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Phase transitions and critical points in rare earth-transition metal ferrimagnets
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1569-1570
of j • 4 leurnal magnetism and inagnellc materials
ELSEVIER
Phase transitions and critical points in rare earth-transition metal ferrimagnets L.R. Evangelista
a,*,
A.K. Zvezdin h
a Departamento de Fisica, UniversidadeEstadual de Maringr, Maringd, Paran~, Brazil b General Physics Institute, Russia Academy of Sciences, Moscow, Russia Abstract The new features of the H - T diagram for ferrimagnets RE (Rare Earth)-TM (Transition Metal) here studied are (1) the occurrence of critical points for which three phases can exist: (a) paramagnetic, ferromagnetic and canted phase (low fields), and (b) paramagnetic, ferrimagnetlc and canted phase (high fields) and (2) the occurrence of reentrant-like phase transitions.
Magnetic-field induced phase transitions from a collinear ferromagnetic phase to an angular (spin-flop transitions in ferrimagnets) have been studied in detail both theoretically and experimentally [1-3]. The most common materials, which have been the setting for most research on these phase transitions so far, are compounds of rare-earth elements and transition metals of the iron group: the iron garnets R3FesO12 and the intermetallic compounds RnTm, where R is a rare-earth element, and T a transition metal [3]. Such systems are generally described in the approximation of a 'rigid' d-sublattice, in which the magnetization of the d-sublattice is assumed to be independent of the external magnetic field and the exchange field exerted on it by the rare-earth sublattice. Much interest has recently been attracted to itinerant f - d megamagnets, which in a sense are the opposite of the materials mentioned above [4]. In them, the d-subsystem is a weak itinerant ferromagnet and can undergo a metamagnetic phase transition with a substantial jump in magnetization. Some typical members of this family are YCo 2, LuCo 2, and RCo 2, which exhibit metamagnetic transitions in the electronic d-subsystem from a paramagnetic state to a ferromagnetic field under the influence of a magnetic field [5-14]. Our purpose is to learn about magnetic-field-induced phase transitions and the H - T phase diagrams in itinerant metamagnets with an unstable d-sublattice. The thermodynamic potential q~ of this system is [14]
CI9= ½am2 + ¼bm4 + ~cml6 -- m ( H 2 + h 2 g 2 - - 2AMH - M H cos 0 - T S ( M ) ,
COS
0 ) 1/2 (1)
where m, M are the magnetizations of the d- and f-sublattices, S(M) is the entropy of f-subsystem, a,b, and c are coefficients whose temperature dependence is determined by the particular band structure and/or by spin fluctuations. Assume, as is customarily done in the theory of itinerant metamagnetism, that the following relations hold: a > 0, b < 0, c > 0. We consider the case in which the d-subsystem is itself paramagnetic down to T = 0. Expression (1) is the thermodynamic potential minimized as a function of the angle between the magnetization of the d-sublattice and the effective field acting on it. Minimizing (1) with respect to m, M, and 0, we find
ho(m) =Hen(M, 0 ) ,
(2)
MH sin 0 ( 1 - Am/Hen ) = 0,
(3)
-T---cos aM
where Hen = H - A M . re(h) in the form m ( h ) = ( Xdh' m 1,
Hen
0,
(4)
Let us approximate the function
h np,
(5)
where Hp is the threshold field for the metamagnetic transition in the d-subsystem. Eqs. (2), (3) and (4), along with the stability conditions (6 2~ > 0), determine the following solutions (phases) and the regions in which they exist:
W:
0=0, m=Xd(H-AM),
Fsl:
0=0, 0='tr, m=ml,
Fs2: * Corresponding author. Fax: +55-442232676; email: Ire [email protected].
0 1-
0 = ' r r , 0 = 0 , m = m 1,
C:
0<0<'rr,
Fe:
0=0,
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01370-5
m = m 1,
0 = 0 , re=m1,
L.R. Evangelista, A.K. Zvezdin/ Journal of Magnetism and Magnetic Materials 140-144 (1995) 1569-1570
1570
metamagnets. The nature of the H - M phase diagrams depends on the relation between Hp and Amr If Hp < Am1/2 , the phase diagram will have, in addition to the 'ordinary' lines of second-order phase transitions between collinear and angular phases (which are determined by the known expressions for the critical fields H¢1 = AIm 1 - M I and He2 = A I m I + M I), lines of a first-order transitions, which are determined by
(a) 1.5
I
C
H M = (Hp - A M ) / ( 1 - 2 M / m ) . 50
T¢
Tc
200
250
Temperature I K )
c
In the interval Am1~2 < Hp < Am1 the phase diagram (Fig. lb) has two interesting features. There cannot only be W-Fs2 phase transitions here, but also first-order transitions from the weak ferrimagnetic phase A 1 to the canted phase C. Transitions of this sort have not previously been seen in isotropic systems. Lines T1Tc and T2B are determined by Eq. (6), and lines A1T1 and A2T2 by
(b)
AI AI
H=A[M+emll,
Fe
. .
.
50
.
.
.
.
T¢ T¢ 150 Tomporeture (K)
.
.
.
.
.
.
.
2OO
.
250
(6)
3130
.
.
.
3OO
H=AIM--emll,
where e = ((2Hp/Am 1 ) - 1) 1/2. The points T1 and T2 have the coordinates M = lml(1 ~ •), H = ½Am1(1 :F e). The second unusual feature of this diagram is that there can be 'inverse transition' from the strong ferrimagnetic phase to the weak ferrimagnetic phase [15].
2.! Fe
References
(c) I
05 . a ~ 'F~I, . , . . . . 5O
~0
150
200
Temperature ( K )
Fig. 1. H - T phase diagrams of an itinerant ferrimagnet with an unstable d-subsystem, a. Hp < Am1; b. Hp < AmI / 2 ; c. Hp > A/2. The dashed line is the line Of the continuous transitions between the ferrimagnetic phase Fsl and ferromagnetic phase Fe. where 0 is the angle between the magnetic field and m. The phases W(Fsl and Fs2) can be weak (strong) ferrimagnetic collinear phases, F e is ferromagnetic phase, while phase C is the canted (angular) phase. To plot H - T phase diagrams, we need to equate the energies of the coexisting phases and fred the lines of first-order phase transitions from these equations. As @d(m) in our approximation we should use the quantity @d = f~'hd(m)dm. The phase diagrams in Fig. 1 give a general picture of the magnetization curves and critical field of itinerant
[1] S.V. Tyablikov, Methods in the Quantum Theory of Magnetism (Plenum, New York, 1967). [2] A. Clark and E. Callen, J. Appl. Phys. 39 (1968) 5972. [3] K.P. Belov, A.K. Zvezdin, A.M. Kadomtseva and R.Z. Levitin, Orientational Phase Transitions in Rare Earth Magnetic Materials (Moscow, 1979). [4] R.Z. Levitin and A.S. Markosyan, Soy. Phys. Usp. 31 (1988) 730. [5] E.P. Wohlfarth and P. Rhodes, Philos. Mag. 7 (1962) 1817. [6] D. Bloch, D.M. Edwards, M. Shimizu et al., J. Phys. F 5 (1975) 1217. [7] M. Cyrot and M. Lavagna, J. Phys. (Paris) 40 (1979) 763. [8] H. Yamada, J. Inoue, K. Terao et al., J. Phys. F 14 (1984) 1943. [9] T. Goto, K. Fukamichi et al., Solid State Commun. 72 (1989) 945. [10] K. Murata, K. Fukamichi et al., J. Phys. Cond. Matt. 3 (1991) 2515. [11] N.H. Duc, T.D. Hien, P.E. Brommer et al., J. Phys. F 18 (1988) 275. [12] W. Steiner, E. Gratz et al. J. Phys. F 5 (1978) 1525. [13] R. Ballou, Z.M. Gamishidze et al., Sov. Phys. JETP 75 (1992) 1041. [14] A.K. Zvezdin, JETP Lett., 58 (9,10) (1993) 719. [15] A.K. Zvezdin and S.N. Utochkin, J. Magn. Mater. 104-107 (1992) 1479.
of j • 4 leurnal magnetism and inagnellc materials
ELSEVIER
Phase transitions and critical points in rare earth-transition metal ferrimagnets L.R. Evangelista
a,*,
A.K. Zvezdin h
a Departamento de Fisica, UniversidadeEstadual de Maringr, Maringd, Paran~, Brazil b General Physics Institute, Russia Academy of Sciences, Moscow, Russia Abstract The new features of the H - T diagram for ferrimagnets RE (Rare Earth)-TM (Transition Metal) here studied are (1) the occurrence of critical points for which three phases can exist: (a) paramagnetic, ferromagnetic and canted phase (low fields), and (b) paramagnetic, ferrimagnetlc and canted phase (high fields) and (2) the occurrence of reentrant-like phase transitions.
Magnetic-field induced phase transitions from a collinear ferromagnetic phase to an angular (spin-flop transitions in ferrimagnets) have been studied in detail both theoretically and experimentally [1-3]. The most common materials, which have been the setting for most research on these phase transitions so far, are compounds of rare-earth elements and transition metals of the iron group: the iron garnets R3FesO12 and the intermetallic compounds RnTm, where R is a rare-earth element, and T a transition metal [3]. Such systems are generally described in the approximation of a 'rigid' d-sublattice, in which the magnetization of the d-sublattice is assumed to be independent of the external magnetic field and the exchange field exerted on it by the rare-earth sublattice. Much interest has recently been attracted to itinerant f - d megamagnets, which in a sense are the opposite of the materials mentioned above [4]. In them, the d-subsystem is a weak itinerant ferromagnet and can undergo a metamagnetic phase transition with a substantial jump in magnetization. Some typical members of this family are YCo 2, LuCo 2, and RCo 2, which exhibit metamagnetic transitions in the electronic d-subsystem from a paramagnetic state to a ferromagnetic field under the influence of a magnetic field [5-14]. Our purpose is to learn about magnetic-field-induced phase transitions and the H - T phase diagrams in itinerant metamagnets with an unstable d-sublattice. The thermodynamic potential q~ of this system is [14]
CI9= ½am2 + ¼bm4 + ~cml6 -- m ( H 2 + h 2 g 2 - - 2AMH - M H cos 0 - T S ( M ) ,
COS
0 ) 1/2 (1)
where m, M are the magnetizations of the d- and f-sublattices, S(M) is the entropy of f-subsystem, a,b, and c are coefficients whose temperature dependence is determined by the particular band structure and/or by spin fluctuations. Assume, as is customarily done in the theory of itinerant metamagnetism, that the following relations hold: a > 0, b < 0, c > 0. We consider the case in which the d-subsystem is itself paramagnetic down to T = 0. Expression (1) is the thermodynamic potential minimized as a function of the angle between the magnetization of the d-sublattice and the effective field acting on it. Minimizing (1) with respect to m, M, and 0, we find
ho(m) =Hen(M, 0 ) ,
(2)
MH sin 0 ( 1 - Am/Hen ) = 0,
(3)
-T---cos aM
where Hen = H - A M . re(h) in the form m ( h ) = ( Xdh' m 1,
Hen
0,
(4)
Let us approximate the function
h
(5)
where Hp is the threshold field for the metamagnetic transition in the d-subsystem. Eqs. (2), (3) and (4), along with the stability conditions (6 2~ > 0), determine the following solutions (phases) and the regions in which they exist:
W:
0=0, m=Xd(H-AM),
Fsl:
0=0, 0='tr, m=ml,
Fs2: * Corresponding author. Fax: +55-442232676; email: Ire [email protected].
0 1-
0 = ' r r , 0 = 0 , m = m 1,
C:
0<0<'rr,
Fe:
0=0,
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01370-5
m = m 1,
0 = 0 , re=m1,
L.R. Evangelista, A.K. Zvezdin/ Journal of Magnetism and Magnetic Materials 140-144 (1995) 1569-1570
1570
metamagnets. The nature of the H - M phase diagrams depends on the relation between Hp and Amr If Hp < Am1/2 , the phase diagram will have, in addition to the 'ordinary' lines of second-order phase transitions between collinear and angular phases (which are determined by the known expressions for the critical fields H¢1 = AIm 1 - M I and He2 = A I m I + M I), lines of a first-order transitions, which are determined by
(a) 1.5
I
C
H M = (Hp - A M ) / ( 1 - 2 M / m ) . 50
T¢
Tc
200
250
Temperature I K )
c
In the interval Am1~2 < Hp < Am1 the phase diagram (Fig. lb) has two interesting features. There cannot only be W-Fs2 phase transitions here, but also first-order transitions from the weak ferrimagnetic phase A 1 to the canted phase C. Transitions of this sort have not previously been seen in isotropic systems. Lines T1Tc and T2B are determined by Eq. (6), and lines A1T1 and A2T2 by
(b)
AI AI
H=A[M+emll,
Fe
. .
.
50
.
.
.
.
T¢ T¢ 150 Tomporeture (K)
.
.
.
.
.
.
.
2OO
.
250
(6)
3130
.
.
.
3OO
H=AIM--emll,
where e = ((2Hp/Am 1 ) - 1) 1/2. The points T1 and T2 have the coordinates M = lml(1 ~ •), H = ½Am1(1 :F e). The second unusual feature of this diagram is that there can be 'inverse transition' from the strong ferrimagnetic phase to the weak ferrimagnetic phase [15].
2.! Fe
References
(c) I
05 . a ~ 'F~I, . , . . . . 5O
~0
150
200
Temperature ( K )
Fig. 1. H - T phase diagrams of an itinerant ferrimagnet with an unstable d-subsystem, a. Hp < Am1; b. Hp < AmI / 2 ; c. Hp > A/2. The dashed line is the line Of the continuous transitions between the ferrimagnetic phase Fsl and ferromagnetic phase Fe. where 0 is the angle between the magnetic field and m. The phases W(Fsl and Fs2) can be weak (strong) ferrimagnetic collinear phases, F e is ferromagnetic phase, while phase C is the canted (angular) phase. To plot H - T phase diagrams, we need to equate the energies of the coexisting phases and fred the lines of first-order phase transitions from these equations. As @d(m) in our approximation we should use the quantity @d = f~'hd(m)dm. The phase diagrams in Fig. 1 give a general picture of the magnetization curves and critical field of itinerant
[1] S.V. Tyablikov, Methods in the Quantum Theory of Magnetism (Plenum, New York, 1967). [2] A. Clark and E. Callen, J. Appl. Phys. 39 (1968) 5972. [3] K.P. Belov, A.K. Zvezdin, A.M. Kadomtseva and R.Z. Levitin, Orientational Phase Transitions in Rare Earth Magnetic Materials (Moscow, 1979). [4] R.Z. Levitin and A.S. Markosyan, Soy. Phys. Usp. 31 (1988) 730. [5] E.P. Wohlfarth and P. Rhodes, Philos. Mag. 7 (1962) 1817. [6] D. Bloch, D.M. Edwards, M. Shimizu et al., J. Phys. F 5 (1975) 1217. [7] M. Cyrot and M. Lavagna, J. Phys. (Paris) 40 (1979) 763. [8] H. Yamada, J. Inoue, K. Terao et al., J. Phys. F 14 (1984) 1943. [9] T. Goto, K. Fukamichi et al., Solid State Commun. 72 (1989) 945. [10] K. Murata, K. Fukamichi et al., J. Phys. Cond. Matt. 3 (1991) 2515. [11] N.H. Duc, T.D. Hien, P.E. Brommer et al., J. Phys. F 18 (1988) 275. [12] W. Steiner, E. Gratz et al. J. Phys. F 5 (1978) 1525. [13] R. Ballou, Z.M. Gamishidze et al., Sov. Phys. JETP 75 (1992) 1041. [14] A.K. Zvezdin, JETP Lett., 58 (9,10) (1993) 719. [15] A.K. Zvezdin and S.N. Utochkin, J. Magn. Mater. 104-107 (1992) 1479.