperature just above tk, with increasing field, we observe a first order phase transition between phases of the same symetry, followed by a second order transition to a paramagnetic phase. (c) The phase diagrams of FeCI2 and FeBr2 are experimental illustrations of these results. FeC12, characterized by J~ -> - J 2 , has a classical metamagnetic behavior and corresponds to the situation obtained for a > 0. The theoretical results obtained for a = - 0 - 2 5 describe very well the experimental behavior of FeBr2: the first and second order transition field is an increasing function of temperature near the critical point and the experimental ratio TdTN is close to the theoretical one. This confirms our conviction, based on our recent experimental results (to be published), that FeBr2 is characterised by a great antiferromagnetic interaction - - J 2 ~ J~ in spite of the values previously proposed by us [5]. H o w e v e r this work does not take into account the possibility of a magneto-elastic coupling and we must recall that it may be present in the study of the behavior of compounds as FeCI2 and FeBr~. We have also neglected the possibility of a spinflop phase though this phase may be observed in the extreme case of a very low anisotropy or of a great antiferromagnetic interaction. Preliminary experimental results obtained for FeI2 show that this compound may be an illustration of this case.
REFERENCES I. Start C., Bitter F. and Kaufmann A. R., Phys. Rev. 58, 977 (1940). 2. Jacobs I. S. and Lawrence P. E., Phys. Rev. 164, 866 (1967). 3. Carrara P., Th~se, Orsay (1968). 4. Jacobs I. S. and Lawrence, P. E., J. appl. Phys. 35, 996 (1964).
Phase transition of an antiferromagnetic system 5. Fert A. R., Carrara P., Lanusse M. C., Mischler G. and Redoules J. P., J. Phys. Chem. Solids 34, 223 (1973). 6. Motizuki K., J. Phys. Soc. Jap. 14 (6), 759 (1959); Kanamori J., Motizuki K. and Yosida K., BuseironKenkyu 63, 28 (1953). (In Japanese). 7. Goter C. J. and Van Peski-Tinbergen Tineke, Physical 22, 273 (1956). 8. Bidaux R., Carrara P. and Vivet B., J. Phys. Chem. Solids 28, 2453 (1967). 9. Garrett C. G. B.,J. chem. Phys. 19, 1154(1951). 10. Ziman J. M., Proc. Phys. Soc. 64A, 1108 (1951). I I. Heap B. R., Proc. Phys. Soc. 80, 248 (1962). i 2. Yomosa S., J. phys. Soc. Japan 15, 1068 (1960). 13. Kenan R. P., Mills R. E. and Campbell C. E.,J. appl. Phys. 40 (3), 1027 (1969). 14. Kenah R. P. and Mills R. E. (To be published). 15. Lanusse M. C., Carrara P., Fert A. R., Mischler G.
271
and Redoules J. P., J. Phys. 33, 429 (1972). 16. Schnatterly S. E. and Fontana M., J. Phys. 33, 691 (1972). 17. Laurence G., Phys. Lett. 34A, 308 (1971). 18. Laurence G., Petitgrand D., Phys. Rev. (To be published). 19. Nasser J. A., Hammann J., Phys. Status Solidi 56 (1973). 20. Beille J., Alberts H. L., Bartholin M., Bloch D. and Vettier C., C.R. Acad. Sci. Paris t. 275, B, 713-722 (1972). 21. Vettier C., AIberts H. L., Beille J., and Bioch D., C.R. Acad. Sci. Paris t. 275, B, 915-918 (1972). 22. Tsallis C., J. Phys. 32, 903 (I 971). 23. Boccal'a N., Lea Principes de la Thermodynamique Classique, Voi. 1, p. 83. Prosses Univgrsitaires de France, Pads (1968).